Welcome back to Math Kos! Today's entry is the beginning of a series on logic, the winner of the poll in the previous installment.

Logic is a wonderful interdisciplinary field, and I hope those of you with some philosophy background will touch on some elements with which I may not be familiar. Applications of logic range from computer science and mathematics to philosophy and rhetoric. Just as mathematics and computers are tools for science, logic is largely a tool for mathematics, although there are certainly elements of it which are subjects of current research as fields in their own right. On the whole, however, because of its nature as a foundation on which the rest of the field builds, it has few elements which are clearly recognizable as "mathematical" and is therefore generally quite approachable for people with a liberal arts background.

Logic Part 1: Definitions
While in most realms of life we are accustomed to dealing with ambiguity, applications of formal logic demand unambiguous terminology. In order to kill the ambiguity, we start by formally defining certain terms.

Statement: A statement is a symbolic expression, in words or otherwise, that is either true or false. It must be sufficiently specific for the purposes for which it is being used; terms in it must be clearly defined so that they can be understood unambiguously by the audience. For purposes of distilling an argument down into its logical form, statements are often represented by single letters or single words. In the definitions below, when I use a single letter, it can be interpreted to mean "some generic statement, regardless of content."

Example Statements:
Barack Obama is President of the United States.
There is a unicorn in my bathtub.

Example Non-Statements:
It might rain. (When? Where? How do you quantify "might"?)
Give me all your kittens. (This is an order, not a statement of fact.)

Boolean Operators
The Boolean operators are ways of stringing statements together to make bigger statements. They're named after George Boole, who invented Boolean algebra (the algebra of logic), which was later applied to electronic circuits because of its convenient natural binariness.

Logical AND: The logical AND has a very similar meaning to the word "and" in English. In general, you can interpret AND as "and." The equivalency usually works the other way too, although there are some rather contrived examples of ways that the English "and" can be used to really mean "or." There are several other English words that also usually mean AND (albeit with various other layers of meaning that must be stripped off in order to distill a sentence into its logical elements): "but," "however," "albeit," and others.

a AND b is true if a is true and b is true. If either is false, a AND b is false.

Logical OR: The logical OR is the inclusive or (and/or) and should usually be translated in a context-sensitive way as either "or" or "and/or.". Again, there are ways in English that "or" can mean AND, and there are a ton of situations (such as in the last sentence) where "or" means the exclusive or, so one should be careful with the meaning of a sentence when translating it into predicate logic.

a OR b is true if a is true, b is true, or a and b are both true. a OR b is only false if both a and b are false.

Logical NOT: NOT means...well, "not." It negates whatever comes after it.

NOT a is true if a is false and false if a is true.

Derived Operators: XOR (exclusive or), NAND (not AND), NOR (not OR/neither nor). These can be derived from the basic three described above.

a XOR b is true if a is true or if b is true but false if both are true or if both are false.

a NAND b is true if neither a nor b is true or if a is true or if b is true but false if both a and b are true.

a NOR b is true if neither a nor b is true, but false if either a or b is true or if both are true.

There are all number of annoying notations for the Boolean operators. Programmers have a couple, logicians have some, set theorists have their own (with different names), and there's even a really horrible graphical representation of them for circuit design. I'll spare you. If you want to learn more about Boolean algebra, I'll direct you to the Wiki intro page.

Conditionals
Conditionals have a subtly different meaning in logic than they do in ordinary English. In English, if-then constructions generally imply some sort of causality or connection between the two events; in logic, this is not necessary. All of the conditionals are logically equivalent to (can be translated to and from with no loss of information) expressions with Boolean operators; however, the Boolean expressions can be quite unwieldy and are somewhat foreign to the human way of reasoning, so unless we're actually building hardware circuits or programming in machine language, we prefer to use conditionals.

If-Then: The basic conditional, if a then b means if a is true then b is also true. It says nothing whatsoever about what happens if a is false; b could be false or it could still be true. If a then b is only false if a is true and b is false. If both are true or if a is false, then if a then b is true. Also remember that no causality is implied.

Example true if-then statements:
If Barack Obama is President, then Canada is north of the United States. (both true)

If Republicans controlled Congress, then the infamous FISA bill passed. (if false then true)

If Obama is a secret Muslim, then John McCain is President. (both false)

Example false if-then statement:
If Barack Obama is President, then racism is dead. (if true then false)

Only if: This one takes some thought to figure out. It's counterintuitive, but a only if b actually means if a then b. The difficulty comes in because in the translation, the causality seems to be reversed - but remember, there is no causality.

True only if statements:
Americans will only have full civil rights if gay and lesbian couples may legally marry. (means: If Americans have full civil rights, then gays and lesbians can marry. Does not mean that if gays and lesbians can marry, Americans have full civil rights; one can easily imagine a future in which same-sex marriage is allowed but transgendered persons are still denied their rights.)

Hillary Clinton will be President in 2009 only if Americans are willing to vote for a candidate who is not a white man.

False only if statement:
Barack Obama is only the President if he had his hand on the Bible when he was sworn in the second time.

If and only if: The biconditional, a if and only if b, means "if a then b AND if b then a." It is true if both a and b are true or if both are false, but false if one is true and the other false. "If and only if" is occasionally abbreviated "iff," which is one of the few conventions of notation in this field that I actually like, but I will refrain from using it here as it can be read as a typo.

True if and only if statements:
Barack Obama is President if and only if the Earth orbits the Sun.
John McCain is President if and only if the Sun orbits the Earth.

Arguments: An argument in logic is a series of statements that are asserted to be true (the premises) followed by a final statement asserted to follow from the premises (the conclusion). An argument is valid if and only if the conclusion does in fact follow from the premises. An argument can be logically valid whether or not anything in it is actually true; the important thing is the structure of the argument.

Example valid argument:
There can only be a unicorn in my bathtub if unicorns exist.
Unicorns do not exist.
Therefore, there is not a unicorn in my bathtub.

Example invalid argument:
If there is a unicorn in my bathtub, then unicorns exist.
There is not a unicorn in my bathtub.
Therefore, unicorns do not exist.
(note: this is the fallacy of the converse. We'll talk about both converses and fallacies thereof in the next installment.)

I'm sorry this part was a little dry; unfortunately, all the definitions are necessary so that when we start talking about more complicated arguments we can be certain that we're all speaking the same language. In Logic Part 2, I hope to discuss the rules for constructing and evaluating formal logical arguments, which should be much more interesting. I'll also try to touch on informal logic (the logic of language) and some of the fallacies unique to it. Thanks for reading!

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#### Comment Preferences

• ##### nails on a chalkboard?(30+ / 0-)

I know you were not asking me, but....

or maybe the sound that a foam ice chest makes?

• ##### Yes, it did!!!(4+ / 0-)

It will take the rest of this year on IR to recover from those sustained head injuries.

Born in Oklahoma Raised in Ohio Escaped to Meechigan!!!

[ Parent ]

• ##### For me, like the sound of a car crashing(15+ / 0-)

into the side of a house, a lot of breaking glass and bent metal.

"I'll be long gone before some smart person ever figures out what happened inside this Oval Office." --George W. Bush, Washington, D.C., May 12, 2008

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### like (5+ / 0-)
Recommended by:
mattman, kyril, RubyGal, Alec82, kjoftherock

licking the wooden spoon that comes with a cup of ice cream.

• ##### What was it like? (10+ / 0-)

I think you're assuming quite a bit.  I made a concerted effort not to listen to that man during his "presidency."

What has been is what will be, and what has been done is what will be done, and there is nothing new under the sun.

[ Parent ]

• ##### Someone has not paid attention(7+ / 0-)

You did not phrase it in a logical frame lattice ; )

When we got into office, the thing that surprised me most was to find that things were just as bad as we'd been saying they were. -JFK

[ Parent ]

• ##### Like what I imagine Hell to be, only forever. n/t(2+ / 0-)
Recommended by:
reflectionsv37, kyril

Until the economy recovers, I'll settle for cheap laughs

[ Parent ]

• ##### I never heard a logical argument NT(3+ / 0-)
Recommended by:
3goldens, Simplify, kyril
• ##### I hate Bush, but honestly, he's(7+ / 0-)

not that much worse than any politician with regards to logic. They do not apply "logic." They apply manipulation. Thus, definitions, construction of logical arguments, etc are secondary to the outcome that they are seeking.

For that matter, the average voter is worse. I can at least pull back the b.s. to see the real reason for Bush's actions or that of any politician.

The average voter- not so much. Remember "What's the Matter With Kansas?"

Often times, there is no logic at all other than "how it makes me feel" or "what I believe." Opinion is all. Definitnions, history, context, etc are all optional. There is no attempt at  a) logic and b) once you have a logical theory testing that logic against facts (empiricism).

Opinion (like the flat earth argument) is all valid. There is no hierarchy of logic by which we are suppose to judge it.

• ##### there are two types of opinion:(2+ / 0-)
Recommended by:
mmacdDE, kyril

objective and subjective.

objective opinion is based on what we clearly know to be true.

subjective opinion is based on what is true for you the individual.

the difference between the two is that one can become fact with time (you'll get a sunburn if you don't put on sunblock), the other will never be fact (pepperoni is the best pizza topping).

it is true politicians do not apply logic, because corruption has become institutionalized over the last fifty years. institutionalized corruption and logical politicians are the antithesis of one another.

americans are logical, their problem is that they are lied to. and no one in mainstream media is allowed to lead the charge to the logical solution to corrupt government: the Article V Convention.

working consciousness to raise consciousness

[ Parent ]

• ##### Disagree, but let me explain in full(2+ / 0-)
Recommended by:
john de herrera, kyril

First, politics has always involved corruption,including at the scale and in some ways worse than what we see today. The dead voting was not a joke. it happened.

Second, people make a lot of excuses for Americans. Politicians and the media can not do to us what we did not allow them to do. We are not victims here.

Americans are not logical. This is not theorectical. This is fact. Even when given facts- they still choose to believe what they want to believe. This is called cognitive dissonance.

I am fond of using this example:

In 2004, Bush repeatedly said that he was against importing drugs from Canada for seniors in America. At the time, it was a fairly big issue because it was being tauted as a measure to reduce cost of meds to seniors. Indeed, Bush mentioned that he was against such programs of importation at the debates.

When people from a southern state (I can't remember if its KY or TN or MS) were asked did Bush support importation of drugs from Canada- what was their answer?

Yes, bush supports importation of drugs. Now, is that the media's fault, the politicians fault, or the American people's fault when no other source has told them that Bush said that?

It's too easy to blame others for what we as a people do. It's too easy ,but it's not exactly the truth. The truth is that we are part of this society, and we choose to let it be what it is.

I see people here engage in this subjective opinion making despite countervailing facts all the time. There is a reason why Karl Rove called Bush's supporters faith based. That's because subjective opinion has won out over objective opinion. If someone subjectively believes gays are not subject to violence, then any evidence to the contrary will be denied in favor of discussing gay marriage (and yes, I have seen diaries here in which people write about violence against gays, and some posters will say "Well, politicians will differ on gay marriage" as if that's a response to the diary's subject matter.).

So I disagree because I don't see any evidence of the clear lines of demarcation or that evidence alone determines outcome regarding the thought processes of the American people.

Does that mean I don't think some of what you say is not true? Yes, what you say is true. But, it's ultimately an excuse. When you push toward them having information  and logical arguments they still do not always care.

• ##### of the bell-curve of consciousness(0+ / 0-)

there will always be those with greater and lesser intelligence. and of course there will always be ignorance, it's part of the human condition.

in the end, intelligent people are aware that those who sit on pharma corporate boards, also sit on media corporate boards, and also fund elected offices. we're lied to (often by ommission), and we've been lied to hard core since truman signed the national security act. this society has been caught up in a military/industrial funhouse for over fifty years.

so the question is, when the political machine has "divide and rule" down to a science, and fear is thrown into the mix, how far can you trust polling results? how far can you excuse the society?

sometimes i wonder if it's that our collective i.q. is too low to break this cycle. other times i wonder if it's simply a question of the bad guys having won critical battles sometime back (loss of the Fairness Doctrine during the reagan revolution, the 1996 telecommunications act), and now they're on the verge of winning the war.

it's close, but still too soon to tell.

working consciousness to raise consciousness

[ Parent ]

• ##### Never before or since(2+ / 0-)
Recommended by:
jayden, kyril

have I witnessed my body react so quickly and forcefully with the worst of profanities shouted in the direction of his voice.

Every time.

(-7.75, -7.69) No matter how cynical I get, I just can't keep up - Lily Tomlin

[ Parent ]

• ##### Great start! (8+ / 0-)

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### Keep it up! Depending on where this goes(5+ / 0-)

...it might turn out to be a good set of diaries to point people to when they are a little weak on logic.

"Certainly the game is rigged. Don't let that stop you; if you don't bet, you can't win." Lazurus Long

[ Parent ]

• ##### Thanks!(8+ / 0-)

I'm afraid I'm starting off a bit slow just to make sure I don't lose anyone. It could be a long-ish series, although it certainly should be less daunting than an actual book.

Math Kos II is open.

[ Parent ]

• ##### You might want to clarify(8+ / 0-)

The explanation of Valid.  The way we learned it, in philosophy, is that an argument is valid is there is no way for the premises to be true and the conclusion false.

This cuts down on misunderstandings later on, like how some weird arguments are considered valid, even though they are pretty much nonsense.

[ Parent ]

• ##### Ah, yes(5+ / 0-)

I'll talk more about validity next time; it really needs an introduction of truth tables, and this was already getting pretty long.

Math Kos II is open.

[ Parent ]

• ##### I really(3+ / 0-)
Recommended by:
mmacdDE, kyril, sjr1

wish I could take another logic class.  Doing proofs on the water board in the philosophy lounge was so, so fun.

[ Parent ]

• ##### I think they should make(2+ / 0-)
Recommended by:
Jules Beaujolais, kyril

logic a required course. The course on logic I took was the best course I ever had. One of the few that actually gave me solid, usable, information that I use all the time.

• ##### I don't(2+ / 0-)
Recommended by:
Jules Beaujolais, kyril

understand why they don't require basic logic for science and math degrees.  Most math degrees have proofs classes, but not basis logic.  Math is so, so much easier once you know logic.  Especially advanced math.

[ Parent ]

• ##### You can teach validity without truth tables.(4+ / 0-)

In fact, I never use them in my intro classes.  One simply has to ascertain that the relationship between the premises and conclusion is a truth-preserving one (and you can do this through counterexamples, e.g.).

"What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet."--Woody Allen

[ Parent ]

• ##### I suppose you're right(5+ / 0-)

but the main advantage of truth tables is that they're an organized way of finding a counterexample; a counterexample is in effect just one line in a truth table.

Math Kos II is open.

[ Parent ]

• ##### I know (3+ / 0-)
Recommended by:
rapala, Jimdotz, kyril

that of the two logic teachers at former college, one used truth tables, the other never taught them.  No one wanted to take the teacher that didn't teach them, it made it a lot harder to understand once you started doing proofs.

[ Parent ]

• ##### The advantage to not using them(4+ / 0-)
Recommended by:
rapala, Simplify, kyril, rfall

is the same as the advantage to not using any algorithmic system; it takes intuition and understanding to see whether the relationship you're after in a problem actually exists.

Truth tables are useful with extremely long arguments of the sort that you'd never find in ordinary language.

One incredibly useful skill is the ability to construct models to show invalidity, and more generally, to understand a problem.  You of course get this with truth tables, but you get this without them, as well.  There's a great Feynman story where he does this (constructing models, a with oranges or some-such.

"What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet."--Woody Allen

[ Parent ]

• ##### "of the sort that you'd never find(1+ / 0-)
Recommended by:
kyril

...in ordinary language."

Agreed--and it's why we engineers love truth tables, because the kinds of problems we deal with in designing circuits, etc. are not easily describable in ordinary language, and much more amenable to description, and manipulation, in truth tables.

At least, when I went to school, they were very useful teaching tools.  Nowadays, I'm not so sure.

"Certainly the game is rigged. Don't let that stop you; if you don't bet, you can't win." Lazurus Long

[ Parent ]

• ##### This Logic teacher loves truth tables.(5+ / 0-)

It unifies the whole subject, and is sufficiently mechanical to allow weak students to make a stab at solving complex problems.

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### As a student in the 70s...(3+ / 0-)
Recommended by:
Danjuma, Jimdotz, kyril

...I learned truth tables in my Analytical Geometry class as a high school senior (basically the first semester was what I later recognized as Symbolic Logic).  In college, I took a Logic class through the Philosophy department.  I don't recall any truth tables (or symbolic stuff) in that one, it was all verbal syllogisms.

• ##### And, fun to use, too, for those us of(2+ / 0-)
Recommended by:
Jimdotz, kyril

...who have a strangely warped sense of fun, solving those classic logic problems, "One car model is blue; Betty has a 2001 model car, yadda, yadda, who owns what color car?"

Could whip those out in no time with truth tables.

"Certainly the game is rigged. Don't let that stop you; if you don't bet, you can't win." Lazurus Long

[ Parent ]

• ##### when you do truth tables(4+ / 0-)
Recommended by:
Danjuma, kyril, vizard, rfall

please make sure you drive home the most important point: complex ideas are subject to the cold hardness of mathmatics. one does not get to go around believing what they want about life. what they believe is either true and valid or it's not, based on the facts as we know them.

i trust you'll do a great job, just don't forget to make clear what truth tables tell us about sapient beings--whether one can draw one up on a chalk board or not.

working consciousness to raise consciousness

[ Parent ]

• ##### I agree and don't agree(1+ / 0-)
Recommended by:
kyril

I found that, as pattern-matching creatures, we can understand some concepts more easily, and certainly manipulate them more easily, in a visual form.

