We count and stop only when we grow tired. We cannot imagine coming to the end of numbers, especially when we have learned arabic numerals. One cannot conceive of a situation where we could not write or say the name of the next number. Being unable to find another piece of paper seems somehow irrelevant.

Young children who have learned to count to 9 or 99 may imagine that there are no more numbers, but this is a brief situation. I suspect that many Romans were uncertain about numbers more than 5000 whereas Arabic numerals provided a hint of infinity.

I suggest that this situation alone is why we postulate integers without end—that and the fact that much of mathematics explores the consequences of such postulates without yet reaching a contradiction.

Peano’s axioms include the axiom of induction which says of the integers that any set of integers M, such that 1 belongs to M and any successor of a member of M itself belongs to M, includes all of the integers. For me that, with a few even simpler axioms, captures the essence of the integers.

Aristotle named the syllogism and brought to our attention rich mechanisms which we had already mastered but which we had scarcely thought about. With his work we began to think, or at least write about, thinking.

From this perspective I can see the integers as merely a fruitful hypothesis. I do not expect to see a contradiction revealed in the next 50 years and if one should be found I would expect some minor adjustment to the postulates to expunge the contradiction, thus leaving the integers sensibly intact.

The systems of our physical world are much intertwined. Physics must disentangle these systems as best it can to glimpse underlying truths which simple logic can grasp—simple logic and some not-so-simple mathematics. Perhaps the preeminent success in this thrust was deciphering the laws by which planets appear to move in the sky. The first great victory was Kepler’s laws which built on previous generations of observers and theorists. Newton explained and perfected Kepler’s success with an even more general and accurate theory. (Newton built on Galileo who built on Aristotle’s

We must however add that Euclid played a tremendous role in this victory. He taught us to think about space quantitatively. (There were fragmentary Egyptian precursors.) He gave us the theory of space that underpinned the first victory.

Newton’s theory of mechanics in almost any other setting must be endlessly qualified by terms such as frictionless. Such qualifications enormously complicate efforts in the laboratory to confirm Newton’s laws.

Something about our universe tends towards patterns and concomitantly exceptions to those patterns. Tracking down these patterns and their limits was necessary for our ancestors’ survival, going back at least to those with even minimal nervous systems. It seems clear that the first means to cope with such patterns in our ancestors were represented directly in the DNA and resulting neural configurations. After many stages of evolution we began to profit from patterns recognized and by individuals as a result of the experiences of that individual. I know of no speculation on these stages. I doubt that concepts were useful before the first elementary logic, and conversely. Somewhat later language evolved to communicate such patterns to other individuals for even greater advantage. The language of this communication was in part isomorphic with formal logic in the sense of predicates and nouns. (Predicates encompass verbs in this perspective.)

There is a great deal to language beyond the conveying of propositions.

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