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Re: Landau letter, Re: Mathematica as a New Approach...
*To*: mathgroup at smc.vnet.net
*Subject*: [mg128051] Re: Landau letter, Re: Mathematica as a New Approach...
*From*: Andrzej Kozlowski <akozlowski at gmail.com>
*Date*: Wed, 12 Sep 2012 02:57:24 -0400 (EDT)
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*References*: <k1n71e$e5m$1@smc.vnet.net> <20120831075725.55C076873@smc.vnet.net> <20120911063409.DBC17684A@smc.vnet.net>
As I have written before, I don't think this is an appropriate space for
explaining the basic ideas on modern philosophy to people who clearly
have never heard of them. Instead I suggest reading:
=
http://www.igewem.tu-dresden.de/die_tu_dresden/fakultaeten/philosophische_
fakultaet/iph/thph/braeuer/lehre/metameta/Quine%20-%20On%20What%20There%20=
Is.pdf
and
http://faculty.unlv.edu/jwood/wm/Quine.pdf
It might (or might not) become clear what I have been hinting at: that
is that the idea of a sharp distinction between "experiment" and
"theory" etc. are illusionary. The results of every experiment have to
be judged by a human mind and interpreted in the light of some theory
and there will always be many alternatives "explaining" whatever
"evidence" is provided by an experiment. One of such theories that can
never be completely discounted is that the experimenter himself is
hallucinating (or is taking part in a group hallucination -such things
have known to happen on a massive scale), or the results are due to some
deliberate deception, etc, etc. In the end the choice between one such
theory and another is not based on yet another experiment (you cannot
refute either the "hand of God" or "evolutionary development" by any
experiment) but on considerations like simplicity (it is simpler to
believe that what you see in front of you when you get up in the morning
is "really there" than that it is the result of very elaborate
deception, involving holography etc., even though the later might
actually be the case), which are based on aesthetics. The Copernican
system was chosen over the Ptolemaic one, not because it was better
confirmed by experiment but because it was simpler. One can "explain"
everything that general relativity explains without using non-Euclidean
geometry - but again the same thing happens. This sort of thing has been
discussed in the philosophy of science since the beginning of this
century so much that it has now become passe (look up Karl Popper, Imre
Lakatos, Paul Feyerabend, Michael Polanyi etc, etc=85). There is no
point repeating all this stuff on a forum devoted to Mathematica.
Andrzej Kozlowski
On 11 Sep 2012, at 08:34, John Doty <noqsiaerospace at gmail.com> wrote:
>> Mathematical Platonism is modern form is no more than a belief that
>> the natural world is governed by "laws", which are discovered by
human beings but exist independently of them and can be expressed in
>> mathematical form.
>
> The natural world is, of course, the domain of science, not
mathematics. The imagined world of Platonic mathematics is most
definitely *not* the natural world, as it is inaccessible to the methods
of science. But mathematics as a product of human thought is most
definitely accessible to cognitive science.
>
>> Like all metaphysics worth its salt, this belief can neither be
>> validated nor refuted. Anybody who thinks that it can be
>> "comprehensively demolished" is either using rhetorics more fitting
to a political than a philosophical dispute or else should catch up on
>> his Hume.
>
> A hypothesis that won't stand up to test deserves little respect. (my
Bayesian colleagues can even argue this mathematically). But mathematics
is a human practice, occurring in the real world, accessible to
experiment. Thus, to a scientist, you are in fact demolishing your view
by insisting that it cannot be demolished.
>
>> Philosophically I am close to Quine, and so I
>> believe that ontologically there is no fundamental difference between
>> the objects studied by mathematicians, such as groups or sets, and
the ones studied by physicists such as atoms or electrons.
>
> There are absolutely fundamental differences. Physical objects are
accessible to experiment. The properties of groups result entirely from
the definition of "group". But no amount of reasoning can tell you much
about the properties of atoms given only the definition of "atom".
>
>> They are all human posits which we use to "explain" the sense data
>> which arise from some independent reality. But as the the actual
>> nature of this reality we can only speculate and in doing so we can
>> rely on nothing more then our aesthetic judgement.
>
> No. I agree completely that aesthetic judgement directs mathematics,
but science is directed by evidence. We often see that aesthetic
judgement undisciplined by real world evidence is wrong.
>
>> So what
>> exactly is the evolutionary path from a near "laboratory animal" to
>> Riemann or Perelman?
>
> I think it's similar to the evolutionary path from laboratory animal
to elite athelete. Sports like ice skating are not really like anything
humans evolved to do, but involve physical and cognitive "modules"
evolved for other purposes, combined in novel ways. Scientific
understanding of this has made it possible to teach atheletes to perform
feats once thought impossible, like "quadruple jumps". I see no reason
this shouldn't also apply to mathematics.
>
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