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View Diary: Math mania: Why teach math? Why sing? (92 comments)

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  •  This is early on a slow morning so I'll risk a (8+ / 0-)

    slightly off topic question.

    Are mathematical truths objective statements about the world or are they merely statements about human apperception?  IOW, Kant or Hegel?

    (Sorry, I knows it's still early.)

    •  My view is that they are objective truths (10+ / 0-)

      about any conceivable universe.

      3 is prime because it is, not because of how we conceive of multiplication.

      The way we state these truths, and which truths we discover, may be due to our apperception ... certainly using 10 as a base, for example, is likely due to our have 10 fingers.  But that's not the mathematical truth, that's just the language those truths are stated in.

      Some other civilization (or species) might discover truths that we haven't discovered, but they would still be true for us

      And that's different from provable.  Godel showed that there will always be statements that can neither be proven nor disproven; but that's not the same as saying whether they are true

    •  they can't be (5+ / 0-)

      mathematical truths are true within the realm of mathematics which is apriori defined. That's it. The fact that real things can, to greater or lesser degree, be modeled with math, is an entirely different point.

      I mean really, MajorFlaw, I thought you were an attorney and hip to such things :)

    •  objective truth is a social construct (3+ / 0-)
      Recommended by:
      MajorFlaw, plf515, jessical

      sayeth the historian.

      surf putah, your friendly neighborhood central valley samizdat

      by wu ming on Sun Oct 19, 2008 at 05:18:44 AM PDT

      [ Parent ]

    •  Not off topic at all. (3+ / 0-)
      Recommended by:
      haileysnana, MajorFlaw, plf515

      The diary brought up logicism.  Russell's logicism, at least, was an attempt to provide an account of objective mathematical truth that avoided the platonism tacitly referred to by some others that have responded to you.  Platonism is just the idea that mathematical objects exist in a non-physical realm, that they are timeless and necessarily existing.  

      Many philosophers have found platonism to be a tough pill to swallow, for a number of reasons.  One reason is that having posited these abstract objects to make true our mathematical statement it becomes difficult to understand how we've come to know anything about them.  Plato believed that human souls existed in the "world of the forms" before birth and that we obtained mathematical knowledge through recollection (the Greek word is "anamnesis").  Others, including myself, have found that implausible and fanciful.

      The logicist project began with Frege.  Frege's system was not without abstract objects.  His Basic Law V posited the existence of "concept correlates".  It would take us pretty deep into the weeds to adequately discuss that law, but it might be understood as giving both an account of what abstract objects are AND how we come to grasp them.  Sadly, Basic Law V falls to the Russell Paradox and Frege's system, indeed his life's work, was shown to be inconsistent.  

      Russell replaces Basic Law V with an axiom of infinity in the system of Principia Mathematica.  As I noted in a post below, he succeeds in deriving the Peano Axioms for arithmetic (and much more) in this system.  However, it now seems that the system is no longer purely logical because it is not clear how an axiom stating the existence of infinitely many objects is a logical axiom.  Also, Goedel showed that the system of PM cannot prove every arithmetical truth.

      •  I have a hard problem with most metaphysics (2+ / 0-)
        Recommended by:
        play jurist, cynndara

        My reaction is usually "Does anyone really think that?"  :-)

        •  Some "traditional" questions have become (1+ / 0-)
          Recommended by:
          plf515

          irrelevant as scientific discovery replaces philosophical inquiry.  Over time, the Empiricists win.

          •  I don't know if I agree with that. (4+ / 0-)
            Recommended by:
            haileysnana, MajorFlaw, plf515, cynndara

            Philosophers remain divided, of course, as ever.  It's said that mathematicians are platonists throughout the work week and formalists on Sundays.  Similarly, I'd bet that most scientists are realists throughout the week and empiricists on Sundays.

            •  Not sure I follow. (2+ / 0-)
              Recommended by:
              haileysnana, plf515

              Scientists are by definition empiricists.  It's the difference between scientific method and philosophical speculation.  Why speculate when there is evidence.  When Plato, Aristotle etc. tried to understand  the cosmos they could only turn inward.  Now we have Astronomy.  

              •  IF (3+ / 0-)
                Recommended by:
                MajorFlaw, play jurist, plf515

                you can get far enough away from the city to actually see the stars at night. Which is not a bad metaphor for clearing out the musty smog of social illusions/delusions before getting down to clear perception and thought.

                There are a HELL of a lot of social illusions centered around Science, too.

              •  We have different ideas of "empiricism". (4+ / 0-)
                Recommended by:
                haileysnana, MajorFlaw, plf515, cynndara

                Of course, science by definition involves subjecting theories to observational scrutiny.  If that's what you mean by "empiricist" then you are correct.  That seems to be the common meaning of the term for a lot of people.  In philosophy specialized meanings have arisen.  The distinction between a realist and empiricist (very, very roughly) is that a realist thinks observation provides evidence that things like atoms, electrons, quarks, etc. actually exist while the empiricist holds that we never have reason to believe anyhting more than that our theory is empirically adequate.  The empiricist thinks of things we cannot directly observe (things like electrons) as theoretical constructs not real entities.

                Theoretical phycists, btw, still do plenty of inward turning.  You are correct that they also turn outward and aim to check their theories by observation (hopefully, eventually, etc--see string theory).  But I think that the contrast between Plato/Aristotle, etc. is not as strong as you might think.  In Physics Aristotle opposede Parmenides on motion and change because the fact of change was evident to the senses, for example.  Plato and the Pythagoreans founded a research program of using mathematics to study the world.  It's true that they were less advanced than we are, but it is not clear that our advances are due to fundamental methodological differences so much as time and technology.

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

          With six billion humans on the planet right now, I think you could find SOMEBODY who believes ANYTHING.

          Even flying spaghetti monsters, may Its Noodly Appendage touch us all.

          •  Really? (0+ / 0-)

            Anyone who believes that 1 + 1 = 17,322,306?

            Or how about someone who believes that we are actually living on one of the moons of Neptune, and that the idea that we are living on Earth is an illusion, fostered by the evil overlords, who have occupied the Earth since their home planet, Zarkon, is no longer big enough?

            :-)

            •  Welll ... (1+ / 0-)
              Recommended by:
              plf515

              The first is easy, given self-replicating biological systems; it just takes a while.  But with bacteria, probably no more than twelve hours.

              The second, you'd have to give me time to write the story, and plant and feed it for a while, then distribute it on the internet through viral propaganda channels working off the old UFO/Men In Black networks.  That could take a couple of years; I lost my main contact for that group when I left the university.  But then, there ARE the Victorian Goth sf fans my apprentice wanted to hook me up with over the holidays ...

              Remember that magicians practice believing six impossible things before breakfast every morning.  It keeps the mind flexible.

            •  It can be.... (1+ / 0-)
              Recommended by:
              plf515

              ... if 1+1=0.

              (Shame I missed this diary yesterday, missed out on really fun conversation :( )

              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)

              by SadEagle on Mon Oct 20, 2008 at 04:36:17 PM PDT

              [ Parent ]

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