Are mathematical axioms arbitrary?

Question

I've been thinking recently about whether or not mathematical axioms are arbitrary.

I'm trying to figure out what axioms in systems are derived from and just how arbitrary they really are.

My main point of concern comes from the idea that physics isn't mathematically rigorous and that there often aren't proofs for ideas, this doesn't seem right as if we have axiomatic systems that can model the reals then surely we can model physics with proofs which are things based in the physical world.

Or am I wrong and naive in thinking this, thinking that axiomatized systems are more powerful (sufficient for modelling physics) than they really are and that they only exist in the abstract and that any set of axioms is just as arbitrary as any other set for modelling the physical world, some just seem to do a slightly better job than others?

Answer 1

To motivate this, consider a sandbox theory: The nature of the shape of the Earth.

We might be pretty sure that the Earth is round-ish. That is, if you were to travel along the surface of the Earth trying to keep a constant heading, dealing as you might with oceans and mountains etc., eventually you would come back to where you started. And that you could make this "all the way around" trip at any angle you wanted. And each such journey would be approximately 40,000 km. A little more or less depending on the breaks.

Without being especially accurate, we could be sure the Earth was a ball. A bit lumpy here and there. But more-or-less ball shaped.

So if somebody came along and said "let us take as our axiom that the Earth is shaped like a school desk" you would not be tempted, except possibly as an amusing diversion. If he said "let us assume it is a perfect sphere with some lumps attached" that would be more difficult to dispense with. But more-or-less ball-shaped is axiomatic.

The axioms of physics are much like that. For example, in gravity, the ideas of General Relativity are usually the default method of analysis. People do get up to trying other things. But the other things they try are nearly always based on some kind of geometric metric ideas. That is, the idea of a metric to explain gravity has become axiomatic.

The reason a metric is axiomatic is because of a history of measurement and attempts to understand those measurements. We have concluded that we know, to the "lumpy sphere" level, how things work in physics for things we see when we walk around. And we have struggled with a lot of careful experiment and careful analysis, and concluded that, to keep that level of understanding, we think a metric theory of gravity is required.

So the details of General Relativity's description of gravity might be wrong. But the idea of a metric to explain gravity seems very difficult to shake. It would require throwing out so much physics that it would be the equivalent of suddenly finding that the Earth was a big flat disk.

There are other aspects of physics with the same general notion. An example is a characteristic of quantum theories called "unitarity." Basically, this is what allows the number of particles in a system to stay the same if you don't do anything to it. (Here you insert a metric tonne of mathematics, and then you have a basic understanding.) The result is, people only look at unitary interactions.

There are other such things that are axiomatic within a background of less confident notions. For example, there is a property called spin. This comes in units of 1/2. The fundamental fields we usually analyze come in spin 1/2 (electrons for example), spin 1 (photons for example), and spin 2 (gravity). There are reasons to think that it is not possible to have a fundamental field with spin larger than 2. People have explored such ideas and concluded that any such field would be contradictory, pathological, or made up of other lower spin fields and so not fundamental. And so "nothing bigger than spin 2" has become an axiom.

These axioms become the "fixed points" in physics. People explore possible physics theories where these are held but nearly anything else is relaxed, modified, put in with the opposite sense, etc.

Then the resulting theories are tested against experiment. And the ones that survive the encounter are taken forward and worked on some more.

Answer 2

Axioms are not in actual practice/history, chosen at the beginning of mathematising, but are part of a specific type of project, to work backwards to find logical grounding. It began with geometry, as any

"proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a. Oxford English Dictionary Online

But as Euclid got superceded, they came to just mean starting point for reasoning from - which in the wider context means, if they don't work well, you pick new ones. It's like choosing the rules for a game, or maybe more accurately - rules of a language. Both logical, and emergent, and also somewhat arbitrary.

The project of seeking fundamental universal axioms had it's last gasp with the Hilbert Programme, & ended with Godel's incompleteness theorems. See also Hawking on implications for physics.

I see Hofstadter's 'strange loop' model as the way to understand axiom choice. We step 'outside' of a system and do metamathematics or metaphilosophy, in which the 'shape' or progress of the way of thinking is judged using heuristics, like parsimony, beauty or other aesthetics applied to their consequences, or making a good 'work space'. These are far from arbitrary. But they can certainly be tested for consequences of neglecting them. We collaborate on building emergent sets of tools for shared endeavours, and adjust the axioms to match achieving them.