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Ryan Yang
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9/16: Is Mathematics Invented? - Ryan Yang

9/16: Is Mathematics Invented?

y.ryan.yang


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9/16: Is Mathematics Invented?


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“Is mathematics invented or discovered?”

This question is purely semantic. But, I have noticed that people’s responses, arguments, and intuition are actually a remarkably good litmus test for mathematical background. The specific answer doesn’t matter[1]. This piece will hopefully serve as a snapshot of my STEM education at age 17. I, internally feel like I’m more mathematically mature than I was before, mainly because I read a ton of papers nowadays [2].

A friend relayed the question to me and then tried to argue that it was discovered. I wasn’t going to write about it, until a New Yorker article, “How Mathematics Changed Me,” popped up in my news feed. And this is where the invention vs. discovery question as a litmus test idea was born.

The article is by Alec Wilkinson, an author of ten fiction books. The premise is that he struggled with mathematics as a kid, and relearns it as an adult. And as often happens when people who haven’t done that much math try to write about math, it becomes vague and poetic. But he has one brilliant point, stating early that “I used to feel that I saw something divine in nature, and now I think that what I saw was an intimation of mathematical structure, of pattern and motion and symmetry and scale, among other things, and that these produced a sense of the divine.” I really like this idea. Unfortunately, afterwards he goes off in a weird direction and betrays his limited math experience by using the concept of “infinity” as the example of divineness. This isn’t an example of mathematical structure, infinity is just a helpful abstraction.

To finally answer the opening question, I believe that mathematics is invented. I will make a stronger claim first, characterizing what math is. I believe that mathematics is a combination of tool development and the study of systems. Early math education teaches tool development [3]. Later math, beginning with olympiads and continuing in research, is about studying systems, roughly correlating to Terrence Tao’s conception of a final post-rigorous stage

Tool development is probably less invented than the study of systems, but I believe the heart of mathematics is the study of systems, so let’s ignore tool development. What I mean by the study of systems is an axiomatic/systematic framework, which then creates structure which is difficult to predict a priori.

And this is why Wilkinson’s concept of nature’s diviness coming from mathematics is such a good idea. Newton was a groundbreaking philosophical figure in the Enlightenment movement because he took the universe, and instead of being things that just moved however they moved, they moved following laws.

Circling back to why I believe that mathematics is invented, it is because so much math, just like the universe, is built by considering some axioms/rules and then seeing what this tells you about what happens.

Example 1 (great): What are the natural numbers? You might think that they’re tight to the number of apples or oranges, but in the modern conception, with Peano Axioms, they’re just a set following 9 axioms.

Example 2 (great): Groups are an excellent example. The modern set of axioms was not always used, and it was only after a while that the current (I think it’s associative + identity & inverse exist) axioms became used.

Example 3 (good): Topology. Open Sets are bonkers. A “topology” is a set of open sets, roughly designed as being a collection closed under union and intersection. You can’t tell me that this, along with the definitions built on top, are natural and inevitable.

One may note that the process of mathematics I’m describing does end up sounding awfully like science. Making observations, synthesizing them into laws/theorems. But then the difference comes since mathematicians can prove theorems, and scientists can verify laws but cannot truly prove them. Even the law of gravity is not 100%, since it exists out in the real world. 

(Corollary 1) This characterization is nice because it also points to why mathematics is fundamentally different from science, and thus more of a humanities. In my mind, the science-humanities continuum is defined by how big of a flex you could do on aliens. Imagine we start communicating with an alien civilization. Great literature (Ulysses by James Joyce?) is unlikely to impress them. Beautiful math, perhaps having a set of axioms that are shorter and more powerful, would be the ultimate flex, but still not quite as much of a flex as space travel. 

You may also note that the examples I give are relatively modern conceptions. Amusingly, when all these new rigorous axiomatic systems were all the rage, the Bourbaki movement caused elementary schools to go from teaching arithmetic to teaching set theory (true story). Thus, I also make the qualifying statement that most modern mathematics is invention, and that early mathematics was probably mostly discovery.

Finally, this interpretation is my favorite one because we have established that mathematics is invented, which gives a possible analogy; Mathematicians’ relationship to math is the same as God’s relationship to the universe. And I think this is my new mental model for what god is, he is the one who initialized the inputs to the universe. And the trend is towards modern mathematics being more invented, so we’re getting closer and closer to godliness?

Or maybe it’s all just a simulation.

Endnotes

This was written on September 16th and posted on the 24th. Feel free to leave comments here, and a google doc copy is at:
https://docs.google.com/document/d/1ZgNQVroKlD65dRHoYuaVaGg_0NhOPQ6N6MLBpDdfnyg/edit?usp=sharing

[1] Something like this (https://twitter.com/taimurabdaal/status/1340343872900239363) could be true, so it would be impossible to say either answer is predictive of being “more mathematically mature.”

One Possibility (I placed myself in the leftmost bucket because IQ is dumb)

[2] One of Choate’s “Seminars in Modern Math” homework assignments is proving that any integer (not necessarily positive) can be represented as the sum of six cubes (also not necessarily positive). My junior year solution was  “(k-1)^3 + (k+1)^3 is nice.” Then, subtracting two “k^3″s gives you 6k, and two more cubes is enough to get you to all integers. (the state of the art is that 5 cubes suffice, and that 4 might suffice). But, motivated by optimistic gradient descent, I instead took the 2nd discrete derivative of k^3 as a solution, which also gets you 6k. 

[3] People say this is a bad thing, and want math projects to have “real life applications.” But I think tool-learning is a necessary evil, and these projects anecdotally look dumb.