Mathematics and Tautologies

To speak of tautologies is to descend into dark pits whilst perceiving shining light everywhere.

As a figure of speech a tautology is a needless repetition, as in an “widow woman”. The notion of “needless repetition” is not as clearcut as it seems. If we grant that a widow is necessarily a human woman (ignoring its use in reference to other species), still the two terms, “widow” and “widow woman”, do not have the quite the same connotations. Any woman whose husband had died would be a widow; however the phrase, “widow woman”, suggests a woman for whom being a widow is an integral part of her social status and image. IOW it is only clearcut if we focus on the formal definition and ignore the tangential implications.

Truth by definition is of the essence. Thus, “corrupt politician” is not a tautology even though all politicians are corrupt; their corruption is a matter of fact rather than one of definition.

A tautology in logic is an expression that is true regardless of the truth value of the propositions within it. Thus “A or not A” is a tautology – if we accept the law of the excluded middle, which is another matter.

When we come to “mathematical truths are tautologous” things get more than a little murky. As a side note, if I am not mistaken, the notion of mathematical truths being tautologies is a modern one, introduced by Wittgenstein. The idea is much as follows: If we consider a formal axiomatic system with the theorems within the system being those statements deducible from the axioms then the theorems are tautologies. This is a thesis with problems.

To begin with, suppose we have an axiom system, call it A, and a theorem, call it B. The condensed proof of B is A=>B, i.e., there is a sequence of steps that are equivalent to “A implies B”. Now what is it that is supposed to be a tautology? Is it B? Not so, because the truth of B is contingent on A. Is it A=>B? Not so, because the inference A=>B isn’t true for all values of A and B. The (logical) tautology is (A & A=>B) => B.

What we are left with is the rather vague notion that the theorems of a system are tautologies because they are “true by definition” by virtue of being deducible from the axioms.

It is a notion further compromised by its identification of theorems within a system with mathematical truths. The formalist programme attempted to capture all of mathematics within a formal (finite) axiomatic system. As is well known, Godel showed that there is no such system; no formal system can capture the entirety of mathematical truths.