Motivation:

Many scientists, including a large number of mathematicians, hold the mathematical sciences in high regard due to our ability to derive profound truths from a small number of postulates. This is consistent with the belief that mathematical insights, such as Newton’s Principia are discovered rather than made.

However, I think we should revisit this common assumption that mathematical axiom systems are not free constructions of the human mind.

Euclidean geometry and Euclid’s parallel postulate:

Euclidean geometry, being the geometry most scientists are familiar with, is often considered by many to be above scrutiny. But, as this geometry is reducible to five axioms it may be worth a closer analysis:

1. A straight line segment may be drawn joining two points.
2. Any straight line segment may be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. Parallel postulate: In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the given line.

However, in the early 1800s Bolyai and Lobachevsky independently discovered non-Euclidean geometry where the parallel postulate does not hold. Other mathematicians, such as Riemann, built upon these findings-which paved the way for Einstein’s theory of relativity.

The relativity of mathematical truth:

As the reader reflects on the significance of the parallel postulate, it is worth noting that any science may be distilled to a finite set of axioms that are necessary and sufficient. So we may deduce that the axiomatic method is inevitable and that any scientific truth is relative to a particular choice of axioms that have withstood the test of numerous experiments. Now, mathematics being one science among others, does the relativity of mathematical truth mean that it is arbitrary?

Not in the least. Any axiom system is ultimately chosen for reasons of convenience.

The case for constructive mathematics:

We have reached the stage where we may assert that the mathematical sciences and their corresponding axiom systems are in some sense free constructions of the human mind. But, we may also assert that they are convenient. How can we make these ideas precise?

A set of axioms is in some sense the minimum description length of a scientific theory. If $\mathcal{A}$ is our axiom system and $T$ is our theory then we have:

$$K(T):= \min_{\mathcal{A}} \{l(\mathcal{A}):U(\mathcal{A})=T\}$$

where $U$ is a universal Turing machine and $K$ denotes Kolmogorov complexity. It follows that choosing a set of axioms amounts to a problem in model selection.

Discussion:

At this point the reader may ascertain that constructing axiom systems is a form of scientific model building and therefore a creative enterprise. The reader may also be inclined to believe that axiom systems are the ultimate representation of a scientific theory and therefore the main activity of theoretical science is reducible to the construction of useful axiom systems. However, this would be a step too far.

With a scientific axiom system we can construct all possible theorems that are true relative to that set of axioms but how many of these theorems are actually interesting? It is worth noting that scientific intuition, the human process for sampling from the space of possible theorems, escapes the deductive process. I think this is a strong indication that any axiomatisation is necessarily incomplete.

But, what then is scientific intuition? That perhaps is a subject for another day.

References:

1. Poincaré, Henri. La Science et l’Hypothèse. Champs Sciences. 1902.

2. Weisstein, Eric. “Euclid’s Postulates.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/EuclidsPostulates.html

3. Marcus Hutter (2007) Algorithmic information theory. Scholarpedia, 2(3):2519.