# Math was discovered, not created

The Mandelbrot argument seems totally unconvincing to me. By the same token, since nobody can hold in their head all the architectural details of the Kölner Dom, it was not created by man, but merely ‘discovered’. What matters is not the complexity of what can be produced using certain rules, but the rules themselves; and there, we haven’t yet made use of any rules too complex to be held in the mind (and won’t for some time, until we find a way to extend our mental capacities). So from this point of view, human capacity provides a strict bound on the kind of math we use; any too complex set of rules would simply be incomprehensible.

There’s also the question of undecidable propositions: both the continuum hypothesis and its negation are compatible with the axioms of Zermelo-Fraenkel set theory (i.e. basically ‘math as it’s used today’) (if it is consistent, that is). You can do consistent mathematics assuming its truth, but also, assuming its falsity. So what is the independent reality of the continuum hypothesis?

But it’s trivial to come up with a formal system in which 2 + 2 = 17 is both a wff and a theorem—that system just won’t be arithmetic. But that arithmetic is special is just a contingent feature of our world: we find it to be populated by individuable objects that happen to provide a model for the requisite axioms. This could be different: we could live in a world that is, in some sense, fluid—such that we’d hold, for instance, that 1 + 1 = 1 (one puddle plus another puddle is again a puddle). If we’re reasoning about Majorana fermions (which are their own antiparticles), we might naturally hold that 1 + 1 = 0. And so on.

I can write a recursive sub-routine. I don’t have to know when it will stop calling itself before it finds the final answer. I just have to set up consistant rules and framework to let it do its job.

The language of math is solipsistic and it is a human invention. Many, certainly not all, of the principles it discovers (e.g. Mandelbrot set) are the result of the invention of the language of math. That these results sometimes lead to patterns similar to what you find in nature (e.g. sunflower) is fascinating. But similar is not the same as identical.

Edify me, folks. Do Mandelbrot sets actually represent something present in nature or are they just numerical patterns that only *appear to be similar *to things found in nature but are not intrinsically copies of those things?

I think, however, the most important problem for Platonism is the problem all kinds of dualism must face: what is the causal nexus that connects two fundamentally different substances? It seems it can’t be of either substance, but neither can it be a new substance, or we would just multiply the problem. So, how do abstract mathematical objects—which have no location in space or time, no mass, no energy, no capacity to exert forces, etc.—produce the concrete physical influences they must exert in order to make physical beings aware of them?

The problem is exacerbated by the fact that we know the physical only through its causal influences: anything that interacts causally with something physical, we generally consider physical. So if mathematical objects interact causally with physical objects, what does it mean for them not to be physical themselves?

Personally, I think the most reasonable philosophy of mathematics is one of a stucturalist bend: mathematics ultimately describes structures of objects, or relations between objects. This does not restrict mathematical systems: just as we can counterfactually conceive of objects that do not in fact exist, physically, we can conceive of relations between them, thus coming up with novel mathematical systems that may not have any direct relation to what we observe in the world.

You misunderstand the thrust of my argument. Math IS a language in that we have constructed an abstract set of names, symbols, and concepts to describe a set natural principles. For example, define “1”. It can mean lots of different things, but refers to a single unit of something. It has no hard definition other than that. The problem is that even the concept of single is an abstract construct until you can observe the atomic level. Math was developed long before that, and now it seems that on the subatomic level some very strange things go on indeed. Since, therefore, that mathematical concepts must rely on the subjective agreement of humans for their very definitions, math is language. Grammar is irrelevant; or at least very different from the languages we use to communicate verbal concepts. It’s veracity is objectively verified by the application of subjectively agreed upon concepts.

I think Ají de Gallina makes a point that goes a great deal deeper. The Pythagorean theorem is a statement that captures a bit of something, and the statement is distinct (it isn’t about the volume of a sphere, for example). But we can imagine that it captures a bit of an underlying truth that is not distinct. I bet you couldn’t jiggle reality to make this theorem false without also changing how sphere volumes work, or how 1/r^2 physical laws work, or how bisecting an angle works. The underlying truth is this continuous carpet that pervades existence. Pythagoras captured a tidy but tiny partial view of it that works well for human consciousness.

I think Pythagoras invented his theorem, which fits human thinking nicely like pliers fit human hands nicely. He invented a tool. I also think the underlying truth of geometry - which itself isn’t broken into tiny partial views that work well for human consciousness - always existed. It is even beyond discovery, only details we invent can be tested against it.

This is more or less how I feel. In one sense I am sympathetic to the idea that to exist, a thing needs to have physical substance (mass, energy, experiencing forces, etc). I’m a physicist and it’s how I view the world.

I also, though, agree that the mathematical tools we have developed/invented are subject to some sort of fundamental truth about the universe, and that if the universe were very different, we would have completely different mathematics.

At the end of the day, I kind of still feel like mathematical models, theories, etc, are invented, but arise from the natural laws of the universe.

Humas compartamentalize reality to facilitate its study.
Of course, we can say that geometric triangles, with perfectly staight lines that are infinitely thin do not exist outside our minds.

Theorems exist within the context of theories, so that they’re “discovered” does not necessarily imply some sort of physical or metaphysical existence.