So, for instance, while describing the XOR function as "either A or B but not both", the pattern I saw in the 2x2 truth table for the XOR function made instant sense, and allowed me to quickly see that similar patterns in larger truth tables could be seen as cascades of XOR functions.

I guess the difference might be one of intent:  while I appreciate pure mathematics, I was never good at it, so my focus was on applying algorithms based on abstract concepts to real-world problems.

Often, I'd spent the skull sweat to "get" the abstract idea then, once I'd done that and filed away the simple note that "it makes sense", I would forget how I understood that abstract concept and focus completely on its practical application.

Difference between an engineer and a mathematician, I suppose.

"Certainly the game is rigged. Don't let that stop you; if you don't bet, you can't win." Lazurus Long

[ Parent ]

• ##### Yup; never used truth tables, but after learning (1+ / 0-)
Recommended by:
kyril

valid in philosophy, was able to slam it out in in two seconds in symbolic logic. It's actually very intuitive, but people don't really see the relationship between logic and real life any more.

Perhaps its near omnipresence through computer programming makes people think it is merely mechanistic and of no use for political discourse;

or perhaps its the denigration of science showing;

Whatever, Valid is such an important concept it ought to be carved in stone in every classroom in the land.

A Patriot

[ Parent ]

• ##### OOOh! I dkos encyblopedia of logical fallacies!(7+ / 0-)
How practical that would be!  Would make one save so much time not to have to resort to long-winded explanations but simply insert a link.

Debaters, myself included, who get entangled on a point of logic do have such a tendency to going back to Adam and Eve!

-7.88/ -7.44

[ Parent ]

• ##### Sorry for the typos(2+ / 0-)
Recommended by:
kyril, Nailbanger

didn't mean to write a surrealistic comment...

-7.88/ -7.44

[ Parent ]

• ##### What a great idea! When we run into a(8+ / 0-)

...commenter who commits a logical fallacy, we just say "that's a number 27 error" and move on.

"Certainly the game is rigged. Don't let that stop you; if you don't bet, you can't win." Lazurus Long

[ Parent ]

• ##### Well, it was Your idea. :)(3+ / 0-)
Recommended by:
Caneel, kyril, Nailbanger
And I agree, it would be great.  Not to mention the coolness factor.  "What?!  You really don't know what error #27 is??!!!  Also great for the snobbishly inclined...

;)

-7.88/ -7.44

[ Parent ]

• ##### Great idea in theory(3+ / 0-)
Recommended by:
Caneel, Jules Beaujolais, kyril

In practice, it will lead to thread hijacks in which 2 or more logicians debate whether it was, in fact, an error # 27, or whether it should be labeled a #31 error.

I can't wait for the moment when Lieberman's vote saves the day.

[ Parent ]

• ##### Then we just insist it's an inclusive OR of these(3+ / 0-)
Recommended by:
Jules Beaujolais, kyril, Nailbanger

...two, and move on.

Or, don't logicians operate that way?

"Certainly the game is rigged. Don't let that stop you; if you don't bet, you can't win." Lazurus Long

[ Parent ]

• ##### Not really(3+ / 0-)
Recommended by:
Jules Beaujolais, Jimdotz, Nailbanger

Any significant exposure to logic tends to encourage a stubborn and pedantic streak in those inclined to develop one.

Math Kos II is open.

[ Parent ]

• ##### Part III: Fuzzy Logic?(3+ / 0-)
Recommended by:
theran, Jules Beaujolais, kyril

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### Hmm(2+ / 0-)
Recommended by:
Jules Beaujolais, Jimdotz

That's one I know just enough about to be dangerously misleading. We'll see what I can do, though, with some research.

Math Kos II is open.

[ Parent ]

• ##### Not sure I understand fuzzy logic, but it has(2+ / 0-)
Recommended by:
Jimdotz, kyril

always struck me as a way to escape the hard work of distilling every ambiguity down to its logical base, so as to clarify the question.

That's from a rhetorical point of view, of course; my liberal arts education in pholosophy & history didn't mention it.

A Patriot

[ Parent ]

• ##### Actually, you do understand some of it, namely(0+ / 0-)

the half-baked idea, one neither proven true or practical, nor proven untrue or impractical. It was generalized in a delightful little book published in the 1960s under the title The Scientist Speculates: An Anthology of Partly-Baked Ideas. The idea is to define a bakedness scale running from 1 (fully baked, like General Relativity) to 0 (astrology), or in my personal version, to negative numbers (Global Warming Denial, Creationism, and Voodoo Economics).

String theory in physics is an excellent example of a partly-baked idea with a steadily-rising bakedeness value that is still not very high.

Like all the rest of mathematics, fuzzy logic more generally starts from a few definitions and assumptions that have to be verified in any proposed application, and then works out the consequences of those definitions and assumptions. The usual Boolean algebra has only two truth values, True (frequently represented by 1) and False (0). Fuzzy logic allows any value between 0 and 1, and redefines the usual And, Or, and Not functions to apply to any fuzzy values with a fuzzy result. One simple set of definitions is

And: Return the smaller truth value. 0.5 and 0.2 = 0.2

Or:  Return the larger truth value. 0.5 and 0.2 = 0.5

Not: Subtract from 1. Not 0.2 = 0.8

There are many fuzzy logics, some of which have turned out to have practical value.

The canonical example of alternative definitions is plane and solid geometry (extended over time to higher dimensions). Euclidean geometry has a unique line parallel to a given line through a given point, while Riemannian geometry has none, and Lobachevskian geometry has many. The surface of the Earth is locally Riemannian (Great circles always intersect in two points), and there is good reason to think that spacetime as a whole is Lobachevskian.

Busting the Dog Whistle code.

[ Parent ]

• ##### HAHAHAHAHAHAHAAAAAA!(2+ / 0-)
Recommended by:
Foxwizard, kyril
Too true!

It may still be shorter though than

Person A: "What's your definition of truth?"

Person B: "First, what's your definition of definition?"

etc, etc, et fucking endlessly cetera.

-7.88/ -7.44

[ Parent ]

• ##### One thing studying philosophy helped me (3+ / 0-)
Recommended by:
Jules Beaujolais, kyril, rfall

understand is the need for an understanding of one's epistemology: how do I know what I know?

Bush seems to have personified the fruition of that American romanticism that teaches feelings are more important than facts, and that perception is truth.

That, of course, is a rejection of the western philosophical tradition that has always maintained that reality is separate and knowable apart from one's interior life. This tradition, upon which scientific thinking is based, recognizes the observability of phenomenon and the possible establishment of facts that are independent in their reality from the observer. Note also that one's perception of the facts may be distorted, but that does not change the essence of the facts themselves.

Romanticism conflates external reality into the interior life of the one percieving it, so in essence no establishment of independent facts is possible. Hence, perception is taken to be complete, since the fact has no essence or reality apart from the one percieving it.

That provides some great seeds for thinking about the silliness of our political discourse.

A Patriot

[ Parent ]

• ##### Nicely said! And, not being a philospher,(2+ / 0-)
Recommended by:
Foxwizard, kyril

...(though my wife was one, and in this exact area of belief systems), I tend to use a fairly simple argument as to why

the observability of phenomenon and the possible establishment of facts that are independent in their reality from the observer.

is a good philosophical position--i.e., why the scientific method, evidence-based medicine, etc. are "good".

It's this:  because it works.

Not much of a philosophically grounded position, I know, but I like it. ;-)

"Certainly the game is rigged. Don't let that stop you; if you don't bet, you can't win." Lazurus Long

[ Parent ]

• ##### Thanks. Be careful, though, for it was(1+ / 0-)
Recommended by:
kyril

David Hume, the father of Utilitarianism ("Because it works") that maintained reality only existed insofar as it was percieved by the observer.

Just one of the cross-currents that can pull one down if you're not aware it's there.

A Patriot

[ Parent ]

• ##### Actually, Mill and Bentham(0+ / 0-)

Hume, my favorite philosopher, taught quite otherwise than the Utilitarians in The Theory of Moral Sentiments and other works.

He did not claim that only our perceptions are real in An Enquiry into Human Understanding. He claimed that our perceptions are all that we know. We do not know what is real. The existence of a material world operating by laws of physics is a tenable hypothesis, as long as we are clear on the extent of our ignorance about it. Of course, our ignorance in matters of religion tends to be much greater, the more so for those who claim more certainty, and complete for those who claim to know everything.

Busting the Dog Whistle code.

[ Parent ]

• ##### There are actually people who do this with movie(1+ / 0-)
Recommended by:
kyril

plots.

Member, The Angry Left.

[ Parent ]

• ##### I teach Logic. This is a wonderful introduction.(7+ / 0-)

The best part about trying to learn Logic is that, when you think about it, it's... logical.

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### Oh My God(2+ / 0-)
Recommended by:
rapala, kyril

Math on Daily Kos, 2 of my most favorest things combined, what next?  Whip Cream on pizzas?

How come when it is a human it's an abortion, but when it's a chicken it's an omelet? -George Carlin

[ Parent ]

• ##### This should be front paged(2+ / 0-)
Recommended by:
LeftOverAmerica, kyril
• ##### Great diary(2+ / 0-)
Recommended by:
Caneel, kyril

and a great example of why progressives have better ideas. This diary is on the rec list at DKos, because progressives by-and-large understand that learning to reason and think clearly is a prerequisite to constructively solving real problems. Good luck ever finding an entry like this at RedState or LGF.

Looking forward to the next installment. Tipped and Recc'd.

"I seek the truth, which never yet hurt anybody. It is only persistence in self-delusion and ignorance which does harm." --Marcus Aurelius

[ Parent ]

• ##### Nice refresher for me, thanks. n/t(1+ / 0-)
Recommended by:
kyril

A Patriot

[ Parent ]

• ##### tip, rec & cheer!(8+ / 0-)
Recommended by:
mattman, theran, Pluto, elwior, kyril, RubyGal, Alec82, sjr1
• ##### How about those aspects of complex reality(6+ / 0-)

that defy formal logic?  Impredicative loops of cause for example?

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

• ##### How bout 'em? (6+ / 0-)
Recommended by:
mattman, Wee Mama, elwior, kyril, dawnt, kjoftherock

T Bartoszynski had some articles about that back in 1996. Somehow I think it might be a bit premature for this logic course, considering kyril is starting at the beginning.

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### Sems that the beginning is where you(0+ / 0-)

spell out the limits of what you are about to do.  Or do you like leading people on?

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### It strikes me as similar to physics(2+ / 0-)
Recommended by:
kyril, don mikulecky

...teaching, in which I remember being told "the proof of X is assumed--we'll show later why it's true" in order not to overwhelm the new student.

And it makes sense to me--after all, lots of people had lots of time to originally puzzle out what I'm being taught in just a quarter or two, so avoiding some of the internal, interlocking details seems reasonable.

Unless, of course, the goal is to either "forget" to provide the later proof of X, or wind up in a self-referential loop in which what we proved by assuming X is used to prove X.

"Certainly the game is rigged. Don't let that stop you; if you don't bet, you can't win." Lazurus Long

[ Parent ]

• ##### The similarities with physics teaching are(1+ / 0-)
Recommended by:
kyril

stronger than you realize.  Modern complexity theory turns classical relationships upside down.  For us, physics is a very limited and special topic, while complexity theory and "soft science" as well are generic.

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### If you are a physics teacher(1+ / 0-)
Recommended by:
kyril

And your post here indicates you might be, what approach would you take in instructing a student in the basics of logic? How do you view the subject? I ask this in all sincerity, since your perspective is different from mine (I am an applied mathematician).

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### I am a teacher of complexity science(2+ / 0-)
Recommended by:
kyril, LRLine

I have taught physics, biophysics, and have written a book on biomedical engineering.  I also taught three sections of philosophy of science to honors students for over 20 years.

It is time we came to our students with these simple facts.

Physics has created a surrogate world and serves us very well to a point.

The real, complex world has presented us with urgent problems that are beyond the scope of classical physics even though classical physics is useful to a point.

The most important aspects of the real complex world deal with things that are not even computable.  They involved closed loops of causality and things classical logic can not handle.

The applied math we use in these instances is not the usual analytical version so helpful in the surrogate world of Newtonian Physics, but category theory.

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### Questions for you, if you get this note(0+ / 0-)

and feel like answering these, of course...my husband and I have some connection with Santa Fe Institute, which pushes complexity science pretty hard for advanced students. I am not totally unfamiliar with it. I do view it as an advantageous system over reductionist systems although there are modeling problems.

At what grade level would you introduce concepts of complexity science if you had the ability to rewrite the science and math curriculums for elementary and high school students? Or do you believe that these things are best saved for college? I know of few schools moving in this direction although it might be a better direction to pursue than the single disciplinary systems we have at present.

Do you believe that the current curricula in biology and chemistry is meeting the needs of students? What about physics?

I did not get from your post here exactly at which level you were teaching, it sounds like college? Or are you teaching honors high schoolers?

Is there any benefit, in your opinion (which is a considered one here due to your experience) in teaching subjects like classical logic to students in preparation for mathematics? Do you advocate teaching mathematics along currently constructed subject lines, or would you rewrite the curricula to push such things as CAS and systems (linear and non-linear) for modeling?

How would you advocate building a background for this emerging discipline in elementary school?

Thanks again.

Thanks!

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### I would tell the truth from day one(0+ / 0-)

Especially for the youngest.  Let them grow up knowing how we use science, what its strengths and weaknesses are.

As for the S. F. I. They are recovering reductionists just like I am.  The twelve step program is a tough one.  Few really get cured.

The biggest casualty from all this denial is the ability for religion to pretend it is science.  When scientists act religious about science they invite this.  It is totally unnecessary.

That is why the young are my hope.

By the way, in 1967-68 I was involved in an open structure K-12 experimental school system in Boston.  My own kids went there and I was Chairman of the Board.  I have some experience in all this.

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### PS ,,,,my honors courses were undergrad.(1+ / 0-)
Recommended by:
LRLine

I taught medical school most of my life.  My experience in Boston was while I was teaching ar Harvard Medical School.  I have published a lot on all this if you go to my web page.  Just click on my name.

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### Thanks for answering. (2+ / 0-)
Recommended by:
kyril, don mikulecky

Certainly your experiences in life and your accomplishments speak to a revision of our educational system (which I personally believe is in order -- top to bottom).

Again, thank you for your time.

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### Thanks. I guess it is never too late. n/t(0+ / 0-)

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### Myself personally? Oh yeah, I love to lead(2+ / 0-)
Recommended by:
Foxwizard, kyril

people down rabbit holes. That's why, in general, I don't write diaries here, preferring instead to publish books and papers and make students whine and groan, etc.

However, it's kyril's diary, so sie calls the shots. So far I like what I see.

The beginnings of "logic" per se would stretch far beyond what the norm would be for most diary lengths, and if kyril breaks it into a couple pieces, I think it's a good plan.

And no, I don't necessarily agree that the beginning is where you spell out the limits of what you are about to do. I certainly don't find that to be the case in broken text, in English, and in a way, outlining in words the very basics of symbolic logic is like using broken text.

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### A bit too complex(7+ / 0-)

for today, although certainly fascinating.

Math Kos II is open.

[ Parent ]

• ##### Where you lead,(3+ / 0-)
Recommended by:
Pluto, Jimdotz, kyril

we will follow.

"We the People of the United States...." -U.S. Constitution

[ Parent ]

• ##### My first three diaries were an attempt to (2+ / 0-)
Recommended by:
elwior, kyril

introduce the ideas.  That was over a year ago.  Maybe I'll try again.

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### Yes, please do!(2+ / 0-)
Recommended by:
kyril, don mikulecky

"We the People of the United States...." -U.S. Constitution

[ Parent ]

• ##### OK It doesn't take much to get me going.(3+ / 0-)
Recommended by:
LynChi, elwior, kyril

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### for today? ummm Russel was stumped(1+ / 0-)
Recommended by:
kyril

by them a long time ago.  most of modern complexity theory rests on these ideas.

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### Yes(2+ / 0-)

Indeed. Which makes complexity theory too complex for today :) (I'm not implying I'll ever get to it; it's not exactly one of the areas I understand well enough to discuss in any detail)

Math Kos II is open.

[ Parent ]

• ##### I understand. Maybe we can do one together(2+ / 0-)
Recommended by:
3goldens, kyril

one of these days.  A sort of back and forthh discussion as a diary.

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### I hope so!(3+ / 0-)
Recommended by:
3goldens, Jules Beaujolais, sjr1

Maybe a good time to bring it up might be after the next logic installment - I'll see how far I can get.

Math Kos II is open.

[ Parent ]

• ##### sounds fine. n/t(1+ / 0-)
Recommended by:
kyril

An idea is not responsible for who happens to be carrying it at the moment. It stands or falls on its own merits.

[ Parent ]

• ##### I am unaware of these complexity theories, being(1+ / 0-)
Recommended by:
kyril

but a poor software engineer who continues to be a student of history, phlosophy and theology. But I would share that I think it is impossible to advance to the higher levels without establishing the foundations; and my observation is that far too many people have not established these foundations.

Let's start at the very beginning.

A Patriot

[ Parent ]

• ##### Yes, we will get to the limits(0+ / 0-)

I plan to write a diary about the proof that you cannot define Truth syntactically in a system with Excluded Middle. The construction in Gödel's theorem would allow you to specify a sentence that was neither True nor False.

