So I was reading Euler's calculus textbook and in it he talks a lot about whether matter is infinitely divisible...

So I was reading Euler's calculus textbook and in it he talks a lot about whether matter is infinitely divisible, or made up of indivisible atoms, and he talks about this as if it's a very important thing to consider before you get into calculus.

It's surprising because nobody talks about this type of stuff in modern math books. What gives? what's the difference between the way old-timers like Euler thought about math, and the way people today think about math?

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Back in the days they actually cared about laying a consistent foundation. Nowadays they sweep it all under the rug for convenience.

The invention of alternative facts (which gave rise to the field of postmodern mathematics) in the early 20th century by Alfred Tarski, though the credit is typically awarded to the more well-known Godel (and occasionally Turing).

fuck off, mathematics up until it was formalised was literally random shit strung together, that worked well enough. Mathematics adheres to ZF Set Theory and is entirely consistent, it had to be formalised as it got more complex.

>Mathematics adheres to ZF Set Theory and is entirely consistent
lol

>Nowadays they sweep it all under the rug for convenience.
Agreed

>Mathematics adheres to ZF Set Theory and is entirely consistent
Prove it then.

>Mathematics is entirely consistent
i will award you a phd if you prove this

>Prove it then.
>i will award you a phd if you prove this
>Requiring proof for common knowledge.

>It's surprising because nobody talks about this type of stuff in modern math books. What gives?

Because Euler was writing 80~ years before Cauchy and Weierstrass came up with (ε, δ)-definition of the limit that put calculus on a firm rigorous foundation and 160~ years before Einstein proved discrete atoms exited with his 1905 paper on Brownian motion (which was hotly debated by chemists before then).

I mean, I don't wanna be That guy, but:
en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#Second_incompleteness_theorem

Because infinity, set theory and a lot of "formal" mathematics was just religious mumbo jumbo passed of as academic work. Once computers became a thing we had to let go of these spiritual notions and come back to what was the core of mathematics for 5000 years, empiricism. Which meant numerical solutions, finite series, no bullshit set theory or "axiomatic proofs".

Of course you still have a lot people reading these works and a few poor souls actually working on it but all of that has no affect on the real world. I just hope that if you do try and understand "the mind of God" through this kind of voodoo math that you're a religious man. Nothing's sadder than a brilliant mind of an atheist being wasted on this pseudo-religious pursuit like Bertrand Russel's was when he wrote 300 pages trying to prove stuff like 1+1=2 axiomatically.

Logic (and later mathematics) wasn't formalized until the 1900s. Mathematical foundations was never truly resolved but the tools to even approach it didn't properly exist until that point.

Euler talks a lot about that because that shit causes paradoxes and was actually fairly difficult to approach from a formal perspective. There are a few different modern solutions to the problem he discusses and the reason they aren't brought up in modern calculus textbooks is because you will eventually get to that stuff in an Analysis textbook.

This is one of the most retarded posts in this thread. Everyone stop for a moment and look at this moron.

Yeah I think that there's truth to what you are saying. Euler never use 'sets' in his book, not even once... But in any modern calculus textbook, everything is a set. A circle is a set of points that are at the same distance from some point, a line is also a set of infinite points, everything is a fucking set. It's so refreshing reading stuff from before everything became a set.

ever heard of Godel, nigger?

Calculus back in the days of Euler was very controversial and nothing like calculus today. Limits as we know them didn't even exist; derivatives and integrals were calculated using positive numbers that were "infinitely close" to zero. There were a lot of philosophical and even spiritual objections to infinitesimals.

Back then, you had to preface anything you wrote about analysis with at least an acknowledgement of the controversy. Nowadays, we have a much deeper understanding of the theory and a formalization that's more acceptable to everyone.

Downloaded the thing to give it a look.

At first it was fine but then I jumped ahead to see more of the content and It really put me off quickly.

What are some other good Analysis books (besides the classics that are the norm such as Rudin)?

>If you assume math is consistent, then there are unprovable true statements

Okay, what is your point?

Hanbook of analysis and its foundations, by Eric Schechter

I'm going through Tao's book now and it's fantastic. I've looked at Pugh's as well and the exercises are great.

That's not what the theorem says.