Guys

Here is an explanation of the difference between a logical and a non-logical axiom, both of which are forms of axioms used in mathematics.

The explanation starts off by referring to the way an axiom is usually understood in epistemology, as well as the controversy inherent in that understanding:

In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. To say the least, not all epistemologists agree that any axioms, understood in that sense, exist.

The point is then made that:

In mathematics, axioms are not self-evident truths. They are of two different kinds: logical axioms and non-logical axioms . Axiomatic reasoning is today most widely used in mathematics.

The explanation on logical axioms makes the point that these types of axioms used in mathematics, come closest to being "true", but (take note, DJP) that the mathematical requirement of offering "proof" of an axiom, contradicts the very notion of the "self-evident truth" of an axiom:

These are formulas which are valid, i.e., formulas that are satisfied by every model (a.k.a. structure) under every variable assignment function. More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values.

Now, in order to claim that something is a logical axiom, we must know that it is indeed valid. That is, it might be necessary to offer a proof of its validity (truth) in every model. This might challenge the very classical notion of axiom; this is at least one of the reasons why axioms are not regarded as obviously true or self-evident statements.

Logical axioms, being mere formulas, are devoid of any meaning; but the point is that when they are interpreted in any universe, they will always hold no matter what values are assigned to the variables. Thus, this notion of axiom is perhaps the closest to the intended meaning of the word: that axioms are true, no matter what.

But it is when we look at non-logical axioms, that it becomes really clear that such axioms can in no way be "true":

Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as algebraic groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate.

Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that in principle every theory could be axiomatized in this way and fomalized down to the bare language of logical formulas. This turns out to be impossible and proved to be quite a story.

This is the role of non-logical axioms, they simply constitute a starting point in a logical system. Since they are so fundamental in the development of a theory, it is often the case that they are simply referred to as axioms in the mathematical discourse, but again, not in the sense that they are true propositions nor as if they were assumptions claimed to be true. For example, in some groups, the operation of multiplication is commutative; in others it is not.

Note that it says "almost every modern mathematical theory starts from a given set of

*non-logical* axioms" and from the discussion above, we know that non-logical axioms are theory-specific. In other words, unlike logical axioms, non-logical axioms are

*not valid in all contexts.* Clearly, something that is not true "in any possible universe, under any possible interpretation and with any assignment of values" cannot be claimed to be objectively "true".

Hopefully it is becoming clearer now why I am saying that

**at the level of axioms, mathematics cannot be said to be true.**
**Edited by Skeptic, 20 June 2005 - 05:09 AM.**