This is an enduring conundrum. “Math,” of the 1+1=2 sort, is straightforward and obviously reflects how reality works. However, “mathematics” builds upon that to spin constructs of vast complexity.
Jim Holt’s 2018 book, When Einstein Walked With Gödel — Excursions to the Edge of Thought, frames the question: “Do the truths discovered by mathematics describe an eternal and other worldly realm of objects . . . that exist quite independently of the mathematicians who contemplate them? Or are mathematical objects actually human constructions, existing only in our minds? Or, more radically still, could it be that pure mathematics doesn’t really describe any objects at all, that it is just an elaborate game of formal symbols, played with pencil and paper?”
That might juxtapose mathematics against physics — our scientific effort to learn how reality works. And while quantum mechanics (part of physics) defies commonsense notions (and Bohr, as Holt notes, considered it merely a tool for predicting observations), most physicists do think it describes an underlying physical reality. That’s why it’s called physics! Different from mathematics.
Returning to math, start with the simple counting numbers: 1,2,3,4,5, et cetera. They readily map onto real world quantities. But add zero to the picture — which actually wasn’t done until quite late in history — and things get dicey. For example, any quantity can be divided by any other, the result reflecting something about reality. But that blows up if you try to divide by zero.
Now take the concept of prime numbers. A prime is a whole number, greater than 1, not divisible by any two other whole numbers. Thus 2,3,5,7,11,13 are primes, and so on. Though for big numbers, primeness is not so easy to discern. And primes give rise to great conceptual problems still bedeviling mathematics. For example, Riemann’s “zeta conjecture,” positing a “hidden harmony” governing where and when primes pop up in the sequence of all numbers.
But again the question — does this have any resonance in nature? Is mathematics (beyond simple counting numbers and operations) a reality that’s “out there” that we have discovered (analogous to, say, Newton’s laws of motion) or is it just a human construct, playing with numbers?
Now, it’s not as though “higher” mathematics is divorced from reality. To the contrary, we have often found that the former yields insights into the latter. Calculus, for example, is pretty advanced mathematics, and while we don’t find any of its equations per se in nature, it does help us get answers to real world problems.
Thus Holt quotes Einstein — “How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?” And Holt paraphrases mathematician Edward Frankel that mathematical structures are themselves objects of reality — not mere products of human thought but, rather, existing timelessly in a platonic realm of their own, waiting to be discovered by mathematicians.
Thus, Holt suggests, mathematics has a reality transcending the human mind. The “strange patterns and correspondences” emerging from it hint “at something hidden and mysterious. Who put those patterns there? They certainly don’t seem to be of our making.”
And he similarly quotes French mathematician Alain Connes that “there exists, independently of the human mind, a raw and immutable mathematical reality” that is “far more permanent than the physical reality that surrounds us.”
This evokes Plato’s concept of “forms” existing, well, somewhere, embodying perfect realities (like perfect circles, impossible on Earth), whose mere imperfect shadows are what we encounter in our everyday existence. Connes is an extreme Platonist. And people like him actually seem to think they have some sort of intuitional access to that ethereal realm. Yet Holt sees no way our physical brains “could interact with nonphysical reality.” I would take issue with even his use of the word “reality” there.
However, Holt does also suggest no need to go as far as Connes (or Plato). We have plenty of purchase on the concept of a perfect circle, even without access to the reality of one, and we can use our logical reasoning powers to do all that’s needed. (In fact, we can make circles damn close to “perfect,” and how much does the discrepancy matter?)
Granted that mathematical truths — like, say, the Pythagorean theorem — are necessarily true regardless of anything in the actual cosmos — even if, indeed, there were no cosmos. But that kind of reality is nothing like what we mean when we speak of reality in the way physics does. A Higgs Boson is an actual physical object (even if too small for us to see).
Once more, a law of physics, like Newton’s third, describes how reality operates. Pythagoras’s describes how mathematics operates. And while the starting premises of mathematics, the counting numbers, and some facts about how they work, do similarly describe reality, human thought has gone way beyond that in constructing ginormous cathedrals of mathematical concepts that have no counterparts in physical reality.
Calling them human constructions might imply we could just as readily have constructed different ones. But off course that’s not so. They are unavoidable, given their ultimate grounding in the basic building blocks, carried to logical conclusions. So in a sense we did discover them. Yet still, such truisms exist apart from the physical reality of the universe — which is again the domain of physics.
The distinction is manifested in how our understandings of physics have led to practical real world applications — like the atom bomb. Whereas it’s hard to see how that can be so for the abstruse theorems of higher mathematics, like Riemann’s zeta conjecture. Thus the latter are ultimately “mind games.” I leave it to the reader whether to say just mind games.
