Let's stop maths: We started at the wrong place.

 I know that nobody else is as interested in mathematical foundations as I am, but hey, I rap about things you aren't interested in all the time. :)

This time it's the succession function.

https://en.wikipedia.org/wiki/Successor_function

It's like succ(n). Sure, I don't think there's anything much wrong with adding 1 to something, especially, but it's just that "maths starts here" thing which makes you go "WHY?". These things always seem to be some arbitrary cop-out like "We can't actually explain why we did this but, well, we felt we had to do something, and this is that thing". It isn't packaged with a perfectly reasonable explanation for why that thing should even exist and should not be discarded or replaced with something else.

You might start arguing against it by saying adding one is just a special case of adding, probably not more magical than adding 5 or 17 but I think even adding is just a special case of something else.

That way it moves up, rather than down, levels of abstraction.

I prefer a top-down approach to forming a philosophy. Beginning with the highest level of abstraction as also the most fundamental and each thing filling in at reduced levels of abstractions and more specificity.

My own starting point is a relator function. This is what I was taught at 11 in the elite-kids maths class of my middle school.

You begin with a transformer shell with no content:

->R->

The R is a label on a box. Inside the box is a mechanism which takes some form of input, does something with it, then produces output having done that thing.

That is a high level of abstraction: The set of things which do things to things.

Then in that box you can place whatever you want. Not only a successor function but addition, multiplication, a random number generator, anything you want. So, at first, the whole set of functions themselves become the most general case.

What you then have is a tool which you can use to make a kit bag of things you need for whatever you are trying to do. The kit bag of things you can make can include an axiomatic basis for mathematics and all the bits you need to do it.

I think it's time people stepped out and admitted there isn't only one possible axiomatic basis for maths and that any such basis is, at some level, quite arbitrary. So, rather than merely have a mathematical basis, I think the starting point should be the acceptance that there is an arbitrariness to mathematics as a thing. Maths is not a fundamentally existing thing but a tool, one of an infinite variety of tools and arbitrary things we could have made out of the more fundamental and highly abstract idea of things which do things to other things.

That "things which do things to things" folds in the crucial concept of cause and effect. Having defined some kind of substance and its properties you then define the things with properties which act on those things.

In that way I think existential matters, such as what are the basic laws of the universe, can escape from being placed inside mathematics. It separates the two. Real world object behaviours exist. Mathematics is something we invented and now maths is the tool we use to quantify and describe observations about real world things and how real things act on other things. However, if you want to do that then why force-fit those real things to mathematics? It means saying "Hey, we got all this real stuff and we have this tool we made so let's take this conceptual tool and use it to describe what manifests in the real world".

How about we don't? How about we first accept maths and real-world physical interactions are two separate realms? We may devise a fundamentally far more appropriate tool for describing reality than mathematics. The tool we devise may be completely foreign to mathematics and incompatible with it. We can create a tool especially for dealing with real-world physics without going via the pre-existing tool of mathematics. Not only do I think we can do this, I think we should. Let's scrub maths except for physics and what we can say about physics using maths and using that body of theory then devise a far better and more appropriate tool for coping with those real world phenomena than mathematics.

A relational operator can do this because real-world physics is still "things that do things to things" (or perhaps more fundamentally something which acts on itself) but I don't think mathematics is its basis. Mathematics is a just a tool we already had which we now use. So I say starting with "R" we can build maths, or physics, without having have both. We don't do enough to keep those worlds apart and we don't do enough to point out both of those worlds fit under the one umbrella of things and the things which change them.