Reddit mentions of Mathematical Thought from Ancient to Modern Times, Vol. 1

This is a very loaded question, but I'll do my best to answer it.

Disclaimer: I'm a lowly undergrad.

Math is made up of many different fields, so it's hard to assess its development in a linear fashion. The main areas are: algebra, geometry, analysis, logic/set theory/foundations, number theory, and applied math.

Now, the calculus that you study was mostly invented by the end of the 17th century, but interestingly, ideas about limits didn't really get formalized until the 19th century. Cauchy was instrumental in formalizing the definition of the derivative, and Riemann in formalizing the equivalent definition of the definite integral.

If you were to continue to take math classes in an undergraduate curriculum, you might take any of the following: linear algebra, abstract algebra, more calculus (multivariable, etc.), differential equations, discrete math, real analysis, complex analysis, basic functional analysis, topology, set theory, non-euclidean geometry, graph theory, number theory, probability and statistics, etc.

But even if you took most courses available to you as a college student, you would still be about 100 years behind mathematics as it stands today. You could argue the exact number of years, as different fields have evolved at different rates, but the point is that math has exploded in variety over the last century.

The last person to truly grasp "all" areas of mathematics of his time is probably someone like Gauss or Euler. Today, you're lucky if you can master a subfield of your subfield.

I'll now try to give you a quick overview of how math has developed since the invention of calculus. Apologies if it sounds vague (that's because it is).

The invention of calculus itself was a gradual process. Leibniz and Newton are credited with the invention largely because they collected the known information at the time and managed to come up with the Fundamental Theorem of Calculus, along with a ton of notations. Newton also applied it to his physics, which began a long relationship between calculus and the natural sciences. Up until this time, most math was geometric, rather than algebraic. Complex geometric proofs were drawn up to solve problems that would later be solved simply with algebra. Analytic geometry had only been invented a few years prior by Descartes, and though it seems obvious to us today, it was revolutionary at the time.

Progress in calculus was made very quickly in the 17th century - before long, all of the calculus you learn (and then some) was invented, though it lacked the formal foundations that it has today. The Bernoullis helped to found the calculus of variations, which is a topic usually reserved for grad school. That gives you an idea of how much math there truly is. In the 18th century, most of the progress was made in further developing the calculus, while other fields of math were being born. Euler and Gauss helped to popularize the notion of complex numbers. Euler considered the Königsberg bridge problem which would later inspire graph theory and topology. Lagrange and others made discoveries in what would become the field of abstract algebra.

The 19th century brought rigor to mathematics. While mathematicians were proving things up until then for the most part, there was often an element of intuition (and sometimes hand-waving) going on, especially back in the 1600s. A discovery by Fourier that functions could be approximated by series of sines and cosines (later, Fourier analysis) made the mathematical community worried. With the foundations of calculus on such empirically-strong, but theoretically-shaky grounds, how could they be sure if these wild claims were really true? This quest to put Fourier analysis on solid ground led to a rigor revolution in mathematics. What we now call "analysis" (the more rigorous version of calculus) was developed, and many other discoveries were made as a result. The other revolution going on in the 19th century was in the field of abstract algebra. Many of the great algebraists lived during this time: Galois, Abel, Noether, and others. Noether actually lived about 80 years after Abel and Galois.

Towards the end of the 19th century, Cantor developed his theory of sets, which led to a second revolution in mathematics. Although they are ubiquitous now, sets were relatively unused at the time. The introduction of sets into mathematics, combined with the revolution in rigor discussed previously, had a huge impact on mathematics. Progress in nearly every field accelerated like crazy in the new set-based more rigorous framework. The 20th century brought a deluge of mathematical information unlike any time before it. Some of the key fields that were invented in the last century -- or really started to get going -- were topology, algebraic geometry, linear algebra (for computers), computational logic, model theory, set theory, graph theory, combinatorics, various applications of math to advanced physics like relativity theory, etc.

But these are just the tip of the iceberg. If you really want to know how far math has progressed in an intuitive sense, I invite you to open a recent math journal (or look at one online) and see how much you can understand. It's truly mind-blowing.

I hope this helped more than it hurt. I'm no historian, but I think I got at least the general shape of things right. If you want a more in-depth look, especially one that considers the more modern developments, try Kline's series of books:* link to the first one

I'm reading Mathematical Thought from Ancient to Modern Times by Morris Kline, and I really like it. It's three volumes, so it's pretty thorough. I've only finished the first volume, but he does a really good job of explaining the way the Greeks thought about mathematics, compared to the modern Western way. It's not too technical to follow, but he tries to explain the important theorems and their proofs in some detail. Volume 3 goes up through Poincare and Goedel, and then stops, so you won't learn about later twentieth century developments from him. But he says it's too soon to know what's historically important there, anyway. I'm looking forward to the next two volumes.

If you're interested check out Mathematical Thought from Ancient to Modern Times by Kline or A History of Mathematics by Suzuki.

Morris Kline isn't a professional historian but he has a book on this that you might be interested in. IIRC, volume 2 has the most stuff on the history of calculus.