Mathematics/Waismann: in our view, mathematics is not tautological, nor is it a mere branch of logic. It rests only on its own determinations.
The belief that mathematics is more securely founded by logic is a misunderstanding.
2 + 2 = 4 does not correspond to a tautology, but to an instruction. It is much closer to an empirical proposition than a tautology. It is just a rule, similar to chess, which is obeyed or transgressed. This would not be possible in the case of a tautology, for what is it to obey or transgress a tautology?
The opinion that the entire mathematics is based on Peano's 5 axioms can no longer be maintained today. Mathematics is a multiplicity of systems.
The theorems of arithmetic are neither true nor false, but are compatible or incompatible with certain determinations.
Thus a certain dualism is overcome:
It was believed that only the natural numbers were eternal, irrefutable truths, or they expressed them, whereas the rational and real numbers were mere conventions. (Kronecker).
WaismannVsKronecker: that is a half measure, and the whole development of arithmetic shows which way we have to go: the possibility of a number series 1,2,3,4,5,... - many have already been mentioned.
E.g. if we think that a distance is divided into parts by points, then it makes sense to say that the distance has 2,3,4... parts, but not: "the distance has a part." One would rather like to count here:
and this corresponds to the sentence series: "The distance is undivided", "the distance is divided into two parts", ... etc. i.e. we do not count here according to the scheme we use, and yet this is an everyday case. ((s) linguistic overvaluation of "consists of." Solution: 1 = fake part.)
But not only the number series, but also the operations we might think of as changed: Suppose, we should carry out additions with many millions of digits. The results of two computers will not match then. Is the concept of probability introduced into arithmetics here? Or a new calculation is introduced.
The error of logic was that it thought it had firmly underpinned the arithmetic. Frege: "The foundation stones, fixed in an eternal ground, are, however, flooded by our thinking, but they are not movable."
WaismannVsFrege: already the expression "justifying" the arithmetic gives us a false picture,...
...as if its building was built on ground truths, while it is a calculus, which proceeds only from certain determinations, free-floating, like the solar system, which rests on nothing.
We can only describe the arithmetic, i.e. specify their rules, but not justify it._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
Einführung in das mathematische Denken Darmstadt 1996
Logik, Sprache, Philosophie Stuttgart 1976