The h Bar

@ACuriousMind, in my mind, discovered is finding out about something with it already being there, and invented is finding out about something via creating it. So then, discovering math, maybe finding the pattern among the chaos of the real world? I love that quote "the natural numbers are the work of God, the rest is the work of man" (to paraphrase a bit). I'd take it further, but that's my thought.

ACuriousMind

(You don't need to give a full definition of "discover", but enough of one to distinguish it from invention)

Sir Cumference

@ACuriousMind Let's talk semantics. Would you say "Newton helped discover calculus" or "Newton helped invent calculus

Ali Caglayan

but they can't do the same thing and arrive at a different result

Physics Guy

@ACuriousMind To find structures and ideas in mental confusion.

heather

11:08 PM

@PhysicsGuy, a most interesting definition, and one I agree with now that I've heard it.

Physics Guy

In things that seem to be "structureless"

ACuriousMind

@heather And what does "being there" mean here? As I said, math is not a physical object, so what does it mean for it to exist?

Sir Cumference

There's also the fact that math is not always agreed upon by certain people

For example, some people accept the continuum hypothesis as true, others don't

Physics Guy

Physics isn't either.

Sir Cumference

This makes it confusing as to whether there is a single truth to math

ACuriousMind

11:09 PM

@SirCumference Since I don't know what it means to "discover" an intangible idea as opposed to inventing it, I'd say he invented it.

Physics Guy

I think the continuum hypothesis can once be proven (or refuted).

Sir Cumference

@ACuriousMind All right, new question. Would 1+1 equal 2 if humans weren't around to declare it does?

@PhysicsGuy That defeats the purpose of calling it a hypothesis

ACuriousMind

@PhysicsGuy That's a flowery description of what math (or most rational thought) does, but it doesn't bring me no closer to understand what "discover" as opposed to "invent" means.

Sir Cumference

Fact is no one knows, it's a matter of accepting it or not at this point

Physics Guy

Yeah.

So, does differential cohomology exist ?

Does it exist or did we define it as being something ?

In our minds.

Sir Cumference

11:13 PM

How about asking like this: does mathematics predate sentient beings, or do sentient beings create mathematics?

ACuriousMind

@SirCumference That's like asking me if a mouse would still be a small rodent if humans didn't exist. I don't see what one statement has to do with the other.

Physics Guy

Or homotopy groups. They actually are something. We name them "Homotopy group" but actually it describes something "real".

A versy interesting thing is that I heard people claiming that mathematics is a kind of Spiritual Science, which isn't false.

ACuriousMind

@SirCumference I think that's a false dichotomy. Mathematical statements are true or false, regardless of whether I know a proof of them or not. Their truth value doesn't depend on us knowing it. Since mathematics is entirely about logical statements without reliance on reality, it doesn't 'exist' in any proper sense. You can't observe it, you can't test it, its truth is utterly unrelated to anything that exists or not.

So I deny the idea that it can "predate" anything, because it is not something that exists within or refers to something within space or time.

Sir Cumference

Huh...interesting point

ACuriousMind

I guess what I'm trying to say is that this is a deep-sounding question that doesn't actually mean anything.

The notions of 'discovering' and 'inventing' refer to physical objects, or at least ideas directly relating to physical objects. You have to already commit to the stance that mathematical objects/math itself "exist" in some sense for this question to make sense, and then you've basically already also defined that humans just discover them, because otherwise you need to explain how humans actually imbue an abstract idea with "existence".

2

Nogueira

11:22 PM

Depends on what you mean by math. A set of axioms and definitions with "logical" implications?

Physics Guy

Yes

Nogueira

Is a path integral a mathematical object than?

Physics Guy

Difficult question.

Yes

It is.

Nogueira

Because, path integral is not defined by a set of axioms and definitions.

ACuriousMind

@Nogueira Only the path integral of QM, and that of some 2 and 3D QFTs.

Physics Guy

11:23 PM

Are there others ?

Higher dimensional QFT ?

Sir Cumference

It's interesting to ponder the idea of mathematics. Mathematics cannot be proven, as shown by Principia Mathematica. It's weird.

Nogueira

And all the other 'beautiful' limits that we physicist does and do not have a sharp mathematical axioms sets!?

ACuriousMind

@PhysicsGuy Hmmm? Standard relativistic QFT describing our would needs to be 4D.

Sir Cumference

It's the only thing I can think of not bounded by the physical realm.

Physics Guy

Ok.

@ACuriousMind And paths integral there are not mathematical objects and the others are, or what ?

I forgot about that.

ACuriousMind

11:25 PM

Yes. Because we can define the low-dimensional versions rigorously, but not a general version.

Physics Guy

Man, that's embarrassing.

Sir Cumference

Actually, that raises another philosophical question: is math the only thing not bounded by the physical realm?

Nogueira

@ACuriousMind yes, my point is that there is something that we can call mathematics but are not (yet?) mathematics

ACuriousMind

(Not that you'll ever see a physicist actually rigorously defining a path integral in any case :P )

Nogueira

I think that this is more a lexical problem than a pphylsophic one

Physics Guy

11:26 PM

All objects used in physics can be lead back to mathematical objects (described by axioms and stuff...)

ACuriousMind

@SirCumference Well now...what does "thing" mean here? If "thing" does not mean "physical object" (which math isn't), what exactly are we talking about?

Sir Cumference

@ACuriousMind Well, what category would math fall into?

Nogueira

We can, if we want, assum that mathematics is the ultimate language of nature and see the axioms and deffinitions as a human beeping adaptation only.