Now what is this about "Impredicative loops of cause"? That phrase is not found in a Google search. In fact, Judith Rosen makes clear the difference between "Causal" and "Impredicative" loops in her father Robert's theory.

Busting the Dog Whistle code.

[ Parent ]

• ##### I just dont know how you can dis those (15+ / 0-)

great Boolean symbols.  I loved them when I was and electronics technician in the Navy.

• ##### They don't communicate a lot of meaning n/t(3+ / 0-)
Recommended by:
mattman, kyril, kjoftherock

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### and you think XOR, NAND, and NOR communicate(5+ / 0-)
Recommended by:
mattman, skrymir, LordMike, kyril, Nailbanger

Meaning? If-Then? Only if? They only communicate meaning after they are defined, just like the poor dissed symbols (which are very suggestive in their construction)

-8.00, -6.87
Hope. Peace. Integrity.

[ Parent ]

• ##### Not really(2+ / 0-)
Recommended by:
AaronInSanDiego, kyril

But I especially think that drawing some complicated graph of gates doesn't really express an algorithm, which is usually the intent.  If the concept you care about is "cyclic shift", then just saying so is easier most of the time.

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### The intent is to translate minterms into hardware(3+ / 0-)
Recommended by:
AaronInSanDiego, rapala, kyril

The symbols used when drawing combinatorial logic schematics represent actual hardware devices that an engineer can buy. They translate the minterms developed using Boolean Algebra minimization techniques into actual devices. One can buy an IC(74XX00) containing NAND gates, or NOR(74XX02) gates, or NOT (74XX06) inverters, etc. That is why learning the symbols is useful and important for engineers.

Also, combinatorial logic is typically not used to express an algorithm - that is a function typical of sequential, clocked logic circuits such as an "algorithmic state machine". And von Neumann machines like microprocessors.

-6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

[ Parent ]

• ##### Kind of(4+ / 0-)
Recommended by:
Garrett, skrymir, rapala, kyril

You need an unlimited, random-access storage and a state machine to make a Turning Machine.  No credit should go to John Von Neumann for the logical construct of computability.  JvN described the first implementation of the machine Turing had dreamed up (what became the EDVAC).

I teach this every day.

`You have a much too willful will. You think that what you do not do... does not happen' -Zen master to E. Herrigel

[ Parent ]

• ##### Aren't all modern uPs based on von Neumann?(1+ / 0-)
Recommended by:
kyril

As in "stored-program computers". I think I was accurate in my statement. Other types of Turing machines, like the "Harvard" architecture, do things differently and are certainly capable of implementing algorithms, but I fail to understand the objection to describing modern uPs as von Neumann machines. That was the description used everywhere I ever worked and at university(a long time ago). Has von Neumann gone out of style?

As far as the "logical construct of computability" goes, if you say Turing owns it, I'll agree. But von Neumann invented the basic architecture of almost all modern computers and uPs.

-6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

[ Parent ]

• ##### As I said(1+ / 0-)
Recommended by:
kyril

"JvN described the first implementation of the machine Turing had dreamed up (what became the EDVAC)."

and he did.

An FSM != a TM was my point.

A TM (or better a UTM) is a state machine plus an unlimited random access memory.

von Neumann proposed one possible implementation, the EDVAC.

`You have a much too willful will. You think that what you do not do... does not happen' -Zen master to E. Herrigel

[ Parent ]

• ##### OK, I see. JvN stole Turing's idea?(1+ / 0-)
Recommended by:
kyril

I hadn't realized there was a controversy over where von Neumann got the ideas for his paper. I'm all for historical accuracy, but the convention has been, during my working life, that Turing invented the computer but von Neumann invented the modern architecture.

-6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

[ Parent ]

• ##### I'd hazard to say that(2+ / 0-)
Recommended by:
skrymir, kyril

the von Neumann EDVAC is not the modern architecture.  In fact, we are closer to the Harvard architecture today with split I and D caches, and non-overlapping text and data paging systems.

Even Intel's modern architectures (P6 defendants, Atom, and Willamette) treat self-modifying code (which was a cornerstone of EDVAC) as a fire drill whenever it is encountered.

`You have a much too willful will. You think that what you do not do... does not happen' -Zen master to E. Herrigel

[ Parent ]

• ##### and my main point in the original post was that(2+ / 0-)
Recommended by:
skrymir, kyril

the progression should be combinational to FSM ** to TM, not JvM, which is a subset of TMs.

(** insert push-down automata in this slot if you want to be entirely pedantic)

`You have a much too willful will. You think that what you do not do... does not happen' -Zen master to E. Herrigel

[ Parent ]

• ##### OK, I see. I was just thinking of the difference(1+ / 0-)
Recommended by:
kyril

between combinatorial(combinational?) logic and logic circuits that execute algorithms, as the commenter above me had claimed that combinatorial logic schematics were intended to describe an algorithm, which is incorrect.

And you're certainly correct about the recent changes in uP architecture, especially concurrent instruction execution, branch prediction, separating I and D streams, and caching schemes. But the bottleneck is still the interface to external memory. Is anyone doing any work on multi-level logic? I've always thought that 4-level logic(every input/output has 16 states) would be a good way to increase throughput without more speed or circuitry. We used to build analog computers that implemented very complex functions very rapidly, and I believe the IC technology(mixed analog & digital) now exists to implement something like this, although speed might be a problem.

I'm afraid I'm not really up to speed on the most recent uP architectures - I got promoted to my level of incompetence many years ago, although I still dabble for fun and to check for senility.

-6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

[ Parent ]

• ##### On multi levels(2+ / 0-)
Recommended by:
skrymir, kyril

It's 'combinational,' and 'combinatorial' would be quite bad news if we were to build it :)

As for multi-level logic, it comes in fads.  Then people realize that because of the resistive drops on wires (no matter how good they are), multi-level logic (where multi > 2) would require tighter noise margins and thus shorter wires.  Process variation is causing undue grief as we plunge towards single-digit nanometer transistor fabrication processes.  Trying to accomodate multi-level logic would add to the grief unfortunately.  The net result is binary is here to stay!

What is rather interesting is that we can no longer crank up the frequency.  We've reached the limits of forced air cooling, or even heat pipes.  But we still have more silicon real estate due to a combination of cleaner processes (leading to larger dies) and smaller transistors.  What to do?  What to do?

Larger caches help, but at some point the entire program working set will be in-cache and thus adding more cache memory isn't going to give a big breakthrough in performance.

This is why we are now putting down more than one processor core on a die.  What's bad about that is it doesn't speed up single-program performance the way that "under the covers" techniques like superscalar have done.  To truly squeeze more performance out of these parallel systems, the programs themselves must change.  That means programmers have to think in parallel.

This is leading to a revolution in computing that is relatively rare in the history of computers.  Fun times indeed!  If you're an old timer like me, though, it might be a good time to retire! :)

`You have a much too willful will. You think that what you do not do... does not happen' -Zen master to E. Herrigel

[ Parent ]

• ##### I keep telling people those extra cores don't(2+ / 0-)
Recommended by:
tmc, kyril

make their computers run faster because current programs are written for only one CPU. You'd be surprised how many people believe they do(advertising). I'm waiting for the first Microsoft OS that uses multiple cores - should be about 500GB of solid bloatware and require about 50GB of system memory.

The noise problem has always been there for multi-level logic. But, as you say, we have about reached the speed limit so I thought someone must be working seriously on multi-level as an alternative method of increasing throughput. I still think it's a good idea - use larger features and diff amps to maintain signal integrity. Of course, the only IC design I've ever done is some simple ASIC stuff and gallium arsenide interfaces, so what the hell do I know. I retired 10 years ago.

-6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

[ Parent ]

• ##### Best. Logic. Sig. Ever! n/t(0+ / 0-)

Busting the Dog Whistle code.

[ Parent ]

• ##### No, frankly, those are pretty icky too.(2+ / 0-)
Recommended by:
mattman, theran

But I'm not quite sure how the symbols are suggestive; how does a rocket-ship-thing connote "or" on first sight?

Math Kos II is open.

[ Parent ]

• ##### maybe I am thinking of different symbols...(4+ / 0-)
Recommended by:
rcd, kyril, MKSinSA, sjr1

my "or" looks like a "V", when I get lazy, I just make a "U".

Logic really is just set theory with different words.

Statement = Set A
OR = Union
AND = Intersection
NOT = Complement ~A
XOR = (A U B) / (A /\ B) blah blah blah...

-8.00, -6.87
Hope. Peace. Integrity.

[ Parent ]

• ##### Ohh, no, those are pretty awesome(3+ / 0-)
Recommended by:
mattman, rapala, sjr1

I don't mind them at all. Quite suggestive. They make me happy. Those are the ones that mathematicians use, and math notation is actually quite good overall (with the exception of function notation).

No, the ones I hate are the graphical representations used in circuit schematics (OR is a weird rocket-ship thing, NOT is a triangle with a circle, AND is a rectangle with one rounded end).

Math Kos II is open.

[ Parent ]

• ##### We used those in formal logics(1+ / 0-)
Recommended by:
kyril

in college.  Here's a table.

[ Parent ]

• ##### I think the symbols are a better shorthand(1+ / 0-)
Recommended by:
kyril

not sure how to do them here, though.

• ##### Which?(1+ / 0-)
Recommended by:
plf515

The v,^,~ symbols? I like those, and the and and not come out nicely, but there's no good or; v doesn't look right.

Math Kos II is open.

[ Parent ]

• ##### You can get them all in unicode(1+ / 0-)
Recommended by:
kyril

Even if they are somewhat ugly:

⋀⋁¬

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### I can make them in LaTeX, save to a file(2+ / 0-)
Recommended by:
Random Excess, kyril

convert to a png, move it to photobucket, and post...
but what a pain

• ##### Great example of solving a math problem(3+ / 0-)
Recommended by:
AaronInSanDiego, plf515, kyril

by reducing it to a previous case which has already been solve.

-8.00, -6.87
Hope. Peace. Integrity.

[ Parent ]

• ##### Just use the mathematician's trick: "Clearly..."(3+ / 0-)
Recommended by:
plf515, kyril, Nailbanger

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### Use(3+ / 0-)
Recommended by:
rcd, kyril, sjr1

'v', '&', '->', '~', for sentential connectives.

'(x)' and 'Ex' for quantifiers.

"What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet."--Woody Allen

[ Parent ]

• ##### If I'm going to use '&",(1+ / 0-)
Recommended by:
Garrett

I might as well just use '|' too, since I won't be getting into divisibility. It'll make the computer programmers happy.

Math Kos II is open.

[ Parent ]

• ##### Not so sure about that...(1+ / 0-)
Recommended by:
kyril

I like the C++ symbols myself, I think it all depends on what you're used to (I'm a computer programmer).

If I'm not an activist, would that make me a pacifist?

[ Parent ]

• ##### Technically, I don't believe there is much (1+ / 0-)
Recommended by:
kyril

meaning in the boolean operators; they are, after all just stating the realtionships of statements. Therefore they gain their meaning from the statements which they relate.

A Patriot

[ Parent ]

• ##### But they make complex statements (1+ / 0-)
Recommended by:
kyril

possible, with a fair economy of effort.

A Patriot

[ Parent ]

• ##### Heh, those tech days for me(7+ / 0-)

were when cores were king and mos-ram was a baby.

• ##### how about (3+ / 0-)
Recommended by:
mattman, kyril, MKSinSA

magamps? i used to have to troubleshoot those babies inn the Navy.

• ##### Which ones?(3+ / 0-)
Recommended by:
tmc, rapala, MKSinSA

The circuit-design ones? They're horrible to draw, and the visual design of them doesn't really communicate their meaning. Once you train yourself to read them they're not that bad, I suppose.

Math Kos II is open.

[ Parent ]

• ##### Well, to draw them I used templates(5+ / 0-)
Recommended by:
mattman, rapala, kyril, MKSinSA, kjoftherock

and to trace circuit logic it is much easier than a string of operations.
And, they were kinda suggestive, as others have noted.

• ##### To each his own I guess(3+ / 0-)
Recommended by:
rapala, kyril, Nailbanger

to me the circuit symbols for AND, OR, NAND and NOR are beautiful. :)

`You have a much too willful will. You think that what you do not do... does not happen' -Zen master to E. Herrigel

[ Parent ]

• ##### i was an ET also(6+ / 0-)

nuclear type, one each.

• ##### Here's some Boolean(3+ / 0-)
Recommended by:
kyril, Nailbanger, MKSinSA

art for your viewing pleasure. : )

• ##### There great when you're writing(2+ / 0-)
Recommended by:
kyril, Nailbanger

stuff out.  But they seem pretty awkward for the internet.  Especially when yo start doing proofs.

[ Parent ]

• ##### I didn't understand this diary until after I'd(22+ / 0-)

smoked some weed. Thanks for bringing back some old memories. Hmmn, I love the smell of maths and philosophy in the morning.

• ##### The fine line between science and art sometimes(13+ / 0-)

needs a catalyst.

The Tutoring Room will always be open and updated (new diary) on Wednesday mornings 1/21 Methods:http://www.dailykos.com/story/2009/1/21/3457/72066/529/686957

[ Parent ]

• ##### Science (and Math) is the Art of the Universe!(7+ / 0-)

But, yea, sometimes a little help in seeing it doesn't hurt.

-8.00, -6.87
Hope. Peace. Integrity.

[ Parent ]

• ##### There are those who argue that the Universe...(2+ / 0-)
Recommended by:
algebrateacher, kyril

must be finite in extent because Logic requires its Universal Set to be well-defined, and the Universe is logical.

I'm not sure I buy it, but it's a cool idea.

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### the universe is not logical(2+ / 0-)
Recommended by:
algebrateacher, Jimdotz

working consciousness to raise consciousness

[ Parent ]

• ##### I disagree. The Universe makes perfect sense.(1+ / 0-)
Recommended by:
kyril

It's just that we are only beginning to understand its true nature. We are living in the Golden Age of Cosmology. What a wonderful scientific time to be alive!

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### Amen to that!(1+ / 0-)
Recommended by:
Jimdotz

(I'm actually hoping to study cosmology in grad school when I get there. Truly fascinating stuff.)

Math Kos II is open.

[ Parent ]

• ##### I'm jealous.(1+ / 0-)
Recommended by:
kyril

This is a time that will be rememebered forever. I see it as the beginning of the Deductive Era of Science, as opposed to the Inductive Era we are currently in.

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### the greatest truth(1+ / 0-)
Recommended by:
Jimdotz

is that there is no "greatest truth."

working consciousness to raise consciousness

[ Parent ]

• ##### No, it's because a finite big bang cannot(0+ / 0-)

expand to infinity in finite time. Pay no attention to the Flatness Problem pseudoscientists.

Busting the Dog Whistle code.

[ Parent ]

• ##### Is this the state in which you learned(9+ / 0-)

it originally?

moderation in everything ... including moderation

[ Parent ]

• ##### That I can't remember. But since I got some of (2+ / 0-)
Recommended by:
algebrateacher, kyril

the suppressed old memories back with the aid of the 'catalyst', it seems likely i was trying to recapture dem good ol' days!

• ##### In the old Perry Mason series(3+ / 0-)

he could often find the answer by "re-enacting the crime."

moderation in everything ... including moderation

[ Parent ]

• ##### You should expand on the idea of validity...(11+ / 0-)

...to include concepts like soundness and cogency.  But to do that, I guess, you have to introduce the difference between deductive and inductive arguments.

This is a cool idea -- there have been many times around here when I thought a good logic lesson would be beneficial...

(-9.62/-6.77) edscan 1/21/09 -- the truth will be revealed...

• ##### "this statement is false"(9+ / 0-)

an excellent example of a sentence that is not a legitimate logical statement.

Nice diary, kyril.  Good stuff.

(-8.00,-7.85) 'The idiot is not our greatest problem. It is the moron type that is our great problem." -- H. H. Goddard

• ##### There is no proof of this statement.(6+ / 0-)

SSDD. EOM.

-8.00, -6.87
Hope. Peace. Integrity.

[ Parent ]

• ##### Paradox (5+ / 0-)

Ah, the joys of logic...

What has been is what will be, and what has been done is what will be done, and there is nothing new under the sun.

[ Parent ]

• ##### The classic case of confusing...(2+ / 0-)
Recommended by:
ATinNM, kyril

Truth, Meaning, and Validity. Truth is applied to statements as a whole, Meaning to its internal language, and Validity to arguments.

The whole "this statement is false" controversy seems important, but it's not.

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### It is rather important(2+ / 0-)
Recommended by:
Jimdotz, kyril

since it is the basis, or maybe the seed, of Godel's proof, which shook the foundations of math.

• ##### Inspirationally, yes, but as a matter of ongoing(2+ / 0-)
Recommended by:
plf515, kyril

logical paradox, it's actually quite easy to get passed it merely by understanding definitions.

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### See Kurt Godel(2+ / 0-)
Recommended by:
bubbanomics, kyril

You have written a sentence that shows the system is incomplete.  It's incomplete because it can express itself.

`You have a much too willful will. You think that what you do not do... does not happen' -Zen master to E. Herrigel

[ Parent ]

• ##### self-referential things(2+ / 0-)
Recommended by:
tmc, kyril

are almost always great counterexamples in mathematics.  Godel's incompleteness theorem is really one of the great intellectual achievements.

