I don’t think it’s a red herring though: I don’t consider it trivial that humans create arithmetic to suit our purposes and needs as a species: I think it’s marvelous.
and that we created it? yes, it exists. It’s an object of the mind and the mind functions on a substrate of very real tangible stuff.
Well, … we’re almost in agreement.
We take what’s in our heads and scratch it out on paper and teach it to offspring, create devices that function in such a way as to visualize our arithmetic using similar sized pieces – and it’s no easy feat to make a machine that counts.
In computers, very specific voltage ranges, highly controlled.
Prior to that, wheels that would turn when stones fell upon them required stones that were of equal weight.
Or water wheels would have to have same sized buckets in order to capture the same amount of water to be able to count in an even way.
In our arithmetic, we use evenly sized pieces. Pebbles, electrons, same sized water wheel buckets, etc.
But nature is logarithmic
and even then, for me to say “nature is logarithmic” isn’t really true-true either, although it is from our perspective (again): natural log is a measure of growth time. How much time does it take to grow? Well, we measure time in evenly spaced (timed) units. SOME things in nature seem to work in evenly timed units: natural clocks exist. Gravity and dripping water can be a natural clock. Cesium atom a famous example we use has a steady fast pulse that’s evenly timed.
Yes, it’s very practical for us. Yet, If we were to encounter aliens though, we would be remiss to declare our arithmetic as TRUTH because they may not have created such a system for their use.
It’s good for us because it’s a good universalizing analogy. Our brains are too small to handle all the uniqueness around us and so we need to compress and simplify the world around us into analogies that we can link to other knowledge so that we can comprehend and function. Arithmetic fits the bill in a way that doesn’t need us to specify which tangible objects if any tangible objects. It works as a general analogy for us.
also, not all arithmetic ties into physical matter — if any!
We map matter with it just as we name things, create grids to partition things, etc.
Concepts we’d call “more” and “less” and “sufficient” seem to span physical and biological objects.
[or “not enough / enough / too much”]
But units are a bit of a questionable thing.
It’s pragmatic: whatever works.
For us, it points to something and it’s useful for us.
I don’t think we’re in disagreement really. We have a knowledge of the world out there that works for us in our bodies and our species.
It’s arbitrary in general but not arbitrary for us. It maps nicely to our brains and how our brains think and how we work. It’s great for our species and its needs.\
For all intents and purposes, since we CAN’T go outside of ourselves, it’s not strictly necessary to imagine any NON human intelligent life and what they ‘might’ do because we only have what humans do and what we imagine non-humans do.
So we can say: colors don’t really exist. But for us, they do exist. So, colors exist.
Music is probably the best analogy I can think of for mathematics generally.