First of all: there is a philosophy of mathematics tag in this SE, so this could be the proper site for my question. I do not know how to evaluate this.

When we assume an axiom to prove something, why can this be done? And by this I mean exactly the assumption of an axiom. What is my ``axiom´´ that allows me to assume axioms? Would that be an axiom of mathematics as any other or would that be a principle, as in the Principle of Identity, which I interprete as an axiom of logic and different than the axioms of a mathematical theory. But then my questions start: would not that make the whole use of logic paradoxical? Why would an axiom of logic be valid per se but an axiom of mathematics not? Where the differences from axioms of logic and axioms and mathematics start?

I know there are a lot of questions and this whole discussion may be too vague, but, come on, I am under the tag of philosophy. If there is a good answer about the title question, I would accept it. Thanks for reading.


closed as too broad by Lord Shark the Unknown, Noah Schweber, Did, Shailesh, Lee David Chung Lin Jan 28 at 4:58

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I would really like the downvoters to explain the downvotes.., $\endgroup$ – creepyrodent Jan 27 at 6:55
  • $\begingroup$ "What is my ``axiom´´ that allows me to assume axioms? " No axioms at all... we live (I hope) in a free world and we are free to assume whatever we want. $\endgroup$ – Mauro ALLEGRANZA Jan 27 at 8:50
  • $\begingroup$ I've downvoted and voted to close since I think that, while interesting, this question isn't really appropriate for MSE. Philosophy.stackexchange might be a better bet, but even there I'm not sure - this question is very open-ended, and I can't imagine what a truly satisfying answer would be. Note that I'm interpreting this question as being substantially more than just, "What's the difference between logical axioms (e.g. law of excluded middle) and mathematical ones (e.g. commutativity of $+$)?" That narrower question would I think be appropriate here, but I also think it's not your question. $\endgroup$ – Noah Schweber Jan 27 at 10:49
  • $\begingroup$ Thanks, Noah. I think I agree with you. $\endgroup$ – creepyrodent Feb 1 at 3:23

"What is my "axiom" that allows me to assume axioms?"

"When we assume an axiom to prove something, why can this be done?"

No "super-axiom" is needed here.

Logic and reasoning are "linguistic techniques" that humans use everywhere (in mathematics, science, philosophy, legal) to argument, i.e. to prove statements from other statements.

In order to do so, some starting points are needed : they are the initial assumptions of the argument.

In the context of a mathematical and scientific theory, they are called axioms.

In addition, we need also "rules of the game" of argumenting : this is logic, made of common principles to all topics and arguments.

The logic principles are rules, like e.g. modus ponens, and logic laws, like e.g. the laws of identity : $x=x$.

In conclusion :

Why is it possible to use logic?

Because it is not possible not to use it : logic is part of human language and activities.

A good overview can be found in SEP's entries about the Foundations of Mathematics.

More details, into : Stewart Shapiro (editor), The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford UP (2005).

  • $\begingroup$ Why is it part of human language and activities? sorry, this is not clear to me. But I am reading your links. $\endgroup$ – creepyrodent Feb 1 at 3:26
  • $\begingroup$ Your answer looks like a good answer to my bad question. But I am not skilled enough to understand your answer. What should I do? Accept it? $\endgroup$ – creepyrodent Feb 2 at 23:27
  • $\begingroup$ since we are talking about policy, is it better for the site if I delete my post? $\endgroup$ – creepyrodent Feb 4 at 8:17
  • $\begingroup$ ^ this is not a rhetoric question, I am asking because I want to learn how to use better this site and for the sake of the site. I don´t understand in the rules if this is the case... $\endgroup$ – creepyrodent Feb 21 at 11:43

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