(-8.00,-7.85) 'The idiot is not our greatest problem. It is the moron type that is our great problem." -- H. H. Goddard

[ Parent ]

• ##### counterpoint(2+ / 0-)
Recommended by:
algebrateacher, kyril

Just as mathematics and computers are tools for science, logic is largely a tool for mathematics, although there are certainly elements of it which are subjects of current research as fields in their own right.

Counterpoint

Just as mathematics and computers are tools for science, logic is a tool for mathematics.  There are elements of it which are subjects of current research as fields in their own right, philosophy and mathematics.

======
Peace. It's cheaper, and more fun. We've already tried everything else.

• ##### Myself, I try not to mix(1+ / 0-)
Recommended by:
kyril

math with philosophy. Not sure Russell did us any favors in that area at all...

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### I think it's impossible to do (1+ / 0-)
Recommended by:
kyril

philosophy without some implicit understanding of the way connectives and quantifiers work.  This is true even if the philosophy is written entirely in English. This is more true in certain areas of philosophy (mostly contemporary stuff) than in other areas, I think, but still true in general.

"What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet."--Woody Allen

[ Parent ]

• ##### You're probably right.(1+ / 0-)
Recommended by:
kyril

I'm pretty hardened at the moment into Wittgenstein, and successively, Avril Strum. Wittgenstein left logic at the Tractatus, and I spend most of my work there still working through his concept of language games. You're totally correct in your remark about "even if the philosophy is written entirely in English" because in different languages, the meaning is either sharper or more dense (depending on which language you are reading in). English has its own problems.

I know about logic from my work in mathematics. As such, I tend to view it as its own subarea. I am in amazement, frankly, that someone here even attempted to diary the subject, and am in admiration of kyril's attempt and it's continuance.

It's a credit to the dKos community that people would come forward and attempt to educate in a diary length forum.

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### I'm tipping and rec'ing(16+ / 0-)

but can't participate -- you folks are way over my head -- I'm math-challenged.  I try, honestly I do, but the powers of the universe just meant me to be an internationally famous and well beloved literary genius.

You'll never have a quiet world till you knock the patriotism out of the human race. - G.B. Shaw, "Misalliance"

• ##### aw (hugs)(10+ / 0-)

I hope you come back next time. We'll talk about informal logic and logical fallacies in written English arguments, which should be a pretty comfortable area for you as a literary type :)

Math Kos II is open.

[ Parent ]

• ##### I always read,(7+ / 0-)

and get educated to the extent my ability allows.  Your diary is one of the many I enjoy here, along with the various and sundry hard science diaries.  I love reading how many truly educated folks are members of the community.

You'll never have a quiet world till you knock the patriotism out of the human race. - G.B. Shaw, "Misalliance"

[ Parent ]

• ##### I myself did not get math at all(7+ / 0-)

until I took logic.

• ##### I actually excelled in logic(9+ / 0-)

which may be hard to believe -- but once anyone starts using actual math terms -- "binary" being one -- my brain freezes over.

I keep reading because every bit of info that breaks through my ignorance is a gift.

You'll never have a quiet world till you knock the patriotism out of the human race. - G.B. Shaw, "Misalliance"

[ Parent ]

• ##### Only computer scientists say binary :) n/t(3+ / 0-)
Recommended by:
gchaucer2, kyril, kjoftherock

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### In mathematics we talk all the time about Binary(6+ / 0-)

As in Binary Operators. It's like the Thunder Dome, Two terms enter, one term leaves.

-8.00, -6.87
Hope. Peace. Integrity.

[ Parent ]

• ##### sure ....all of the usual arithmatic (3+ / 0-)
Recommended by:
theran, gchaucer2, kyril

operations are binary operations.

Of course there are operations where two items enter and the output is a dissimilar object (e. g. inner product of vectors)

When liberals saw 9-11, we wondered how we could make the country safe. When conservatives saw 9-11, they saw an investment opportunity.

[ Parent ]

• ##### Those are getting to the fun stuff...(5+ / 0-)
Recommended by:
Garrett, theran, onanyes, Jimdotz, kyril

Coordinate-free linear algebra. Yay! No matrices, just operators!!!

-8.00, -6.87
Hope. Peace. Integrity.

[ Parent ]

• ##### My linear algebra students won't see it that way(4+ / 0-)
Recommended by:
Garrett, theran, Jimdotz, kyril

at least, not most of them.

Those that do....well...hopefully I'll see them in the second course. :-)

When liberals saw 9-11, we wondered how we could make the country safe. When conservatives saw 9-11, they saw an investment opportunity.

[ Parent ]

• ##### How about DKos Eigen-Analysis? (2+ / 0-)
Recommended by:
onanyes, kyril

What would that look like?

Finding the un-Principaled Components of Dick Cheney's value system?

• ##### eigen analysis(1+ / 0-)
Recommended by:
kyril

So, we need to find something whose "direction" is kept constant after our transform is applied...or something whose direction is kept the same by our effort.

I'll have to think about that one....

When liberals saw 9-11, we wondered how we could make the country safe. When conservatives saw 9-11, they saw an investment opportunity.

[ Parent ]

• ##### Good point(2+ / 0-)
Recommended by:
gchaucer2, kyril

For some reason I rarely write the word though.

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### Don't get me started.(4+ / 0-)
Recommended by:
theran, 3goldens, elwior, kyril

I'm just happy that I know how to login to the intertoobz.

You'll never have a quiet world till you knock the patriotism out of the human race. - G.B. Shaw, "Misalliance"

[ Parent ]

• ##### Don't get thrown by a word!(3+ / 0-)
Recommended by:
Jules Beaujolais, gchaucer2, kyril

"We the People of the United States...." -U.S. Constitution

[ Parent ]

• ##### There are 10 kinds of people...(3+ / 0-)
Recommended by:
Caneel, kyril, LRLine

those who understand binary numbers and those who don't.

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### It is not so bad(6+ / 0-)

The problem with "logic" is that you're interested in "truth," which is hard to describe.  The big advance in the early-mid part of the last century was a standard program:

1. Define the kinds of things you can say (what we have here)
1. Define a way of interpreting these statements (formal truth, not here at all)

Together these give you a "logical theory", at which point people ask questions like:

1. Are all the statements you can make either true or false?
1. Can you prove everything that's true?
1. Can you prove anything that is not true?
1. How fast can you do it? (This is known as computer science and combinatorics, in normal language.)

But if you want to take away anything from this discussion, it's that philosophers came up with some ideas about how to understand "truth" using mechanical manipulations.

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### Nonsense. You too can learn the (8+ / 0-)

ways of math. It just has to be done in short shots...15 to 20 minutes per day, one small step at a time. It takes roughly 100 hours before the lightbulb will come on, but when it does, it becomes unbelievably transparent -- complete with head rush.

Overcoming "math anxiety" and "being math-challenged" is the thing I do best. Kids, adults, it doesn't matter. It's like learning a new language in a way...more complex than say, Spanish (for an English speaker), and less complex than say, Arabic (for an English speaker). <-- just an example, insert language of choice here.</p>

Please come back, and follow kyril's great diaries. You'll learn something for sure and no one will be the wiser...you'll still be our well-beloved literary genius. We won't tell a soul.

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### I always read the diary(6+ / 0-)

and comments -- is I learning?  You betcha!

You'll never have a quiet world till you knock the patriotism out of the human race. - G.B. Shaw, "Misalliance"

[ Parent ]

• ##### (learnin')(1+ / 0-)
Recommended by:
kyril

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### Exactly. I teach HS math(6+ / 0-)

Geometry to be exact.  Thank you for saying what you said.

When kids get frustrated, throw up their hands and tell me "I just can't get math" or "I suck at math" or something to that effect, I give them a similar pep talk.  I work closely with them through the year and it is so cool to see them slowly progress and begin to "get it".

What is frustrating to me is that their initial bad attitude towards math is way too often reinforced by adults in their lives.  If adults dismiss math as impossible, why should kids think they can do it?

And we wonder why our math skills suck in this country.

Labor unions - the people that brought you the weekend.

[ Parent ]

• ##### Yep, you've expressed our national ill(4+ / 0-)
Recommended by:
Garrett, TexH, Mother of Zeus, kyril

here very well. If I had a dime for every person, man, woman or child, who said that very thing, I'd be rich beyond my wildest dreams.

The trick is this: if the kids didn't get whatever support they needed for basic math and reading skills by 5th grade, it gets to be a real mind meld problem to get them over the resultant anxiety that results as the material gets more and more complex. This is the single biggest reason why the No Child Left Behind Plan doesn't work. In fact if anything it reinforces and shores up failure memes -- where remedial skills should have been slowly and methodically applied without pressure on any child to succeed beyond the level at which they were comfortable! Talk about immediate FAIL!

This idea here isn't new, it happens to be borne out in the work of the Kumon people -- they are a hugely successful supplemental educational system started some 50 years ago and are now international in scope, and (as far as I know), by other people involved in educational endeavors (the old Sylvan learning system), not to mention the great texts by John Saxon (whom I consider the last word in how to study math). I have all sorts of textbooks over here from fifty years ago, and the difference in how work is structured and measured is enormous.

Why fifth grade? I wish I knew, but I have a feeling it has a very great deal to do with neurological development, and an integration that occurs at that time into peer networking. From that point on it's one tough struggle for even the math aptitude child to make headway.

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### I was placed in a crazy "academically(2+ / 0-)
Recommended by:
kyril, LRLine

talented program" in 1st grade, based solely on an I.Q. (and, I think, school psych eval) given in kindergarten.  The goal of the program was to get all kids 2 grade years ahead in math and science and 1 year ahead in all other subjects by the end of elementary school.  I was totally not gifted enough in math to succeed.  My first really bad grade was a "D" in algebra in 5th grade.  It was crushing and I was convinced I could not do math.  Thank God those kinds of segregated, accelerated programs do not really exist anymore.

When we changed school districts after the next year, I refused to take any accelerated classes and went BACK two years in math.  I was always able to do fairly well after that since a lot of it was review, but I never challenged myself in math again and I stopped taking it as soon as I was allowed to in, I think, 10th grade.

And most math teachers that I've ever had really are not very good.  It is a real gift to teach math well.  I think our friend kyril here seems to have the gift.

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### You're far too kind(1+ / 0-)
Recommended by:
LRLine

I'm only a student. I've had the privilege of having some truly astoundingly talented teachers, however, on whose lecture techniques I've been able to model a number of my explanations.

Math Kos II is open.

[ Parent ]

• ##### It happened to my son in 3rd grade(2+ / 0-)
Recommended by:
kyril, LRLine

He started faltering in math in a big way, said he hated it, and convinced himself he was no good at it.  We got him into a small group remedial group.  He thrived, and by the end of that year, it was his favorite subject.  Better yet, his reward if they finished the lesson early was that he could do his homework in class.  (My son, you must understand, is opposed to homework in principle).

Then it happened again this year, in eighth grade.  He nearly failed the second quarter.  This time, there were different reasons.  He had missed some things due to a long, but temporary change.  I told them how he responded to small group/individual remedial math back in elementary school, so they agreed to try it again.  It's working beautifully.  He's getting 90s and 100s again.

If teachers were able to get the timely, remedial help for kids in math classes, it would do a world of good.  Often it isn't needed for very long either, as in my son's case, less than one school year.

I am (perhaps overly) vigilant with my kids and math.  My brother fell behind in math somehow, in the early grades, and it caused him so much trouble for the rest of his school years, and I believe it caused many other problems as a result.  Somewhere along the line in adulthood he changed all that, and now he teaches.

"The true measure of a man is how he treats someone who can do him absolutely no good." --Samuel Johnson

[ Parent ]

• ##### Still in her 50's, I heard my mother(2+ / 0-)
Recommended by:
kyril, LRLine

make idiotic jokes about how she flunked math because she asked the teacher if she should use blueberry or cherry "pi"!  She still thought the joke was cute all those years later.

She still counts on her fingers.  It makes me very sad.

I'm no math whiz, but at least I see the value of trying.  And so with each generation, we can hope to improve, I guess.  I have at least the confidence to try.  I'm hoping my kids will have both the confidence and the concrete support to actually succeed. That'd be nifty.  To see them really and truly "get it."  I'd love that.

And I'm counting on our resident genius, kyril here, to help me with the foundation to give them better concrete support.

No pressure, here, kyril. It's not like future generations are counting on you, or anything.

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### It makes my head itch(2+ / 0-)
Recommended by:
Mother of Zeus, kyril

and my brain reel. I wish I could "get it" because I am the only one in my family that just has never been able to "get it".

I have a Phd Dad (organic chemistry) and a Phd brother (Climatologist). I always feel like I have been smoking pot around them.

The meaning of life is to live it.

[ Parent ]

• ##### Always, sometimes, never. Inductive vs. deductive(10+ / 0-)

Hypothesis vs. conclusion.  Counterexamples.

I save these subjects for my eighth grade Algebra students until the end of April to give their brains as much time as possible to mature before testing.

Talking about functions is bad enough.  Every year, I have students who are "funny" because they refuse to say "functions" correctly.  And function notation?  A day for great patience.  The eff of ex?  Hey, David, eff your ex?

The Tutoring Room will always be open and updated (new diary) on Wednesday mornings 1/21 Methods:http://www.dailykos.com/story/2009/1/21/3457/72066/529/686957

• ##### I feel for you. n/t(2+ / 0-)
Recommended by:
algebrateacher, kyril

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

That's all.

• ##### Epistomology(6+ / 0-)

I'm looking for part II to cover induction and the relevence of logic to modern science.

We're all just monkeys burning in hell. SmokeyMonkey.org

• ##### Wow. This Should be in the FAQ(8+ / 0-)

(oh, just ignore my cat)

• ##### "Truth is beauty(5+ / 0-)

Beauty is truth
That's all ye know
That's all ye need know"

Who said it?

"We the People of the United States...." -U.S. Constitution

• ##### Keats!(3+ / 0-)
Recommended by:
bdizz, john de herrera, elwior

Math Kos II is open.

[ Parent ]

• ##### Very good Professor!(3+ / 0-)

Actually it's:

"Beauty is truth, truth beauty,—that is all
Ye know on earth, and all ye need to know."

Part II - What is it from?

"We the People of the United States...." -U.S. Constitution

[ Parent ]

• ##### Hmm...(2+ / 0-)
Recommended by:
Jules Beaujolais, elwior

Um...that, I don't know. The Google would tell me, but that would be cheating.

Math Kos II is open.

[ Parent ]

• ##### Well, here's a hint then from the riddle(1+ / 0-)
Recommended by:
kyril

What's a Greek Urn?
(I dunno, about 10 bucks an hour.)

"We the People of the United States...." -U.S. Constitution

[ Parent ]

• ##### Oh!(1+ / 0-)
Recommended by:
elwior

Ode on a Greek Urn. Or something like that.

Math Kos II is open.

[ Parent ]

• ##### Hmmm, so close. Judges?(2+ / 0-)
Recommended by:
joanneleon, kyril

Yes, we'll give it to you. The actual answer is Ode on a Grecian Urn, but we've got some liberal judges here.
I thought of this though earlier when someone was commenting about Logic being the link between Arts and Sciences, how this phrase explains that link.
Thanks for this wonderful diary Kyril.

"We the People of the United States...." -U.S. Constitution

[ Parent ]

• ##### I feel right at home here(8+ / 0-)

Takes me back to my days as an assembler programmer.

• ##### Yep. I speak 8008,F8, 8080, 6800,6502,Z-80, etc.(6+ / 0-)
Recommended by:
tmc, LordMike, rapala, kyril, Nailbanger, LRLine

Along with a bunch of others. The difference between now and then is that the line between hardware and software was a lot less defined then and we had to understand both. Programmers today never even heard of Booth's algorithm.

-6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

[ Parent ]

• ##### OMG what a great comment! n/t(2+ / 0-)
Recommended by:
LordMike, kyril

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### But, do you speak the binary language of(3+ / 0-)
Recommended by:
theran, kyril, Nailbanger

...moisture vaporators?  Do you speak bocce?  Is it like a second language to you? ;-)

I agree with you... these young whippersnappers nowadays don't have enough foundations in the basics, that's for sure!

If you've never done 6502 programming, you've never lived, and are probably way to young!

Go grab an Apple I emulator and get to work youngling!

Comments that say "GM workers should get retraining" without SPECIFIC DETAILS about those "new jobs" that never come are trollworthy

[ Parent ]

• ##### Actually, I go back long before that. (4+ / 0-)
Recommended by:
LordMike, Mother of Zeus, kyril, cs

I got out of grad school in 1964. I worked for a subcontractor on the Apollo program designing some of the first digital telemetry. I've designed things with vacuum tubes, with J/K flip-flops made of discrete components, had an old PDP8 as my first home computer, an IMSAI 8080 as my second, and have more old electronic junk piled up in CA than any man living.

I didn't really like the 6502 - I much preferred the Z80 because of the extra register set for use during interrupts.

-6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

[ Parent ]

• ##### You only do 8-bit arithmetic? You are OLD! nt(2+ / 0-)
Recommended by:
skrymir, kyril
• ##### Truth be told-I coded an Intel 8051 in assembler(2+ / 0-)
Recommended by:
skrymir, kyril

In 1983...

{:-(

• ##### I also used the 8035/39 uPs in force measurement(1+ / 0-)
Recommended by:
kyril

devices that interfaced between the strain gauges on load cells and the display/controls. These were the predecessors to the 8051. Try doing linearity correction math while maintaining a display with only 64 bytes of internal register that also holds the stack contents.

