[nyc111]

"If the axioms are true, and the subsequent reasoning is sound, then the conclusion is irrefutable. What we now have is a proof: something we can know for sure."

... if the axioms are true. We still don't know for sure absolutely.

"The idea of self-evident truths goes all the way back to Euclid’s “Elements” (ca. 300 B.C.), which depends on a handful of axioms—things that must be granted true at the outset, such as that one can draw a straight line between any two points on a plane."

Strictly speaking, Euclid does not state axioms. He starts with 23 Definitions, 5 Postulates and 5 Common Notions. Drawing a straight line from any point to any point is stated as Postulate 1.

I realize this is a newspaper article.

[eru]

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.)

[woopsn]

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.

[zzo38computer]

There is philosophy of mathematics. The below are my ideas of my philosophy of mathematics.

The axioms are true within the system that has those axioms. Therefore, the theorems that result from those axioms and rules, are also true in any system that has those axioms and rules.

It does not make it 'true' in an absolute sense (since it is 'true' within the system and any others (including the Platonic realism, and others too) that includes them), but absolute Truth is inexpressible (this is my conclusion from my study of mathematics and of philosophy of mathematics, but it applies to other stuff too).

However, you should avoid to be confused by such a thing, since some people apparently are. For example, just because some specific sequence of symbols has some use in some system, does not mean that it is the same in a different system (even if they can be mapped to them, which they often can be). Furthermore, even if "X OR NOT X" is true (regardless of what X is, as long as it is well-formed), that does not mean that either "X" must be true or "NOT X" must be true. And, just because values can be assigned to the symbols of classical (or other kind of) logic, does not make it necessary to assign those or any other values.

(Principia Discordia also has some things about "Psycho-Metaphysics".)

[js8]

> 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).

[vouaobrasil]

> 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.

[Scarblac]

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.

[mathgradthrow]

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

[nyc111]

"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.

[voidhorse]

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)

[eru]

Even if you are a platonist, the conditional nature of axioms is still useful.

Eg look at group theory. It basically says, if you have a set of elements and some operation on this set that satisfies certain criteria (= the axioms of group theory), then you can draw all these conclusions.

I don't think anyone ever argued about whether the axioms of group theory are 'true' in the abstract, because everyone recognises that it depends on your application. Eg they are satisfied for the operations you can do on a Rubik's cube (especially a Platonic ideal of a Rubik's cube), but they aren't true for moves in Sokoban (even a Platonic ideal of Sokoban) nor Tetris.

More famously, look at Euclidean geometry: even setting aside curved spacetime of general relativity, even the ancients knew that Euclidean geometry isn't 'true' on the surface of a globe.

[redczar]

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.

[eru]

Though to be honest, by modern standards Euclid sneakily uses some extra assumptions in his proofs that you actually need to add as axioms.

See https://math.stackexchange.com/q/1901133/1051561 for some examples.

[bheadmaster]

> 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.

[johnp314]

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

Well we only know that if we 'think'.

[bravesoul2]

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

[bowsamic]

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

[thesuitonym]

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.