From Jordana Cepelewicz at Nautilus, on mathematician Harvey Friedman:

The foundations of mathematics is also a field—in stark contrast to the casual and light tone of Friedman’s emails—that has been in crisis for nearly a century. In 1931, the Austrian mathematician and philosopher Kurt Gödel proved that any logical system adequate to develop basic arithmetic gives rise to statements that cannot be proven true or false within that system. One such statement: that the system itself is consistent. In other words, no system can ever prove itself to be free of contradiction. The result seemed to present an insurmountable problem for mathematicians, not so much because it prevented them from ever knowing whether the system their work is built on is consistent (so far there haven’t been inconsistencies), but because it meant their fundamental logic had significant limitations.

So mathematicians got by with a system called ZFC:

And so something odd happened: Mathematicians chose to move on. Incompleteness, they decided, had no direct bearing on their own work. The axioms commonly known as ZFC (the Zermelo-Fraenkel axioms plus the axiom of choice) that constitute today’s most commonly used foundation of mathematics provides a rigorous framework for proving theorems.

…

Friedman proved that for any set in the rational cube (from three to an arbitrary number of dimensions), there is a maximal emulation with drop symmetry between specific pairs of points. To prove that theorem and identify the points for which it holds, he had to rely on a system stronger than ZFC. That is to say, it cannot be refuted, nor can it be proven, in ZFC.

Showing the theorem is not refutable is a pretty standard (although certainly not simple) process: Demonstrate that it logically follows from the consistency of large cardinal axioms. Showing it’s not provable, on the other hand, is more difficult. He did this with a proof by contradiction: He began with the assumption that he could prove his theorem in ZFC, and then constructed from it a system of objects in which ZFC holds. Which means that if his theorem holds true, then ZFC is consistent—and, transitively, that ZFC has proven its own consistency. But by Gödel’s incompleteness theorem, that cannot possibly be the case. And so, the theorem cannot be proven in ZFC. He’s working to extend the theory to other types of symmetries, other definitions of “maximal,” and other types of objects.

…

With a broadened foundational diversity may come new opportunities to solve old problems. In his 1960 essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” physicist Eugene Wigner recalls a student asking a perspicacious question: “How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nonetheless, explains just as many phenomena as the present theory.” Wigner goes on to note that the idea is a valid one—or, at least, that there’s never been any evidence to suggest this wouldn’t happen. More.

Mathematics is probably safe from Harvey Friedman if 2 + 2 continue to make 4.

*See also:* How was mathematics woven into reality?

and

Mathematics challenges naturalism, says math prof

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