In Mysticism and Logic, Russell says that :
"Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."
I understand in which sense the theorems of a mathematical system can be considered as conditionally true statements: they are true if axioms and definitions are assumed as hypothesis.
But, is Russell's claim correct regarding axioms?
If Russell's claim were correct, then the axioms of a mathematical system could not be considered as true in this system, for, apparently, they do not play the role of consequent in any conditional statement belonging to this system.
Does not this consequence contradict the standard thesis according to which the axioms of a deductive system are true in this system?
Remark. can one say that axiom A is true because : A --> A is true?
But in a mathematical system, an axiom has not the form : A --> A, but the categorical form : A.