Is Math Invented or Discovered?

 

The answer, as usual, is both, and the answer, as usual depends on how we define the terms of the question.  

Mathematics is a formal structure, or more accurately it is several formal structures - a tremendous number, in fact.  An infinite number?

Some of these formal structures are very closely related and resemble each other closely.  Others are more distant.

Some formal structures are close enough to each other that it can be very difficult to distinguish between them.

The progress of human mathematics can be understood as humans getting better at distinguishing between formal structures.

Indeed, that is what is meant by "formal": informally, we may use concepts in a loose way, which may change from one meaningful context to another, and people will still generally be able to understand each other about a great variety of topics.  Indeed, a poet might borrow terms from different areas of life and use them freely and creatively, in combinations that have never occurred to anyone before, and still convey a meaning.  To impose a formal structure does not in any way deny or negate that.  It merely specifies one specific context in which a term will have a formalized - that is to say, clarified, rule-determined - meaning.  For instance, most people tend to use words with meanings that only have a "family resemblance".  In everyday speech, for instance, we tend to use the word "tree" to refer to both palm trees and maples.  A biologist, however, may use words in a more technical and restrictive sense, attempting to construct well-defined clades.  (Cladistics is not mathematics, but it is another kind of formal structure.)  

Generally, nowadays, human mathematicians tend to distinguish between formal structures as follows. Typically, a mathematician will try, to the best of her ability, to define her terms in such a way that there is a clear rule to determine what does and what does not count as being an example of a specific concept.  That is, mathematicians try to reduce ambiguity to the greatest degree that they can.  Nowadays, one formal structure is distinguished from another by a distinctive set of axioms and a set of rules for deriving theorems from axioms and from other theorems.

"Formality" is not an all-or-nothing proposition.  It's a question of degree.  As mathematics has developed, over the millennia, it has gradually become more and more formal - "sharpened," so to speak.  But it is still not perfectly formal, and probably never will be.  "Family resemblances" still remain, and it is the mathematician's job to eradicate them or minimize them to the best of their ability.  When I say that "family resemblances" still remain, I mean that our concepts are still, to some degree, "fuzzy" - there are still marginal elements, such that according to the rules that define them, it remains undecidable - or, at least, undecided - to which category they belong.  This should be our credo: formalize family resemblances.  Never give up.

Some formal structures are consistent, and others are inconsistent, to varying degrees.  To say that a formal structure is consistent means that it is not inconsistent.  To say that a formal structure is inconsistent means that there is at least one derivable theorem that contradicts another derivable theorem.

The axioms and transformation rules for any given formal structure are ultimately arbitrary.  (I suppose, in an informal manner of speaking, one could thus claim that humans "invented" math.)  But within each consistent formal structure, precisely because it is consistent, each theorem is a theorem and thus, so to speak, cannot be denied.  (I suppose that in this informal manner of speaking, there comes to be a sense of "necessity" that feels "intrinsic" to the structure, and thus one might claim that humans "discover" math.)  Of course, you can deny them - but that just means you are stepping outside of the formal structure.

Notice that a formal structure could be consistent, but nonetheless arbitrary.  Indeed, all formal structures are ultimately arbitrary.  Notice also that just because a structure is formalized and consistent, that does not mean that it is true.

Consider the case of lunar arithmetic, also known as dismal arithmetic, developed by David Applegate, Marc LeBrun, and Neil Sloane.  According to this formal structure, when one adds two digits, the answer is simply the larger digit - so, 2+3=3 - and when one multiplies two digits, the answer is simply the smaller digit - so, 7x2=2.  This is a perfectly well-defined, consistent formal structure, which happens to produce very silly results.  It's totally arbitrary, but so is standard arithmetic.  Mathematicians have worked it out in impressive detail, deriving many theorems, calculating thousands of prime numbers in this system, and so on.  But they're just having fun.  So far as I know, as of this writing, no one has found any practical use for this consistent formal structure, and yet it "exists."  Did humans invent it?  Well, yes, in a manner of speaking.  Did humans discover it?  Again, yes, in a manner of speaking.  I haven't found any people who claim that lunar arithmetic existed before humans ever evolved.  But you could say so, with just as much justice as you can claim that normal arithmetic existed before humans.

So what's the difference between lunar arithmetic and standard arithmetic?  The difference is that standard arithmetic, the kind you learned as a kid in school, as it turns out, has tons of practical applications.  And so we humans have devoted a lot more of our time and resources to exploring the possibilities of this particular formal structure than we have any of the oodles of other possible arbitrary formal structures, in both consistent and inconsistent varieties.

