Let us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jelly-fish, buried deep in the depths of the Pacific Ocean. It would have no experience of individual objects, only with the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count.
I loved this thought experiment because it’s the first instatiation I see of what a truly different kind of math would be like. Just imagine, a math without integers! As Jameson Graber elaborates here, we started with integers and only through calculus first started to truly grasp the continuous. What if there were other paths?
Having thought about this question a good deal, I believe that math is a human construct in that the Math that is possible is far Vaster than we imagine, and from that gnarly Vastness we choose only one thread. That’s what Atiyah’s quote illustrates to me.
Without beings to think it Math exists only in a combinatorial, potential form, just like all that we’ll ever write already exists in a latent form in the alphabet.
As to its universal truth, validity, applicability…, perhaps all that can be said is that empathic nonhumans might be able to get and accept some of it, just as exotic stories start to make sense to us only after we understand the exotic sensibilities that gave rise to it.
Math is not a special, magical kind of thought but simply the ever more sophisticated, ever more rigorous thought that we have. That it is, as it is famously said, “unreasonably effective”, is just an endorsement of thought itself.