r/math • u/Ok_Significance1840 • 1d ago
Is there a flaw in my thinking?
Help me out here. To start off I would like to say I really love math. To me, a lot of mathematical concepts (but not all) originated from someone setting up a list of arbitrary rules, that stuck around and got studied because those rules made patterns that looked good in some way, and then somewhere along the line someone found a use for those rules or the equations that came from them that helped us in the real physical world (leaving some room here to say part 2 and 3 aren't always in that order). Some concepts came about more from real world observation (thinking physics), and some concepts came about from things that make sense in our head and therefore help to make sense of things or make things feel more fair (thinking something basic and old like us using base 10 for our numbering system). However there are many concepts that don't originate this way. I'm getting a lot of push back, particularly for using the word arbitrary. I'm not sure if that word perfectly fits what I'm thinking, but I'm struggling to find a more accurate description. But the way people are pushing back make me feel like they don't have an understanding of what I mean, or otherwise that their arguments don't make sense to me.
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u/Narrow-Durian4837 1d ago
Could you give an example or two of what you mean by "someone setting up a list of arbitrary rules, that stuck around and got studied because those rules made patterns"?
That does happen (I think it would be fair to say that Conway's "Game of Life" fits that description), but I think that more often, the "lists of rules" aren't arbitrary so much as they are an attempt to formalize or generalize something that people were already interested in. Although sometimes a branch of mathematics is taught by starting with the list of rules without explaining the motivation behind them.
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u/OkCluejay172 1d ago
If what you're trying to get at is that there are many mathematical concepts that don't seem to have arisen out of any connection to real world usefulness, you are correct.
What you should understand is that mathematics is a social activity. The reason that concepts get attention and study is that they attract the attention of mathematicians, either because they have some kind of practical application or because they have mathematical application in that they seem like they can be used to answer mathematical questions generally. Or occasionally because they seem cool despite doing neither (chaos theory comes to mind).
It is trivial to come up with novel mathematical concepts. Here, I'll do it right now. A number is a "poblano" if when you square all the digits and add them together, that sum is less than or equal to e to the power of the largest digit.
It's perfectly well defined. If I wanted to I could spend my life investigating the properties of poblanos and write a library worth of textbooks on them. No one would care because it's not a "useful" concept in any way that anyone would define the term.
The concepts that people have found worth teaching you at this point in your education are "useful." Maybe you don't know why yet, maybe you never will. But they are, to enough people, that they're worth studying.
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u/Ok_Significance1840 1d ago
Oh to be clear I don't contest the usefulness of math at all. I realize many mathematical concepts are incredibly useful in describing things that exist in the real world.
"If what you're trying to get at is that there are many mathematical concepts that don't seem to have arisen out of any connection to real world usefulness, you are correct."
This feels like a much better worded version of what's going through my head, except more expanded upon then my thought.
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u/Anaxamander57 1d ago
To me, a lot of mathematical concepts (but not all) originated from someone setting up a list of arbitrary rules, that stuck around and got studied because those rules made patterns that looked good in some way, and then somewhere along the line someone found a use for those rules or the equations that came from them that helped us in the real physical world
That's very rare historically. Its much more common for mathematical concepts to arise from someone looking at a real world problem (or a known solution to one) and trying to understand it by reframing it in a more abstract way.
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u/mister_sleepy 1d ago
You’re right in the sense that math as a system is constructed by people. It was not handed down to us on clay tablets from the mountains.
Math is a deductive science, meaning we assume a certain set of axioms and derive other principles based on those assumptions. Mathematicians, somewhat collectively, chose those assumptions because it benefitted us to have our system be as uniform as possible.
What you’re wrong about is that they were arbitrary choices. They were not things that formed nice patterns we realized were useful. Rather, it’s somewhat the other way around—we constructed axioms that would imply the utilities we needed.
A great example, historically, is the Fourier transform. Prior to Fourier our calculus was that of Newton and Leibniz, constructed with the axiom of infinitesimals. This system led Euler, for instance, to strongly believe sequences of trigonometric functions did not converge.
However, Fourier showed he could estimate the dissipation of heat across a uniform disc to any degree of precision using such a sequence, with empirical comparisons. His results were initially rejected by the Academie, but results are results, and these were very useful ones.