-6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

[ Parent ]

• ##### I wrote a speech coder with the 8051(2+ / 0-)
Recommended by:
skrymir, kyril

converted from mu-law to linear PCM, then built a kind of block floating-point to save bits (32k/sec vs. 64)...It was a (lousy) predecessor to ADPCM...

A lot of 'double-precision' (16-bit) arithmetic though...pain in the ass...

• ##### I have never seen the need to count beyond 255.(3+ / 0-)
Recommended by:
Garrett, kyril, LRLine

Hey, Booth's algorithm was waay cool at the time.

-6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

[ Parent ]

• ##### The PDP-8 was a 12-bit machine(3+ / 0-)
Recommended by:
Garrett, skrymir, kyril

You CAN count to 4095!

• ##### This one got me an F on a logic quiz in HS:(10+ / 0-)

No cat has 9 tails.
All cats have one more tail than no cat.
Therefore all cats have 10 tails.

• ##### Ah, it's spot the fallacy!(10+ / 0-)

One has to translate the statements into formal terms first:

There does not exist a C which is an element of the set "cats" such that C has 9 tails.
For all C that are elements of the group "cats" and all N that are elements of the set "not cats," C has one more tail than N.
Therefore, for all C that are elements of the group "cats," C has 10 tails.

Clearly invalid :)

Math Kos II is open.

[ Parent ]

• ##### The only thing I spotted was a(8+ / 0-)

teacher with no sense of humor.

Speaking of fallacies, my pet peeve is how "begging the question" has come to mean, "requires one to ask" ...or something along those lines. You see it used like that all the time now and it drives me nuts.

• ##### I'm with you on that(4+ / 0-)
Recommended by:

Drives me bonkers. So does the misuse of "strawman" when people really mean "reductio ad absurdum," especially when the reductio is actually used in a valid way.

Math Kos II is open.

[ Parent ]

• ##### I still don't get this.(1+ / 0-)
Recommended by:
cs

And this is the kind of thing that always just pisses me off and makes people fail math classes because it is intentionally trying to be misleading.  What is the point of that?

You could also "translate" the statements into formal terms as follows:

There exists an object N which is called "no cat" and N has 9 tails.

For all C that are elements of the group "cats," every C has 1 more tail than N

Therefore, all C have 10 tails.

I mean, why can't it be translated that way?  What is invalid about reading it as trying to define an object called "no cat" that has 9 tails?  It is obvious from the way it is written that it is trying to lead you to that: "All cats have one more tail than no cat" is a completely unnatural way to use language.  It's so strained and twisted, in fact, that it sets the minds of many people to work trying to figure out if there isn't a way to alleviate the awkwardness of that statement.  And lo and behold, there is a way, and it's not a logically invalid way.  On what possible ground can one insist that the first sentence be translated as meaning ~C rather than as defining a unique element "no cats"?  I'm not being cute here.  I don't see how it is possible to insist that ~C is the "correct" translation.  ~C may be a more natural translation in the first statement, but "no cats" is a more natural translation in the second statement.  Why is one "right" and one "wrong"?

Why do the creators of math problems do things like this?  Are they just trying to keep a certain type of brain from succeeding in the field?  Just trying to hurt kids' feelings?  I mean, I'm being serious here, because this is the kind of bullshit that keeps people from succeeding in math.

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### Well(2+ / 0-)
Recommended by:
theran, Mother of Zeus

The problem comes in because "no cat" can be interpreted in one of two different ways. When you read it in English, you naturally interpret the first sentence with one meaning and the second with another. It's clear (how, I'm not sure, but it is clear) that one is intended to read the argument with those dual interpretations. If that's the meaning you intend in the argument, then you have to carry over that intended meaning into the symbolic interpretation.

There is a way to force the argument to become logically sound, but if you do that, it loses its meaning. An argument in English is only sound if, when translated into symbolic logic with the essentials of its meaning preserved, it is valid. If we didn't make that rule, then people would be able to "prove" all manner of silly nonsensical things like all cats having ten tails (because the premises are true in English and the argument can be made to be symbolically valid).

Math Kos II is open.

[ Parent ]

• ##### But logic assesses(1+ / 0-)
Recommended by:
kyril

the validity, not the truth, of arguments.  I certainly don't view this syllogism as a proof that cats have 10 tails, because in fact there is no such thing as an object called "no cat" with 9  tails.   (false premise --> false conclusion) Nonetheless, one of the premises appears to ask me to assume its existence for the purpose of the argument, and I'm willing to do so.

So many examples in logic class are composed of amusing, made up objects like unicorns, and we students obediently set to work evaluating their validity without a thought in the world for their truth.  So exactly how is one supposed to know when it is time to become all serious about preserving  intended meaning.  Truly, the "intended meaning" of these statements is not at all clear.  In fact, it is intentionally unclear.

It has often seemed to me as though I can understand  math halfway decently, but can only rarely second-guess the teacher's whimsy on the test.  So many times I have found myself taking a math test and unable to figure out if the test-maker expects me to zig or to zag on that particular question.  Sometimes I feel as though math teachers make up tests with little regard for the fact that they sort of weave back and forth between expectations of pinched formality and more holistic natural language but never really let on that they're doing it.  There seems to be some sort of instinct about when to zig and when to zag, and I've never had it, and I'll tell you that you won't find it anywhere in the textbook either.

Reminds me of a question on my stats final where the prof asked us to "illustrate" something.  I wrote a lengthy explanation of the concept, with formulae and examples and whatnot.  Got zero credit for the answer.  Why?  He actually wanted us to draw a diagram or picture, to "illustrate" it in that way. That's what he had in mind.  And even though my answer was correct, I got zero - not even partial - credit for it, because I didn't "illustrate" it, and he just could not fathom how I would ever interpret the instruction to "illustrate" in any way except to draw a picture.  He seemed to think I was being cute or something.  Nope.  Just zigging when apparently I was supposed to zag again.  Sigh!

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### You do have a point(1+ / 0-)
Recommended by:
Mother of Zeus

I'm not quite sure how to respond to it. Do you also underperform on standardized tests?

Math Kos II is open.

[ Parent ]

• ##### No. I overperform on them actually.(1+ / 0-)
Recommended by:
kyril

People are usually shocked to learn my SATs and similar scores, because I don't really seem all that smart!

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### Are you sure?(1+ / 0-)
Recommended by:
Mother of Zeus

It's evident from your writing that you are definitely highly intelligent and I would have bet pretty heavily that you'd score in the top decile, which is actually why I asked: I have a pet theory that what standardized tests test, at the highest level, is neither intelligence nor aptitude nor mastery of material but rather a specific personality type.

Differences among top scorers seem to be due primarily to differences in the way we interpret questions; the very top scorers usually have a natural tendency to attempt to read the test-writer's mind through the wording of the question and thus intuitively know how they're supposed to approach the problem. Smart or well-practiced people who don't have that tendency will still score very well, but will rarely approach the maximum score because there will be a few questions on every test that are tricky or worded vaguely and have to be interpreted.

There's also the possibility, however, that you've learned a specific sort of mind-reading that works well for deciphering the intent of standardized-test makers but is not as effective on test-making amateurs like your average college math professor.

Math Kos II is open.

[ Parent ]

• ##### Don't forget that the intent of the(1+ / 0-)
Recommended by:
kyril

test-maker is often encoded in the four answer choices.  There never seem to be more than a small handful of questions that are really tricky or ambiguous once you've looked at the response choices.  So I do very well on those fill in the bubble standardized tests.  Maybe with math tests it's more about anxiety.  I don't really know.  I just think math teachers ought to try to be as straightforward as possible and make sure to be very precise in the question they are asking to be answered.  If that makes the average scores too high, then give some extra credit opportunities so the real math stars can shine!  But I hate those questions that are obviously just trying to be tricky, like the "no cats" one.  I mean, come on, what does guessing which of two tricks the test-maker has in mind have to do with logic?

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### I heard it(5+ / 0-)
Recommended by:

No car is better than my car
My car is better than any car
Any car is better than no car.

Therefore

No car is better than no car.

• ##### No, that's not it(6+ / 0-)
Recommended by:
theran, LordMike, plf515, Jimdotz, kyril, Nailbanger

Nothing is better than sex
Masturbation is better than nothing
Therefore, masturbation is better than sex

Q.E.D. (Queerly Erotic Dissertation)

"Our Founding Fathers...drafted a charter to assure the rule of law and the rights of man.... Those ideals still light the world...." -- President Barack Obama

[ Parent ]

• ##### Invalid Syllogism Form(2+ / 0-)
Recommended by:
Mother of Zeus, kyril
The EAA Form is invalid across all Figures.

You can have an EAE or AEE form under Boolean Interpretation; toss in the EAO and AEO under Aristotelean Interpretation.

The reason being if one of the Premises is Negative (No cats ...) the conclusion cannot be Affirmative (All cats ...)

"...[one] must still have Chaos in oneself to be able to give birth to a dancing star." Nietzsche

[ Parent ]

• ##### Wait. This is interesting, but (0+ / 0-)

I don't understand what the hell you are saying.  What are these EAA, EAE and AEE forms?

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### Apologies ... Decoding(2+ / 0-)
Recommended by:
Mother of Zeus, kyril
A, E, I, and O is Logic shorthand for the four Categorical Statements:

All S is P -- The Universal Affirmative: A
No S is P -- Universal Negative: E
Some S is P -- Particular Affirmative: I
Some S is not P -- Particular Negative: O

"...[one] must still have Chaos in oneself to be able to give birth to a dancing star." Nietzsche

[ Parent ]

• ##### So is this universally acknowledged,(1+ / 0-)
Recommended by:
kyril

that all EAA arguments are invalid?

I find this interesting, because the way I interpreted this syllogism, the first statement isn't a universal negative.  That is, I could not make any sense of it that way - it just wouldn't compute - so I preferred to interpret the first statement as defining something - called "no cats" - that has 9 tails.  As convoluted as that is, it still seemed preferable to the EAA form, because it can be made to be valid, albeit false.

Why is the EAA form always considered invalid, or is this too hard to explain?

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### No Problem (I hope :-)(1+ / 0-)
Recommended by:
kyril

In Categorical Logic, which is what we are discussing, there are either Affirmative or Negative statements.

Affirmative statements, A and I, say something belongs in a class:

Socrates is a man.

This says SOCRATES belongs in, is a member of, the class MEN.

Negative statements, E and O, say something does not belong in a class:

No Martian is a man.

This states MARTIAN is not a member of the class MEN.

When we construct a Syllogism we are looking at which class things belong to -- among other things, but let's not Go There.  ;-)  In the statements above we're talking about how Socrates and Martians relate to the class MEN.

When we put this into an actual Syllogism we get:

Socrates is a man.
No Martian is a man.
Therefore, Socrates is no Martian.

You see?  You can't get away from that NO.  That's how Martians relate to the class MEN: they ain't.

And that follows in every single case.  Anytime one premise of a Syllogism is Negative the conclusion MUST be Negative.

Make sense?

"...[one] must still have Chaos in oneself to be able to give birth to a dancing star." Nietzsche

[ Parent ]

• ##### And, dudes and dudettes...(6+ / 0-)

...the order of words matters.

All republicans don't care.
Not all republicans care.

The first statement means that no republicans care.  The second means that some do not care.

The confusion between the two is driving me nuts.

• ##### Some care? Now my head hurts. n/t(3+ / 0-)
Recommended by:
Jules Beaujolais, rserven, kyril

The Tutoring Room will always be open and updated (new diary) on Wednesday mornings 1/21 Methods:http://www.dailykos.com/story/2009/1/21/3457/72066/529/686957

[ Parent ]

• ##### Was only a hypothetical example...(3+ / 0-)

...and no, some not caring does not require that thre are any that care.

• ##### Some go this way and some go that way.(3+ / 0-)
Recommended by:
bdizz, rserven, kyril

And some go both ways.

-Scarecrow
-Oz

The Tutoring Room will always be open and updated (new diary) on Wednesday mornings 1/21 Methods:http://www.dailykos.com/story/2009/1/21/3457/72066/529/686957

[ Parent ]

• ##### Well, no(3+ / 0-)

The second one just says there exist some that don't care. It makes no assertion about the existence of ones that do; the sets republicans" and "people who don't care" could be equal, or the set "Republicans" could be a subset of "people who don't care." Or they could just overlap.

Math Kos II is open.

[ Parent ]

• ##### When your head is full of straw, you can make(3+ / 0-)
Recommended by:
Jules Beaujolais, elwior, kyril

The Tutoring Room will always be open and updated (new diary) on Wednesday mornings 1/21 Methods:http://www.dailykos.com/story/2009/1/21/3457/72066/529/686957

[ Parent ]

• ##### But they could both be true, right?(1+ / 0-)
Recommended by:
kyril

I'll be kind and say the second is probably most accurate.

• ##### logic(2+ / 0-)
Recommended by:
algebrateacher, kyril

that went the way of the dinasours, didn't it?

• ##### Yay! Thanks...(5+ / 0-)

Of all the math classes I took in college (and I took as few as possbile, trust me) the only one I really did well in and enjoyed was logic. It didn't hurt that I had a wonderful instructor, but the subject itself was just plain fun.

• ##### Yeah, same here. (3+ / 0-)
Recommended by:

I'm a total mathphobe but symbolic logic was fun, like the way music theory was fun. I don't know what it is about math. The numbers maybe?

If you see mistakes in this post, it's because my editor's on vacation. Sorry.

[ Parent ]

• ##### Ha-probably(2+ / 0-)
Recommended by:
TKinVT, kyril

I've always been better with words, and word puzzles. Math word problems, though.....forget it.

• ##### Numbers aren't math.(3+ / 0-)
Recommended by:
Garrett, Jimdotz, kyril

They're engineering. Real math is all about symbols. And arrows. Must have arrows!

Insofar as I may be heard by anything, which may or may not care what I say... (from "Creatures of Light and Darkness", R. Zelazny)

[ Parent ]

• ##### One of my best subjects(9+ / 0-)

I really like this stuff.

Well, there's a reason I'm doing a whole series on Godel Escher Bach!

Oh, by the way, this sentence is false.

• ##### Logic is a systematic method(15+ / 0-)

of coming to the wrong conclusion, with confidence

:-)

• ##### Are you going to get into the limits of logic(6+ / 0-)

and the Peano statements, and all that stuff, and eventually Godel?

This statement cannot be proven in any formal system - translated into a REALLY HUGE number, which has properties .... ooooh fun!

• ##### I'll see what I can do(4+ / 0-)
Recommended by:
Jules Beaujolais, plf515, Jimdotz, elwior

but unfortunately, I'm just a rank beginner in this department. This of course has its advantages in that I have a much easier time explaining it to other beginners than an expert might, but it means I have to do a pretty significant amount of research if I want to go into any of the more abstruse aspects.

I was hoping I could start with algebra or calculus, but the audience has spoken (quite loudly - math is on the rec list! Weird.)

Math Kos II is open.

[ Parent ]

• ##### all math is cool(6+ / 0-)

Godel is freaky cool ....  Hence my diary series on GEB.

I'd love diaries on calculus.

• ##### I have a Calculus Pet Peeve... (4+ / 0-)
Recommended by:
theran, plf515, Jimdotz, kyril

When an instructor or book of basic calculus says "the derivative is the slope of the tangent line", this is usually shown with a graph without a rigorous definition of what a tangent line is.

Or they back track, start showing secant lines that approximate the tangent line. At that point they say the secant lines converge to the tangent line...

So, here I am trying to learn about basic calculus and you expect me to understand what it means for a sequence of functions to converge???

The Tangent line is the line whose slope is the limit of the slopes of the secant lines. (limits of sequences of scalars is assumed before the derivative is introduced.)

Pet Peeve, it's just me.

-8.00, -6.87
Hope. Peace. Integrity.

[ Parent ]

• ##### You're in no better shape that way(3+ / 0-)
Recommended by:
plf515, Jimdotz, kyril

Since you'd want to know why the secant lines have slopes that are converging, wouldn't you?  Formally you'd define the derivative from high school calculus as something like:

If f(x) = f(c) + (x-c)g(x) and g is continuous at c, then g(c) is called the derivative of f at c.

This is a bit opaque, so you have a geometric intuition, which is that if f is sufficiently smooth, then close enough to c, it's well approximated by a line with the slope g(c).

This is why combinatorics is better than calculus for introducing formal things.

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### Well...(3+ / 0-)
Recommended by:
theran, plf515, Jimdotz

The formal definition as it was introduced to me was that df(x)/dx=lim(delta x -> 0) (f(x + delta x) - f(x))/delta x
Which works just fine algebraically, since limits have already been introduced. The geometrical interpretation is what Random is talking about, and it's honestly just there to provide some sort of visual escape from the sea of algebra.

Math Kos II is open.

[ Parent ]

• ##### To me, offering me geometry in a sea of algebra(5+ / 0-)

is like throwing me an anchor that slices into my life vest.

Geometry makes me nuts ... in fact, in general, attempt to visualize things make me nuts.

• ##### Yes, I agree entirely(4+ / 0-)

I'm basically incapable of visualizing in the traditional sense; my geometric intuition is really a subset of my verbal and logical intuition. But the overwhelming majority of students have very nearly the opposite problem.

Math Kos II is open.

[ Parent ]

• ##### I'm the opposite...(4+ / 0-)
Recommended by:
Garrett, theran, Mother of Zeus, kyril

I love the geometric explanations of Calculus.

Actively zoom in on the sketch of a smooth function at a point and you see the deriviative appear before your eyes. If the function is discontinuous or not smooth, you can see it fail and why it fails.

