No. They are largely a human invention.

(non-)Euclidiean geometry

Euclid’s Elements, the best book in maths, proposed this situation where propositions were proved to be true via deductive reasoning starting with axioms (and definitions).

As for the axioms, the belief was that they represent truths, why would one question it for “the whole is larger than the part”?

It turns out that most questioning was for the 5^th postulate (axiom), because of its convoluted phrasing. The question was how many parallels exist in a plan through a point outside a straight line, and the 5^th said that one precisely.

It took more than two thousands years to find that other options exist for the 5^th, such that the option that there is more than one parallel (hyperbolic geometry), or the option that there is no parallel (elliptic geometry).

So then we moved around the 19^th century from the situation where the geometry axioms were absolute truths, to one where they were just choices. They were not entirely arbitrary, because they work extremely well in certain human-scale situations (such a designing a building), but eventually break down at very small scale (such as when lines can’t be divided further when we reach atomic scales), or very large (such as cosmic) scales.

Personal detour on Euclidean axioms

As a fun fact, when I introduced to my child the 2^nd postulate, “a straight line can be made arbitrary longer at both ends”, they giggled and said “well, that’s only until it meets at the side of the Earth” and I explained how in Euclidean geometry a line would be tangent to the surface of the Earth, it’s not the 2D geometry on the surface of a sphere, but they should remember that thought.

They nodded to understand and took the 2^nd postulate as a truth.

But eventually we had the discussion that elliptic geometries describe exactly the situation she thought of initially (the 2D geometry on the surface of a sphere) where straight lines do meet eventually and even such apparent self obvious truths that “a straight line can be made arbitrary longer at both ends” are questionable.

Natural numbers

Around the middle of the 19^th century to the first half of the 20^th century mathematicians turned their attention to natural numbers due to their fundamental role and tried to put it on a clear axiomatic basis. Several systems emerged, some based on the Peano approach (start at zero and apply a successor function to generate the natural numbers), some based on set theory (the empty set is zero, the set containing the empty set is 1 and so on). Such systems share surprising properties such as the ability to express problems that cannot be proven.

But the question the is: are natural number real and the surprising properties unescapable or are they just one of the many possible human inventions just like geometry?

And the answer is that natural numbers are a human invention. The systems of natural numbers have many assumptions such as the ability to recognize entities as being unique or the same, and the ability to go on forever. When that is a good approximation for the problems we look at, using the natural numbers and their derivatives is very successful. But they fail for problems such as counting the grains of sand on a beach and establishing the length of the coast of Great Britain.

What’s not purely human invention are mechanisms that we share with other animals. The evolutionary basis of the natural numbers is the ability of animals to use the nervous system to for example recognise enemies: none (I’m safe), one (I might be able to deal with), two (I might need to run), three (I should run), a few (help), some, many, lots (no chance). While we have an innate basis to distinguish between 2 and 3 entities, we don’t have near the same confidence to distinguish without a lot of analysis between 246 and 250 (with reference to the 1987 movie “Rain Man”).

Personal detour on natural number thresholds

Long time ago I had to investigate why a list in a product has lots of entries. I picked one such case. It had about 2000 entries. That’s a lot, but decided to go through them and classify them to try figure out what’s the underlying cause. Are certain types of entries more common? What’s the cause? In the process of going through the 2000 entries one by one I soon discovered another issue.

The list displayed the entries page by page, but pagination was broken and the first 50 entries were repeated instead of displaying 50 different entries each time. So in my case, the first 50 entries would have been repeated about 40 times.

And this issue was not reported. Thousands of customers complained by the large number of entries, but none complained that only 50 are displayed, the rest are repeats.

My empirical observation is that somewhere between 20 and 50 our brains stop having a intuitive perception of precise numbers and use heuristics based on “there’s lots”, rather than precise numbers.