I think there's a reasonable argument to be made that mathematical constructs are part of the universe.

David Deutsch has a method for ascertaining whether something can can be said to exist or not - ask whether it "kicks back" when you interact with it, in the sense that simulating the response of the thing you're considering in a totally convincing way would involve an effort as large as building a new universe for that thing to exist in.

Rocks "exist" because, if you wanted to build a totally convincing simulation of a rock, you'd need to include all of modern physics as we currently understand it.

Other minds "exist" because simulating them convincingly would basically require you to construct a true articificial intelligence (strong AI).

Mathematics "exists" because giving someone the genuine experience of doing mathematics when they really weren't would involve a simulation of almost unfathomable complexity.

There is a very real truth in the statement that '1 + 1 = 2', or the statement that there are arbitrarily long arithmetic progressions of prime numbers, or any number of other mathematical results. The world of mathematics is a very real part of the universe, that "kicks back" by constantly surprising us when we think about it.

So I don't think the article, which is very well written and researched, deserves the middlebrow "Um, no" scorn that you treated it to.

> I think there's a reasonable argument to be made that mathematical constructs are part of the universe.

Well, one could say that we "create" them using our minds, which certainly are, but that's deep into philosophical territory.

> Mathematics "exists" because giving someone the genuine experience of doing mathematics when they really weren't would involve a simulation of almost unfathomable complexity.

Sounds to me like a misapplication of the method - the complexity arises from simulating the response of a person to math, not from simulating math. By that standard, all fiction is "a very real part of the universe".

> So I don't think the article, which is very well written and researched, deserves the middlebrow "Um, no" scorn that you treated it to.

That concerned on one statement, not the entire article (which I found fascinating as well). Yes, I have to admit that this is rather smartassy, but I actially feel that, quite independant of this article, it is an important and amazing realization that few people make.

"By that standard, all fiction is 'a very real part of the universe'."

Strangely, I believe that statement supports crntaylor's argument, being the point of fiction as far as I can see.

> I think there's a reasonable argument to be made that mathematical constructs are part of the universe.

What if there are 0 universes? Does 0 exist without any universes?

"I think there's a reasonable argument to be made that mathematical constructs are part of the universe."

oh, sure, but everything is part of the universe. human thought, for instance, is part of the universe. and i'd say that's what mathematics is about.

You are wrong. It is reality what drives mathematicians to work, not a pure abstraction. It is because of reality that maths is interesting. It is because of Geo-metry that algebra is so relevant.

Not to speak of Calculus...

> It is reality what drives mathematicians to work, not a pure abstraction.

In general, no.

> It is because of reality that maths is interesting. It is because of Geo-metry that algebra is so relevant.

Perhaps to you, but not to most mathematicians, certainly not those working in academical settings. Have you ever looked at group theory or topology?

I don't think you've ever done the kind of math that mathematicians do. What you learn in school is not math, it's calculating. What mathematicians do is to invent constructs that have no basis in reality and prove statements about them, then come up with more constructs based on those statements, ad infinitum.

Sometimes those constructs may be designed to model real world problems, and getting funding is probably easier in those areas, but just as often the applicability is only discovered afterwards - or not at all.

The best example (because it's something we've actually all learned about) is complex numbers. They were first invented in teh 16th century and considered pointless and irrelevant at first. People soon discovered that they could be useful in proofs about non-complex numbers as well, but it took several centuries before they were found to be directly applicable in electrical engineering (many more applications have been discovered since).

> invent constructs that have no basis in reality

Regardless of how abstract you get, how far you go, math is always tied to our physical reality. The basic operations are reflections on properties of our universe. The "kind of stuff" mathematicians do" allows us to model and reason about our world in ways that wouldn't be possible any other way (that we know of).

>math is always tied to our physical reality

I'm sorry but even if this is objectively valid, this not a claim that you can support. There are entire sub-fields of maths in which there are no known physical attachments.

Now, that's not to say that some time in the future we won't discover the relationship between every mathematical concept and some physical system. But at this moment, the claim you are making has a numerable set of counter examples, with a very large order. For one example, take the Banach–Tarski paradox:

>Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball.[1]

This is, as far as we know, physically nonsense.

