Swapping Theorems with definitions

Question

My question is motivated from the following passage of Gian-Carlo Rota's Indiscrete Thoughts,

'Suppose you are given two formal presentations of the same mathematical theory. The definitions of the first presentation are the theorems of the second, and vice versa. This situation frequently occurs in mathematics.'

The quote appears as a thought experiment to demonstrate the difference between mathematical theories and their presentations, but I'm wondering whether such a swap has happened in the teaching of a particular area of mathematics. It seems plausible that this might have occurred at some point in the history of mathematics, and because the presentation of a theory can vary over time. Is anyone aware of an example?

Answer 1

This occurs so often that there is now a field studying this phenomenon called "reverse mathematics". Our tag Reverse mathematics defines it as "the study of which axioms are required to prove mathematical theorems" but the scope of the field is larger. For example, it is known that the countable axiom of choice is required to prove the $\sigma$-countability of the Lebesgue measure. One might ask, conversely, whether the hypothesis of $\sigma$-additivity implies countable AC (assuming ZF).

A more elementary example would be the theory of the real numbers. They were defined in the 16th century by Simon Stevin in terms of infinite decimal expansions. Ultimately Dedekind (and Cantor) showed that they can be obtained as the "Dedekind cuts" (or equivalence classes of Cauchy sequences). Nowadays it is customary to adopt the reverse approach: one initially defines the reals a la Cantor/Dedekind, and then proves that they can also be expressed by infinite decimals.

Answer 2

It occurs constantly. Trying to think of the earliest examples I remember conics. Defined as intersections of a cone with planes. Then you deduce they satisfy quadratic equations.

However in many modern books more interested in the algebra of the topic conics are defined as the solutions of quadratic equations, and then one finds they are sections of a cone.

The phenomenon is bound to happen. Mathematics is not the study of definitions, but of the way they interact. For that reason, the status of a definition as such is not absolute. They can change from being definitions to being theorems, but they can even change in nature (e.g. adding, removing, or changing conditions to see what happens, or to get the theory written in other ways).

Answer 3

In my experience it goes on all the time. Starting with certain hypotheses can make it very difficult to prove a theorem. Starting with a different definition can often yield an easy solution.

An immediate example is to prove that $e^x$ is differentiable. You can start by defining all exponential functions for the integers, then for the rationals. Then you can bootstrap your way to the reals, if you are a little careful, and you will get continuity. Now try to prove they are differentiable.

If you start with the definition of a derivative, you will quickly run into problems. I have found/developed 3 different direct proofs of this differentiability, none of which is really easy.

However, by turning things around, we can have a very easy proof indeed, and avoid all the detailed fiddling that is needed if we define exponential functions directly. To do this define

log x = $\int_1^x (1/t)dt$ for $ x \in (0, \infty)$

From this definition you can immediately and simply prove the usual properties of the logarithm. By this definition log x is monotonic and differentiable.

Then define $e^x$ to be the inverse of log x as defined above. It is immediately monotonic and differentiable because log x is; and will inherit from log x the usual properties of exponentials. You have also defined e as being the base of this exponential function, without an elaborate derivation of the number or what it may mean. Showing this e matches up with our usual idea of e is pretty trivial.

Thus a complex proof with the direct approach is reduced to a few simple lines with an indirect approach.

This is a simple example, but some of the best proofs I have seen began by turning the problem on its head.