ACuriousMind

@SirCumference That is difficult to answer because if we take a broad definition of math, the very logic I need to use to say what a category (or class or set or whatever you actually mean here) is is itself part of math.

Sir Cumference

@ACuriousMind So how could one define mathematics?

Physics Guy

11:30 PM

As a spiritual science.

In some way.

Dealing with self made objects

ACuriousMind

@SirCumference One usually hopes the listener more or less accepts the notions of an axiomatic system and deduction rules on faith ;)

Physics Guy

and discovering new aspects of these objects.

Sir Cumference

@ACuriousMind Screw faith! Let's get hypothetical

Or philosophical, whatever you'd call this

The fact is, we find paradoxes in mathematics that seem to imply it has holes

Like indeterminate form

ACuriousMind

@SirCumference We can't - the point is that we cannot start "from scratch" because we always start from something that is assumed to be understood, like the meaning of words in natural language. In every sentence with which I could start to define math (or anything else fundamental) you could ask for the meaning of each word until I either stopped giving explanations or became circular.

@SirCumference We don't find "paradoxes in mathematics". Or well, in the form of things like Gödel's incompleteness we might, but I think you mean something else.

Sir Cumference

@ACuriousMind How about the fact that indeterminate form is something we cannot get a value out of? That we must avoid?

Differentiation isn't as easy as setting $\Delta x$ and $\Delta y$ to $0$. That produces indeterminate form.

ACuriousMind

11:38 PM

@SirCumference I don't understand the question. Firstly, there are perfectly reasonable ways to get the value of limits of indeterminate form (e.g. l'Hospital) and secondly an expression being undefined or a sequence diverging are perfectly fine things that can happen. "Math" is not disturbed by those. In the case of an undefined expression you violated the rules of what you were allowed to put in, and sequences can just diverge. Nowhere is a paradox in sight.

Physics Guy

"Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8]

Mathematicians seek out patterns[9][10] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about natur…

That's it.

heather

@ACuriousMind, I mean the pattern is there to be described, just like the Fibonacci sequence is there in nature, the fundamental idea of there being one, one, two, three, and so on, though I would say our symbols for one, two, and so on are invented. Yes, it isn't a physical object, but the basic concepts we have and manipulate with are physical. Further, stuff we've originally thought was purely abstract has been found to be not purely abstract.

So I still stand by it being discovered, and this question being a legitimate question. Would you say physics is "invented"? No, the concepts are there in the world; we just write them in a language we invented. And math is described in the world, we just write that math in a language we invented. I don't know if that quite makes sense, though.

Physics Guy

Look at the golden ratio or at pi. Numbers or structures that are realized in nature.

To be discovered.

heather

@PhysicsGuy, yes, exactly, or the way quantum mechanics operates based on probability.

Physics Guy

Yes.

heather

11:45 PM

Stuff like that. Physics is described by mathematics, and physics is certainly carried out in the real world. So math is "in" the real world.

Physics Guy

Or the whole mathematical description of QM

+

Sir Cumference

@heather Math isn't necessarily limited to the real world though.

ACuriousMind

@heather 1. Not all mathematics describes patterns in nature. Where "is" the line with two origins, or the Banach-Tarski 'paradoxon' as a pattern in nature, for instance?

2. The basic concepts of mathematics are certainly not physical in nature. What is physical about the von Neumann construction of the natural numbers, for instance, or about the very idea of the empty set or the axiom of choice?

Physics Guy

@ACuriousMind The Banach-Tarski paradoxon is garbage.

Sir Cumference

@ACuriousMind Pretty much all supertasks serve as examples

ACuriousMind

11:48 PM

I will not deny as math as it was started/done by humans was motivated by physical patterns and practical use.

Physics Guy

@ACuriousMind I mean, would you really consider it as a paradoxon ?

Sir Cumference

I like Zeno's paradox of the dichotomy. Achilles must reach the finish line in 10 seconds, but it seems impossible if we weren't dealing with a physical plane

ACuriousMind

But math is it is conceived of today, as an axiomatic system, carries no longer any intrinstic relation to reality, although it is of course of interest to us because it can be used to model reality so well.

@PhysicsGuy No, it's not a paradoxon, it's just a counterintuitive result. More importantly in this context, it's a true result (in ZFC) for which I don't see any model physical pattern.

Physics Guy

Yeah.

Sir Cumference

@heather If you haven't heard of Zeno's paradox, it demonstrates the difference between physical limitations and mathematics.

Say we have Achilles running to finish a race. He's going to finish it in 10 seconds.

In 5 seconds, he runs half the distance of the track. In the next 2.5 seconds, he runs 1/4 the distance of the track.

The distance continues to halve and halve. Question is, how can the distance ever reach 0?

How can he ever reach the finish line in 10 seconds? Or any time, for that matter?

ACuriousMind

11:54 PM

All Zeno's paradoxa show is that if you allow for infinite sequences within your mathematical model of relality (i.e. physics), you better have some notion of limit :P

Sir Cumference

According to the physical plane, he can increase by each Planck length, and so he'll reach the end. But mathematics isn't limited. There's no "smallest length" in mathematics, only in physics. We can continue to divide the distance as much as we want, only physically are we limited. Math doesn't have to deal with the physical plane, but physics must deal with mathematics.

Physics Guy

Aaaah, be careful with Planck length.

Sir Cumference

@PhysicsGuy Yes, yes, I know

ACuriousMind

It's clearly, utterly evident that the distance must reach 0 because things are obviously able to move finite distances. Such a "paradox" only shows that your model is flawed - in this case, you're allowing an infinite division of motion without having defined how to deal with that, since normal addition, etc. is only defined on finitely many numbers.

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