Studying Mathematics Thread

[polymath257]

[Jehanne]

[polymath257]

[Jehanne]

Problem is that this stuff is really, really hard, not only to learn but to actually improve upon the work of others.  My problem was that I had a love of physics and math; just was not particularly good at either subject.  Still, got through 3-semesters of calculus and taught myself ordinary and partial differential equations with a little tensor calculus over the years.  They are beautiful subjects, and I see no reason to believe that what modern science observes in the heavens and on earth is just the product of invisible angelic beings who are the true cause of particle motion, whether it be atomic, microscopic or macroscopic.

The Universe can be modeled just fine with higher math, although, just because something is correctly mathematically does not mean that such corresponds to physical reality.

Yes, research in math is hard.

The concept of a fractal dimension goes back to Hausdorff and is around a century old. Using various forms of iteration to produce rather complicated sets is even a bit older (the Cantor ternary set is, perhaps, the first example of a set now recognizable as a fractal).

All I can say is that a lot depends on your definitions. And there are several competing definitions for what constitutes a 'fractal'. But iterative methods involving self-similarity are widespread in math.

It is interesting because what is taught at the entire undergraduate level is mostly 19th-century math and science with some early 20th-century stuff tossed in for the junior and senior level students.  I have read that PhD students in math start learning the 1950s material around their 2nd or 3rd year of graduate studies.

[polymath257]

Yes, research in math is hard.

The concept of a fractal dimension goes back to Hausdorff and is around a century old. Using various forms of iteration to produce rather complicated sets is even a bit older (the Cantor ternary set is, perhaps, the first example of a set now recognizable as a fractal).

All I can say is that a lot depends on your definitions. And there are several competing definitions for what constitutes a 'fractal'. But iterative methods involving self-similarity are widespread in math.

It is interesting because what is taught at the entire undergraduate level is mostly 19th-century math and science with some early 20th-century stuff tossed in for the junior and senior level students.  I have read that PhD students in math start learning the 1950s material around their 2nd or 3rd year of graduate studies.

Actually, most of the undergraduate math is 18th century or before. The one exception at the 'lower' levels is linear algebra. If you get to the junior/senior classes, you get into the 19th century with groups and rings as well as some of the material on epsilon-delta proofs.

And yes, getting past 1950 or so is pretty much limited to PhD students, although some of the material relevant to theoretical computer science gets past that mark.

[Jehanne]

It is interesting because what is taught at the entire undergraduate level is mostly 19th-century math and science with some early 20th-century stuff tossed in for the junior and senior level students.  I have read that PhD students in math start learning the 1950s material around their 2nd or 3rd year of graduate studies.

Actually, most of the undergraduate math is 18th century or before. The one exception at the 'lower' levels is linear algebra. If you get to the junior/senior classes, you get into the 19th century with groups and rings as well as some of the material on epsilon-delta proofs.

And yes, getting past 1950 or so is pretty much limited to PhD students, although some of the material relevant to theoretical computer science gets past that mark.

I have read that the modern-day definition of the limit in calculus was near the last quarter of the 19th-century, with the last book (other than the University of Wisconsin's foray back in time) on infinitesimals being around 1915.

[polymath257]

Actually, most of the undergraduate math is 18th century or before. The one exception at the 'lower' levels is linear algebra. If you get to the junior/senior classes, you get into the 19th century with groups and rings as well as some of the material on epsilon-delta proofs.

And yes, getting past 1950 or so is pretty much limited to PhD students, although some of the material relevant to theoretical computer science gets past that mark.

I have read that the modern-day definition of the limit in calculus was near the last quarter of the 19th-century, with the last book (other than the University of Wisconsin's foray back in time) on infinitesimals being around 1915.

Cauchy did his stuff in the early part of the 19th. By mid-century, the epsilon-delta definition was standard among mathematicians (if not among others).

[Kitan Etos]

[Jehanne]

[polymath257]