That being said, there is quite a lively debate among mathematicians as to whether or not math is tied to reality.

[1] http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

> The basic operations are reflections on properties of our universe.

No, this is most definitely not true for all branches of mathematics.

He's not wrong.

To many mathematicians reality is largely irrelevant. For example G. H. Hardy, an extremely prominent British Mathematician, believed that true Mathematics is an art form and is not useful. He dismissed applications of mathematics as dull and boring.

And I can relate to him. It's incredible how complex, beautiful structures arise from a couple of simple axioms. It doesn't matter if what you study will be relevant or not, what matters is that it's fun and stimulating to explore.

A (somewhat tongue-in-cheek) question to ponder: If Hardy were alive today, would he turn his back on Number Theory as too useful, and study something more theoretical like the existence of long-time solutions to Navier-Stokes?

As someone with a degree in applied math, the pure abstract is more interesting than the applied. Applied math is like building really amazing and intricate sand castles on the beach. Pure math is like building the same sand castle, but in the sky and it's kept aloft purely by how beautiful it is, freed from constraints like "touches the ground" and "can support itself under gravity".

A lot of my friends feel the same way, with some of them specifically avoiding having "real world" applications of their work, as if that makes it an even better sand castle.

As to why I have an applied degree instead of doing pure math, numerical analysis makes a weird intuitive sense to me, and I figured building decent sand castles on the beach was better than making terrible sand castles in the sky that could barely hold themselves up. It also gets the grant money.

Look, I am an Algebraic Geometer and have done Schemes and whatnot. It is REALITY and this has nothing to do with "applied" or "pure". The fact that it is abstact has nothing to do with its being unreal.

Just to clarify: I am an expert too.

Pure math evangelist here. To be clear, it is only reality insofar as the axioms in which the theorems are derived are reality, and only insofar as our understanding and application logic is an objectively valid construct of reality.

When it all works out as beautifully as it does, in say Euler's Identity, it's hard to remember the possibility that the axioms could turn out false, or logic as we know it flawed.

But assuming (heh) that the axioms are true and that our understanding of logic is valid, "pure" math is as much apart of reality as "applied" math. And I'll choose proving the Fundamental theorem of Galois theory over number crunching in Matlab as my exercise in experiencing reality every time.

This is a philosophical position that almost no one engaged in serious math holds.

This book has an extremely modern discussion of this

http://www.phil.cam.ac.uk/teaching_staff/potter/staip.html

Only lay people and arm chair philosophers passing through math land have this unfortunate view.

The statement's a little hyperbolic, but I think you're misunderstanding the author's point. The point is that math is aimed at trying to build understanding in contrast to producing logically airtight proofs. The proofs are important to make sure that our understanding is right, but there's more to it than that.

I agree that's not the literal sentence you quoted, but it's definitely the theme of the article: according to many of the people cited, Mochizuki's shirking his responsibility by not explaining how to understand the result, regardless of whether the proof is correct or incorrect.

Caveat: I Am Not A Mathematician (IANAM)

> The point is that math is aimed at trying to build understanding in contrast to producing logically airtight proofs. The proofs are important to make sure that our understanding is right, but there's more to it than that.

Understanding is necessary to be certain the proof is correct. After that is ensured, many mathematicians will happily use the proof and base their work on it without understanding it.

> Mochizuki's shirking his responsibility by not explaining how to understand the result, regardless of whether the proof is correct or incorrect.

The main problem caused by his "shirking" is that it leaves the correctness of the proof uncertain and trying to fully understand it is a lot of work that could turn out to be wasted. People want to confirm the proof, and Mochizuki could make it much easier and less costly for them, but for some reason declines to do so.

From TFA: "Even an incorrect proof is better than no proof, because if the ideas are novel, they may still be useful for other problems, or inspire another mathematician to figure out the right answer."

edit: I can see taking issue with the word, "universe" in the original quote. I have trouble seeing how the word "understand" is debatable.

Here's an amusing counterargument:

http://pauli.uni-muenster.de/~munsteg/arnold.html

That's not an argument, that's an almost incoherent rant.