We've been wrong about math for 2300 years

https://davidbessis.substack.com/p/weve-been-wrong-about-math-for-2300

by gmays • 1 year ago

38 30 1 year ago

davidbessis.substack.com

taeric• 1 year ago

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!

jl6• 1 year ago

Mathematics is the study of words that have precise meanings.

pharrington• 1 year ago

Ironically enough, that's why we can't put it onto words. That's what the math's for!

tantalor• 1 year ago

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.

Galanwe• 1 year ago

If you ask them, mathematicians may give opinion one way or the other, but they do not actually care very much

Indeed, this is more of a meta scientific, or epistemogic, question (edit: though it seems the French "épistemologie" is not really translatable to "epistemology" in English, at least not in its original meaning)

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

I think this is the question encompassed by constructivist epistemology.

https://en.m.wikipedia.org/wiki/Constructivism_(philosophy_o...

viraptor• 1 year ago

Reminds me of the scientists arguing a long time ago whether the egg or the sperm is more important for creating new life. They wasted lots of time that could be dedicated to learning more.

magicalhippo• 1 year ago

is math invented or discovered?

Yes and yes. Or to put it another way, why does it have to be an "or" question?

To me some things in math are discovered, like prime numbers, pi, Euler's number.

Other things are more like an invention, like the Runge-Kutta methods[1] or the Finite Element Method[2].

[1] https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods

[2] https://en.wikipedia.org/wiki/Finite_element_method

tim333• 1 year ago

True. There are a lot of articles like this that waffle on for ages when it seems kind of obvious that both are true. I even bought Hersh's book that waffles on for ~400 pages trying to argue for invented by studiously avoiding the obvious truth that pi would have the same value in alien civilisations and so isn't just a human invention. Instead he tries to politicize it amazingly enough. Apparently invented is lefty, discovered is right wing and right wing bad so it's not discovered. Ridiculous argument really.

ianmcgowan2• 1 year ago

I love the mental picture from Greg Egan's Diaspora of the truth mines, literally hacking the math like chunks of coal:

"If ve ever wanted to be a miner in vis own right -- making and testing vis own conjectures at the coal face, like Gauss and Euler, Riemann and Levi-Civita, deRham and Cartan, Radiya and Blanca -- then Yatima knew there were no shortcuts, no alternatives to exploring the Mines firsthand. Ve couldn't hope to strike out in a fresh direction, a route no one had ever chosen before, without a new take on the old results. Only once ve'd constructed vis own map of the Mines -- idiosyncratically crumpled and stained, adorned and annotated like no one else's -- could ve begin to guess where the next rich vein of undiscovered truths lay buried."

krikou• 1 year ago

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.

rectang2• 1 year ago

Such a vapid clickbait title.

dredmorbius• 1 year ago

If you feel strongly about this, and can come up with a better alternative, preferably from the article itself, email your suggestion to the mods at hn@ycombinator.com.

Guidance: <https://news.ycombinator.com/item?id=40770024> <https://news.ycombinator.com/item?id=9908533>

Banal snipes within threads are not productive: <https://news.ycombinator.com/item?id=42489399>

rectang2• 1 year ago

The article's flagged now, but the standard approach of using the subtitle instead of the title would have yielded an improvement: "A radical conceptualist take on the foundations of mathematics".

vlovich123• 1 year ago

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).

CobrastanJorji• 1 year ago

Yeah, "You can't do <thing we've been doing successfully for a thousand years>" is a weird claim.

Secondly, I imagine Taoism would like to have a word on the subject of whether one can lead students towards something that cannot be defined.

empath75• 1 year ago

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

bjornsing• 1 year ago

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.

JohnKemeny• 1 year ago

Just as long as you don't call your predictor on itself as input, you should be good to go!

readthenotes12• 1 year ago

And perhaps even good to halt:)

klik992• 1 year ago

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.

ianmcgowan2• 1 year ago

That doesn't explain why math is so amazingly effective at describing reality - all the pure mathematical results seem to eventually end up being useful at describing some new aspect of the world (to the chagrin of the pure mathematician).

Maybe it's the other way around? Our minds work this way because that's the rules and we're just emergent from that reality. It's hard to argue that symmetry is not a mathematical ideal first, and a biological approximation to that ideal second.

klik992• 1 year ago

Yeah I agree our minds evolved to think that way because it was useful, we succeeded because the way the brain works fits very nicely with how things operate at human scale. My point is that the scale matters, as we get further and further away from human scale, the maths get more esoteric and even start depending on probability. Compare the math for bodies in motion versus subatomic particles. There's no question math exists in nature, but given the potentially infinite scale both in time and scale of reality, of which we occupy a small sliver, it's not impossible that the tools we use to model reality all the way back to the big bang may not be fundamental to nature but an emergent property.

Biological symmetry in particular likely emerged as a way to half the data needed to encode life, needing less resources and therefore more likely to reproduce.

But anyway, for all purposes, I do agree. Math obviously exists in nature and it's our best tool at predicting things. I just find it interesting to think that math itself is an emergent property of some other thing. Otherwise, why does anything exist at all? If it was simply maths, then why did anything first pop into existence?

simpaticoder• 1 year ago

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.

wolfi1• 1 year ago

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

w10-1• 1 year ago

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 ...

justonceokay• 1 year ago

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.

readthenotes12• 1 year ago

Since you mentioned "reality", I prefer the four value logic system where you have true, false, unknown, and unknowable.

speak_plainly• 1 year ago

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.

amai• 1 year ago

Math is the opposite of „not math“.

naasking• 1 year ago

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.

We've been wrong about math for 2300 years

Related Stories