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Comment by mkl | Hacker News Reader

I am a mathematician. Nature is of no consideration whatsoever in some fields of maths. But nature or applications to it are the primary focus in many other fields of maths. In still others, nature would be a source of analogies, or applications of a few special cases, etc. Muddying the waters, some mathematicians would expand the definition of "nature" to include completely abstract ideas - anything that feels "discovered", for example.

Mathematics is the study of patterns. Any kind of pattern you can imagine, in anything, including relationships between things. What things? Any things. That covers a lot!

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Whatever people may think what mathematicians think, this comment describes the real situation best as I've experienced it.

Most pure mathematicians I've met/worked with actually look down (in a jocular way) on applied mathematics/physics. When Lagrange reformulated Newtonian physics, he was very proud of the fact that he didn't use any diagrams and arrows showing forces in his paper. In fact, of all the Physics I've seen, I found Lagrange's work to be the most beautiful and elegant.

I love how the commenter put it as "Nature is of no consideration whatsoever in some fields of maths". I'd restate it as "Nature is of no consideration whatsoever in pure mathematics" and I'm quite sure that the pure mathematicians would agree.

I agree this is a common sentiment among mathematicians, but this is a very modern perspective. If you look back 100 years ago to Hilbert, there was less distinction between physicists and mathematicians, much less the pure/applied rift that now exists. Arnol'd (who is referenced above) was one of the mathematicians who tried to keep this unity alive.

https://xkcd.com/435/

>I am a mathematician. Nature is of no consideration whatsoever in some fields of maths.

It's not about actual nature (the universe etc) being into consideration.

It's about many mathematicians coming to see maths as exploration (physics-style) of a mathematical universe, so to speak, rather than a simple constructive process.

So, they come to see mathematics as a kind of physics in this regard, no in the sense that they concern themselves with the outside nature. But in that math work appears to them as exploring a natural landscape (just one made of patterns and numbers).

Yes, that's what my comment says (but maybe you read it before I edited it to be clearer):

>> Muddying the waters, some mathematicians would expand the definition of "nature" to include completely abstract ideas - anything that feels "discovered", for example.

Yes, that.

Though I wouldn't necessarily consider it "muddying the waters", but taking another criterium as important in the distinction of physics-like or not.

Namely, not whether it concerns the study of the material universe, but whether it involves experimentation/discovery of in place structures, and other such physics-like processes (which they think it does).

This is muddying the waters though, though I think you’re in good company.

I’m not sure if mathematics belongs in the sciences or art; it really has hallmarks of both.

It a modeling language that can be used to describe the universe. You don’t have science today, without the math.

Yet some of the proofs and mental exercises in pure math are almost divine; inspired in a way that resonates like a beautiful work of music.

>This is muddying the waters though

In a way, yes, as it extends the casual/conventional understanding of the term. I'm just saying it's not done to intentionally muddy the waters, but to introduce an alternative understanding.

So, yeah, we agree!

The controversy between assuming a point of view of "creating" vs "discovering" things is as old as mathematics.

> rather than a simple constructive process.

This requires some more distinction. 'constructive' can mean very different things. Some non-intuitionists would consider their counterparts definition of 'constructive' as possibly OK, but simple - and held other cases still for construction. Anecdotally, Ramanujan received his results as an inspiration from his household deity. Thinking about it probably brings up 5 different opinions among two people.

You could say that math is the part of laws of physics which humans can't imagine being different. You can pretty easily imagine a world where newtons laws are different, but I'd argue it is impossible to imagine a world where 1 + 1 is not 2. However being impossible for us to imagine doesn't mean that such worlds can't exist, it would just have completely ridiculous consequences we can't imagine, so those rules are a part of our universe and not a fundamental logical truth.

> I'd argue it is impossible to imagine a world where 1 + 1 is not 2.

Actually, you've probably done that yourself, in a programming setting: integers modulo 2, where 1+1 = 0. It's useful in places and the consequences aren't too ridiculous in this case.

