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In my own past research with young children, it seemed to me that their thinking about numbers was more closely related to Giuseppe Peano's basic concept of “successor” than cardinality or quantity.

For my money, the elucidation of the foundations of mathematics a century ago, by Cantor, Frege, Peano, Hilbert, Russell, Zermelo, Gödel, Turing, and others, still stands as one of the greatest triumphs of human thought, up there with evolution or quantum mechanics or anything else.

In it Turing, considered a forefather of computer science, wrestles with questions about mathematical notation and phraseology, analyzing and referring to the work of mathematicians and thinkers including Giuseppe Peano, René Descartes and Louis François Antoine Arbogast.

But in 1931, Gödel showed that no such proof can be found within Peano arithmetic itself.

To turn Peano’s axioms into music, Marthinsen assigned each logical symbol to a musical note to create short musical phrases, which he calls gödelings.