Just finished a fantastic book, Mathematics: Is God Silent? by James Nickel. So I will be posting a review of the book. But first, my own observations about mathematics and math education.
Mathematics is a subject area for which most people, even many Christians, hold the notion that it is somehow absolutely true or "set in stone" or everyone would agree it is true.
I'll get Nickel's book in a later post. First, my own thoughts on the subject. And by "my own thoughts" I mean it is a combination of some other readings I've done and my own critiques and distillations of those readings. I'm not so smart that I thought all this stuff up entirely on my own but I really wouldn't be able to tell you which thoughts I read elsewhere and which ones were the result of those thoughts fermenting in my mind.
So, first a refutation of some commonly held notions about mathematics.
1. Mathematics is a science, not an art.
It's both.
Here's an analogy. Let's say you are standing on one side of a large room and you want to get to the other side. Unfortunately, you're stuck in a maze. In order to get where you are going, you need tofollow some part of the maze. However, simply following the maze won't necessarily get you to your goal. You could wander in a maze for years, always following the corridors but never to your destination. In other words, simply following the rules isn't enough. There is something more. Getting through a maze requires ingenuity, cleverness, and sometimes genius. And this kind of mental acuity is not developed simply by studying rules.
Mathematics is much the same way. You must learn how to think logically and learn the rules of proofs, like following the corridors of a maze correctly. In order to become a skilled mathematician, you can't simply know the rules, you can't just follow the corridors in a maze. You have to possess a mental agility that allows you to get where you want to go given a certain set of constraints. How did McGyver (boy I'm dating myself now) get out of the stickly situations he was in?: imagination leaning against the constraints of the situation he was in. You must possess imagination. One of the beset way to develop this is to memorize proofs of other mathematicians and be able to rewrite them from memory. In other words, you want to learn how other mathematicians have solved their problems.
2. To become a good mathematician you should only study the modern stuff. Studying older stuff is interesting but useless.
Incidentally, this is a good argument for studying historical mathematics, like Euclid, Archimedes, Newton, Gauss, etc. Is there a tough math problems that's so far escaped modern mathematicians? Try studying mathematics of the past for awhile then. The reason those mathematicians are so famous is they often solved problems in a very clever manner, approaching the problem in a way that previous mathematicians hadn't even considered. If you want to solved a difficult math problem, then it would be useful to learn that cleverness.
3. In order to be a productive mathematician, you only need to study mathematics, old or new.
This is a lot like saying that in order for a flower to grow all it needs is water; sunlight is for nancies. Studying more mathematics, historical or not, isn't enough. To develop one's imagination in different directions, it's also useful to study good literature and history. To think that all this knowledge is totally unrelated and that Shakespeare will have no effect whatsoever, however subtle, on your study of geometry is implicitly assuming that all knowledge is unrelated. Knowledge is not simply a collection of facts. All of this stuff is stored in one mind and if you study geometry and develop your logical thinking skills then turn around and try your hand at writing good poetry, you're disciplining that same mind, albeit in different directions.
It's simply historical fact that some of the largest contributions to mathematics came from mathematicians who just made stuff up. One must follow the rules of logic to develop a branch of mathematics but the beginning of that branch must (usually) be simply invented. There are a number of instances of entirely new branches of mathematics growing out of other mathematics. Differential equations naturally grows out of calculus. Topology grows naturally out of set theory. And many branches cross paths. Differential geometry is (what else?) calculus and geometry crossing paths.
But logarithms, calculus, and set theory are all examples of mathematicians creating mathematics out of thin air. That kind of genius does not come out of an undeveloped imagination. A mind merely drilled in the machination of logic doesn't know how to think outside that mindset and won't produce the kind of genius required to invent mathematics of that caliber. Napier, Newton, and Leibniz all participated in an educations era when college education was primarily liberal arts. Cantor may not have, but that doesn't invalidate my point. A fully orbed, inventive, productive mathematician will also have read Herodotus, Homer, Aristotle, Anselm, and Aquinas.
The state of mathematics today should give us an indication of this truth. Mathematics today has been called a "mass of details without focus". Supposedly, there have been more theorems produced in the last century than all previous centuries combined. But mathematicians have been largely only educated in mathematics, and only modern mathematics at that. And you can be productive with an education like that; all you'll really be able to produce are theorems that don't too far from the branch whence they came and furthermore they won't be anything too profound. They'll be so specific and exacting and narrow as to be virtually worthless, though will be in the strict sense "true" but wont' really say much of anything of value.
