Calculus without limits # Mathematicians who did not die

As Hardy once pointed out in his apology, “languages die but mathematical ideas do not”. Hardy added that a mathematician “]has the best chance of becoming immortal whatever the word immortal might mean”. Let me illustrate this with three mathematicians, all teachers of mine, who have earned more than immortality. Specker is the S from Kochen-Specker, Moser is the M from Kolmogorov-Arnold-Moser and Lanford is the L from Dobrushin-Lanford-Rulle. Being entombed in an acronym is probably the closest one can get to become a God. But the main reason to include this here in a blog on quantum calculus is that all three had strong interests in the connections between the discrete and the continuum. Maybe for different reasons: Specker was more fundamentally interested in the foundations of mathematics and discrete approaches like Kustannheimos, Lanford was interested in what it means when we simulate physics and mathematics with computers which are finite entities, Moser through his background with Siegel was a rather classical and conservative mathematician with strong interests also in number theory and algebraic geometry like simulations of twist maps on finite fields as the astronomer Rannou did first.

## Ernst Specker Illuystrating the Kochen-Specker theorem. One of the key insights of Kochen and Specker was that one can use a purely combinatorial argument to show the limitations of quantum mechanical measurements. In this case, nodes are vectors in a Hilbert space. They are used to build a finite set of basis vectors which are used to show that there can not be a classical value function (a function v on the set of self-adjoint operators with the property that it is linear, satisfies f(v(A))=v(f(A) for all continuous functions f and is also multiplicative v(AB)=v(A) v(B). )

## Oscar Lanford

Oscar Lanford is the L of DLR, the Dobrushin Lanford Ruelle equations. He was also the first to prove the Feigenbaum conjecture (with a remarkable computer assisted proof). Lanford was my PhD Dad and a link to an other extremely fascinating community. Lanford’s PhD dad was Arthur Wightman who had other students important to me like postdoc mentors Barry Simon or Rafael de la Llave. The “father of Wightman” John Wheeler is immortal not only through notions like Black holes but also because Wheeler had many important students, like Hug Everett, Robert Geroch, Warner Miller, Charles Misner, Kip Thorne , Robert Wald or Richard Feynman. This is a community, which has strong ties to physics. While Oscar Lanford grew up as a quantum field theorist, he moved to pure mathematics and got later more and more interested also in discrete mathematics and the statistical mechanics related to number theory. Later work on discrete approximations of dynamical systems by periodic transformations is not so well known. The interest of Lanford was certainly also motivated by computer arithmetic, whichcomes in when doing computer assisted proofs like rigorous in analysis, where one replaces everything with rational numbers so that no rounding errors matter and the tedious estimates could still be done by hand if one had the time. It is important when working with rigorous mathematics with the computer that one only uses integer arithmetic so that one can use rational numbers as interval boundaries. In other words, instead of the field of complex numbers, one works in number fields. Naturally, this has ties to very classical topics in number theory. When working with computer simulations, one also gets naturally interested in the foundations of mathematics as one has to contemplate what it means to make computations, whether it is acceptable to make proofs in which silicon replaces brain cells, and how we value such proofs. What does it mean to prove a theorem with the help of a computer? Can such a proof be considered correct? A bit independent of this is the question what does it mean if we simulate an ordinary or partial differential equation with a computer. On a computer, any deterministic system is periodic. What happens if we add randomness? Is this justified as a numerical scheme? In that case, one gets close to an other mathematical field in which Lanford was working in: statistical mechanics. The Bolzman equations for example are a stochastic model approximating much more difficult partial differential equations. Stochastic systems are more accessible to mathematics as one can then use tools from probability theory. In some sense, probability theory smooths out singularities as exceptional cases have now probability zero. Such tools are always used by physicists but much of the physics literature which deals with stochastic assumptions, like mean field theories are not mathematically rigorous. To do rigorous mean field theories, means to do renormalization type approaches which in general invokes heavy analysis. Already establishing existence of solutions of stochastic multi-particle systems like the Bolzman equation requires heavy analytic tools. The transition of using a stochastic model to simulate a deterministic system is similar than using a finite model to simulate it. In the later part of his work, Oscar Lanford worked on finite discrete approximations of dynamical systems. When a computer iterates a map, it is a map on a finite set of sets. Here we see what happens with a discrete Standard map of Chirikov. To the right, we see all the orbits (we just plotted the graph where the vertices are the points and where two vertices are connected if T(x)=y or T(x). The largest cyclic orbit is the “chaotic sea”. In this case we have a 101 x 101 torus (which is a finite field) on which the standard map works. It seems that nobody really got a good way to estimate the size of the largest cycles or give even estimates about the sizes of the cycles.