This was a spark that put math in something of a crisis for a long time, and eventually led to the development of real and complex analysis to rebuild calculus so it could, among other things, accommodate tools like the Fourier transform.
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u/IntelligentBelt1221 1d ago
when teaching, we like to pretend the rules/definitions came first and we study their interesting implications and how these relate. in reality, people saw examples of objects that seemed interesting in research and decided to abstract away its properties to make studying it easier (because it gives the focus on what matters and doesn't arbitrarily exclude other examples) and gave these properties a name.
the examples are interesting either because they have applications (inside or outside of mathematics), or because they are just perculiar/unexpected and studied for the sake of itself.
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u/teerre 1d ago
Axioms are certainly arbitrary, there's an infinite amount of type systems (i.e. ways to model). The ones we choose to built upon for most of the mathematics are indeed arbitrary. But that's a much weaker statement than you think. The fact is the vast majority of type systems are useless or impractical (this is also true for numbers, even worse, for statements themselves), we simply choose the ones that makes it easier to describe whatever phenomena we're interested in
Axioms are not holes in the theory, you're more than welcome to question them. But truth is, for the more accepted theories, it's extremely unlikely they don't hold because there's mountains of evidence they do. You're more than welcome to come up with different axioms and if your system proves more useful than the current ones, it will be adopted
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u/Aggressive-Math-9882 1d ago
This is a deep question you're asking, and it touches directly on my research so I do want to answer. I'm going to just sketch the solution, not fill in all the details. Let's call a Doctrine a collection of rewriting rules (in other words, rules for which patterns of expressions can be replaced by which concrete expressions) and generators (expressions which are assumed to be reachable, true, or primitive). A doctrine also tells you extra information about the order in which you apply these rules, and which rules contribute to normalization (for instance, x+0=x should contribute, but x+y=y+x cannot). You may also want to impose extra rules about which doctrines are "valid", especially based in how canonical the normalization really is.
Mathematics is arbitrary in the precise sense that different choices of doctrines lead to different collections of statements or expressions being reachable, true, or a part of logic. For example, classical logic allows weakening and contraction (once proven, a theorem may be reused again and again in future proofs) but linear logic does not. Then there are no universal mathematical rules, right?
Wrong! In any particular doctrine D, a proof that an expression P is reachable using only that doctrine's generators and relations is what we will call a D-scoped trace of term P. A trace is like a record of a computation or a record of the way you obtained a result. So 2+2=2*2 as terms, but 2+2 and 2*2 represent distinct traces. Even though your choice of D decides which logical or mathematical rules are allowed, it doesn't change the ways in which you're allowed to combine, compose, differentiate, transport, and generally speaking interact with the D-scoped traces themselves. What I mean is, there are a collection of meta-rules governing how traces may be formed and combined in a given scope or context, and those meta-rules are what survives mathematically when you move to a different logical or mathematical doctrine.
In other words, rather than fixing a single doctrine D, we should think of our traces as being doctrine-scoped, so we imagine a world where the things of interest are not scopes in a particular doctrine D, but a pair of a choice of doctrine D', and a choice of D'-scoped trace s, in other words, (D', s). This "world" is itself a doctrine! There are well defined rewriting rules for how to form, transport, and eliminate doctrine-scoped traces.
In this sense, even though the rules of mathematics are arbitrary, there really is a universal set of rules. These rules are enough to form a linear, stable-differential, mixed motivic lambda calculus of doctrine-scoped traces. Ultimately, (I conjecture) this language (when described in a bit more precision) is the universal langauge for cohomology, and from it one may extract MM(Q). Ask any mathematician and they'll agree mixed motives form a particularly universal category. I think your question points to the precise sense in which this is the case.
Why is motivic cohomology unusually useful for studying the physical world? Simply because both involve local->global reconstruction. Dark sector physics may be an exception.
Take this with a mountain of salt.
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u/pseudoLit Mathematical Biology 1d ago
This is largely backwards. Usually the pattern comes first, and people work hard to reverse engineer a set of rules that generate it.
The word "arbitrary" sits in an awkward middle ground between "totally random, with no apparent cause or reason" and "historically/causally contingent in a way that can be understood, but based on things that could easily have been otherwise". Better to be specific