Great mathematical ideas

Normblog has a regular feature, Writer’s Choice, where writers give their opinions of books which have influenced them.   Seeing this led me recently to think of the mathematical ideas which have influenced my own thinking.   In an earlier post, I wrote about the writers whose  books (and teachers whose lectures) directly influenced me.  I left many pure mathematicians and statisticians off that list because most mathematics and statistics I did not receive directly from their books, but indirectly, mediated through the textbooks and lectures of others.  It is time to make amends. 

Here then is a list of mathematical ideas which have had great influence on my thinking, along with their progenitors.  Not all of these ideas have yet proved useful in any practical sense, either to me or to the world – but there is still lots of time.   Some of these theories are very beautiful, and it is their elegance and beauty and profundity to which I respond.  Others are counter-intuitive and thus thought-provoking, and I recall them for this reason.

  • Euclid’s axiomatic treatment of (Euclidean) geometry
  • The various laws of large numbers, first proven by Jacob Bernoulli (which give a rational justification for reasoning from samples to populations)
  • The differential calculus of Isaac Newton and Gottfried Leibniz (the first formal treatment of change)
  • The Identity of Leonhard Euler:  exp ( i * \pi) + 1 = 0, which mysteriously links two transcendental numbers (\pi and e), an imaginary number i (the square root of minus one) with the identity of the addition operation (zero) and the identity of the multiplication operation (1).
  • The epsilon-delta arguments for the calculus of Augustin Louis Cauchy and Karl Weierstrauss
  • The non-Euclidean geometries of Janos Bolyai, Nikolai Lobachevsky and Bernhard Riemann (which showed that 2-dimensional (or plane) geometry would be different if the surface it was done on was curved rather than flat – the arrival of post-modernism in mathematics)
  • The diagonalization proof of Gregor Cantor that the Real numbers are not countable (showing that there is more than one type of infinity) (a proof-method later adopted by Godel, mentioned below)
  • The axioms for the natural numbers of Guiseppe Peano
  • The space-filling curves of Guiseppe Peano and others (mapping the unit interval continuously to the unit square)
  • The axiomatic treatments of geometry of Mario Pieri and David Hilbert (releasing pure mathematics from any necessary connection to the real-world)
  • The algebraic topology of Henri Poincare and many others (associating algebraic structures to topological spaces)
  • The paradox of set theory of Bertrand Russell (asking whether the set of all sets contains itself)
  • The Fixed Point Theorem of Jan Brouwer (which, inter alia, has been used to prove that certain purely-artificial mathematical constructs called economies under some conditions contain equilibria)
  • The theory of measure and integration of Henri Lebesgue
  • The constructivism of Jan Brouwer (which taught us to think differently about mathematical knowledge)
  • The statistical decision theory of Jerzy Neyman and Egon Pearson (which enabled us to bound the potential errors of statistical inference)
  • The axioms for probability theory of Andrey Kolmogorov (which formalized one common method for representing uncertainty)
  • The BHK axioms for intuitionistic logic, associated to the names of Jan Brouwer, Arend Heyting and Andrey Kolmogorov (which enabled the formal treatment of intuitionism)
  • The incompleteness theorems of Kurt Godel (which identified some limits to mathematical knowledge)
  • The theory of categories of Sam Eilenberg and Saunders Mac Lane (using pure mathematics to model what pure mathematicians do, and enabling concise, abstract and elegant presentations of mathematical knowledge)
  • Possible-worlds semantics for modal logics (due to many people, but often named for Saul Kripke)
  • The topos theory of Alexander Grothendieck (generalizing the category of sets)
  • The proof by Paul Cohen of the logical independence of the Axiom of Choice from the Zermelo-Fraenkel axioms of Set Theory (which establishes Choice as one truly weird axiom!)
  • The non-standard analysis of Abraham Robinson and the synthetic geometry of Anders Kock (which formalize infinitesimal arithmetic)
  • The non-probabilistic representations of uncertainty of Arthur Dempster, Glenn Shafer and others (which provide formal representations of uncertainty without the weaknesses of probability theory)
  • The information geometry of Shunichi Amari, Ole Barndorff-Nielsen, Nikolai Chentsov, Bradley Efron, and others (showing that the methods of statistical inference are not just ad hoc procedures)
  • The robust statistical methods of Peter Huber and others 
  • The proof by Andrew Wiles of The Theorem Formerly Known as Fermat’s Last (which proof I don’t yet follow).

Some of these ideas are among the most sublime and beautiful thoughts of humankind.  Not having an education which has equipped one to appreciate these ideas would be like being tone-deaf.

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