Is mathematics discovered or invented?

[Asbo Zaprudder]

My intuition is that it is invented. As evidence, I offer the result (possibly first derived by Euler) that the sum of all the positive integers to infinity is -1/12. Yes, really -- although the infinite series does not converge and the sum also diverges according to the term test, the result can be obtained by several different summation methods.

1 + 2 + 3 + 4 + ... = -1/12

Srinivasa Ramanujan (the subject of the movie The Man Who Knew Infinity) wrote the following to the mathematician G H Hardy in 1913:

Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + ⋯ = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …

Although the result seems nuts, it's been used in Physics to compute the lowest energy levels of bosonic strings and the Casimir force for a one-dimensional scalar field.

Similarly,
1 − 1 + 1 − 1 + ⋯ = 1/2
and
1 − 2 + 3 − 4 + ⋯ = 1/4

These results remind me of the Ted Chiang story "Division by Zero":

https://web.archive.org/web/20111121100139/http://www.fantasticmetropolis.com/i/division/full

 

[Robert Maxwell]

It's constructed, so strictly speaking it's invented. Mathematics is not in nature, it can only describe nature.

 

[Kor]

Yes, it's our way of notating something that we observe in the natural universe.

Kor

 

[Spider]

Some animals, including chimps (and even some birds!), have the ability to count in small numbers.

 

[Ronald Held]

Does that depend if you are a Platoist/AR or physicality?

 

[Asbo Zaprudder]

I used to be a Platonist but then I saw what was outside the cave.

 

[diankra]

As RM said, mathematics is a description of something inherent. The maths varies on counting systems, but the basic binary system exists: 1 and 1 equals 2, or rather 10, and so on however you express it.
More complex notions then have to 'invented', but are inherent, so they are descriptions of discoveries.

 

[PurpleBuddha]

I used to be a Platonist but then I saw what was outside the cave.

Bright light. Hurts my eyes. Fk this shit, back to the cave.

 

[Asbo Zaprudder]

Bright light. Hurts my eyes. Fk this shit, back to the cave.

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[YellowSubmarine]

I don't think there's a difference.

There are physical inventions that can be made in a limited number of ways, they are discovered, yet still invented. If you're overly pedantic, you invent the axioms, and discover the theorems, which together give you a great tool set, which you did both invent and discover. But I don't think you need this division either, and it's not completely accurate – you may come up with the result first.

On an semi-related note, we may think of the universe as a mathematical system – we have yet to observe anything that's not describable¹ with the mathematical toolset, so everything we experience should be isomorphic to a certain mathematical system with certain initial parameters². In particular, in that mathematical system we would be having the same pointless conversation³, and the same crazy submarine would swim by for this verbal faux-jazz improv. Under that premise, whenever you're making physical inventions, you're also making mathematical ones,⁴ so even if some mathematics is discovered, other is invented.


¹ As opposed to already described, as holes in our understanding are still there.
² The damn indeterminism would have you include future events in the initial parameters, but it's still only a finite random sequence you need in advance.
³ I wonder what Gödel would say for that kind of self-reference. Can we access all the axioms and parameters of that system if we're in it?
⁴ Which is hell of a lot easier to demonstrate than the premise. But I also set my house on fire to boil tea.

 

[publiusr]

Some think that, ultimately, there is only math
https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis

 

[Crazy Eddie]

I mentioned this in one of the AI threads, but the fact of the matter is human minds and human perception do not operate numerically or digitally. From our point of view, we have evolved to view everything we interact with as a set of sensations and stimuli that have some meaning we assign to it. Symbols and shapes are specific visual stimuli we assign meaning to, and numbers are "special" symbols that we have been trained from a very young age associate with a quantity of discrete objects.

The thing is, even the concept of a "discrete object" is also just something we have been trained to visualize. I am, for example, sitting at a desk on which I can identify 5 discrete objects: a cup, a watch, model of the Normandy SR-1, a nerf gun, and a cupcake. With the cup it's simple enough to say "this is one thing" but if the cup is full of coffee, is the liquid part of the cup or a separate component? What about the nerf gun: it's not a single component, it's actually a few dozen different parts locked together (the magazine, the scope, the stock, the slide for the air pump, etc), some of which are easily removable; so is the nerf gun one object or is it twelve objects?

In reality, every discrete object is really a SET of objects we have arbitrarily assigned together by some criteria we understand. Down to the core, every set of objects is just a stack of atoms and molecules that happen to be held together by electromagnetic forces and the only reason we see them as different from the objects around them is because they're not stuck together. If my cup somehow melted and fused to the top of the desk so the boundary between them vanished, it would still be a cup, but we have a tendency not to think of it as a separate object. By the same token: there are little pieces of me and of my son lying all over my house, literally EVERYWHERE. Flakes of hair and skin and bits of keratin from fingernails. It's literally everywhere. But we don't say "there are a lot of microscopic pieces of my body in here." We say "there's dust in here." And then we clean up the room and sweep all that "dust" into one place and we say "One pile of dust."

Mathematics is a human invention. More than that, it's also an illusion. It's a way of thinking about the world and figuring out how the world works, but as anyone who ever had to grasp the concept of Pi understands, many things in the world can't accurately be represented by numbers.

 

[Asbo Zaprudder]

^This.

Kurt Gödel was a Platonist and believed that the human mind could make contact an absolute mathematical reality. As evidence, he took his incompleteness theorems and his proofs that neither the axiom of choice nor the continuum hypothesis can be disproved. However, the poor guy starved to death because he didn't feed himself when his wife went into hospital. She used to prepare his meals because he had a fear of being poisoned. My interpretation of his work is that it showed that mathematics is the manipulation of human-invented symbols given a chosen set of axioms. There is no reason to connect it with the real world apart from the pareidolic correlations that our brains observe.

