The beginning of mathematical thought

Last night, I almost wrote a blog entry to commemorate an entire day without snow falling from the sky. However, snow was falling again gently yesterday evening as we returned to our hotel. My former advisor started a snowball fight. I managed to stay out of it somehow. :-)

It's looking as if it wants to snow again. I should be preparing a brief presentation on my research for several people, but I might as well write a few remarks on the opening talk. This was given by Bruno Buchberger himself, inventor of the first algorithm to compute a Gröbner basis, and organizer of this conference.

The first thing I have written down is a story about the different between mathematics and physics. "Imagine a woman," he began (this is obviously a paraphrase), "who goes for a walk in the park with her child. They see a white dog."

Here he drew the outline of a dog (better than the outline I drew), and wrote W(x) on the board.

"Then they see another white dog." Another outline. "Then they see a black dog." (Outline, filled in.) "Here, the mother remarks, 'Look!'" and he wrote:

∃x ¬W(x)

In the notation of symbolic logic this means, there exists a dog that is not a white dog. I don't recall if he actually said that.

"'As you see,' the mother continues," and he wrote again:

¬∀x W(x)

Not every dog is a white dog.

"A few days later," he continues, "they are walking in the park again. They pass two pine trees, then a tree that is not a pine tree. And the mother says again to her child:"

∃x ¬N(x)
¬∀x N(x)

That is, There exists a tree that is not a pine tree. Not all trees are pine trees.

"Up until this point," Dr. Buchberger explained, "what has gone on is physics: strictly the observation of what goes on in the world.

"Some time passes, and one day the mother goes into the park but the child stays home. The mother sees a chlid that is not behaving very well. She returns home and tells her own child,

∃x ¬B(x)"

There was a child in the park who was not behaving well. "This is still physics.

"Then the child answers the mother, 'Yes, mother, this is something we must accept:

¬∀x B(x).'"

Not all children behave well. "This," Dr. Buchberger explained, "is the beginning of mathematics."

Unfortunately, the example wasn't all that clear to me. I wrote down, "But how is this different from logic?" Surely he didn't mean to say that mathematics and logic are equivalent. It turns out I was not alone, because someone else actually asked that question. It is a sign of my dullness of mind that I don't remember the answer...

However, reflecting on it later, I have some thoughts on how this is different. (These may have been in the answer. In any case, the thoughts are not original; I've heard them somewhere before.)

(1) The mother's child did not see the misbehaving child. Thus, it was not direct observation, but some abstract thought was involved. This is mathematical. In mathematics, things can be understood as being true even if they have no obvious correspondence with an observed object. Number theory, for example, was prized by some mathematicians precisely because it had no obvious correspondence with reality. Another example would be group theory. Nowadays you couldn't use an ATM machine, or transmit information via the internet, without depending on the results of number theory; and several profound discoveries of 20th century physics relied on group theory... of all things.

In physics, however, observation of phenomena becomes necessary before a theory can be accepted. "Truth" in physics relies on a posteriori correspondence to an external observation; "truth" in mathematics relies on a priori, internal consistency. The miracle is that the two work together.

It is true that the child's abstraction was little more than a logical restatement of the sentence. That's because mathematics is often indistinguishable from logic. It's just that mathematicians work with special sorts of sentences.

(2) The child engaged in pattern matching. The development of mathematical theory rests on matching patterns and discovering order in the apparent chaos around us. The result in my PhD dissertation, for example, came from looking at and analyzing several thousand polynomials that satisfied certain constraints, and discovering a marvelous pattern that I had not seen before — in fact, it was so marvelous that I couldn't bring myself to believe it.