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