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### The development should capture both (2+ / 0-)
Recommended by:
Jimdotz, kyril

since if you can't see the geometry in the case of a smooth curve in the plane, then understanding smooth manifolds will get completely out of hand.  (Since the formal definitions capture what you'd see from the smooth curve with tangent lines moving...)

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### Oh, GOd. I can't even imagine(2+ / 0-)
Recommended by:
theran, kyril

It was the only thing in math I was ever, really undeniably good at -- visual imagination, geometric intuition, and now you're basically saying that's not really a math skill anyway.  No wonder!!!

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### No, no, no!(2+ / 0-)
Recommended by:
theran, Mother of Zeus

No, I have a nonverbal learning disability. It's actually extremely rare for people with NVLDs to have any success with serious math because we have trouble with visualizing and our geometric intuition is quite weak. I have found workarounds for that because I find the subject fascinating, but my way of thinking about these things is the hard way. It has some very minor advantages in that sometimes I have insights or see patterns or connections that others may not, but those instances are greatly outnumbered by my "wtf" moments when others quickly and easily gain insight into something that takes me months of thinking to truly grasp.

No, the visual skills are an integral, critical part of math learning for the vast majority of people, including effectively all top mathematicians and physicists. I'm swimming upstream here. You're fine.

Math Kos II is open.

[ Parent ]

• ##### Oh.(1+ / 0-)
Recommended by:
kyril

I can't believe you have the persistence to study something that comes that much easier to people around you.  I'm such a damned quitter that I always gave up on anything if it was obvious that I lacked the abilities that made it come really easy to other people.  Wish I hadn't given up on some of those things, actually!

Anyway, good on you.  But I wonder if sometimes relying on one's ability to visualize and imagine isn't kind of a crutch.  At the highest levels, math is far too abstract for anyone but the most outrageous genius to have helpful intuitive, visual representations, isn't it?  So in some ways, if you rely a lot on the ability to visualize things, when you reaches the end of the line on that skill set, you kind of say, "oh, well, guess that's it for me."  So maybe it's better in some ways to always have to think about these things the hard way, since at some point pretty much everyone has to once they reach a certain level of sophistication.

Or not?

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### Actually, I almost didn't(0+ / 0-)

My mother was a mathematician and a highly visual person. I went through my childhood convinced that I was terrible at math because I never understood her explanations of anything, and she had me somehow convinced that one couldn't progress in math if one didn't understand the foundations of what one was learning - and by her (highly visual) standards, my verbal explanations didn't exhibit real understanding.

It took me nine years after leaving home to realize that my way of learning was just as valid as her way, and that I could and did develop a stronger understanding of the basics by progressing to more advanced material much better than by repeating the same material with the same failed explanations. So I went back to school, jumped right into Calculus 3 (after a 9 year break) and did it my way and excelled. Now, I do have a slight advantage in that my way of going about things has led me to be extremely good at symbolic manipulation, so I can do well without really knowing what I'm doing, but after a bit of thinking and working with an idea I do get it as evidenced by the fact that I can go on and use it creatively later.

As far as the top levels of math, I don't know; to my knowledge, nobody who's not a visual person has ever gotten there. From what I gather from my reading, it appears that the definition of "visualizing" loosens somewhat at that level, but geometric intuition and visual analogies are still an integral part of graduate-level mathematics.

Math Kos II is open.

[ Parent ]

• ##### You mean there are two of us?(1+ / 0-)
Recommended by:
kyril

I also have NVLD and like math.

But I learn it differently from most people, and find different parts hard and easy.

• ##### Evidently :)(1+ / 0-)
Recommended by:
plf515

And yes, that's been my experience too.

Math Kos II is open.

[ Parent ]

• ##### Those definitions are the same(2+ / 0-)
Recommended by:
Jimdotz, kyril

I just wrote it differently to give the difference quotient a name (g) and packed the limit into the continuity of g.  The main point was to unravel the mysterious different quotient to the point where the geometric intuition could at least be seen.

(From my perspective, this is also the interesting thing about smoothness: the ability to linearize nearby.  That's the general pattern that comes back and also explains why you would care about it, which I never really got from high school calculus.)

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### Ah(2+ / 0-)
Recommended by:
theran, Jimdotz

I don't see it because it's the delta x that makes it make sense to me; it's exactly like a slope calculation, except you're taking a limit.

Math Kos II is open.

[ Parent ]

• ##### But that's not very sensible!(2+ / 0-)
Recommended by:
Jimdotz, kyril

Which is what Random was complaining about in the first place.

If you think of a differentiable function f(x) as being like:

f(x) = f(c)+(x-c)g(x)

with continuous g, then near c, f is very close to

f_1(x) = f(c)+(x-c)g(c)

which is a line.  (For easier visualization, make c zero and f(0)=0.)

So the algebra and the geometry coincide.  The pedagogical problem is finding a way to explain both correctly.

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### Hm(2+ / 0-)
Recommended by:
theran, Jimdotz

Perhaps the unfortunate problems with typed-math notation are obscuring it, but honestly what you're saying makes far less sense to me. It appears you're using the derivative as an approximation of the function, which is indeed one of the things we can apply it to, but I'm not certain how it leads to a deeper understanding of the derivative.

Math Kos II is open.

[ Parent ]

• ##### I dunno(2+ / 0-)
Recommended by:
Jimdotz, kyril

You say that the derivative at c, if it exists is the limit of

(f(x)-f(c))/(x-c)

as x -> c.  Instead, I just call this thing g(x) and rearrange to get the form I had.  Remember that g is not f'(x).  For example, if f is x^2 and c is 2, then g is x+2.

Then, if you have the derivative, it must be the case that g is continuous at c (since the limit exists), which is why the definitions are the same.

I got the line f_1(x) from just plugging in c instead of g(x).  f_1 is clearly a line from the algebra.  Then if you subtract:

f(x)-f_1(x) = (x-c)(g(x)-g(c))

which by continuity of g says that the line f_1 is a good approximation of f near c.

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### I've always seen it desribed (2+ / 0-)
Recommended by:
theran, kyril

as a limit, in one form or another.

OK, it is a tangent line ... but a rigorous definition of a tangent line to a curve is, I think a lot harder than the derivative!  Rather, one can say it is an instantaneous velocity.

• ##### Here is the subtlety that I explained badly above(2+ / 0-)
Recommended by:
plf515, kyril

Think about a PL curve.  You can define a tangent line at c by saying that the line touches the curve at c, and there's a neighborhood of c where all the points of the curve are on one side of the line.  But you don't have the tangent.

The main point of high school calculus seems to be that for a differentiable curve, you can talk about the tangent line at a point, and that it's a good local approximation to the curve.  (I wrote down the equation above, but getting it from an algebraic point of view.)

Of course you can make these kinds of definitions without any reference to geometry, but both views are nice.

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### I found this recently:(4+ / 0-)
Recommended by:
ATinNM, plf515, Jimdotz, kyril

http://www.bbc.co.uk/...

The guy who does logic is the guy who best explains what Godel is up to.  In a short audio bit, it's the best I've heard.

Oh, the evils of Socialized Media.

"What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet."--Woody Allen

[ Parent ]

• ##### You might want to look into Wittgenstein(3+ / 0-)
Recommended by:
plf515, kyril, LRLine
for a bit of fun with the abstruse aspects!

-7.88/ -7.44

[ Parent ]

• ##### Agreed. n/t(1+ / 0-)
Recommended by:
kyril

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### You can go there via countability(3+ / 0-)
Recommended by:
plf515, Jimdotz, kyril

Somehow the result is more interesting than the proof (to me), since the main step is to prove that if you have enough number theory, then you can encode "programs".  (I have to admit that I find logic a little dry, and therefore don't know anything about it, so I am the wrong person to ask.)

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### Is it a really, really big number?(3+ / 0-)
Recommended by:
joanneleon, plf515, kyril

Some will be awed by it.
Others will deny its existence.

The Tutoring Room will always be open and updated (new diary) on Wednesday mornings 1/21 Methods:http://www.dailykos.com/story/2009/1/21/3457/72066/529/686957

[ Parent ]

• ##### It's HUMONGOUS(2+ / 0-)
Recommended by:
algebrateacher, kyril

I am not sure if Godel ever actually PRODUCES the number... but he proves things about it.

• ##### Scarecrow represented farmers who kept putting(3+ / 0-)
Recommended by:
Jules Beaujolais, plf515, kyril

more and more acreage into production because the prices kept dropping because there was such an overabundance of crops.

Over and over, on into the distance of numbers, never to end...and incomprehensible.

The Tutoring Room will always be open and updated (new diary) on Wednesday mornings 1/21 Methods:http://www.dailykos.com/story/2009/1/21/3457/72066/529/686957

[ Parent ]

• ##### Free Republic does something like this, too(13+ / 0-)

Actually, no, they fucking don't.

Truth is what most contradicts itself in time.

• ##### Yes they do too!!! I took a screenshot to prove(1+ / 0-)
Recommended by:
kyril

it to you:

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### I love this place(10+ / 0-)

and all the nerds and geeks it. Thanks, kyril. Logic is hard to make fun, and you've done a bang up job.

• ##### Thanks! :)(5+ / 0-)
Recommended by:
mcjoan, joanneleon, Jimdotz, elwior, COwoman

Proud to be a nerd :)

Math Kos II is open.

[ Parent ]

• ##### Nerds and geeks have become cool.(7+ / 0-)

Like our brand new President.
Who'd a thunk it?

"We the People of the United States...." -U.S. Constitution

[ Parent ]

• ##### Insert Star Trek reference here.(3+ / 0-)
Recommended by:
Jimdotz, kyril, LRLine

Extra points if you know where your old pair of pointed ears are out in the garage.

No red shirts, please.  Green alien girls are fine.

The Tutoring Room will always be open and updated (new diary) on Wednesday mornings 1/21 Methods:http://www.dailykos.com/story/2009/1/21/3457/72066/529/686957

[ Parent ]

• ##### Not at all(4+ / 0-)
Recommended by:
algebrateacher, dirtfarmer, Jimdotz, kyril

in case you thought I was making fun. My geekitude lies in other directions, but my admiration is sincere.

• ##### History-based geekitude(2+ / 0-)
Recommended by:
mcjoan, kyril

I'm impressed.
It takes a fearless geek to go in that direction.  The facts you have to know can be actually verified.

The Tutoring Room will always be open and updated (new diary) on Wednesday mornings 1/21 Methods:http://www.dailykos.com/story/2009/1/21/3457/72066/529/686957

[ Parent ]

• ##### I stuck pretty close(1+ / 0-)
Recommended by:
algebrateacher

to the primary source, so felt comfortable with it. I still haven't had a chance to retrace the whole trail, which is my primary goal.

• ##### Do "communicator" pins count?(2+ / 0-)
Recommended by:
algebrateacher, kyril

I've got a dandy one from the 1960s if they do.

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### Interesting stuff(3+ / 0-)
Recommended by:
algebrateacher, Jimdotz, kyril

Having studied it myself, I appreciate this diary.  But from years of experience I know most people's eyes glaze over when you try to explain it to them.

I suggest a good approach might be, "How to win a debate with a wingnut".  The logic should be a shortened version of what you've got here.

• ##### Only way to win an argument with a wingnut:(3+ / 0-)
Recommended by:
Jules Beaujolais, Jimdotz, kyril

Say, "The fact that you don't understand my arguments shows that my arguments must be true.  All I can say now is...I won."

Then run away.

The Tutoring Room will always be open and updated (new diary) on Wednesday mornings 1/21 Methods:http://www.dailykos.com/story/2009/1/21/3457/72066/529/686957

[ Parent ]

• ##### I made straight A's in logic in college(8+ / 0-)

and used it to substitute for two higher math classes.

Of course, it baffled my prof, because he said if you don't do well in math, you won't do well in logic. At the end of each semester (regular and symbolic logic), I had the highest grade in the class.

I can't add 2 + 2 and always get 4, but logic speaks to a part of my brain that numbers cannot reach.

I wish logic were part of the high school curriculum as well as being taught in college. Maybe it could make actual math easier if it were taught at the lower grade levels.

Thanks for the diary!

"It always seems impossible until it's done." - Nelson Mandela

• ##### Thanks for reading!(1+ / 0-)
Recommended by:
Jimdotz

I do think that some logic could be introduced before algebra to great effect. Even if it didn't benefit students in later math courses, I still think it would be invaluable; the value of math education above simple arithmetic is largely in the logical thinking skills it develops, and these can be taught directly.

Math Kos II is open.

[ Parent ]

• ##### Do not confuse Arithmetic and Mathematics.(4+ / 0-)
Recommended by:
theran, plf515, kyril, LRLine

They are NOT the same thing. Arithemtic is "house painting"; Mathematics is art.

If my students can't explain it, they don't understand it.

[ Parent ]

• ##### I need to talk to you about the Theory of(2+ / 0-)
Recommended by:
Jimdotz, kyril

Arithmetic. hehe. That's some nice stuff. You don't see it around much anymore.

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### Nice post(4+ / 0-)
Recommended by:
Garrett, algebrateacher, Jimdotz, kyril

I hope you will do something about the ad hominem fallacy and soon.  Its unchecked and growing presence, even here on Kos, is ruining American life as we know it -- that is, as us logicians know it.  IMHO, the  ad hoimenem is a crucial, or maybe even a structural part of postmodern thought.  Scary to think about?  Yes.  Yes it is.  But think about it we must.

• ##### I will(1+ / 0-)
Recommended by:
Jimdotz

and soon! Next week, I think. The ad hominem fallacy is actually one of the easiest to demonstrate formally.

Math Kos II is open.

[ Parent ]

• ##### Of course.(2+ / 0-)
Recommended by:
theran, kyril

Someone like you would think that.

• ##### Wow. I loved this. Difficult but.....(2+ / 0-)
Recommended by:
Jimdotz, kyril

...educational. Lawyers will recognize boolean logic from Lexis and Westlaw.

Tonight I'm going to party like it's 1929.

• ##### Of course i am assuming you will(1+ / 0-)
Recommended by:
kyril

eventually move to the limits of logic?

• ##### You're not the first to ask that :)(2+ / 0-)
Recommended by:
theran, Jules Beaujolais

While I'll certainly try to touch on some of them, I'm not personally equipped to discuss the topic in detail. I'd like to invite anyone else with more expertise to do so, though!

Math Kos II is open.

[ Parent ]

• ##### I am equally not equipped to address(1+ / 0-)
Recommended by:
kyril

the subject in a manner that will help others.

• ##### the limits of logic(1+ / 0-)
Recommended by:
kyril

The best writeup I've seen on this was in Charles Tart's essay on "state-specific sciences," that was originally published in Scientific American and later in his book Transpersonal Psychologies, and is widely available online.

Tart (who was an electrical engineer before he went into experimental psych) argues persuasively that our entire set of scientific method is built from within a particular state of consciousness that can be described as "the rational waking state."   Further, that given sufficient familiarity with other states, other systems of logic arise, and from them, other potential sets of scientific method.

The test of a methodology, of course, is in its results: the ability to arrive at conclusions that enable humans to interact with various phenomena in a predictable manner.

So.

One example with which we are all familiar, is the kind of reasoning that occurs in the dream state.  It tends to be highly nonlinar and simultaneous as distinct from sequential.  Another familiar example is "stoned logic," that occurs when a person is high on marijuana, and another type occurs while drunk on alcohol.  Psychodynamic logic is observable with patients in psychoanalysis, and in the research lab with psychedelics and with hypnotic trance.

Tart argues that the common conception of "higher" states of consciousness (for example in meditation) or "lower" states of consciousness (for example certain kinds of intoxication) is a value judgement rather than empirically true.  As I was just writing to a friend via email, the issue is, what state of consciousness is best for what purpose?   The state you want to be in while driving on the freeway, is different from the one you want for doing the bookkeeping, and the ones you want for composing a symphony or reading scripture are also different.

A fairly decent example concerns "shamanic pharmacology," whereby tribal cultures in Central America developed a fairly advanced grasp of the medicinal uses of local plants.  A tribal shaman was typically an individual who was easily capable of going into altered states, often with the use of sacred plants, and would spend quite a bit of time observing what went on in the natural world around him/her.  By observing the subtle interactions between animals and plants, s/he would become aware of plants that could have value to humans, and then use these plants on other tribe members.  Over time, knowledge would accumulate: this one brewed as tea brings down fever, this one burned in the hut prevents a disease (possibly by the smoke repellling mosquitos), this one added to food keeps it fresh longer, etc.

Whatever intuitive or "rational" or altered-rational process the shaman used, the results converged and knowledge was gained over time.  Western pharmaceutical companies have spent quite a bit of research money going into the woods to learn from these individuals and translate their knowledge into something usable to western science.

Another obvious example is acupuncture, which arose under the Chinese theory of medicine, and produced repeatable results that stood up empirically in the context of western medicine.  Currently, the western theory of medicine is seeking explanations for the means by which acupuncture produces anaesthesia and other effects.  (I'm willing to predict that the outcome here will be that western medicine develops a usable theoretical understanding of this, and that this theory will also be translatable back to the Chinese theory, such that the Chinese terminology will continue to be used clinically.)

In the case of Chinese medical theory, what we have here is not an "altered state" as such, so much as a different system of reasoning and set of understandings that arise in conjunction with an entire culture.

Anyway, that's a brief scratch of the surface and now I have to get back to work...

• ##### A not OT pimpage(4+ / 0-)
Recommended by:
nota bene, joanneleon, dirtfarmer, kyril

of my series: Godel Escher Bach

last week's diary

another one tomorrow!

