Heuristic: (adj.) Enabling a person to discover or learn something for themselves.
The next handful of blog posts will all be notes and observations from Mathematics and Plausible Reasoning, Volume I by the great math pedagogue George Polya. The book was lent to me by a good friend and mentor at work. Ironically, I spent two years working for a teaching lab named after Polya that (despite it’s other merits) didn’t actually follow Polya’s philosophy in nearly any regard.
This is also the first book I’ve ever read that is nearly 50% equations. Many of these I had to skim through as they involved higher math than I am not readily familiar with (differential calculus, analytical geometry, etc.). Some of the examples involving solid geometry and probability were more accessible. I didn’t care about the math though – I was interested in what Polya had to say about the nature of problem-solving in general. Fortunately, there was a lot to be gleaned from his examples. The bits of history sprinkled here and there were also fascinating. I was reminded of the Newton biography I read last year. My respect for Newton continues to soar.
I love a book that starts out by clarifying word definitions. This is what captivated me in N.T. Wright’s big books. It seems that in many of my favorite works, the introduction or preface is often the most helpful section of all. Polya begins with some great comments on the nature of explanations and conjecture:
Strictly speaking, all our knowledge outside mathematics and demonstrative logic (which is, in fact, a branch of mathematics) consists of conjectures. There are, of course, conjectures and conjectures. There are highly respectable and reliable conjectures as those expressed in certain general laws of physical science. There are other conjectures, neither reliable nor respectable, some of which may make you angry when you read them in a newspaper. And in between there are all sorts of conjectures, hunches, and guesses.
We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning.
The difference between the two kinds of reasoning is great and manifold. Demonstrative reasoning is safe, beyond controversy, and final. Plausible reasoning is hazardous, controversial, and provisional. Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs. Demonstrative reasoning has rigid standards, codified and clarified by logic (formal or demonstrative logic), which is the theory of demonstrative reasoning. The standards of plausible reasoning are fluid, and there is no theory of such reasoning that could be compared to demonstrative logic in clarity or would command comparable consensus.
-George Polya, Mathematics and Plausible Reasoning, Volume I, p. v
Biologists, climatologists, archeologists, economists, psychologists and especially sociologists have got to stop pretending that what they do is rigid “demonstrative” reasoning. They imagine themselves to be hard scientists like physicists (to some degree) or mathematicians. It’s not the same. That’s OK, but be honest about it. The standards are fluid (ever changing!) and even the best work is riddled with guesses, some supported substantially and others not so much. Biologists often laugh at theologians for not being real scientists, but in reality they are both on exactly the same playing field.
I do not believe that there is a foolproof method to learn guessing. At any rate, if there is such a method, I do not know it, and quite certainly I do not pretend to offer it on the following pages. The efficient use of plausible reasoning is a practical skill and it is learned, as any other practical skill, by imitation and practice. I shall try to do my best for the reader who is anxious to learn plausible reasoning, but what I can offer are only examples for imitation and opportunity for practice.
In what follows, I shall often discuss mathematical discoveries, great and small. I cannot tell the true story how the discovery did happen, because nobody really knows that. Yet I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it, in short, everything that deserves imitation. Of course, I shall try to impress the reader; this is my duty as teacher and author. Yet I shall be perfectly honest with the reader in the point that really matters: I shall try to impress him only with things which seem genuine and helpful to me.
I have to come to really enjoy the way that authors humble themselves (or not!) in the first few pages of their books. The authors that do not are immediately suspect to me. Someone that can’t laugh at themselves or immediately recognize how large insignificant their work is are not, in my opinion, spiritually or psychologically healthy enough to be thinking clearly (or gracefully) about the topic at hand.
“I shall try to make up a likely story.” If only so many other books came with this much-needed disclaimer! I love it!
Most parts of this book have been presented in my lectures, some parts several times. In some parts and in some respects, I preserved the tone of oral presentation. I do not think that such a tone is advisable in printed presentation of mathematics in general, but in the present case it may be appropriate, or at least excusable.
I’m going to disagree a bit here. I think the oral tone IS acceptable in print. I have found it to frequently be a clear mode of writing. This is maybe because I am a slow reader that “reads aloud” to myself in my head. It may be inefficient, but I actually prefer it in many cases. Transcriptions of speeches can be messy and superfluous, but they can also be crystal clear and de-gunked in a way that unspoken prose cannot rise to.
The advanced reader who skips parts that appear to him too elementary may miss more than the less advanced reader who skips parts that appear to him too complex.
A good warning! (And encouraging to this reader, who definitely skipped some of the complex parts.)