We've been wrong about math for 2300 years
36 comments
·March 10, 2025vlovich123
CobrastanJorji
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.
tantalor
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.
magicalhippo
> 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
tim333
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.
Galanwe
> 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...
ianmcgowan
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."
viraptor
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.
taeric
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!
krikou
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.
w10-1
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 ...
naasking
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.
speak_plainly
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.
wolfi1
IIRC the term mathematician was first mentioned as a faction of the Pythagorean sect (or church or whatever) (the other faction was the 'akusmaticians')
empath75
His new explanation is just formalism, as far as I can tell.
simpaticoder
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.
jl6
Mathematics is the study of words that have precise meanings.
pharrington
Ironically enough, that's why we can't put it onto words. That's what the math's for!
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bjornsing
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
Just as long as you don't call your predictor on itself as input, you should be good to go!
readthenotes1
And perhaps even good to halt:)
> 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).