[FOM] New Proof of Fundamental Theorem of Arithmetic

[Thomas Lord]

On Fri, 2009-09-25 at 14:23 -0400, joeshipman at aol.com wrote:
> The second point of view here has been taken to extremes by Doron 
> Zeilberger, and I find it hard to gainsay him; yet the new proofs of 
> results like Heron's formula somehow seem to improve our
> understanding. 
> It is a puzzlement.

This prompts me to ask if people on the list are familiar
with and, if so, what they think of the book "Where 
Mathematics Comes From" by George Lakoff and Rafael E.
Núñez?

That's my main question and if perhaps I would be 
wiser if I just stopped there but I'll (surely) embarrass 
myself and say what I take from them.  If the editors
think the following stuff is "just dumb" but the question
an interesting one, I hope they will please just forward
the question an omit the rest.

This is  my reading and inference drawing from Lakoff
and Núñez, not something to be blamed on them:


On the one hand, robust proofs are generally things
that, if suitably expressed, can be mechanically checked.
We have grammars of formulas and diagrams, plus axioms
and axiom schema expressed as essentially syntactic 
transforms, these define formal languages in which
proofs can be recognized by machine.

On the other hand, there is the sense of "something beyond"
a purely syntactic understanding.  There is *something* being
modeled.   Lakoff and Núñez locate that feeling in 
the innate capacities of our embodied brains.  For example,
like many animals, we are essentially born with a capacity
to do some simple arithmetic with small numbers, or to imagine
some aspects of an iterated process, or to imagine certain
kinds of spacial transforms.

When formal mathematical constructions "feel meaningful"
it is because they give rise to more detailed imaginings
that use those innate capacities.  

In that view, alternative proofs of things are... forgive me
for being unable to be quite precise here but... something
like a form of induced synesthesia:  one kind of (abstract)
sense perception, a kind of extended proprioception, crosses
over to another.   By analogy, if we saw the first proof as
a visceral fact about colors, the second proof helps us perceive
the same stimuli (the formal text of the proof) as sound.
We're left "feeling" a single theorem in two different ways
and "feeling" a complex of visceral metaphors that make it
feel obvious the two are related.

The innate capacities (like a sense of low-number arithmetic
or of an iterated process, ec.) are the unreliable but productive
source of mathematical creativity.   As we play around with
these imagined things we can't count on our imagination not
making mistakes - hence the invention of "proof" - but our 
visceral imagination is what guides our search through the 
possible space of new theorems.

So a second proof - one that doesn't trivially invoke the
same "monkey brain" imaginings but instead invokes a different
set - expands our capacity to imagine possible new theorems.

I don't think that Lakoff and Núñez really have much
to say about questions like Platonism.   I think they
are talking "orthogonally" to issues like that.  They don't
directly contribute to foundations per se, at least as I understand
this list to contemplate foundational questions.   But what
they write *does* seem to have implications for how far our 
capacity to reason about foundations can possibly go.   And I
think it does have some retrospective perspective to offer
on why certain big controversies / challenges in math grab
our attention so much:  They make for compelling stories.

-t