But that may change.  In the 19th century, mathematicians were exploring the possibilities of non-Euclidean geometry, simply out of what amounts to idle curiosity.  Then, in the 20th century, Albert Einstein and Emmy Noether developed the General Theory of Relativity and showed that our universe, at large enough scales, operates more according to Riemann geometry than Euclidean geometry.  What had been a purely speculative mathematical abstraction now suddenly became something useful and real.  Perhaps lunar arithmetic may prove useful as well, for instance in computer science and cryptography.

There is the possibility that any given formal structure is inconsistent, but that we do not yet realize that it is inconsistent, simply because we haven't derived every possible theorem from the given set of axioms.  So when I have been using the word "consistent" so far in this essay, this really meant "consistent as far as we know."  Somewhere "out there" in the space of all possible theorems there might "be" a theorem that contradicts one of our previously derived theorems.  There is no perfect guarantee that this is not the case.  If that should ever "happen," I would hope that this would be an opportunity for distinguishing between two formal structures that previously had been blurred together, and that at least one of these formal structures would be internally consistent, though they contradict each other - and, preferably, both formal structures would be internally consistent.  But there's no absolute guarantee of that, either.  All we can say is, so far, math seems to work pretty well.

Q: Yes, but why does it work well?  Are you saying it's just a coincidence?

A: I suppose you could put it that way.  That's probably not the word I would use.

Q: So we just happen, by chance, to have randomly stumbled upon an internally consistent system that perfectly matches how the universe actually works.  Out of all the - as you yourself admit - potentially infinite possible systems that could have been constructed, we just randomly selected the one that is absolutely true.

A: I'm not sure that I would endorse the idea that it's absolutely true, or that it perfectly matches the way the universe actually works.

Q: So you think that math might be wrong?

A: Of course!  Then again, it depends on how you define "wrong".

Q: Oh boy, here we go.  You're going to split hairs on the definition of "wrong"?

A: Yes.  I can think of two relevant definitions of "wrongness" here.  On the one hand, "wrong" might mean that of the formal structures that we call math doesn't match the way the universe actually works.  On the other hand, "wrong" might mean that one of these formal structures is, itself, internally logically inconsistent - perhaps in a way that we haven't noticed yet.  Actually, either of these things might be true.  

Q: What would we do f we ever found out that math was wrong?  Wouldn't it be a catastrophe of unimaginable proportions?

A: No, it would be an ordinary day.  It's already happened, several times.

Q: What are you talking about?  Can you name a time math has ever been wrong? 

A: Sure!  We've already been talking about one example: for thousands of years, we used Euclidean geometry, and we thought the universe obeyed the laws of Euclidean geometry.  Then we found out that, because of General Relativity, the universe obeys rules that are closer to Riemann geometry than Euclidean geometry.  So it turns out that Euclidean geometry does not perfectly match the way that the universe actually works - in that sense, I suppose, Euclidean geometry could be said to be "wrong."  But it comes very, very, very close for most of our practical, everyday lives.  It's not absolutely correct, but the difference between what it predicts and what actually happens is so itsy bitsy teeny tiny that it might as well be correct.  And in any case, (so far as we know) Euclidean geometry remains a perfectly consistent formal structure - so in that sense, it's still "right."  

Q: Okay, so Euclidean geometry turned out not to be right - but still, Riemann geometry - that's still math.  So it didn't turn out that math was wrong, just one specific kind of math.

A: That's right.  So we developed - and are continuing to develop - another kind of math to deal with it.  Maybe one day, it will turn out that even Riemann geometry is not an absolutely perfect analog for the way the universe actually works, either, and we need an even more subtle and complex system of geometry to deal with certain problems.  There might be places where even Riemann geometry "breaks down".  

Q: Like the "singularity" at the center of a black hole? 

A: Sure, maybe, or maybe something else.  Who knows?  Who knows what weird stuff nature is going to fling in our faces?

Q: So then what?

A: In general, I'd say, whenever math breaks down, there are two possibilities.  Possibility number 1: we figure out some new kind of math to deal with problems that we couldn't deal with in our old system.  Here's hoping that's the case.  Possibility number 2: we're just not smart enough to do that.  There are some things we'll simply never understand, or at least which we won't understand mathematically.  I think we can live with that.

