[FOM] Re: Shapiro on natural and formal languages

[Jeffrey Ketland]

Rob Arthan wrote:

>Even better in my opinion are purely algebraic
>things like:
>
> 1 + 2 + .. + n = 1/2n(n+1)
>
>proved visually by thinking of an n x (n+1) rectangular array of points
>divided into two similar triangles. In examples like this, visual
>insight bypasses an inductive argument very directly.

Also, maybe a visual proof of an instance of the pigeonhole principle, say
when n=3:

                   *   *   *   *
                   |    |    \  /
                  o   o    o

There must be a "converging" pair of links.
One interesting question concerns whether we merely see that the principle
holds for the specific case indicated (and that generalizing to "for all n"
is illegitimate), or perhaps the visual representation above helps us grasp
that the principle holds for all n.

An example from logic I sometimes use is a visual proof of the validity of
the sequent

     ExFx, Ax(Fx -> Gx)  |- ExGx

Draw a Venn diagram with F as a subset of G, and put an object inside F.
Automatically, it's inside G too.
We do see the requisite generality (it holds whatever the sets F and G are).

Related to such questions, in his discussion of mathematical intuition,
Goedel commented on the "the abstract elements contained in our empirical
ideas":

      In should be noted that mathematical intuition need not be
      conceived of as a faculty giving an immediate knowledge of
      the objects concerned. Rather, it seems that, as in the case
      of physical experience, we form our ideas also of those objects
      on the basis of something else which *is* immediately given.
      Only this something else here is *not*, or not primarily, the
      sensations. That something besides the sensations actually
      is immediately given follows (independently of mathematics)
      from the fact that even our ideas referring to physical objects
      contain constituents qualitatively different from sensations or
      mere combinations of sensations, e.g., the idea of object
      itself, whereas, on the other hand, by our thinking we cannot
      create any qualitatively new elements, but only reproduce
      and combine those that are given.
      Evidently, the "given" underlying mathematics is closely
      related to the abstract elements contained in our empirical
      ideas. It by no means follows, however, that the data of this
      second kind, because they cannot be associated with
      actions of certain things upon our sense organs, are
      something purely subjective, as Kant asserted. Rather
      they, too, may represent an aspect of objective reality,
      but, as opposed to the sensations, their presence in us
      may be due to another kind of relationship between
      ourselves and reality.
      ... the question of the objective existence of the objects
      of mathematical intuition ... is an exact replica of the
      question of the objective existence of the outer world.
      (Goedel 1964, Supplement to "What is Cantor's
      Continuum Problem?", Collected Works, Vol 3, p. 268.)

Regards --- Jeff

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Jeffrey Ketland
School of Philosophy, Psychology and Language Sciences
University of Edinburgh, David Hume Tower
George Square, Edinburgh  EH8 9JX, United Kingdom
jeffrey.ketland at ed.ac.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~