What makes mathematics certain knowledge?

Why can mathematical knowledge not be doubted? What gives it the characteristic of certainty? No one has ever coherently answered this question, as far as I am aware. Everyone just seems to take for granted its certainty like they take for granted causality's existence, for instance. Please explain, in full detail, why it is certain. Preferably without utilising formulas or proofs. To wit, referencing formulas or proofs is fine but they cannot be used to elucidate this matter for uneducated individuals.

The only way we can be certain of a particular conjecture is through a mathematical proof, so I'm afraid there's no real way to convey this through words.

Well, yes, language is limited, in a way. You can't use words to prove something beyond all doubt. When you make an argument through words, you're using logic. While logic is all fine and good, and is an essential skill to have, you can't prove something through logic. There are things that go against logic. You can show that something is highly likely (or unlikely), but that's about the extent of it.

Similarly, you can't prove science through words either. That takes years of experimenting, etc.

Because everything in mathematics is based on something else. We can use basic mathematical axioms to prove simple theorems, and simple theorems to prove complicated theorems, and therefore if the axioms are always true then the rest of mathematics is always true as well.

The proofs themselves are infallible, because they are made up of many smaller steps which, assuming the axioms are true, are also infallible. Therefore, if mathematics can ever be wrong, it will be because the assumptions which underpin it, and the whole of logic and philosophy, are also wrong.

You are utterly incorrect. Deductive logic proves something beyond a doubt - it is inescapably true. The second emboldened part is a loose definition of inductive logic. All mathematics may be based on logic - a branch of philosophy. However, I am willing to concede that perhaps I too have assumed logic's certitude as mathematicians assert the certitude of mathematics.

This is an uncomfortable dilemma that has no easy solution. You cannot readily assume a particular method gives you objective proof if you cannot prove it. I cannot conceive of a way in which an infinite regress can be avoided.

Using logic to argue that logic is possibly incorrect seems very odd to me.

Sorry, I think you've misunderstood me here. What I was trying to say was that you can't prove something in mathematics with logic through words, not in general.

You must realize though, that mathematics differs from science and philosophy in the sense that it's a human construct, rather than a study of what already exists. You can conclusively prove conjectures within the realm of this human construct.

Philosophy and science are human constructs too. Perhaps they are internally valid or perhaps not.

Mathematics is a language used to express rather complicated concepts, you'd find it difficult expressing numbers without it. These concepts are certain, mathematics has no affect on certainty just like the english language does not affect what a car is for example, the word just expresses the concept of a car.

Why can mathematical knowledge not be doubted? What gives it the characteristic of certainty? No one has ever coherently answered this question, as far as I am aware. Everyone just seems to take for granted its certainty like they take for granted causality's existence, for instance. Please explain, in full detail, why it is certain. Preferably without utilising formulas or proofs. To wit, referencing formulas or proofs is fine but they cannot be used to elucidate this matter for uneducated individuals.

Why can mathematical knowledge not be doubted? What gives it the characteristic of certainty? No one has ever coherently answered this question, as far as I am aware. Everyone just seems to take for granted its certainty like they take for granted causality's existence, for instance. Please explain, in full detail, why it is certain. Preferably without utilising formulas or proofs. To wit, referencing formulas or proofs is fine but they cannot be used to elucidate this matter for uneducated individuals.