Q: Alright, but still... how could it be that math matches up with the way the universe works - maybe not "absolutely" or "perfectly" but with such differences that are "itsy bitsy teeny tiny" as you put it... can that really just be a coincidence?  We just randomly selected a formal system that is this close to reality?

A: Okay, I take it back.  It's more than a coincidence, because it wasn't just randomly selected.  I think a process like natural selection must have occurred.  So every formal structure is ultimately arbitrary - in a manner of speaking, "random."  But, as with random mutations in animals and plants, most of these will produce random garbage that is unhelpful or even harmful and will quickly be forgotten.  But if an arbitrary set of axioms and rules of inference of transformation produce ideas that are useful, then they will be remembered and expanded upon.  Thus we got closer to the math we know and love.

Q: Well, that's all very well and good, but do numbers really exist?  Like pi, for instance: is pi just something in our human minds, or does it really exist out there in the world?  

A: No and no.

Q: What?

A: No, pi is not in human minds, and no, pi does not exist out there in the world.  At least as far as I know.

Q: Huh?

A: What is pi?

Q: I mean, that's what I'm asking you.  Pi is a number, but what is a number? 

A: No, I just mean, recite it.  

Q: Oh.  It's 3.14159... something.  I think the next digit is a 2?

A: So, that's not pi, exactly?

Q: No, just an approximation.

A: So you don't have pi in your mind.

Q: I used to know someone who could recite the first 100 digits.

A: Akira Haraguchi has the record: he memorized 111,701 digits of pi.

Q: Wow, that's really impressive.

A: So, he memorized all of pi, right?

Q: I mean, no... haven't they worked out billions of digits of pi?

A: So pi isn't even in Akira Haraguchi's mind.  So pi isn't in any human mind.

Q: I mean, I guess, technically....

A: And what about those computers, that calculate hundreds of trillions of digits of pi - soon, probably quadrillions - but even they haven't calculated all of pi.  So not only does pi not exist in any human mind, it doesn't exist in any computer, either.  

Q: Just approximations of pi.

A: So where else might we expect to find pi - pi exactly - if not in human minds, and not in computers?  Does pi exist anywhere out there in the world?

Q: I mean, yes - it's everywhere!  It keeps popping up in the most amazing places!  Didn't you see the video where two blocks bump into each other and it turns out they bump into each other pi times... or something like that?  There's pi in Heisenberg's Uncertainty Principle!  In Einstein's field equation!  In Coulomb's Law!  And Euler's Identity!  And stuff!

A: Okay, slow down.  That's a lot to tackle all at once.

Q: Well... I mean, this gets to a big problem with what you're saying here.  Okay, humans don't have the complete decimal expansion of pi in our heads.  But that's just the way we write pi - one way of expressing pi.  We could also use the greek letter, or we could spell it out in English, "pi".  What's important is, we have the concept of pi in our heads.

A: Oh? And what is that?

Q: Pi is the ratio of the circumference of a circle to its diameter.

A:  Okay, let's talk about circles.

Q: Yes! Let's!

A: Do circles exist out there in the world?  Or do they only exist in human minds?

Q: Hey, I'll do the questioning around here!

A: Alright.

Q:  Do circles exist out there in the world?  Or do they only exist in human minds?

A: No.

Q: No, what?

A: No, neither.  Circles do not exist in human minds, and circles do not exist out there in the real world.

Q: How can you say that?  There are circles everywhere!

A: Where? I don't see any.

Q: That clock over there is circular - and then there's the sun, and the moon, and the Earth, and all the atoms in your body, and -

A: You think atoms are circular?

 

 

 

Q: Again, these are just the drawings of circles.  But we have the concept of a circle in our heads.

A:  Oh? What's that?

Q: A circle is the locus of points, within a 2-dimensional plane, equidistant from a single point.  It's a 2-dimensional sphere. 

A: How many points is that?

Q: Well, it would be an infinite number of points.

A: And you're saying you have this concept in your head?

Q: Yes....

A: What does that mean?

Q: It means I can imagine it.

A: You can imagine an infinite number of points?

Q: Well, no....

A: How many points can you imagine?

Q: Um, hmmm....

 

A: To be honest with you, I don't know if you can imagine any points.

Q: Hey!

A: Oh, I'm not insulting you.  I'm just saying, I doubt whether any human being is capable of forming the concept of a point. 

Q: Sure we can.  If not, geometry would be impossible.

A: Well, what is a point?