# The Deductive Method

"Readers of crime fiction will be familiar with two types of detectives. One adopts the card index method of Francis Bacon, collecting all relevant information piece by piece. The other follows a hunch, like Newton, and like Newton, abandons it at once when it comes into conflict with observed facts. From time to time the philosophers of science emphasize the merits of one or the other, and write as if one or the other were the true method of science. The unity of science resides in the nature of the result, the unity of theory with practice. Each type of detection has its use, and the best detective is one who combines both methods, letting his hunch lead him to the best hypotheses and keeping alert to new facts while doing so." - Lancelot Hogben, Science for the Citizen, 1938, p. 747

## Mathematics - the Deductive Method

Geometry was organized by the Greek mathematician Euclid, and the structure given to Mathematics by the Greeks is still used by mathematicians today. It is called the deductive method. The Greeks immediately recognized the power and utility of Euclid's method of inquiry, which came to be called the deductive method. Mathematics courses do not generally emphasize the deductive nature of mathematics much these days, so it is entirely possible that a high-school mathematics major would not have heard of it - yet it is the structure of mathematics!

To use the deductive method, here is what you need to do:

1. You have to start somewhere, and you start with undefined terms. You pick undefined terms to be very common and self-evident, then you just have to assume that everyone will be "on the same page". For Euclid, undefined terms were things like point, line, etc. You can discuss what you mean by an undefined term, but you can't define everything.
2. Once you have agreed on some undefined terms, you can use them to create definitions. Euclid, for instance, could give a precise definition of a triangle in terms of points and lines.
3. Next, you need to pick some simple, obviously true statements about the undefined terms and definitions. These statements are called axioms or postulates. You want to keep the number of axioms to a minimum - Euclid had 5 axioms for all of geometry. (One was "Two points determine a line.")
4. Now, things can get interesting. You can combine your axioms, definitions, and undefined terms with the rules of logic to prove that other statements must be true. These statements are called theorems. (Oh, yes, you remember trying to prove theorems!)
5. Once a theorem is proven, you can use it, along with other proven theorems, axioms, definitions, and undefined terms to prove other theorems.

Using the deductive method, you start with a few true statements (the axioms) and use them to prove dozens, hundreds, thousands, or millions of other statements (the theorems).

Here is a simplified diagram of how the deductive method works. It omits undefined terms and definitions, and it only shows two axioms, but it tries to show how a logical deductive system is constructed.

## What's Really Scary

It's not a big deal that Aristotle made some physics mistakes. People - even very intelligent people - make mistakes all the time. What I think is REALLY scary is that the mistake was perpetuated for almost 20 centuries! Why?

Here again, I think that it is a definite mistake to suppose that everyone between 300 B.C. and 1600 A.D. was just not very smart. Certainly, people dropped things. They had eyes and ears. It doesn't take a genius to make a simple observation.

I think it was mostly "because Aristotle said so." How many times have you said or written something like "I don't really understand this, but the book says..." Oops! I think that there is a moral lurking here...

"Simple people, like children, love security more than freedom; they worship authority blindly, and swallow its teaching whole. You may smile at this and say, "We are civilized. We don't behave like that..." We now condemn "Aristotelian dogmatism" as unscientific, yet there are still people who would rather argue from a book than go out and find what really does happen. The modern scientist is realistic; he tries experiments and abides by what he gets, even if it is not what he expected." - Eric Rogers, Physics for the Inquiring Mind, p. 8

## In All Fairness...

This discussion might tempt you to think that mathematics is a plodding, step-by-step ritual - theorem, proof, theorem, proof, theorem, proof, etc., etc., etc. This impression is entirely false. Even though the product of mathematics is a deductive system, the process of mathematics can be extremely intuitive and creative. In the words of Paul Halmos:

"Mathematics - this may surprise or shock some - is never deductive in its creation (emphasis added). The mathematician at work makes vague guesses, visualizes broad generalizations, and jumps to unwarranted conclusions. He arranges and rearranges his ideas, and he becomes convinced of their truth long before he can write down a logical proof.... The deductive stage, writing the results down, and writing its rigorous proof are relatively trivial once the real insight arrives: it is more the draftsman's work not the architect's." - Paul Halmos

Also, certain aspects of science were practiced in antiquity. The ancients made careful observations of the heavens, and were able to make some amazing hypotheses - for instance, the Greek Aristarchus described a model of the Solar System that is essentially the one we use today.1 The ancient Greeks also argued that air was a real substance based on their observation that an inflated pig bladder resisted compression.2

1see Arthur Koestler, The Sleepwalkers, 1959, chapters 1 and 2

2 see Arnold B. Arons, Teaching Introductory Physics, 1997, p. 47