We've been wrong about math for 2300 years

> If mathematicians hadn’t skipped Marketing 101, they would have taken the problem more to heart: you can’t teach a subject you can’t define, just like you can’t market a product you can’t explain.

Well, we’ve been doing math for 2300 years so I think we can actually teach it well enough and use it to marvelous effects considering it got us to the moon, invented computers, probes leaving the solar system, & now AI (and numerous other things that would be impossible to list in a short post).

Another way to put it: is math invented or discovered?

If you ask them, mathematicians may give opinion one way or the other, but they do not actually care very much, because this question has nothing to do with their work.

I was fully expecting to be triggered by this article. I was pleasantly surprised to find I was not.

I think there is definitely something to it, if there is far more begging of the question that we can strictly define any field. Something that is largely not true. Our categories and topics in schools are affordances made to make it easier to teach and to learn.

To that end, what is it to write, but to have imaginary conversations with others about a topic. Communication through written material, then, is largely sharing of these imaginary communications with others. Some of the sharing is so that others can take part in the conversation. Some is so that they can critique the conversation itself, regardless of how imaginary it was.

Math easily fits there. The critiques go over not just if the idea was communicated, but expands to offer if it agrees with a lot of other rules we have added. And note that sometimes it doesn't, necessarily, while still being valuable!

Highly recommend David Basis "Mathematica: A Secret World of Intuition and Curiosity" or "Mathematique, une aventure au coeur de nous-meme" in french.

For me, it has been a refreshing and profound way to reflect (and possibly better understand) on my own way to "think" (for instance when I build software architecture), and explore what might be happening in my head while doing so.

Aristotle says we should only require of a discipline what the domain affords (when he is establishing principles not by induction).

Indeed, a perfectly serviceable philosophy of math would address what enables it to be taught and used consistently by different practitioners, so they all get the same results (or better, by some agreed metric thereof).

But I think the real question is what makes one mathematical approach better than another: economy? insight? accuracy? transparency? composability with other approaches? usability? cultural value? economic relevance?

Then, if you want to traverse 2,300 years, does the historical evolution of math evidence tension between the math we want and the math we got, and how (TF) to get what we want?

Realizing we're wrong about math is the essence of math: it's how we got 365 days instead of 360, irrational numbers distinct from ratios ...

The problem I have with conceptualism is that it doesn't address the unreasonable effectiveness of mathematics in describing the natural world. Clearly reality embeds and preserves many mathematical properties.

This is why I prefer some kind of structuralism, ie. that math is the study of structure. Clearly reality has coherent structure, therefore it's no surprise that math would be so successful in the sciences.

If you look at Frege’s original logical notation, all logical operators (apart from negation) reduce to conditional (‘if–then’) statements. Perhaps mathematics mirrors the cause-and-effect structure of reality, similar to Hume’s idea of causation as empirical regularity. Numbers could then be understood as convenient fictional tools, while logical operators capture something genuinely real about how the world itself works. I'm no mathematician, but it's just a thought.

IIRC the term mathematician was first mentioned as a faction of the Pythagorean sect (or church or whatever) (the other faction was the 'akusmaticians')

His new explanation is just formalism, as far as I can tell.

If you zoom out a bit, any human activity (like "doing math") can be characterized by "societal impact" and analyze whether any given activity (or underlying concepts) have utility. Take for example the concept of "nation" - why does this exist? Because as soon as anyone invents it, it will spread until resisted by another "nation". Why do we need money? Because a society with money is stronger than one without. In the same way, we can imagine a society with and without math. To a first order, the society with math is (far) stronger through its application to technology (and therefore industrialization, and therefore warfighting). Of course, pure math has had some profound impacts, far beyond what you'd expect (it discovers the tools that science later requires).

Math is yet another example of what we do with free-time when existence is not "nasty, brutish and short", which historically maintains and grows that free-time. Eventually math discovery may "peter out" and reach 0 contribution asymptotically, but even this behavior is acceptable: as the background of teaching students what is already known; as a peon to the concept of artistic patronage; as a dividend paid on math's incredible legacy; and to the always non-zero possibility that these new tools with eventually become useful.

Platonism vs nominalism is a bit of a meta rabbit hole, which most mathemeticians wisely ignore.

Mathematics is the study of words that have precise meanings.

One of my hobby projects is to construct a new foundation of mathematics: mathematics is the art of predicting if a Turing machine will stop or not.

I love wittgensteins take on the reality of math. He thought that there were true statements, false statements, nonsense, and tautologies.

True and false are easy enough. “The cat is on the mat” is true or false depending on where the cat is in the room. It’s verifiable. Nonsense is what he would call any value statement, such as “the flowers are beautiful”. By using the word nonsense he doesn’t disparage, it just isn’t a verifiable statement.

Tautologies are where math comes in. He thought that constructions of language were like symbolic pictures that had relation to states of the real world. Math however is statements about statements themselves. So “1+1=2” isn’t “true” in the same way that “the cat is on the mat” is true. But it is a tautology; a declaration that when you have two cats you can say there are “2” cats or you can say there are “1+1” cats. It’s the same thing.

He likened our knowledge of math to our knowledge of chess. Just like we humans invented the game of chess to pas time, we invented the game of math to better understand what statements make sense.

Just because a model is useful for describing/predicting the world doesn’t mean it’s a fundamental part of reality. To be very broad, I feel math is at the intersection between how our minds work and how the physical world roughly at human scale works. You can see this in how at extremely smaller and extremely larger scales math gets more and more complex at describing things. This is probably an unpopular opinion, but I think math in the physical world is itself an emergent property at certain scales, and not the “rules of reality” that all other emergent properties arise from. Still very useful for us, but in the way solar flares can sometimes flip a bit in an otherwise reliable computer, math (as a tool to describe reality) is an abstraction that things can leak through. Pure math, and don’t downvote me just because you disagree, is a property of how our minds work, it’s a thing humans do.

Math is the opposite of „not math“.

Such a vapid clickbait title.