No distinction in kind rather than degree between ourselves and the chimps? No distinction? Seriously, folks? Here is a simple operational test: the chimpanzees invariably are the one behind the bars of their cages.

The world of shapes, lines, curves, and solids is as varied as the world of numbers, and it is only our long-satisfied possession of Euclidean geometry that offers us the impression, or the illusion, that it has, that world, already been encompassed in a manageable intellectual structure. The lineaments of that structure are well known: as in the rest of life, something is given and something is gotten; but the logic behind those lineaments is apt to pass unnoticed, and it is the logic that controls the system.

More than sixty years ago, mathematical logicians, by defining precisely the concept of an algorithm, gave content to the ancient human idea of an effective calculation. Their definitions led to the creation of the digital computer, an interesting example of thought bending matter to its ends.

No distinction in kind rather than degree between ourselves and the chimps? No distinction? Seriously, folks? Here is a simple operational test: the chimpanzees invariably are the one behind the bars of their cages.

Arithmetic is where the content lies, and not logic; but logic prompts certainty, and not arithmetic.

However good an argument in philosophy may happen to be, it is generally not good enough.

An axiomatic system establishes a reverberating relationship between what a mathematician assumes (the axioms) and what he or she can derive (the theorems). In the best of circumstances, the relationship is clear enough so that the mathematician can submit his or her reasoning to an informal checklist, passing from step to step with the easy confidence the steps are small enough so that he cannot be embarrassed nor she tripped up.