Is math discovered or invented? | Po-Shen Loh and Lex Fridman

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GUEST BIO:
Po-Shen Loh is a mathematician at CMU and coach of the USA International Math Olympiad team.

PODCAST INFO:

SOCIAL:

[ericfurze]

Mathematical truths are discovered, the techniques used to find them are invented.

[marin]

I'm glad I discovered this channel, underrated af.

[brianwestphal]

I’ve now ended up watching this whole podcast but in clip form.
I truly love this format for your podcast

[salman-]

The produced work of mathematics is completely invented (Notations, Syntax, Axioms, Number Systems, etc). The rules by which the math operates are apparently discovered.

[TyyylerDurden]

As every language, it is invented in order to tie up discovered concepts, measurements and logical patterns of EXISTING nature around us.

[kakha]

Every new recommendations from this channel is unique and interesting

[jonathanhenderson]

I'd say math is invented to model what is discovered. The fact that the map accurately matches the territory doesn't mean we discovered or didn't invent the map.

[ubberJakerz]

Interdisciplinary parallels are super cool, my favorite is that while Darwin was developing his theory for evolution and discussing the idea of genes, Gregor Mandel was pioneering the workings of DNA, without really realizing the implications. If only they could have met. They were each doing their work in the same timeframe as well.

[Timesend]

I'm going to treat to myself to a takeaway pizza tonight

[Yzjoshuwave]

Really want to come back and read everyone’s responses. It’s honestly a question I think I could spend many hundreds of hours on. My natural inclination is toward a renovated version of Platonism, which is to say that Math is discovered, but I also love the notion that formalism, or the notion that it’s created, could paradoxically be compatible with it - that there could be an intrinsic symmetry to Platonism and formalism. I like the analogy of a Möbius strip, where looking at them locally gives the illusion of being on opposite sides, but if you follow it around there’s only one side, so Platonism and formalism are fundamentally the same thing. In what sense can Math exist outside thinking? If we are imagining some sort of “data bank” of mathematical truth, what could that possibly mean? Are the truths embedded in it all Being thought at once, or do you think they have a substantive existence outside thought? But what could that possibly mean? What does it mean to “BE” outside of Being? It doesn’t seem like a tenable proposition? But if it somehow IS, we have to realize that ALL the information in the Universe, or even an arbitrarily large metaverse, would be an infinitesimally small subset of the truths it would need to hold. But if what it means for it to BE is the same as for it to be THOUGHT, we have to suppose a mode of thought that isn’t constrained to actual beings. We’re supposing an infinite realm of God in which the foundations of logic exist by their presence in a transcendental form of Cogito… But isn’t this starting to sound a bit like some sort of formalism? It’s a pretty trippy concept.

[Marwan-fznx]

Discovered, it's been always there. We're just learning from what we reveal.

[MADARAISEPIC]

I think the axioms are invented and everything within that axiom realm is discovered.

Although if math really is the Premier language of the universe then maybe there is some intrinsic thing about the universe that leads us to make those axioms in first place. In that sense, the axioms are discovered as well.

[LandscapeJoe]

Discovered. There are clips of Feynman stating that at a young age he was discovering formulas himself that had already been discovered centuries before him by scholars.

[johnh]

I think it’s a false distinction. Invention is actually a form of discovery. It’s discovering how to use, and combine, already available knowledge and objects to make something useful.

[mustang]

Did we physically discover that things add up or did we consciously invent that things add up? Which has primacy?

[shlurrpie]

The relationships we discover in nature are communicated through the invention of mathematics

[yevgeniygrechka]

Going further, anything that is "invented" is actually just discovered. From prehistoric times, there was always a way to harness electricity; the light bulb is a discovery of one such process.

[simranthiara]

I wonder if Terence Tao would ever come on to talk with Lex

[captainvonkleist]

I think you can discover conclusions from a set of assumptions.
But, the assumptions must, at some point, be invented.
Infinity, for example. How could that be anything but invention?
And if mathematics isn't invented, then what is? I think there may be some differences in what's considered invention and discovery, rather than simply a disagreement on the nature of mathematics.
As for the relationship between mathematics and reality, which I think the question is hinting at...
Mathematics includes notions (such as infinity) which have no physical analog. A determinist may make the case that a random variable is such a notion. Really, the mathematical ideas of arbitrary scale or resolution are problematic in practice.
It may be possible to conceive of a mathematics that's based on assumptions with physical analogs, but I don't believe our current mathematics is it.
Because our minds perceive patterns in nature on an unconscious level, it may be natural to assume that such patterns exist in a physical way. However, in many cases such an intuition is simply an act of projection.
Even when the pattern is empirically verifiable, producing a model using multiple patterns still produces incomplete and contradictory analyses, except when applied to the most simplistic of physical systems.

[gecg]

when it explains something that already exists it's discovered, a discovery, like a treasure in the sand.