I don't like how they give and example of a geometric axiom and then give a number theory result. This makes it seem that the number theory result follows from the geometric axions.

> I don't like how they give and example of a geometric axiom and then give a number theory result. This makes it seem that the number theory result follows from the geometric axions.

This isn't some weird gloss on their part; there are number-theory results in Euclid's Elements, even if you and I would nowadays think of them as belonging to a different discipline.

Well, in modern mathematics we don't presume that the axioms are 'true' in any meaningful sense. All of modern mathematics is conditional.

So _if_ you find a system where the axioms of eg group theory hold, you can apply the findings of group theory. That doesn't make any statement about whether the axioms of group theory are 'true' in any absolute sense.

They hold well enough for eg the Rubik's cube, that you can use them there. But that's just a statement about a particular mental model we have of the Rubik's cube, and it only captures certain aspects of that toy, but not others. (Eg the model doesn't tell you what happens when you take the cube apart or hit it with a hammer or drop it from a height. It only tells you some properties of chaining together 'normal' moves.)

When you accept axioms randomly and reject even logical semantics, you wind up working with something like the so-called rado graph of Erdos. Call the first set of assumptions/structure meaningless and choose a new one, also having no grounding in reason - it turns out that would make no difference. "Almost certainly" you wind up with the same structure (the rado graph) in any case.

https://en.wikipedia.org/wiki/Rado_graph

So you can reject meaning (identifying the axioms with simple labels 1, 2, 3, ...) and truth (interpreting logical relations arbitrarily), but in doing so you ironically restrict mathematics to the study of a single highly constrained system.

I only disagree somewhat though - it is all contingent. We do say something like

    IF { group axioms } THEN { group theory }

and introduce even more contingencies in the application of theory. In a real problem domain these assumptions may not hold completely - but to that extent they are false, not having a meaningless truth value. Crucially, we then search for the right set of assumptions, and the conviction of modern science that there is a right set of assumptions is "effective" at least.

In the previous century there was an attempt to identify mathematics with formalism, but it defeated itself. We need to see truth in these systems - not individual axioms per se, but in systems composed from them. This is justified by the body of mathematics itself, to the non-formalist, the richness of which (by and large) comes from discoveries predating that movement.

> All of modern mathematics is conditional.

True, and there is some variation in the axioms. But for the record, pretty much all systems keep the logical rules of AND elimination and OR introduction for example: if A and B are true, then A is true. If A is true, then A or B is true. However, the law of excluded middle is sometimes excluded for constructivist reasons.

> All of modern mathematics is conditional.

I am not sure this is true for "all" mathematics. You're still using some metalogical axioms that are always true. In particular, laws of untyped lambda calculus (or any other model of universal computation) are "axioms" that you consider unquestionably true (just like you can consider unquestionable that objective shared mathematical reality exists).

> I am not sure this is true for "all" mathematics. You're still using some metalogical axioms that are always true. In particular, laws of untyped lambda calculus (or any other model of universal computation) are "axioms" that you consider unquestionably true (just like you can consider unquestionable that objective shared mathematical reality exists).

Although, if you informally asked me if the rules of logic were true, then I would say that of course they are, if you asked me formally I think I would say that they are not unquestionably true, only unquestionable if you want to do classical mathematics. If you're willing to grant basic rules of logic, then certain consequences follow. If you're not, then you're not doing classical mathematics, although you might still be doing interesting mathematics—for example, if you decide not to accept the law of the excluded middle.

Which axioms you take as true is a free choice. They aren't true or false by themselves.

What's irrefutably proven is that if you take this particular set of axioms, then these conclusions hold.

But you are free to choose other axioms, that will lead to other conclusions.

Some statements people use as axioms are equivalent (you can include one, and then derive the other and vice versa). Some are contradictory: you can include the axiom of choice or the axiom of determinacy, but not both as that will lead to a contradiction and thus an unsound system.

In a sense it's a matter of taste, mathematicians choose a set of axioms that leads to interesting things to think about.

you cannot prove the consistency of a system of proof within that system, ie at all.

"What's irrefutably proven is that if you take this particular set of axioms, then these conclusions hold."

This is what I tried to say in my comment. It's the author who talks about the truth of the axioms. I'm objecting to his claim that we end up with "something we can know for sure". No. Your truth depends on your assumptions.

We do end up with something we know for sure: the whole proposition "if we take these axioms as true, then these statements hold."

Axioms and postulates as Euclid uses that term are the same thing. In modern times we have gotten rid of the idea of statements that are self evidently true. So we don’t use the term postulate. We’d call Euclid’s postulates axioms today.

> We still don't know for sure absolutely.

If we go down the skeptics' route, we can't know anything absolutely (except that we exist yada yada). But we still have to function in the real world, so we assume the most consistent observations will never change. From those observations, we extract the axioms, on which we build the tower of conclusions.

We can know much more than that. Read Hegels Science of Logic

> we can't know anything absolutely (except that we exist yada yada).

Well we only know that if we 'think'.

Mathematics diverges from reality with infinites... That is where the trouble with axioms starts.

I dare you to formulate a version of quantum mechanics without it.

Quantum mechanics makes very good predictions.

That talks more to how humans think than to what nature is. It is almost a philosophical debate but I feel gut instinct infinities are not really in nature.

Sure you can draw a good circle and say the ratio is Pi and that number encodes infinite information (albeit at high entropy) but to me that Pi is an algorithm for computing better and better approximations to a perfect circle, an object absent from nature.

I don't know QM but I suspect it is the same. It is our mental model. Using infinities is our concept. To me infinite things are algorithms (i.e. as X tends to infinity... Tends being an important word)

Infinity is fully real, in fact it’s the very nature of the finite to pass over into infinity

Not a mathematician but my understanding is that the axioms aren't some universal truth that we discovered, but rather the foundation of our language of mathematics.

As others have noted, axioms are more akin to a line in the sand—they are either so "obvious" as to be true or constitute such a useful and economic basis for further development that we decide to use them.

Mathematics is more about coordinating human observation and discussion than it is about capital T truth (unless you are a Platonist)