Today's Sunday reflection, a passage about the role of intuition in mathematics (from "

**How Mathematicians Think**" by William Byers):

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*"For the mathematician, the idea is everything. Profound ideas are hard to come by, and when they surface they are milked for every possible consequence that one can squeeze out of them. Those who describe mathematics as an exercise in pure logic are blind to the living core of mathematics -- the mathematical idea -- that one could call the fundamental principle of mathematics. Everything else, logical structure included, is secondary.*

"The mathematical idea is an answer to the question, 'What is going on here?' Now the mathematician can sense the presence of an idea even when the idea has not yet emerged. This happens mainly in a research situation, but it can also happen in a learning environment. It occurs when you are looking at a certain mathematical situation and it occurs to you that 'something is going on here.' The data that you are observing are not random, there is some coherence, some pattern, and some reason for the pattern. Something systematic is going on, but at the time you are not aware of what it might be…

"The feeling that 'something is going on here' can even be brought on by a single fact, a single number. A case in point happened in 1978, when my colleague John McKay noticed that 196884 = 196883+1. What, one might ask, is so important about the fact that some specific integer is one larger than its predecessor? The answer is that these are not just any two numbers. They are significant mathematical constants that are found in two different areas of mathematics. The first arises in the context of the mathematical theory of modular forms. The second arises in the context of the irreducible representations of a finite simple group called the Monster. McKay intuitively realized that the relationship between these two constants could not be a coincidence, and his observation started a line of mathematical inquiry that led to a series of conjectures that go by the name 'monstrous moonshine.' The main conjecture in this theory was finally proved by Fields Medalist winner Richard E. Borcherds. Thus the initial observation plus recognition that such an unusual coincidence must have some deep mathematical significance led to the development of a whole area of significant mathematical research….

"…But still it is possible to say 'we do not really understand what is going on.' Understanding what is going on is an ongoing process -- the very heart of mathematics."

"The mathematical idea is an answer to the question, 'What is going on here?' Now the mathematician can sense the presence of an idea even when the idea has not yet emerged. This happens mainly in a research situation, but it can also happen in a learning environment. It occurs when you are looking at a certain mathematical situation and it occurs to you that 'something is going on here.' The data that you are observing are not random, there is some coherence, some pattern, and some reason for the pattern. Something systematic is going on, but at the time you are not aware of what it might be…

"The feeling that 'something is going on here' can even be brought on by a single fact, a single number. A case in point happened in 1978, when my colleague John McKay noticed that 196884 = 196883+1. What, one might ask, is so important about the fact that some specific integer is one larger than its predecessor? The answer is that these are not just any two numbers. They are significant mathematical constants that are found in two different areas of mathematics. The first arises in the context of the mathematical theory of modular forms. The second arises in the context of the irreducible representations of a finite simple group called the Monster. McKay intuitively realized that the relationship between these two constants could not be a coincidence, and his observation started a line of mathematical inquiry that led to a series of conjectures that go by the name 'monstrous moonshine.' The main conjecture in this theory was finally proved by Fields Medalist winner Richard E. Borcherds. Thus the initial observation plus recognition that such an unusual coincidence must have some deep mathematical significance led to the development of a whole area of significant mathematical research….

"…But still it is possible to say 'we do not really understand what is going on.' Understanding what is going on is an ongoing process -- the very heart of mathematics."

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I think, in essence, this is very much what Eugene Wigner's famous notion of "

*the unreasonable effectiveness of mathematics*" revolves around… how is it that the human brain is capable of such seemingly successful intuitions about the world around us….

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one you submit, I'll cite the contributor.]

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