Aesthetics and Mathematics:

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Aesthetics and Mathematics Some
Observations Juliet Floyd Boston University
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• In what sense are these terms of criticism, when
  they occur within working mathematics, truly
  aesthetic?
• Can they be considered epistemologically
  relevant?
• 3. How systematic are mathematicians uses of
  such terms? Are there anything like objective
  standards of taste at work here?
• How important is it to focus on the patter
  surrounding proofs? Should consideration of such
  appeals play a role in philosophical discussions
  of the nature of mathematics?
• 5. What would it mean to treat mathematical
  structures as aesthetic objects?

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Frege to Russell, June 22, 1902 Your discovery
of the contradiction caused me the greatest
surprise and, I would almost say, consternation,
since it has shaken the basis on which I intended
to build arithmetic. Gödel, Russells
Mathematical Logic By analyzing the paradoxes
to which Cantors set theory had led, Russell
freed them from all mathematical technicalities,
thus bringing to light the amazing fact that our
logical intuitions (i.e., intuitions concerning
such notions as truth, concept, being, class,
etc.) are self-contradictory.
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Wittgenstein to Schlick, 31 July 1935 As to the
application of what I've said to the case which
you quote I want now only to say this If you
hear someone has proved that there must be
unprovable propositions in mathematics, there is
first of all nothing astonishing in this vorerst
gar nichts Erstaunliches, because you have as
yet no idea whatsoever what this apparently
utterly clear prose-sentence says. Until you go
through the proof from A to Z, down to the last
detail, you cant see what it proves. To one who
would wonder at the fact that two opposite
sentences are provable I would say look at
schau an the proof and then you will see "in
which sense" the one and "in which sense" the
other is proved. And before you have studied the
entire proof precisely you have no reason to
wonder.
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All that you can learn from "my teachings" is
that about such a proof with its result nothing
can be said before you have investigated the
determinate proof. That is the philosopher is
always wrong who wants to prophesy a
quasi-something in mathematics and say, "that is
impossible", "that cannot be proved". Why not?
That which is supposed to be proved is nothing
but a word expression and the proof gives it its
particular sense and with how much warrant we
then call this proof the proof of this prose-
sentence is partly a matter of taste that is, it
is a matter of our judgment Ermessens and our
inclination whether we want to apply the
structure expressed here in this prose-sentence,
or not. How the matter of our inclination is,
whether we want to speak of imaginary points or
not or of invisible light, or not.
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--The proper genau investigation Untersuchung
of a complicated proof is extraordinarily
difficult. That is, it is extraordinarily
difficult to organize the structure of
gestalten the proof perspicuously
durchsichtig and to obtain complete clarity
about its relation to other proofs, its position
in certain systems, and so on. You have only to
try properly to investigate untersuchen a proof
such as that of the sentence that the square
root of 2 is irrational and you will persuade
yourself of this. This does not however mean
that there is something mystical in this proof
before this investigation, but only that we have
not yet clearly taken in überschauen the proof
and especially its position among other proofs.
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McGuinness, Asceticism and Ornament (1985)

• For Wittgenstein style, the way something was
  put, was of enormous importance, and that not
  only in the artistic sphere. He said once, it
  wouldnt matter what a friend had done but rather
  how he talked about it. There are dangers, if a
  feeling for style becomes the supreme
  commandment. It is not to be thought of that
  this was a risk for Wittgenstein in the moral
  sphere, but in aesthetics as he himself
  suggested, he perhaps incurred it.

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McGuinness, Asceticism and Ornament

• The excessive frequency of accidentals in
  Wittgensteins manuscripts and typescripts
  reflects an almost pathological insistence on
  finding the correct distribution of emphasis in a
  sentence It is almost as if he regarded
  something as false as soon as it was written
  down

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McGuinness, Asceticism and Ornament
False philosophy must be exorcized. But that is
an operation best performed viva voce and through
personal contact. One false notion is driven
out, and immediately the next false notion that
threatens to take its place must be corrected.
.. Was his philosophy bare asceticism without
positive content to make it worth the effort and
the abnegation? This difficult question must be
resolved in any attempt to assign Wittgenstein a
place in the history of ideas.
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Gödel, Note added to his 1931, August 1963

• In consequence of later advances, in particular
  of the fact that due to A.M. Turings work a
  precise and unquestionably adequate definition of
  the general notion of formal system can now be
  given, a completely general version of Theorems
  VI and XI is now possible. That is, it can be
  proved rigorously that in every consistent formal
  system that contains a certain amount of finitary
  number theory there exist undecidable arithmetic
  propositions and that, moreover, the consistency
  of any such system cannot be proved in the system.

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Added Note

• In my opinion the term formal system or
  formalism should never be used for anything but
  this notion. In a lecture at Princeton I
  suggested certain transfinite generalizations of
  formalisms, but these are something radically
  different from formal systems in the proper sense
  of the term, whose characteristic property is
  that reasoning in them, in principle, can be
  completely replaced by mechanical devices.

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Kant, KU 62, On Merely Formal, as Distinguished
from Material, Objective Purposiveness
All geometric figures drawn on a principle
display a diverse objective purposiveness
Zweckmässigkeit, often admired (bewunderte)
they are useful for solving many problems by a
single principle, and each of them presumably in
an infinite variety of ways.
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The mentioned properties of geometric figures,
and presumably of numbers as well, are commonly
called their beauty, because they have a
certain a priori purposiveness for all sorts of
cognitive uses which is unexpected in view of how
simple it is to construct these figures. For
example people will speak of this or that
beautiful property of the circle, discovered in
one way or another.
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Yet it is not by an aesthetic judging that we
find such a property purposive, not by a judging
without a concept, a judging that reveals to us a
mere subjective purposiveness in the free play of
our cognitive powers rather, it is by an
intellectual judging, according to concepts, and
this judging reveals distinctly an objective
purposiveness, i.e., a suitability for all sorts
of purposes (of infinite diversity). Instead of
calling such a property of a mathematical figure
its beauty, it would be better to call it the
figures relative perfection.
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It would be more plausible to call a
demonstration of such properties beautiful,
since such a demonstration makes understanding
and imagination, the powers of concepts and of
their a priori exhibition, respectively, feel
invigorated, so that here at least the liking is
subjective, even though based on concepts,
whereas perfection carries with it an objective
liking. (That invigoration of understanding and
imagination, when it is combined with the
precision that reason introduces, is called the
demonstrations elegance.)
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Wittgenstein, On Certainty 139 Not only rules,
but also examples are needed for establishing a
practice. Our rules leave loop-holes open, and
then the practice has to speak for itself.
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I had never made a garden before, so I said, Why
not? Mel Bochner
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Hermine, Wittgenstein hausgewordene Logik
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Wittgenstein, OC 564 A language-game bringing
building stones, reporting the number of
available stones. The number is something
estimated, sometimes established by counting.
Then the question arises Do you believe there
are as many stones as that?, and the answer I
know there are Ive just counted them. But
here the I know could be dropped. If, however,
there are several ways of finding something out
for sure, like counting, weighing, measuring the
stack, then the statement I know can take the
place of mentioning how I know.