I recently read a wonderful book by William Byers called How Mathematicians Think and it really had an impact on my thinking. Byers’ central thesis is that mathematics is about more than just formal logical rules and algorithmic thinking, it’s actually about much more. He claims that mathematics is the most creative of all human endeavors and that ambiguity, contradiction and paradox are essential to creating new mathematics.
Some people find the philosophy of mathematics titillating, and for others, it causes great despair (even for some mathematicians). At certain times, I could be classified as fitting in both categories. The philosophy of mathematics is the kind of subject that is likely to alienate people. Writing about it can be a great way to get people to quit reading your blog (I hope that isn’t the case though). However, it isn’t necessary to have a deep technical background in mathematics (although it may help) in order to philosophize about it.
Let’s face it, mathematics can be intimidating for a host of reasons, namely because it’s hard to understand what it really is. My formal mathematical education largely consisted of memorization and manual calculation, which was often boring and dry. As a student learning mathematics, I didn’t always understand what I was doing. Even worse (gasp!), I didn’t understand what the subject matter itself really was.
When you think about it, I mean deeply think about it, mathematics is beautiful, yet deeply perplexing. What is a real number, really? Where do mathematical ideas come from? Does our sense of logic help us create mathematics or does our mathematics help us understand our sense of logic? Did humans invent mathematics or discover it?
On the surface of it, there appears to be this strange relationship between the natural world and our mathematical intelligence. As the physicist Eugene Wigner put it, what are we to make of “the unreasonable effectiveness of mathematics in the natural sciences?” That’s certainly an interesting question. However, it’s a bit of a stretch to assume that the very fabric of the entire natural world is coded in a mathematical language which we are capable of understanding just because some aspects of our mathematics seemingly match up with some aspects of the natural world. To even consider this argument, we must first assume that there is some absolute objective reality out there — but is there? There was an excellent passage in How Mathematicians Think, which elucidates this point so I am going to quote it in its entirety.
Strangely enough, in order to understand mathematics and the subtle nature of mathematical truth it may be necessary to give up our attachment to the idea of “absolute objectivity”. The existence of such an objective domain is an assumption that we all unthinkingly make, especially in science. But a moment of thought will reveal that it can only be an article of faith — it can never be proved. The only contact human beings have with reality is through the impressions that are received by the senses and the mind. The “objective” world is not, as far as human beings will ever be able to tell, completely objective. We know it through acts of perception and cognition. Postulating an absolutely objective realm is just that — an assumption — not something that can be empirically validated.
Whatever the reason, I’ve always been sympathetic to the belief that mathematics was invented. The way I look at mathematics is that it’s a language, albeit a very special type of language, that requires human ingenuity to create. It’s a creative process which attempts to map reality into concepts the human brain has evolved to understand. Reality isn’t necessarily algorithmic, rather algorithmic thinking helps us to understand certain aspects of reality. Or as the physicist David Bohm put it: “I think people get it upside down when they say the unambiguous is the reality and the ambiguous merely uncertainty about what is really unambiguous. Let’s turn it around the other way: the ambiguous is the reality and unambiguous is merely a special case of it, where we finally manage to pin down some very special aspect.”
So what exactly is mathematics anyway? Describing mathematics is very difficult and perhaps it would be easier to say what it is not, i.e., mathematics is not simply a body of facts arrived at by deductive reasoning from a body of definitions and axioms, although it’s tempting to think that way. At its core essence, mathematics is a creative process that is often poorly taught as if it were some stringent logical system I just described. Ultimately, I think it’s both difficult to teach and learn something if you don’t really understand what it is.