• ##### see my reply to kyril just below her reply to you(1+ / 0-)
Recommended by:
kyril

RE. the limits of logic...

• ##### When it gets too dry...(1+ / 0-)
Recommended by:
kyril

Take a break with this. I guess it was all that talk about unicorns...

"You can't get something for nothing...It's time to stop being stupid." Bob Herbert

• ##### dry?(4+ / 0-)

i think it's fantastic!!!

"Well we don't rent pigs and I figure it's better to say it right out front because a man that does like to rent pigs is... he's hard to stop" Gus McCrae

• ##### Thanks! :)(1+ / 0-)
Recommended by:
Jimdotz

I did my best to spice it up with unicorns and issues. But this stuff gets a whole lot more interesting.

Math Kos II is open.

[ Parent ]

• ##### This is a very necessary diary(4+ / 0-)
Recommended by:
Cartoon Messiah, Jimdotz, kyril, dotalbon

Unfortunately, I'm betting that most people reading it either won't understand it or won't think they need to even read it.

• ##### Thank you, but I'm not sure about that(5+ / 0-)

It seems like a lot of people are reading, though of course many are doing so out of pre-existing interest.

Math Kos II is open.

[ Parent ]

• ##### When you said...(3+ / 0-)
Recommended by:
TKinVT, Jimdotz, kyril

I'm betting that most people reading it either won't understand it or won't think they need to even read it.

Was that an "and/or" or... or an exclusive or?

I'm confused.  :)

I guess the real question is:  Is our children learning?

• ##### Quite a few of us who are reading(5+ / 0-)

could, largely, have written it ... :-)

It's cool to see someone else teach something you already know, and one always learns something new, doing so. Or, at least, I always do.

• ##### It's the rest of the people who need it(3+ / 0-)
Recommended by:
plf515, Cartoon Messiah, kyril

The ones who skim the opening paragraph and then move on, thinking that it's not something they need. IMO, the single biggest failing of our educational system is that we don't learn how to think critically.

• ##### Curious statement, if you don't mind me saying(2+ / 0-)
Recommended by:
kyril, Fixed Point Theorem

Unfortunately, I'm betting that most people reading it either won't understand it or won't think they need to even read it.

If they read it and don't understand it, have they really read it?
&
If someone who reads this doesn't think that they need to read it, why would they have read it?

"Superstition, idolatry, and hypocrisy have ample wages, but truth goes begging." - Luther

[ Parent ]

• ##### You're right. It's not clear(1+ / 0-)
Recommended by:
Cartoon Messiah

It's even "illogical", you might say. I intended to suggest that most people would simply read without comprehending, or read the introduction and skip the rest because it was irrelevant to them.

• ##### I got your intented meaning, and I agree 100% (2+ / 0-)
Recommended by:
dianem, kyril

I guess I was sort of playing off of your comment to make an epistemological statement.

"Superstition, idolatry, and hypocrisy have ample wages, but truth goes begging." - Luther

[ Parent ]

• ##### Interesting diary. thank you.(3+ / 0-)
Recommended by:
3goldens, kyril, dotalbon

The Shape Of Things "Beware the terrible simplifiers" Jacob Burckhardt, Historian

• ##### Logic according to Tweedle Dum(5+ / 0-)

"I know what you're thinking about," said Tweedledum: "But it isn't so, no-how."

"Contrariwise," continued Tweedledum, "if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic."

Intriguing start to a series. I'm a set theorist. I look forward to seeing where you take this.

The big guy in the commercials would not approve of my use of the High Life.

• ##### Carroll was a great logician.(7+ / 0-)

It's not a coincidence he came up with such philosophically interesting scenarios in his works.

"What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet."--Woody Allen

[ Parent ]

• ##### But is there an IF....ONLY(1+ / 0-)
Recommended by:
kyril

operator?

Behind every great man, there's a woman saying "Stand up straight"

• ##### Oh, but If Only...(1+ / 0-)
Recommended by:
kyril

There were!

• ##### Hmm...(2+ / 0-)

"If only I were a cat, I would have a tail."
I can't construct an "if only" statement without a subjunctive, and logic doesn't deal terribly well with subjunctives...but it appears to me that the above sentence is only true if I am not a cat and I do not have a tail. Which means "if only" statements are NOR statements, I think. But there's some added meaning in thare that I'm not sure how to capture.

Math Kos II is open.

[ Parent ]

• ##### If only(2+ / 0-)
Recommended by:
freespeech, kyril

x were 0, then my students would be happy, because then (x+a)^2 = x^2 + a^2 for all a.

• ##### Mmm, yes(0+ / 0-)

My dear husband would be happy too if only that were the case.

Math Kos II is open.

[ Parent ]

• ##### Maybe if you're Montague.(2+ / 0-)
Recommended by:
kyril, LRLine

But it's probably just pragmatic and gets no operator.

"What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet."--Woody Allen

[ Parent ]

• ##### Killing ambiguity is dangerous business...(2+ / 0-)
Recommended by:

If ... a is false, [then] if a then b is true.

--Thanks kyril, good stuff!!

• ##### LOL(3+ / 0-)
Recommended by:

Yes, I'll fix that :)

Math Kos II is open.

[ Parent ]

• ##### True if not falsifiable or not applicable? (0+ / 0-)

Logic and applicability are too different.
To say a statement is "true" in logic really seems to mean "consistent with", "not in conflict with" other given statements.
If so, it's really mean of logicians to insist on using the word. Kind of a nasty streak in the professional sense of humour.

• ##### Not falsified(1+ / 0-)
Recommended by:
seriously70

is the best interpretation I can give. It's certainly still falsifiable; all you have to do is provide an instance where a is true and b is false, and if a then b is falsified.

Math Kos II is open.

[ Parent ]

• ##### Great job!(2+ / 0-)
Recommended by:
kyril, yellow dog in NJ

You've given me a lot to think about.  Thanks for your efforts to put this together.  I took a Logic class waaaaaaay back when I was in college as part of my Philosophy requirements for graduation and I really didn't appreciate it as a freshman in college.  Over time, however, I really came to understand why that course was a requirement at the Liberal Arts college I graduated from.  I can see that I badly need this "review"---it's been so long it's almost like starting all over again.  LOL!  Off to go "study" and re-read this diary.

• ##### logic tutor?(1+ / 0-)
Recommended by:
kyril

If you can find a good automated logic "tutor", that may be a good addition to your next post.

IN College, I took a self-paced logic course, and the main activity was doing proofs using a nifty little program...it gave the student a premise and a conclusion, then asked the student to fill in the gap. It was actually pretty fun.

Unfortunately, the program is not available to the public and I haven't seen anything else like it.

"Avoid extremes; forbear resenting injuries so much as you think they deserve." -Benjamin Franklin

• ##### Good thought(0+ / 0-)

I'll see what I can dig up!

Math Kos II is open.

[ Parent ]

• ##### The two directions of math(3+ / 0-)
Recommended by:
Jimdotz, kyril, dotalbon

1 + 1 = 2.

now, there are two directions.  One is the one that we are all at least a little familiar with:
Subtracting
Multiplying
Dividing
Fractions
Exponents
Algebra and on into the wilds of calculus, analysis, and so on.

The other is to look to see what it actually means to say 1 + 1 + 2.  Why are we so sure it is so?  Do we really understand it? Probably not....

1 cup water + 1 cup sugar = less than 2 cups of sugar water.

hmmmm

1 tablespoon of salt + 1 teaspoon of salt = ????

What does + really mean?
Not to mention 1 and 2

Deep questions.

• ##### Very(4+ / 0-)
Recommended by:
plf515, Jimdotz, Mother of Zeus, dotalbon

although the escape from most of those is just to make sure that when you're doing math on real things, you keep your units straight :)

Math Kos II is open.

[ Parent ]

• ##### Straw Men, Red Herrings, False Equivalencies...(5+ / 0-)

Please get to these asap, because they get thrown around rather irresponsibly on here.  (And only the charge itself.  There's never a demonstration of, say, how someone just constructed a straw man and a new argument.  Nah, just lob the charge and move on!!)

People know what they do; they frequently know why they do what they do; but what they don't know is what what they do does. -Michel Foucault

• ##### But it sounds so good to say them!(2+ / 0-)
Recommended by:
eamonsean, kyril

Do I have to give them up just because I don't understand their proper use?  Oh, okay.

I'd also love to have clarified the difference between a solipsistic argument, a circular argument, and a sophistic argument (presuming they're not all the same thing) --

Electing a Republican is like hiring a carpenter who thinks hammers are evil.

[ Parent ]

• ##### I'll give it a shot(5+ / 0-)

"Sophistry" is generally used to describe an argument that is crafted to appear logical, or occasionally simply to seem so complex and intimidating that the audience doesn't feel competent to evaluate its logical structure, but is fatally flawed in some way. The term connotes that the flaw is intentionally included and concealed.

A circular argument is another term for the logical fallacy called "begging the question." A circular argument takes the conclusion (usually in some sort of disguise) as a premise and then uses it in the argument to prove itself. A circular argument may be used in a sophistic argument.

"Solipsistic argument" can mean a number of things, from "an argument in favor of solipsism" to "an argument from a solipsistic premise." Here's the wiki entry on solipsism. It's not closely related to the other two.

Math Kos II is open.

[ Parent ]

• ##### Thanks!(1+ / 0-)
Recommended by:
kyril

My brain is now in gear.  I'm off to look for examples of all three.

Electing a Republican is like hiring a carpenter who thinks hammers are evil.

[ Parent ]

• ##### Many of the arguments for Intelligent Design(0+ / 0-)

are excellent examples of circular sophistry. Behe and the like have written entire books best described as "the tale of an idiot, filled with sound and fury, signifying nothing."

Math Kos II is open.

[ Parent ]

• ##### Actually many here will(2+ / 0-)
Recommended by:
john de herrera, kyril

not like hearing this but I see quite a bit of circular logic regarding Obama's actions. Now, some of which I have agreed with , and others not so much. But when I read someone say well Obama intended the action because the action occured that's begging the question of how can you then determine intent and decide accountability for what someone said previously. maybe he did intend the result, but there can be no certainty, and yet people will run with the argument that begs the question.  I do not expect Obama as a politician to admit to these inconsistencies, but I would hope that we as his supporters would sothat we can obtain the best result.

• ##### I have also seen to be honest(1+ / 0-)
Recommended by:
kyril

quite a bit of sophistry here as well.

• ##### You won't have to look(1+ / 0-)
Recommended by:
kyril

far, on this site!

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

[ Parent ]

• ##### We should have a workshop (1+ / 0-)
Recommended by:
kyril

on all of those!  (including the strawman, red herring and false equivalency)

Commenters can post comments that they believe are valid examples, and the experts can judge.

Gad, how strange is it that I think that would be a fun exercise?  :)

"The true measure of a man is how he treats someone who can do him absolutely no good." --Samuel Johnson

[ Parent ]

• ##### 100 percent agree(3+ / 0-)
Recommended by:
Garrett, joanneleon, kyril

Someone will say "you are arguing a false equivalency" even when they really just mean they don't like the analogy.

I made an analogy to Gillibrand and Lieberman yesterday, and the poster kept saying I made a falsely equivalent statement.

I was stunned I had to point out that I was using an analogy, and the the intent of the analogy was not to say they were exactly the same , but instead represent of the idea of two politicians who have views that do not relate to the general political centers of the electorate of their repsective states.

What they really mean, I suspect, is that they did not like comparing someone they like to someone they don't. But that does not make the analogy false. It just makes it an uncomfortable check on what we feel.

A false equivalence would be my saying Gilliband is exactly like Lieberman without context, qualification, degree, meaning etc add into the mix. A = B. Rather than A on this point I am specifically analogizing is similar to B.

Don't get me started on the misuse of red herrings and strawmen. Again,that's a product way too often of "I don't like your argument, but I don't have a response."

• ##### The value of intuitive thought(1+ / 0-)
Recommended by:
kyril

It's fast.

Make a best guess. Be proven right. Feel vindicated. Acquiring a new ability to predict correctly is more pleasant than just plain reward.

Make a best guess. Be proven wrong. Feel regret. Regret is unpleasant and long lasting.

Repeat the guessing again and again and again, on ever changing circumstances.

You get a well tuned system with a lot of ability to say "I don't like your argument, but I don't have a response." Or to respond with shoddy argument that is still quite convinced it is right. Or to dismiss seemingly well constructed argument as being wrong anyways.

It sucks for discussion between people with differing views, but it's a built in system. It's not going away.

• ##### Here's my view of life(3+ / 0-)
Recommended by:
Garrett, joanneleon, kyril

My shit smells just like everyone else. I may not like hearing my shit smells, but I want people to tell me when I am wrong, and provide me with some understanding of how I am wrong.

I may not even admit to it at the time, but, ultimately, when my ego goes away- I prefer it. My best experiences have come when I pushed to think better or more creatively.

A preacher once told me that it's better to have a man yelling at you who annoys you, but prevents you from crossing the street into oncoming bus than it is to have a man who allows you to have a pleasant time until you are hit by the oncoming bus.

There is nothing wrong with guessing. There is something wrong with rote responses and groupthinking and a lot of other things that makes us not understand one another. That's assuming, of course, that everyone is here to communicate with everyone else. That's an assuumption that maybe flawed.

• ##### Groupthink and the like are important aspects(2+ / 0-)
Recommended by:
joanneleon, kyril

of modern political discussion.

Just really elaborate social dynamics that seem designed to ward off and prevent the regret we should feel on discovering we are wrong.

• ##### Sometimes, I think we do not live on Animal Farm(2+ / 0-)
Recommended by:
Garrett, kyril

or in 1984, but instead  in a Brave New World, in which our desires are how the world secretly controls us.

i.e., our desire for being in the middle class to meet our basic needs pushes us into debt far beyond our means because our society in other ways does not provide a real safety net. The desire to be economic safe is not wrong, but the manipulation of that desire can lead to bad things.

In a healthy society, nothing should be wrong with regret in response to the specific stimuli so long as its reflective of what is actually happening- that we are wrong. But, that's not likely in this Brave New World because regret is one of thos emotions that may force us to think too much.

• ##### Question(2+ / 0-)
Recommended by:
Garrett, kyril

Can you explain your last sentence?  I was absorbed in your comment, nodding my head, until I got to the last sentence.  I think you may have thrown a bit of sarcasm in there, or maybe I just didn't get your final point.

"The true measure of a man is how he treats someone who can do him absolutely no good." --Samuel Johnson

[ Parent ]

• ##### Not sarcasm. Just poor explanation on my part(3+ / 0-)
Recommended by:
Garrett, joanneleon, kyril

It's complicated so don't quote me.

I will try to explain my point, but I am not sure I can do a good job of explaining.

The best kind of control  and manipulation works through keeping us solely focused on our desire or fear. Anything that breaks the feedback loop  (or spell) of desire and/or fear can break the control and manipulation.

In other words, control and manipulation only work if Dorothy does not pull the curtain back long enough to realize that the Wizard is pulling the levers in Oz.

If you are the wizard (the controllor or manipulator) you do not want Dorothy noticing the Wizard. Anything that tends to promote thinking too much may result in the spell being broken.

Regret may require reflection on history. It may require a kind of introspection (if its the good kind of regret).  Instrospection, and the subequent, regret you may feel is bad for the control and manipulation business if it's linked to things you really need to address.

I have a hard time believing, for example, that the Iraqi War would have been possible if we truly were introspective and regretted VIetnam as a people.

I also have a hard time believing that we would buy into what politicians tell us if we truly were introspective about our history and regretted the mistakes we made. We spent the last 30 years repeating the same economic mistakes we already made in previous generations.

On a micro-level, here we see people repeat some of the behaviors we have from recent memories about how to support Obama. To me, support does not require blind support because I am not willing to allow my fears and desires to be manipulated by the political system (including Obama).  Thus, I am going to hold him to some standard beyond my fears and desires. I hold all politicians to that standard including those in Congress.

It does not mean I am not wrong. I often am but it provides me a standard that's not easy for others to manipulate. If I regret my previous thinking (which was laissez faire liberalism in the 90s), then I must realize when those thinkings are coming into play again and respond to them rather than say to myself that I never held those thoughts.

• ##### I don t know if that last post makes any sense(3+ / 0-)
Recommended by:
Garrett, joanneleon, kyril

I am trying to explain something that's a littel hard for me to explain since this is a developing view of life

• ##### Yes, it does(2+ / 0-)
Recommended by:
Garrett, kyril

I thought that was what you were saying, but I lost the connection at some point.

When you wrote this:

The best kind of control  and manipulation works through keeping us solely focused on our desire or fear. Anything that breaks the feedback loop  (or spell) of desire and/or fear can break the control and manipulation.

I immediately understood.  Thank you for taking so much time to explain!

"The true measure of a man is how he treats someone who can do him absolutely no good." --Samuel Johnson

[ Parent ]

• ##### P.S.(2+ / 0-)
Recommended by:
Garrett, kyril

I have spent a lot of time thinking about the manipulation and distraction that goes on in this country.  But I had never thought about the fact that part of the manipulation might be to suppress regret because it is a powerful emotion that interferes with the "programming" and also prevents us from learning lessons, thereby making it easier for a manipulator to accomplish the same goals again and again.

Thanks again.

"The true measure of a man is how he treats someone who can do him absolutely no good." --Samuel Johnson

[ Parent ]

• ##### No problem(3+ / 0-)
Recommended by:
Garrett, joanneleon, kyril

The reality is that I did not think of this by myself.

