We try to ease the flow of exposition by putting some details in appendices, which you can consult as you wish. We also try to provide ample resources for the reader who finds something interesting enough to dig deeper. Either way it is a very nice reading.

## Rational Number Theory in the 20th Century

Twelve Landmarks of Twentieth-Century Analysis [3]. I enjoyed Allen Hatcher's "Algebraic Topology" very much. It's free.

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He's been retired for a while now. Silvanus P. Thompson's Calculus Made Easy [1] is a good read. Per the reviews, the paperback and Kindle versions should be avoided in favor of the hardcover, although my softcover copy predates this by decades. Manfred Schroeder's book entitled Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity is a wonderful introduction to applied discrete mathematics, or concrete mathematics, to use Knuth's phrase. I bought this book on a whim and then I couldn't put it down.

Very good elementary overview of some deep mathematics. It is quite delightful. Good "casual" advanced math books. The Grothendieck-Serre correspondence contains an exchange of letters of who might be the most influential postwar mathematicians. Villani has his book "the birth of a theorem" or something like that.

These seem very interesting, thanks for the suggestion! It's a very casual read, but walks you through a lot of "advanced" mathematics. For me "casual mathematics" would mean a causal, hand-waving, light-on-the-proof-details description of mathematical results.

## Narkiewicz | Rational Number Theory in the 20th Century | | From PNT to FLT

It's really short too. It's outdated now in light of more recent discoveries but still a great book. It's more encyclopedic in nature, but I believe it's intended to be fairly accessible. It might be something to dip in and out of. It's full of real maths — full-blown results and detailed proofs — but explained very clearly and easy to read. Accessible, enjoyable, fascinating, and educational! Cannot write an answer, but this MAA review indicates you might like M. Spivak's 'Comprehensive introduction to differential geometry', especially volume 2: "Once one starts talking about surfaces, however, Spivak really gets interesting.

He takes a quasi-historical approach, meaning that he follows the historical order of progress when it is helpful and ignores it when it is not. After a brief chapter on the theory of surfaces before Gauss, what we get is a careful reading of Gauss's famous Disquisitiones Circa Superficies Curvas. One of the innovations in this third edition is that [cont. Jumping off from Gauss, the chapter reworks the theory in modern terms. Then he does it again: starting from Riemann's famous essay "On the Hypotheses which lie at the Foundations of Geometry," Spivak develops the basic ideas in the geometry of manifolds, including a brilliant section called "The Birth of the Riemann Curvature Tensor.

enter site I have not slogged through much of it yet, but "Visual Complex Analysis" Needham seems like an interesting approach. Here's a great online textbook that covers the math behind deep learning. It's a good mix of background and math, and it's a great into to the technology. Marcel Berger, Geometry revealed, D. I like the list. I would add: the discovery of the positional number system, zero also negative numbers , and the ensuing efficient algorithms for basic arithmetic addition and multiplication.

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Yuval, yes positional number systems and zero are great human discoveries which are largely mathematical. Can we think about positional number system as a new, for its time, database for numbers? I would be reluctant to regard it a milestone within mathematics because of its larger scope. This may also apply to the discovery of natural numbers, the wheel, and digital computers. And, thanks Scott.

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Can you think of effective numerical analysis or any form of computation without a positional number system? I will try, but I am not sure these paradigm shifts will be always beneficial in the retroaspect. One simple example is the concept of number. After the real breakthrough that counting and measuring are two sides of the same coin, there was a short outburst of enthusiasm until Pythagoras discovered irrationality of the square root of two. This turned the brightest mins of the next millennium to the problems that were well beyond the scope of the mathematics of that time, lacking the adequate language.

Historically very near was Archimedes, who came very close to legalizing infinite constructions and introduce the real numbers as we know them. The Ptolemean system at the time was more accurate and, what is more important, contained in itself an infinite potential for improvement: no matter what are the laws determining a periodic motion, it can be always expanded in a Fourier series which is what the Ptolemean system was all about with its complicated system of circles.

Newton adopted the Kepler model and derived from it the Laws of Nature in the form of differential equations. One could imagine a genius who would make a step in the different direction and develop the Fourier analysis.

This would be a very unusual mathematics, with integrals preceding derivatives, probably variational principles preceding the Euler-Lagrange differential equations etc. When Descartes and Fermat developed what today is called analytical geometry it became possible to use algebraic tools to get insights into geometry and ask new geometrical questions based on algebraic considerations.

New coordinate systems were developed, including homogeneous coordinates which results in being able to use algebra to investigate projective geometry which gave new insights into the conic sections and many other parts of geometry. Analytical geometry also spurred interest in the function concept and proved useful in exploiting the evolution of ideas related to Calculus.

Very often exciting new developments in mathematics are born in geometric form e. I think of analytical geometry as an important milestone. Hi Joe, analytic geometry is described in the text in the geometry section. But not the connections with calculus. Alas, not knots. Gil I would add with PNT the theorem of Dirichlet on prime numbers in arithmetic progressions as a milestone of using analytic methods in number theory. The Greeks left a few challenges.

Trisection, circle squaring and cube duplication problems was a challenge that was finally responded with creation of the Galois theory which equations can be solved in quadratic irrationals and the analytic number theory transcendence of pi e. The Fifth Postulate challenge was met with development of the concept of Riemann manifold. The Fermat equation challenge was responded by creation of the algebraic number theory and the modern Grothendieck-style algebraic geometry.

Very often the formal solution of the problem, albeit duly noticed and celebrated, turns out to be just a step in fast progress driven by the accumulated momentum. What should count as a milestone in such situations? The formulation of challenges? Sometimes challenge and response follow each other relatively fast, within decades like with most of the Hilbert problems , sometimes it took millennium or longer.

Hi there! Three papers in There is a pdf on my webpage. Dear Gil, dear Nati, Discovery number 7 only deals with the laws of classical probability — perhaps it could be expanded to include the discovery of quantum-mechanical laws and of a way to formally represent them. Along similar lines, Discovery n.

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