[tux3]

Trivial in math is a term that refers to anything you've already learned.

You sometimes hear people say that math is tautological. But regardless of whether it's all just an elaborate rephrasing of the axioms, it's quite beautiful.

[math_dandy]

Historically, mathematicians have spent a huge amount of time and effort formulating optimal axioms and foundations so that theorems would follow naturally from structure. Theorems following “trivially” from a theoretical framework that took years to develop isn’t an indictment of the theorem, but an endorsement of the incredible effort expended to develop an optimal context for expressing and understanding the theorem.

[mathgradthrow]

Half of the work of mathematics is in correct definitions. Groethendieck referred to the division between mathematical labors as hunting and farming.

This is not my most popular opinion, but probably the most consequential invention of the last 400 years was the set. Suddenly all mathematical knowledge could be verified in one framework. Physicists had a target in which to state their models.

If you could state your hypothesis in the language of mathematics, "everyone" knew exactly what you meant by it, and how to go about testing your claims, or proving them, if they happened to be about mathematics itself.

Calculus was invented in 1690ish, physicists like to claim that this was the most important advance in physics, but quantum mechanics and relativity didn't happen until dedekind invented the real numbers, 200 years later.

It turns out that knowing what you're talking about matters.

[gjm11]

Do you have any evidence that Dedekind's formalization of the real numbers was an essential step on the road to quantum physics and relativity? This seems very doubtful to me.

A more plausible claim: the general move towards greater rigour in mathematics, one of whose expressions was Dedekind's formalization of the real numbers, improved the state of mathematical understanding in ways that were necessary for the arrival of quantum physics and relativity. E.g., to do quantum physics you want the notion of "vector space"; to do general relativity you want the notion of "Riemannian manifold"; to do special relativity maybe you want to have encountered the "Erlangen programme".

But I'm not 100% convinced. It's not unusual for physicists to make use of mathematical notions that they don't have precise definitions of. E.g., I'm not sure anyone has an entirely satisfactory formal account of "path integrals"; string theory may or may not turn out to have anything to do with how the universe actually works, but if it doesn't it probably won't be because we don't have a complete account of what it actually is. Newton managed to do pretty impressive things with calculus before anyone had a really convincing definition of such advanced notions as, er, "derivative".

[mathgradthrow]

My evidence is just the timeline. Mathematics blew up right before physics did.

[Sharlin]

Post hoc does not imply propter hoc.

[mathgradthrow]

Yeah, you're right, my evidence would be better if we did a randomized controlled trial.

[chermi]

Lol the authority you speak with is so hard to reconcile with the absurdity of your claims.

[mathgradthrow]

I didn't clain to have a popular opinion.

[chermi]

I know. But you should realize that sometimes opinions can be wrong.

[mathgradthrow]

the best I can do is assure you that it is informed by experience. Calculus could not progress without analysis. einstein needed poincare, who needed topology, which needed sets.

[aleph_minus_one]

> Trivial in math is a term that refers to anything you've already learned.

According to a professor, "trivial" means: "If this is not trivial for you, you should see this as a clear signal that you should take this course seriously instead of slacking of, or even that you simply are in the wrong course."

[kevinventullo]

I dunno, a common refrain I heard across all fields of math in grad school was “This is obvious. Wait, is this obvious…? Y… yes yeah it’s obvious. ”

[xelxebar]

It's actually an interesting observation. If you know where your keys are, finding them is trivial, but if you don't, then even the refrigerator becomes plausible.

Math does feel like that a lot of the time. Once you've tree-searched proof space and found the connection, you can usually spend way less time proving it the next time around.

[tekla]

This. It's always a good sign you've fucked up somewhere.

[gosub100]

My pet peeve math term is "clear". A long time ago I thought could teach myself group theory by buying the Springer group theory book and reading it from chapter 1, 1 page at a time. But I was blocked within the first 5 pages because the axioms and first few proofs kept saying how "clear" it was that all the results followed. Unfortunately, it was not "clear" to me :(

[mwcremer]

I had a calc prof who was in the middle of a lecture, "...and as any fool can see, X is..." He stopped, turned around, and said, "You know, sometimes when I say, 'It is intuitively obvious', or, 'As any fool can see', I realize it may not be intuitively obvious, and any fool may not be able to see. But as any fool can see, X is..."

[JadeNB]

> the Springer group theory book

I am skeptical that this uniquely identifies a book (unless you mean the book "Linear Algebraic Groups" by the author called Springer, rather than the publisher called Springer, in which case it's definitely not the way to start learning group theory!).

[gosub100]

It was the yellow Springer publishing book. Happened 20+ years ago now, cannot recall the author. IIRC the title was "a course in the theory of groups".

[layer8]

This one, probably: https://link.springer.com/book/10.1007/978-1-4419-8594-1

It’s a graduate-level text, to be fair.

[xelxebar]

Indeed. Famously, though, figuring out if something is a tautology is undecidable!

[zem]

ironically the article itself uses "trivial" in its other, more mathematical sense :)