I came to to this over time. First, there was this guy right after 09/11 (forgot his name) who point blank in an interview said, " America will fuck up because we lack introspection as a cultural trait."

Then, recently, I heard the same on Bill Moyers in the fall. Moyers ask this guy if he thought Obama would matter. He said, "no, because Americans do not like looking in the mirror." To truly changes things requires reflection.

• ##### Hmm(2+ / 0-)
Recommended by:
Garrett, kyril

You've got me thinking :)

It's late, so I'm going to stop thinking real soon, but I'm wondering how guilt figures into this equation.  It's a powerful tool, and it should trigger introspection and perhaps regret.  But it's kind of warped emotion so much of the time.  And it seems to be plentiful in this country, at least in some segments of the population, or perhaps all, I'm not sure.  Anyway, something to think about for another day.

"The true measure of a man is how he treats someone who can do him absolutely no good." --Samuel Johnson

[ Parent ]

• ##### Actually(4+ / 0-)
Recommended by:
Garrett, LordMike, rapala, kyril

NAND is logically complete, although a PITA to deal with.

Same is true of NOR, but the NAND completeness (and its affinity for CMOS circuit implementations) makes the computer you're using possible... :)

`You have a much too willful will. You think that what you do not do... does not happen' -Zen master to E. Herrigel

• ##### AHHHH(1+ / 0-)
Recommended by:
dotalbon

I'm all of a sudden back in PHIL 170...MAKE IT STOP...TOO BORING...NOOOOOO

• ##### Woot for logic. Computer Science Prevails!!(4+ / 0-)
Recommended by:
rapala, kyril, dotalbon, user 174092

I can't wait for Axioms!!!

/geekness

• ##### Old school(2+ / 0-)
Recommended by:
LordMike, kyril

reminds me of working through the Boolean to troubleshoot floating point reciprocal functional units, and such, when they were composed of a dozen, or more, modules and 100's of chips and discrete components. Now its all "in the chip".

Liberalism is trust of the people tempered by prudence. Conservatism is distrust of the people tempered by fear. ~William E. Gladstone, 1866

• ##### Perhaps(3+ / 0-)
Recommended by:
LordMike, rapala, dotalbon

I don't see it as a bad thing, though, to understand (at least at a simplified level) what's going on "in the chip." The basics are needed for any kind of coding work anyway, and they're absolutely necessary for proofs in math and philosophy (and I think they're good to know even for constructing verbal arguments!)

Math Kos II is open.

[ Parent ]

• ##### Fascinating.(2+ / 0-)
Recommended by:
rapala, kyril

I have especially found logic to be beautiful at the point where having internalized its principles, it becomes possible to use logic to analyze practical conditions and solve practical problems.

It doesn't appear as though the political process can easily stand this type of functioning. But I think and hope that Pres. Obama will apply this ability to logically assess issues, and to hone in on the best possible policies.

Logic demands that the so-called Drug War be ended, except for those who benefit from it, and they are very few.

Nice topic, thanks... will tip.

PEACE

• ##### LOAD "WHITEHOUSE",8,1(4+ / 0-)
Recommended by:
rapala, jnhobbs, kyril, LRLine

LIST

10 OBAMA = 77
20 GOP = 22
30 IF GOP > OBAMA THEN PRINT "ERROR--LOOK AT MY APPROVAL NUMBERS AND GET BACK TO ME."
40 IF OBAMA >= GOP THEN PRINT "I WON! I TRUMP YOU!"
50 GOTO 30

RUN

I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!
I WON! I TRUMP YOU!

BREAK IN 30
_

Comments that say "GM workers should get retraining" without SPECIFIC DETAILS about those "new jobs" that never come are trollworthy

• ##### I'm sorry, Kyril...(2+ / 0-)
Recommended by:
bruh1, kyril

I'm afraid I can't do that...  ;-)

Comments that say "GM workers should get retraining" without SPECIFIC DETAILS about those "new jobs" that never come are trollworthy

• ##### here's to logic(1+ / 0-)
Recommended by:
kyril

though I must say that this really just touches the surface and that some of the more in depth applications of logic on the surface don't seem to work.

Nice intro though.

• ##### Where are you getting all these definitions?(0+ / 0-)
Or are you making them up yourself?

For example.

"It might rain."

Depending on the context, this could be a statement that is either true or false.

In space, it will not rain... so the assertion that it could possibly rain seems to be a statement... more about the possibility than the rain itself. In that case, it's false.

Additionally, if there is a discussion about the possibility of rain occurring and someone merely makes the claim that it might rain... that is a truth. It might.

Is there any further distinction then that's helpful? Links to sources? Or is this just going to be a purely semantic exercise with no resolution?

If you're trying to set up a datum from which people can base their understanding of these concepts...

I'm not sure I'm with you yet.

Thanks.

• ##### I'm using my discrete math textbook(2+ / 0-)
Recommended by:
ReneInOregon, noweasels

(Epp, Discrete Mathematics with Applications) as my primary resource, although the explanations and examples are my own. If you'd like an online-accessible source, Wikipedia has several articles on symbolic logic.

As far as where I'm going with this, I'm not entirely certain myself; I started the series because after I lamented my lack of ideas of things to contribute in one of the Daily Kos University threads, plf515 was kind enough to encourage me to start a series on what I know, which is mostly math and basic physics. I'll figure out where I'm going when I get there :) My main hope in the process is to provide some food for thought and a space for discussion of math-related ideas.

On "might" and statements: There's a legitimate argument to be made that "There is a non-zero probability that it will rain in some location at some time in the future" is a statement. Now, if you don't specify a location, it can be argued that it's a tautology, but tautologies are valid statements. So perhaps that wasn't the best example. I do think it conveys some of the problems with English vagueness, though. Probabilistic statements are best handled with fuzzy logic anyway.

Math Kos II is open.

[ Parent ]

• ##### ?(2+ / 0-)
Recommended by:
Garrett, kyril

Probabilistic statements are best handled with fuzzy logic anyway.

I don't understand this.  I'm not really a foundations person, but the standard approach to probabilistic inference is that what it represents is degrees of certainty about events that are either true or false.

In fuzzy logic, you have statements that are partially true.

This would be an interesting topic for a diary.

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### It would be indeed.(2+ / 0-)
Recommended by:
theran, kyril

If kyril were dealing with probabilistic methodology. What is being covered here is not specific to that however, but rather to symbolic logic independent of application.

I like your suggestion though. If you wrote on this topic, I would tip and rec it.

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### I have basically nothing to say about it(2+ / 0-)
Recommended by:
kyril, LRLine

I'm not a foundations person or even an applied statistician or machine learning person.  I don't really have anything coherent to say.  (If I were going to write a diary on this kind of thing, I'd write something fun from combinatorics or geometry or algorithms.)

The philosophical issues about uncertainty are rather deep, though.  Causality too.  (Judea Pearl's book seems to be well liked in certain circles.)

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### If anyone else needs a rigorous set(1+ / 0-)
Recommended by:
kyril

of examples of what kyril has covered today, check out Irving M. Copi, "Symbolic Logic", Prentice Hall.
I have the fifth edition ...might be later ones... up through page 21 will get you comfortable with "refutation by logical analogy" (which is what kyril's used in her examples), and get you warmed up for part II of what logically must be coming.

In addition to kyril's source and the above, there is the more modern Discrete Mathematical Structures, by Malik and Sen, Thompson Course Technology, c. 2004 (complete with MAPLE) (the first 29 pages).

No doubt there are also many online sources that cover this material.

"I love it when a plan comes together" -- Hannibal Smith, A-Team.

[ Parent ]

• ##### Please explain logical fallacies...ASAP(1+ / 0-)
Recommended by:
kyril

But be careful---

You may cut the numbers of comments on this website by 90% --- and kos will have you banned...

• ##### In particular, reductio ad absurdum(1+ / 0-)
Recommended by:
kyril

When someone makes a reductio ad absurdum argument, it doesn't mean that you (the Kos public) should automatically HR the comment....

Sheesh...

• ##### Right wing logic:(3+ / 0-)
Recommended by:
thenekkidtruth, jnhobbs, kyril

Conservatism cannot fail
Conservative economic philosphy has brought about a depression
Conservative economic philosophy has failed...

Wait.. try again...

Conservatism cannot fail
Conservative economic philosphy has brought about a depression
Conservative economic philosophy has...

Awww.. F U!  We don't need logic, we have FAITH!  Friggin' liberal heathens and your athiest logic!! You'll all burn in hell!!

Now, that's what we call logic!!!

Comments that say "GM workers should get retraining" without SPECIFIC DETAILS about those "new jobs" that never come are trollworthy

• ##### Provided(3+ / 0-)
Recommended by:
rapala, joanneleon, kyril

I once refuted a textbook and won!

It was a book on logic by Stephen F. Barker.  In it, he demonstrated the logical equivalent of the word provided.  He stated that (okay I think I remember it this way) the logical equivalent of "Jones is not a Senator if he is not over 35" is "Jones is over 35 provided he is a senator", which I countered in my college class was not.  The logical equivalent is "Jones is a Senator provided he is over 35."

My teacher, a fossil of a man with interesting stories of his own (he was rumored to be an ex-CIA agent - he used to study Russian books for fun), submitted my analysis to Barker, who relented, which effectively cemented my "A" for the course!  He gave me an extra test grade of 140, LOL!

I loved Logic class!  This was about 25 yrs ago.

• ##### Coooooool!(1+ / 0-)
Recommended by:
kyril

Could you perhaps do a segment on set theory as well?
Keep up the good work.

"Superstition, idolatry, and hypocrisy have ample wages, but truth goes begging." - Luther

• ##### Awesome(2+ / 0-)
Recommended by:
theran, kyril

Combine this with set mathematics and you can become a Sudoku expert in under two weeks!

You can call me "Lord Bink Forester de Rothschild."

• ##### I think sudoku is NP-complete(2+ / 0-)
Recommended by:
bink, kyril

(Or at least if it's easy then something similar to P=NP would be true.)  So there are no experts in some sense. :)

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### Interesting(1+ / 0-)
Recommended by:
theran

I don't know that Sudoku is NP-Complete actually, though I see on the Internet now that people are asserting that it is.

You can call me "Lord Bink Forester de Rothschild."

[ Parent ]

• ##### It doesn't surprise me(2+ / 0-)
Recommended by:
bink, kyril

Since similar problems for Latin squares are.  I think that solving Sudoku assumes that there's a unique solution, in which case you'd get something weaker: that RP=NP (if I remember properly) if there's an efficient algorithm for it.

"Dream for just a second and then do it!" -- Kolmogorov

[ Parent ]

• ##### Illogic.(0+ / 0-)

John McCain is President if and only if the Sun orbits the Earth.

This is incorrect.

Even if the sun were to orbit the earth, John McCain would still not be president of the United States.

The statement commits the fallacy of connecting two false propositions, when these two false propositions have no causality connecting them, and no conditionality connecting them, and are only appealing to the reader because they are both similarly unlikely, implausible, or false, yet remain logically independent, or "orthogonal."

• ##### No(0+ / 0-)

I'm sorry. I sympathize; I argued this point for a good 15 minutes with my math instructor when I encountered it. But he was right and I gave up.

A logical "if-then" statement says nothing about what happens if the "if" portion is false. In Boolean terms, "If a, then b" is equivalent to "(a AND b) OR (NOT a).

The following are true statements:
If true, then true
If false, then true
If false, then false

An "if and only if" statement is an intertwining of two "if-then" statements. "John McCain is President if and only if the Sun orbits the Earth." evaluates logically as:
(If John McCain is President, the Sun orbits the Earth) AND (If the Sun orbits the Earth, then John McCain is President)
which evaluates as (if false, then false) AND (if false, then false)
which is equivalent to True AND True
which is equivalent to True.

Math Kos II is open.

[ Parent ]

• ##### I'm still thrilled with this series and(0+ / 0-)

glad I caught the next installment.  Also very happy to see this up on the rec list.  You are a good teacher!

Of course, it is also true that this is probably the only diary in the series that is a breeze for me, as I was a philosophy major.  (I also had Stats in a graduate psych program I was in at one time, so that will be familiar territory, too.)

But I'm waiting with gritted teeth for those trig and calculus diaries so I can finally face down my nemeses!  So you'd better be in this for the long haul! :D

The festive scenes of liberation that Dick Cheney had once imagined for Iraq were finally taking place -- in cities all over America -- Frank Rich

• ##### Thanks!(0+ / 0-)

Don't worry, I'll get there :) though you may have to read through a fair amount of other stuff first.

Math Kos II is open.

[ Parent ]

• ##### Spock said it best:(1+ / 0-)
Recommended by:
kyril

"Logic is little yellow bird, singing in meadow;
Logic is a wreath of pretty flowers... which smell bad."

I'm not a Democrat, I'm a liberal. Democrats go to meetings.

• ##### My college logic class awarded only 2 grades(2+ / 0-)
Recommended by:
rapala, kyril

'A' or 'F'.

I took it early in my matriculation at Miami University in Oxford, Ohio. Yes, I tell my friends I'm Oxford educated :-)

I needed permission to take this particular Department of Philosophy class - as a sophomore, it was far above my grade level, being almost a graduate level course.

The first day, the prof told us that there's only two grades - you'll either get an 'A' or an 'F'. He said it wasn't his choice - that this fact was simply driven by the subject matter. Some people, he explained, have a "knack" for this, and others will simply fail to grasp it no matter how hard they work at it.

Sure enough, fully a third of the class dropped out after only 2 weeks, and another dozen or so departed shortly after that. I couldn't understand why - I thought it was one of the most straight-forward classes I ever took.

An easy 'A' and a subsequent thirty-year career in Information Technology informs me both that the professor was right, and that I was seemingly one of the lucky ones.

Can you imagine what I would do if I could do all I can? - Sun Tzu

• ##### How to brush up(2+ / 0-)
Recommended by:
rapala, kyril

or maybe even relearn?

In college I did four semesters of engineering calculus, and in most other subjects used it constantly because of course, it's a basic tool in those courses.  But since I haven't used it in many years, I feel like I've forgotten it all.  I can't understand how something I could do often for four straight years nearly every day is now like greek to me, or so it seems.

Any suggestions for where to start brushing up or relearning?  I think I'd first have to sharpen the algebra skills.  I once found some books on Amazon that seemed to be for this purpose but I didn't order them at the time, as some health problems were causing me so much brain fog that I thought it would be futile.  Things aren't perfect now but they're definitely better and I'd really like to get my math skills back.  Maybe it would be an exercise in futility now?  Not sure but I don't think so.

"The true measure of a man is how he treats someone who can do him absolutely no good." --Samuel Johnson

• ##### Did you keep your textbooks?(0+ / 0-)

If not, I'm partial to the Schaum's Outlines for low-level math review purposes. There are also a fair number of interactive "math tutor" computer programs that can help you brush up on the process without quite as much tedium.

If you've done calculus, though, I'm not sure I'd recommend going all the way back to algebra; one of the best ways to recover algebra is to work with it in calculus (assuming you still remember the most fundamental rules of linear-equation-solving, fractions, and exponents). If you are serious about doing a good review, I'd suggest a strong calculus textbook rather than a review book; the one I've liked the most is Larson, Hostetler, and Edwards, Calculus: Early Transcendental Functions. Any edition should do fine, and there are some older editions relatively cheap. Combine that with just about any algebra reference for when you get stuck, and you should be able to self-study through as much as you want without much trouble.

Math Kos II is open.

[ Parent ]

• ##### I think I did keep them(1+ / 0-)
Recommended by:
kyril

There were only two.  One book was used for three semesters, and then another one for Differential Equations.  I do believe that Schaum's was the one I saw on Amazon.  I'm pretty sure the algebra will come back.  I'm fine with fractions and exponents but I'm sure I've forgotten some of the rules for solving equations.

I wonder how much of it is really lost, in a "use it or lose it" kind of way.  I hope I'm able to pick it up without too much trouble.  It bothers me to no end how much I've learned and forgotten in math and the sciences.

"The true measure of a man is how he treats someone who can do him absolutely no good." --Samuel Johnson

[ Parent ]

• ##### If it makes you feel any more confident(1+ / 0-)
Recommended by:
joanneleon

keep in mind I just came back to school two years ago after a nine-year break (wherein I used absolutely none of my math); I didn't lose a thing. I jumped right into Calculus 3 and scored over 100% on everything, right from the beginning. It tool a little jogging of my memory to get back some of the details like integration and differentiation formulas, but the techniques and understanding were still there.

I don't know how long it's been for you, but I doubt you've forgotten the essential how and why; you've probably just lost the details, and even those will probably look really familiar.

Math Kos II is open.

[ Parent ]

• ##### Here I am, last one again in your series(1+ / 0-)
Recommended by:
kyril

but, how interesting! They really should teach logic in law school...but noone I've talked to, nor my school, had it in the cirriculum weirdly enough.  Fantastic work, kyril!

"The object of life is not be on the side of the majority, but to escape finding oneself in the ranks of the insane." Marcus Aurelius

• ##### Great diary...refreshing that you can present the(0+ / 0-)

principles without coming across as a condescending dick like Paul Krugman, who I think deserves all the accolades, but don't appreciate his communication style.

It is best to read the weather forecast before we pray for rain. Mark Twain.

• ##### You wrote "it has few elements which are clearly(0+ / 0-)

recognizable as 'mathematical' ". But all of logic is mathematical, whether written in symbols or in words. Did you mean that it has few elements that have to be written in symbols? That would be true for the most elementary parts, but not once we have to get the heavy machinery out.

Busting the Dog Whistle code.