Well, I've stopped reading philosophy, say, ever since I got a "real" job. I guess it must be the case that the older one gets, the less valuable thing one reads and the more garbage one writes. And, this is what this "real" job wants you to do; very unfortunate. So, let me set up a goal for myself:

By 2010, finish reading Wittgenstein's Remarks on the Foundations of Mathematics and Philosophical Remarks .

Mathematicians and Philosophers

by Chung-Chih Li

Rene Descartes:

"I am thinking, therefore I exist."

Descartes was the most important figure in modern mathematics and philosophy, and the quote above probably is the most famous say of Descartes; but what does it really mean? Well, I do have problem to understand the English word therefore here. Logically speaking, "I'm good, therefor I do it" seems equivalent to "being good implies being doing it", but it seems to me that Descartes didn't really mean "thinking implies existence". It's more like the inverse -- "not thinking implies not existing". The existence is established necessarily via mankind's awareness, which seemed self-evidence to Descartes, but some obvious objections followed immediately. I myself probably like the idea from the school with an acute view against the material world known as idealism that the true world is made of ideals, but I'm not sure I'm strong enough to maintain my existence free from physical world's intervention. You may want to see the excellent debate in " Meditations on First Philosophy " edited by John Cottingham --, which is a collection of Descartes' Meditations, their objections, and Descartes' replies.

"The brain is a machine."

This is controversial, and certainly is when I extract this single phrase and put it here without context. A word " Cartesian " was invented to shine the philosophy of Descartes maintaining that mathematics is the only way to understand the universe. Tons of people proudly announce themselves as Cartesians, e.g., Noam Chomsky called himself a cartesian linguist. Anyhow, that "the brain is a machine " is fine with me if it comes from medical people who necessarily have treated the brain as an organ ; but it does not suggest that we can go one step farther. It rings funny if a psychologist insists that the love I attach to my wife is just some chemical material at work. Well, the desire of having sex with women may be chemical, which with one or two little pills can be either boosted or busted as we wish, but the love I have is not. Is doing mathematics chemical? Machines do not love machines. No,no,no.. They don't. A machine that walks and talks like a woman is still a machine. (Have you seen the movie, Bicentennial Man? I love the movie, anyway)

What behind the brain is indeed the central question of Cartesian philosophy. It is called mind. Cartesian dualism can't help to make the picture clearer. The difficulty is that how does the mind cross the border to control the brain, and therefore the body? Descartes could not answer it. Neither can any dualists.

G. Kreisel:

"The principal interest is philosophical: not to confine oneself to what is necessary for (current) practice, but to see what is possible by way of theoretical analysis."

-- Hilbert's programme and the search of automatic proof procedures,
Springer Lect. Not. Math 125, (1970) pp 128-146
-- quoted from Classical Recursive Theory
by Piergiorgio Odifreddi

David Hilbert:

"The state that we are in now vis-a-vis the paradoxes is in the long run unendurable. Just think of it: in mathematics, this standard of trustworthiness and truth -- the forming of concepts and of inferences, as learned, taught, and used by all of us, can lead to nonsense. Where is one to find reliability and truth if mathematical thought itself can fail? .."

-- from In search of infinity
by N. Ya. Vilenkin, Abe Shenitzer
(Translator) Hardy Grant
(Editor) Bohdan Mykytiuk

"Nobody can expel us from the paradise created for us by Cantor! "

-- from Realism in Mathematics
by Penelope Maddy

Kurt Godel:

"Wait a minute! Sir!"

said Godel to Hilbert.-- Well, I just imagine what Godel might reply to Hilbert's program , but due to his introspective personality he was not likely to rise a challenge like that. Godel is the greatest logician since Aristotle , who, in 1912 at the age of 6, was not so sure about 4-1=3 in his arithmetic workbook.

What the astonishing work of Godel teaches us is that: we should, unlike Hilbert, live with paradoxes, for we cannot get rid of them even in the mathematical world. He practiced this in his life. He became a citizen of the USA after he found a contradiction in the American constitution.

However, Godel had never accepted the idea that the incompleteness of every formal system suggests the limitation of human mind. For many years, he had been bothered by Alan Turing's claim that the power of human reason cannot go beyond the mechanical limitation, which is justified by the fact that the human brain is necessarily limited. Godel finally claimed that he had found a way to refute Turing, in which Godel considered the human mind, although limited, is constantly developing; and there is no reason to conclude that the process must converge to a finite number.