Following through figuring out the consequences of rule changes is a key thing mathematicians do. E.g. do we need this rule? What if this was weaker? What if this was reversed? What if we had this extra restriction?

That is a number system where 1 + 1 isn't 2, not a universe. At least I can't imagine a universe where the concept of 1 + 1 equals 2 doesn't exist.

Numbers don't exist in any real sense, so we're clearly not talking about the actual physical universe. The universes we're talking about are the spaces of possibilities that arise from sets of rules. Examples include number systems and physics models built on them. Newtonian physics, built on Euclidean space; Einsteinian physics, built on space distorted by mass; quantum circuits, where modular arithmetic can show up.

> Numbers don't exist in any real sense

I'd argue they do, numbers arise when counting and counting is definitely a part of our reality. It is pretty hard to imagine a universe where you can't count things.

We can count things, because we use the concept of numbers. Numbers aren't a physical thing, they are all imagined. How I see it is that physics models can be described using number systems, but the numbers aren't part of what's being described. E.g. the numbers describing properties of particles are categorically different from the particles themselves, and only the particles can interact with other physical objects. An electron can never bang into a 7.

I agree. Counting might not have any intrinsic relationship to the physics of the universe, but it's a strange universe where thinking beings can exist, yet are incapable of constructing the mathematical rules that would allow them to count.

For a while, people thought that a universe in which Euclid’s postulates would be wrong would be very strange indeed. In my opinion it is short-sighted to go from “it’s weird (to us, right now)” to “it’s impossible”.

On the contrary, it’s very interesting to explore the consequences of something we take for granted being actually wrong or unnecessary.

In a universe where Euclid's postulates are wrong, like ours, you can still construct them. You're positing a universe where it's impossible to construct counting – i.e., set theory is impossible to imagine, Peano arithmetic is impossible to imagine, Church numerals are impossible to imagine…

In such a universe, how is conscious thought even possible?

> You're positing a universe where it's impossible to construct counting

Not quite. Your parent post was about thinking beings being incapable of counting, unless I misinterpreted, not about the universe making it impossible for anyone to count. My analogy is that for a while our universe was one in which non-Euclidean geometry was unfathomable for at least one thinking species, although it clearly can be observed in the universe.

Counting is something that is deeply embedded in our evolutionary tree (some fish and frogs have a primitive ability to count). So of course it seems fundamental to us. But to me this is not a proof that you cannot think without being able to count in our familiar way.

For example, you could perhaps build a logical system using uncountable quantities and still get something out of it. Like some fish which are able to see which school is bigger and base decisions on this without being able to count.

That's because (imo) numbers aren't intrinsic to the physical universe. They are (imo) an abstraction we humans invented to describe certain phenomena. You may not agree but hopefully you can at least see why some people would have this PoV, and especially the more general PoV that mathematics is not about (physical) nature, but about abstract ideas.

To me this is a non-example though because in this context 2 = 0. So 1 + 1 = 2 = 0.

That isn't a situation where 1+1 isn't 2. It's a situation where 1+1=2 but also 2=0.

A universe where 1 + 1 != 2 is a universe filled with self-contradictions and so cannot exist.

It's like imagining a universe where True == False. It's not a hypothetical, it's a logical impossibility.

1 + 1 = 2 is based on conservation of particles. You put a marble in a bowl, then another marble in the bowl, you now have two marbles in the bowl. If you remove conservation then there is no reason why 1 + 1 should equal 2. 1 + 1 being equal to anything could just be nonsense in that universe, such a construct wouldn't exist and there would be no way to reason about quantities. That isn't a logical inconsistency, so such a universe could exist.

In modulo arithmetic, 1+1 != 2 can work just fine. We're not talking about the literal universe, but the space of possibilities that opens up when you change the rules. E.g. all the amazing power and complexity that comes from imagining the existence of a number i with the property that i^2 = -1. This was initially thought to be a logical impossibility.

True == False does seem pretty broken though. Not sure that can go anywhere.

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