Some mathematical theories are extremely similar in that they broadly speaking have the same applications, but they are not compatible because some “theorems” that are true in one are false in the other. For example ZFC set theory (Zermelo-Fraenkal set theory with the axiom of choice) and ZFD set theory (Zermelo-Fraenkal set theory with the axiom of determinacy) are not compatible.

Let’s try this analogy: let’s say I design and solve a chess puzzle by proving that, for example, white can mate in six in a particular set-up. I have discovered something about chess that perhaps wasn’t known before. However chess itself was created by people, so that I can discover things about it does not automatically imply that I am digging up nuggets of ultimate truth.

This article may be relevant. It illustrates the way in which an intellecutal activity can consist in both invention and discovery of the same thing at the same time–and says something about what it means to find some invention/discoveries more significant than others.

The fact that both the continuum hypothesis and its negation are compatible with the axioms of Zermelo-Fraenkel set theory was discovered, not created.

If, as Bertrand Russell has famously said, “Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing,” the fact that there is more than one possible starting point doesn’t make the process of discovering what follows from that starting point any less a process of discovery.

This can be a lot simpler. Was hydrogen invented or discovered?

To throw another mathematician’s perspective into the mix, here’s a relevant quote from G.H. Hardy. This is from a 1922 address, so I believe it’s not copyrighted and I can safely quote several paragraphs here. The bolding is mine, to highlight a few key points.

Yes: given a set of axioms, what those axioms entail is determinate. The same is true, for instance, for chess: given the rules, whether or not a particular position is a winning position is also determinate. But why should that entail that the rules for chess have some separate nonphysical existence? And how, if they did, could they ever make themselves known to us?

The answer to the question lies in contemplating the ‘first division’ of the universe. It can be imagined in the universe in its primordial state ‘before’ the Big Bang, when it was an extensionless, timeless point. No separate things of any kind; the reflexive property A = A does not apply. I mean, maybe it does- maybe it still is not the case that A != A in this state, the point is that it is a moot question because there is no not-A for any individual thing to be, distinct from anything else, in this state. Without A = A, you can’t have any mathematical system at all beyond the bare assertion, “A”.

To move from this to our physical universe, we have to introduce particular things; we have to introduce A = A. This is the transition from non-duality to duality. All the properties of math follow as consequences of duality and are there for us to discover. I guess it simply can’t be any other way- if it can, my pea brain won’t tell me how it could. Anyway, the physical universe has to ‘make sense’ just like correct math has to, and all the ways and means of ‘make sense’ are ostensibly expressible in terms of correct math. Ultimately correct math is the structure of sense ie the consequences of duality.

Note the wording you used: it does not apply. That doesn’t mean it isn’t true or doesn’t exist. It just means it doesn’t apply to the physical world, because there is no physical world for it to apply to. “A mathematician, on the other hand, fortunately for him, is not concerned with this physical reality at all.”

Well yeah, but then I go on to say that it is a moot question. The math analogue is to start with the math system “A”. To move on, you have to introduce A = A, or you can’t do any operations at all. And math follows as the consequences of that.

The universe is small in the realm of all mathematical possibilities. Whether or not the universe exists, there is the definition of the circumstances by which 1+1=2. The universe is just one set of circumstances, not all that exist in theory. You can’t determine that it’s possible to travel backwards in time just because it’s written about in science fiction, and conversely such science fiction can exist even it is impossible to travel backward in time. Not everything expressed must conform to the universe. Another thing to consider is the weirdness of quantum physics. It doesn’t conform to our conventional concept of how the universe works, yet it exists, and our conventional concepts still work as well.

You read a lot of pieces in New Scientist and various other pop sci, that it is astonishing that mathematics works (i.e. can be used to make correct predictions about our physical reality), and therefore infer that the universe is in some sense mathematical.

But I agree with the OP in considering mathematics to be something invented, and, while the fact that maths works tells us something, it’s not all that surprising.

Basically maths IMO is a set of formal techniques for manipulating information. Ways of teasing non-obvious facts out from obvious facts. Essentially augmenting our own thinking process.

Maths works because of some of the rules by which we define maths. If I invented a new area of maths that was inconsistent and ambiguous, few mathematicians would accept it (and it would not “work” either). Real maths is constrained to be self-consistent and rigorous, and, it would seem, our universe is self-consistent. This is the reason why maths can be useful, but more than that, it’s why any level of reasoning is possible at all.

Having said that, I am in the same situation as the OP in finding things like the set of prime numbers to be a strange thing to call invented. But for the whole mathematics shebang, “invention” is closer than “discovery”.

Mijim, who invented 1+1=2? I don’t mean the symbols, I mean the way things seem to work in the universe. Did 1+1 equal something else before someone invented the math?

Not all things that are discovered have any kind of real form until they are discovered - processes are like this - before anyone first made strawberry jam, not only did strawberry jam not exist, the process by which it is made didn’t exist either.

The process wasn’t ‘out there’ - it was just a possible configuration of actions - someone invented/discovered the process, and now we have strawberry jam, and we have a process for making more.

Maths is the same. The processes weren’t ‘out there’ in any real sense - they were just possible. Maths is the discovery not of what is - it’s the discovery of what is possible.