This goes beyond what is strictly math, but I don't think it's reasonable to say that properties of the real numbers (say, roots, pi, e, and so forth) are invented. In some sense they seem like the simplest thing that fits a few properties (and not that many). Similarly with Euclidean Geometry, and the natural numbers (with primes and their structure, etc.) I really actually think something akin to Occam's Razor applies in math!

You can of course tweak your starting points and get really interesting things too, say non-standard analysis or non-Euclidean geometries. And you can derive (I would say "discover") some rich and surprising properties and patterns in them.

But I don't think any of these things are accidental. The real numbers (and pi, e, etc) would surely be "invented" in the same (modulo shifts in convention) ways by other advanced civilizations, I think. Could you imagine this not being so?

This is also extra-mathematical, and I admit this may be hooey, but I believe (as a non-mathematician) that math gives hints at large patterns and tells you when things fit well or are funny. Like, I can argue whether Pi or 2*Pi is the more "natural" constant, and it's not a discussion completely devoid of content! Also, I think math tells me something is funny about 0^0 (because of the limit 0^x) but that 1 is the more natural fit.

I realize this puts me quite firmly into the category of people being belittled here!

I hope I'm not participating in a discussion where people are being belittled! I think all the viewpoints here are fascinating.

Yours is certainly interesting and valid, I'm simply offering that the "full stop" at the end of your previous comment doesn't reflect the kind of deeper metaphysical questions that underly the whole progression of the history of math.

I believe this to be a bad counterpoint, because you could ask the same thing about natural numbers as well. I've personally never seen a natural number. Sure, I have seen and worked with lots of representations of natural numbers, but the numbers themselves are - as far as I understand it - not physical objects. There is no qualitative difference between natural numbers and any other mathematical objects in that respect. They are all on the same "plane of existence".

At least that's a valid world view or ontology. I know that not everybody thinks like that, but besides clarifying what the parent poster probably meant I think it's not a very meaningful discussion.

On the sheet of paper in front of me, as the hypotenuse of the isosceles right-angled triangle with unit length I drew a moment ago. Irrational numbers are probably a bad example of what you're talking about.

No, they're the perfect example and you're observation about the unit square is glib.

Trying to reason about the diagonal of the unit square basically destroyed the Pythagorean world view of integers as the fundamental building blocks of reality.

If mathematical entities are simply discovered and that-is-that, then why did it take nearly two millenia after this observation for western math to accept irrationals as numbers?

From M. Stifel, 1544:

"Now, that cannot be called a true number which is of such a nature that it lacks precision. Therefore, just as an infinite number is not a number, so an irrational number is not a true number, but lies hidden in a kind of cloud of infinity"

Doesn't the quantised nature of matter mean that 2^½ exists as a real measure of a material object only as much as a perfect circle actually is existent in our universe?