> I would argue that this is wrong. That's what physicists do, not mathematicians.
Notice that many mathematicians would not agree with you, here (but probably, a majority would). As the mathematician V.I.Arnold famously said, "mathematics is a branch of physics where experiments are cheap". So, yes, in the minds of lots of mathematicians, what they do is precisely to study and understand nature.
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!
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.
>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).
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.
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.
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).
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.
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?
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.
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.
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.
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.
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.
I see mathematics as completely opposite to studying nature.
Mathematicians work on extremely simple objects as a basis. For example (since we're here), give them a 0 and 1, and they will spend 200 years building a whole theoretical world from that, an artificial system they will describe through thousands and thousands of pages of theorems, getting more and more complex as time goes.
Nature is an extremely complex system from the start. Trying to understand and describe it is not at all the same approach. You take a complex system (the complexity of the system is given, fixed externally by the nature) and try to simplify it.
You can see this difference in the software world too.
Mathematicians (CS) will build and favour the use languages which are based on a single simplistic axiom: "everything is a list" for the most famous example, "everything is a function" for others, and then you are supposed to build all the rest on those simplistic bases, and in practice that will mean twisting, bending, squeezing the problem world (i.e. the nature) to make it fit in your model.
On the other side, you have programmers, which will more often favour practical, pragmatical languages, which do not exhibit the clean regular, symmetrical simplicity of the former ones, but which are more adapted to describe a complex, irregular world.
In that sense, though, programming would be a branch of physics that's concerned with making brute-force experiments cheap (thus saving the mathematical work needed to find clever tricks to do that same experiment by manipulating a small bunch of symbols).
Physics is experimental model building of phenomena which are not yet understood and are being explored.
Engineering is experimental model building of phenomena which are mostly understood, albeit sometimes with some quirks and unexpected edge cases.
Applied math is the toolset used in both physics and engineering.
Pure math is the abstract and philosophical exploration of symbolic relationships within all of math.
Academic CS - Wirth and Dijkstra-style - is the tiny subset of pure math used to explore theories of computing.
Practical CS is mostly just relatively trivial puzzle solving using a combination of cookbook academic CS with a bit of invention and innovation with influences from user psychology, marketing, and business design.
The most academic and mathematical parts of practical CS is ML and AI, which are genuinely exploratory. The second most academic part is probably processor architecture, where you may be applying statistical modelling to cache design and instruction pipeline outcomes.
Most of the rest is pretty basic compared to engineering modelling - never mind academic physics.
Even if a lot of mathematicians think math is about studying nature they are demonstrably wrong. One can do maths that has no known relation to physical reality, has no basis in physical reality or sits in direct opposition to how physical reality actually works. That is precisely the opposite of physics to say that "well, if I can imagine it it must be nature". So by counterexample, since the field of maths contains things that are anathema to the science of physics maths cannot be a branch of physics.
Math is the study of abstract patterns and those cannot be escaped. But just because individual mathematicians dedicate their lives to finding abstract patterns inspired by physics doesn't mean that either physics or math are branches of the other.
You are making a lot of ontological and epistemological assumptions that are contentious in the philosophy of math. Not saying you are wrong in thinking this, metaphysical questions don't necessarily have answers, but many would not agree with you.
> As the mathematician V.I.Arnold famously said, "mathematics is a branch of physics where experiments are cheap".
I think that is not so much subsuming mathematics under physics as a cheeky way of avowing mathematical Platonism, where eternal mathematical truths reside in some Platonic realm of ideas and wait to be discovered (not invented or proven) by mathematicians.
As you say, it is written in a playful and cheeky manner, but it is just a rhetorical device; the meaning is certainly very deep. Even deeper and longer, but in a similar spirit, you have this text:
It starts with a famous quote, replicating Caesar's gallic war:
All mathematics is divided into three parts:
cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA).
Math, like philosophy, is only interested in using concepts, structures, or otherwise rigorous ideas to pry into other ideas. Like philosophy it is in some senses self referential, and not really interested in discovering some idea of "truth", but rather exploring the avenues by which truth itself is defined.
Then, Arnold also says that mechanics is geometry in phase space (which subsumes physics in mathematics, like in the later “mathematical universe hypothesis” metaphysicians).
Notice that many mathematicians would not agree with you, here (but probably, a majority would). As the mathematician V.I.Arnold famously said, "mathematics is a branch of physics where experiments are cheap". So, yes, in the minds of lots of mathematicians, what they do is precisely to study and understand nature.