topics: |  mathematics | cognitive | philosophy

Some interesting mathematics, but banal remarks on cognition.

However, Byers does succeed in highlighting the key question of how
mathematics is actually done - the role of intuition - as earlier discussed
in the work of Poincare, Polya, Rota, and many others.

Mathematical creativity

Proving the Fermat Theorem

Andrew Wiles gave the final resolution to the Fermat conjecture.
The solution to Fermat was Wiles's life ambition. "When he re-
vealed a proof in that summer of 1993, it came at the end of
seven years of dedicated work on the problem, a degree of focus
and determination that is hard to imagine."

He said of this period in his life, "I carried this thought in my head
basically the whole time. I would wake up with it first thing in the morning,
I would be thinking about it all day, and I would be thinking about it when I
went to sleep. Without distraction I would have the same thing going round
and round in my mind." In an interview, Wiles reflects on the process of
doing mathematical research:

	Perhaps I can best describe my experience of doing mathematics in
	terms of a journey through a dark unexplored mansion. You enter the
	first room of the mansion and it's completely dark. You stumble
	around bumping into the furniture, but gradually you learn where each
	piece of furni- ture is. Finally after six months or so, you find the
	light switch, you turn it on, and suddenly it's all illuminated. You
	can see exactly where you were. Then you move into the next room and
	spend another six months in the dark. So each of these breakthroughs,
	while sometimes they're momentary, sometimes over a period of a day
	or two, they are the culmination of-and couldn't exist without-the
	many months of stumbling around in the dark that precede them.

This is the way it is! This is what it means to do mathematics at the highest
level, yet when people talk about mathematics, the elements that make up
Wiles's description are missing.

Immediate Certainty

Take the following famous quote from the great French mathematician Henri
Poincare. Poincare had been working on proving the existence of a class of
functions that he later named Fuchsian:

	Just at this time I left Caen, where I was then living, to go on a
	geologic excursion under the auspices of the school of mines. The
	changes of travel made me forget my mathematical work. Having
	reached Coutances, we entered an omnibus to go to some place or
	other. At the moment when I put my foot on the step the idea came to
	me, without anything in my former thoughts seeming to have paved the
	way for it, that the transformations that I had used to define the
	Fuchsian functions were identical with those of non-Euclidean
	geometry. I did not verify the idea; I should not have had time, as,
	upon taking my seat in the omnibus, I went on with a conversation
	already commenced, but I felt a perfect certainty. On my return to
	Caen, for conscience's sake I verified the result at my leisure.

Poincare goes on to say:

	Then I turned my attention to the study of some arithmetical
	questions apparently without much success and without a suspicion of
	any connection with my preceding researches. Disgusted with my
	failure, I went to spend a few days at the seaside, and thought of
	something else.  One morning, walking on the bluff, the idea came to
	me, with just the same characteristics of brevity, suddenness, and
	immediate certainty, that the arithmetic transformations of inde-
	terminate ternary quadratic forms were identical with those of
	non-Euclidean geometry. 

Poincare's "immediate certainty" is an essential but often neglected
component of mathematical truth. Truth in mathematics and the certainty that
arises when that truth is made manifest are not two separate phenomena; they
are inseparable from one another-different aspects of the same underlying
phenomenon. p.329

Is  Euler's formula, V - E + F = 2, for polyhedra correct or incorrect?
It all depends-on what you call a polyhedron...

the naive belief in the existence of an "objective truth."  "But," some will
argue, "isn't this inner certainty that you are talking about a merely
subjective phenomenon whereas the truth is surely objective?" On one level
this is definitely the case.  Certainty arises in the mind, and one
definition of "subjective" is "existing only in the mind and not independent
of it."

[On this point, we can quote Euler in his original paper presenting this
idea: 

	I for one have to admit that I have not yet been able to devise a
	strict proof of this theorem. As however the truth of it has been
	established in so many cases, there can be no doubt that it holds
	good for any solid. Thus the proposition seems to be satisfactorily
	demonstrated.

]

Immediate sense of Certainty in Intuition

However, the "immediate certainty" Poincare refers to is also objective. It
does not change from situation to situation. Poin- care, in the above
excerpt, talks about two different instances of this sensation, and it is
clear that it is the same certainty that he feels on both occasions. It may
well be that "immediate cer- tainty" that one experiences in such situations
is the same for all people at all times.

Even animals may experience such certainty. Take, for example, the
experiment with a talented chimpanzee named Sultan by the psychologist
Wolfgang Kohler. 

	Beyond the bars, out of arm's reach, lies an objective [a banana]; on
	this side, in the background of the experiment room, is placed a sawn
	off castor-oil bush, whose branches can be easily broken off. It is
	impossible to squeeze the tree through the railings, on account of
	its awkward shape; besides, only one of the bigger apes could drag it
	as far as the bars.  Sultan is let in, does not immediately see the
	objective, and, looking about him indifferently, sucks one of the
	branches of the tree. 

	But, his attention having been drawn to the objective, he approaches
	the bars, glances outside, the next moment turns around, goes
	straight to the tree, seizes a thin slender branch, breaks it off
	with a sharp jerk, runs back to the bars, and attains the
	objective. From the turning round upon the tree up to the grasping of
	the fruit with the broken-off branch, is one single quick chain of
	action, without the least "hiatus," and without the slightest
	movement that does not, objectively considered, fit into the solution
	described. 


	a video of a chimpanzee experiment similar to the Sultan
	action described by Kohler.   Similar experiments on
	a number of other animals - e.g. the New Caledonian crow: 

https://www.youtube.com/watch?v=URZ_EciujrE
(metatool use)

---


Sultan not only used a stick as a tool, he created the tool by breaking off
the branch. His actions, after he had arrived at what can only be called an
insight, were characterized by an economy of action and a sense of purpose
that are reminiscent of what I have been calling "immediate certainty."