I don't know how much did Godel satisfy his self-claimed victory. A recursive operator, (N -> N) -> (N -> N), a pure mechanical device, goes that way on and on if we take its enumeration as the "mind developing", but it is a far cry from the human mind. Well, there are many guys I happen to know do converge to finite numbers.

On the other hand, as an archetypal of Platonism, in sympathy with Intuitionism, Godel remarked the following to Hao Wang in 1971,

"I don't think the brain came in the Darwinian manner.... Simple mechanism can't yield the brain.... Life force is a primitive element of the universe and it obeys certain laws of action. These laws are not simple and not mechanical."

Another piece of Godel's remarks interesting to me is in a letter to a student in elementary education major, which never sent off. Godel wrote:

"It seems to me that the worth and beauty of mathematics should not be pointed out to beginning students... because one has first to know some mathematics before one can appreciate such a statement. What should be pointed out is the truly astonishing number of simple and non-trivial theorems and relations that prevail in mathematics."

-- From Logical Dilemmas
by John W. Dawson, JR.

So, Godel's approach is very different from Dirac's, one of the top physicists of this century. Here is remarks of Steven Weinberg, a Nobelist in physics:

Steven Weinberg: "I once heard Dirac say in a lecture, to an audience which largely consisted of students, that students of physics shouldn't worry too much about what the equations of physics mean, but only about the beauty of the equations."

-- Steven Weinberg, "Towards the Final Laws of Physics"
from Beauty and Revolution in Science
by James W. McAllister

Without a doubt, we enjoyed the beauty of the laws, and that's what we are obsessed with, but to some degree I have to agree with Godel; what does the beauty of Turing Machines mean to an undergraduate? It's not impossible, but it would be very difficult to explain. At least, I didn't get a chance to have a teacher when I was an undergraduate who could reveal the beauty to me.

Hermann Weyl:

"God exists since mathematics is consistent, and devil exists since its consistency cannot be proved."

-- I forget where I read this quotation

Blaise Pascal:

Reason is the slow and tortuous method by which those who do not know the truth discover it.
The heart has its own reasons which reason does not know.

Karl Weierstrass:

It is true that a mathematician, who is not somewhat of a poet, will never be a perfect mathematician.

I think Pascal had the same finding some 200 years before Weierstrass.

Ludwig Wittgenstein:

Some says that Wittgenstein is the philosopher of the 20th century; some says that he is the two philosophers of the 20th century, and one denies the other. Godel once complained that Wittgenstein pretended he couldn't understand Godel's incompleteness theorem. Indeed, how could that be? Probably admitting the theorem, Wittgenstein felt, would make his entire philosophy trivial. And, HORROR!! This can be proven by mathematics!!! which Wittgenstein latter denied!!

Should I say there is an entire new chapter of philosophy of mathematics created solely by this strange philosopher? May be not. Early Wittgenstein meant to finish the philosophy for good, and he didn't bother to create any alternative on the subject. Thus, he said:

"Whereof one cannot speak, thereof one must be silent."

This is the last sentence in his Tractatus.

"Perspicuity is part of proof"

Well, it's not just a slogan that every mathematics teacher should pronounce to her students as I thought it is when I first time read this quote in Feasible Mathematics V.2. and then put it in my own book Problems on Discrete Mathematics . In fact, it is the very reason by which Wittgenstein convinced himself that any axiomization approach to provide a foundation for mathematics must end up with a failure; mathematics, he believed, needs no foundation at all. Clearly, this strongly intimidates the entire school of formalism, from which Wittgenstein departed his thoughts. Bertrand A. Russell, Wittgenstein's teacher, gave no idea, according to Wittgenstein's perspicuity principle, to prove that 10^10+1 is not equal to 10^10 in his gigantic Principia Mathematica. Thus, I will not be surprised if Wittgenstein could live long enough, he will simply say that the 4-color problem is still open.

"Turing Machines, these machines are humans who calculate."

If what Wittgenstein said is accepted, then, CM5 (Connection Machine 5, one of the most powerful super computer in early '90s) is definitely nothing better than the Turing Machine. At least we do not consider CM5 as a human being (doing calculation only, though), no matter how well do we equip it. However, I'm strongly wondering if there is a Turing Machine that can pass the so-called Turing Test, while Turing himself tried to construct one but left the job for us to fulfill; probably an unfulfillable dream. I have few words on the so-called Church-Turing Thesis.

Do you see A.I. the movie? A dumb movie.