Are all mathematical laws tautologies?

Question

I'm reading a book by Tarski and he says: "Every scientific theory is a system of sentences which are accepted as true and which may be called laws" then after some pages he gives somme laws:

for any p and q:

1) if p, then p

2) if p, then q or p

3) if p and q, then p

etc

All of these are tautologies, but are "laws" in general tautologies? Could I say law = tautology? When I read, for example, Leibniz's law, symmetry law, associative law etc should I see them all as tautologies?

Answer 1

NO.

Tautologies are formulas that are universally valid, i.e. true in every interpretation.

Mathematical axioms and theorems are not so.

Consider e.g. the arithmetical theorem: $\forall n (n \ge 0)$. This "law" is true for natural numbers, but it is false for integers, rationals, etc.

Thus, it cannot be true in every interpretation.

The same for the Associative property:

Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups and categories) explicitly require their binary operations to be associative.

However, many important and interesting operations are non-associative; some examples include subtraction, exponentiation, and the vector cross product.

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"Every scientific theory is a system of sentences which are accepted as true and which may be called laws."

We have axioms, i.e. sentences assumed as true: they are the starting points of the theory.

And we have theorems, i.e. sentences deduced from axioms by way of valid arguments.

We may call axioms and theorems: "the laws of the theory".