 

[JasonJ]

Ya know... I'm rather intelligent, but OW, my head! Mathematics is incredible, and I respect it for what it can describe to us and how we use it to advance ourselves; but I gotta go real slow with this subject to grasp more than the underlying concepts.

 

[YellowSubmarine]

Some think that, ultimately, there is only math
https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis

I'm not sure that's what I mean.

Mathematics provides you with the tools to describe everything and anything, and you can add any tools as you go (it's a useful generalisation of string theory if you like), so with some word twisting, that statement becomes a truism. For that, however, you need to define the subject of the mathematical description itself as mathematics. That's obviously true for abstract stuff (natural numbers, modular forms, C++, etc.) as the subject itself doesn't exist within our definition of existence. But the universe is a whole different beast. For starters, it's impossible to mathematically prove that it is the subject of any mathematics describing it (you cannot mathematically prove that the current physical theories are 100% accurate), so from our point of view it has to be a distinct concept, as the mathematics of it – even it actually is only that – are literally inaccessible. If I ask you to prove that rocks fall down, you cannot ever prove that in real universe math, only in some approximate Newton or Einstein math.

However, from the PoV of a different universe completely isolated from us, it gets funky. Let's say they cannot observe us in any way, yet somehow, by some incredible chance, their version Star Trek describes us in frightening detail. Their laws of physics are so crappy that inter-continental travel and powered flight are incredibly impractical. Their over-the-top mad science fantasy involves aeroplanes and ships travelling between continents in days and hours, and laughable magic like radiowaves. Of course, there would be superfans who go to extreme lengths by dissecting that insane fictional tech in such great detail that you could put it together in a math model.

But no matter if that that model is like our universe or not (hint: it's not), it's still only math – we don't exist. In their definition of existence, anyway.

Which makes a nice insult. ‘You're not real. You're not even math.’

Oh, wait. That insult already exists – ‘You're not making any sense!’

 

[Asbo Zaprudder]

It has been postulated that all of Physics is actually based on observer-dependent metric correlation, which could be taken to imply that the physical world has no mathematical base. See, for example, "Physics from Fisher Information" by B Roy Frieden:

This book defines and develops a unifying principle of physics, that of 'extreme physical information'. The information in question is, perhaps surprisingly, not Shannon or Boltzmann entropy but, rather, Fisher information, a simple concept little known to physicists. Both statistical and physical properties of Fisher information are developed. This information is shown to be a physical measure of disorder, sharing with entropy the property of monotonic change with time. The information concept is applied 'phenomenally' to derive most known physics, from statistical mechanics and thermodynamics to quantum mechanics, the Einstein field equations, and quantum gravity. Many new physical relations and concepts are developed, including new definitions of disorder, time and temperature. The information principle is based upon a new theory of measurement, one which incorporates the observer into the phenomenon that he/she observes. The 'request' for data creates the law that, ultimately, gives rise to the data. The observer creates his or her local reality.

https://en.wikipedia.org/wiki/B._Roy_Frieden

 

[Crazy Eddie]

It has been postulated that all of Physics is actually based on observer-dependent metric correlation, which could be taken to imply that the physical world has no mathematical base. See, for example, "Physics from Fisher Information" by B Roy Frieden:

https://en.wikipedia.org/wiki/B._Roy_Frieden

That's pretty much a given, though. Everything we know about the universe and the laws of physics assumes that our observations would be true in all possible contexts. This assumption has been proven wrong in the past, and the known laws of physics have been modified to accommodate new situations in which our predictions about the world, based on our previous observations, would no longer be valid.

So you can empirically state that the laws of physics are derived and not inherent, just by the fact that no model, theory or law can ever be 100% accurate.

 

[Asbo Zaprudder]

Thanks, @Crazy Eddie, I believe we're on the same page on this. However, some otherwise brilliant physicists/mathematicians do seem to get almost mystical on this stuff, eg, Tegmark and Penrose.

 

[Crazy Eddie]

Thanks, @Crazy Eddie, I believe we're on the same page on this. However, some otherwise brilliant physicists/mathematicians do seem to get almost mystical on this stuff, eg, Tegmark and Penrose.

That's because they believe in a non-deterministic universe where free will is absolute and insoluble. This is actually the whole point of the Schroedinger's Cat thought experiment: mathematically, both states are equally correct, but the real world doesn't reduce to math and is deterministic: given two contradictory states, only one of them will ever REALLY be true. That truth exists independent of our observations, our choices, our illusions or delusions; the wave function tells us what is POSSIBLE, but it doesn't tell us what IS. This is why so many physicists will speak of the "many worlds" hypothesis with a straight face, in the genuine believe that all possibilities MUST occur because if they don't occur then there is no room in the universe for choice. The truth is, there isn't: it's just the quantum domino effect set in motion from the Big Bang on an inconceivably long chain of causality, and the initial conditions that determine everything that will ever happen, everywhere in the universe, were already set billions of years ago and there's nothing we can do about it.

It comes to the point that our perception of the world and our projection of meaning is a different thing from the world itself. We perceive the world through five senses and gather what limited information we can from it, but we perceive very little about what is actually going on or how the things in the world around us interact with each other. And of what little we do perceive, we understand even less, and we fill in the gaps of our knowledge with simple explanations that fit what little information we have even if we know, on some level, that it's mostly wrong.

Very educated people like to think that they've got the world mostly figured out -- or at least, more figured out than the average person -- because they have more information than most people do. Very wise people recognize how little information they actually have and eventually learn to just go with it.

 

[φ of π]

Mathematics is an innate property of the Universe and is discovered.

The number systems we use to describe mathematics are invented.