When reading the Axiom page on Wikipedia, I encountered this sentence at the end of the lead:
Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be "true" is an open question[citation needed] in the philosophy of mathematics.[5]
I would appreciate if minds more knowledgeable than me in this direction, could elaborate what the above sentence means? (especially the part in bold).
Being very macroscopic... Doing mathematics consists in:
A proof consists in doing this deduction the right way. I mean ensuring that the set of axioms is used properly and the same for the deduction rules.
The question of understanding if the axioms are the right ones is not a Mathematical question.
This opens by the way an interesting question... What means being the right ones? This question is closely related to the question are those axioms true or not that you raised.
Well, I read it as questioning whether mathematics is really just a human construct, and questioning as to whether or not mathematics is "real" and "meaningful", which I'll admit is vague wording. But I think what the line is questioning is whether or not mathematics is in existence, so to speak, outside of our minds. People argue both ways on this; I personally believe that there is "more" to math than just human scribbles on a piece of paper.
It also somewhat questions whether or not our starting "principles" if you will, are correct, which is another fair question that is more difficult. However, I think this one depends directly on the first question. If math isn't "real" then our axioms can (theoretically) be whatever we want them to be. If math is indeed, more than just a human construct, then again, theoretically, there could be a "right" way and a "wrong" way, so then we wonder whether our axioms are correct.
Another thing to consider: how do we define "true"? One could define it as self-consistent, that is, no part of math contradicting itself, which math certainly is. We don't write that 2+2=4 over here and 2+2=5 over there. Or we could define it as something beyond just self-consistent, but having some meaning beyond writing in books and papers. It is the latter that this sentence is questioning, I believe. This then brings us back to the first paragraph of this answer.
I hope that makes sense; if it doesn't, let me know and I'll try to clarify.
I think the debate is does the truth of whether or not the axioms of math are absolutely "true" matters as much as how useful they are in describing the world and predicting things. Even if the axioms of math were proven to be false, they can still be useful as approximations to reality, just as Newtonian Gravity is a useful approximation to General Relativity. As long as the axioms of math are consistent, can be used to model reality (not just Physics), and there is no better system in place, does it really matter if the axioms are not justified? Also, I forgot to mention the case in which doing math is pleasing, like pure math, in which case I would argue that it still serves a purpose in that it pleases people. Also, pure math often ends up having applications in the real world, much to the chagrin of pure mathematicians. Hardy, for instance, bragged about how nothing he did had any value. His work, however has been used in both evolutionary biology and quantum physics.
"Open question" in the above statement is not in the sense used in mathematics (e.g. "Collatz conjecture is an open question."), rather it means "open to debate".
I believe the etymology of the word "axiom" leads one in the correct direction. Even though we have axioms precisely because we can, saying that they appear out of the blue is not completely accurate.
I recommend you to take a look at Wigner's essay "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" and Mac Lane's Mathematics: Form and Function.
First of all, just to cover all my bases: an open question is a question that is not yet considered resolved by the academic community.
As for the rest: The "truth" of an axiom is a sticky subject. A famous example is the Axiom of Choice; I won't go into the details of the axiom (Wikipedia can give you a pretty good sketch of it) but the important thing is that it really only talks about infinite sets.
With many axioms, we can say they're "true" because they accurately match what actually happens - for example, consider the axiom "every integer has a successor". We could argue that it's "true" because whenever we have a collection of objects, we can add one more. The problem is, when an axiom only talks about infinite objects, we can't point to any such physical objects.
It seems like in order to say that an axiom is "true", we have to suppose that the things it talks about actually physically exist. This is a wildly problematic supposition, since according to modern developments in set theory there are infinitely many possible mathematical "universes", all of which behave quite differently but are indistinguishable when it comes to the everyday finite sets we actually encounter. Which one are we in? If they all have the same consequences in the finite world, why does it even matter which one we're in?
Which means that the best way to classify an axiom as "true" or "false" is probably just "does it seem sensible?" The problem is, the Axiom of Choice is something that most people, on a first reading, would consider obviously true; but it has truly bizarre consequences, like the Banach-Tarski paradox, which the average person on the street would say is obviously false. So while the Axiom of Choice "makes sense" and so should be "true", its consequences don't "make sense" and so should be "false".
The solution to this whole business that has been generally agreed-upon among modern mathematicians is to just ignore it - who cares if an axiom is "true"? Instead, we concentrate on "consistent" or "inconsistent", which are qualities we can precisely define, or we just look at the interesting consequences of the axiom.
From your comments on other answers I think you are too focused on one type of knowledge. In mathematics (and deductive logic in general) when we take axioms like:
And deduce a conclusion, in this case :
Of course these axioms are very questionable, maybe Max is a human or maybe some dogs have no tails. The point is that mathematics isn't telling us the "Max has a tail" is in itself true, mathematics is telling us "If you accept these two axioms then the conclusion is true". This is a different form of knowledge but one that is still valuable.
It's true that in some cases these axioms aren't true and we can't know that the conclusion is true. This is like when we develop mathematics for $x$ being a natural number but later we learn that $x$ is a real number. Even so, what we know is the conclusions given the axioms.
What you highlighted is not an independent part of the sentence. The sentence says that philosophers disagree about what it means to say that statements like “0 is a number” or “two plus two is four” are true. There are indeed many competing schools of thought.
You ask what this sentence from the Wikipedia article Axiom means:
Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be "true" is an open question in the philosophy of mathematics.
In the version as of 2016-11-22, this statement occurs at the end of the introductory section, i.e. the end of the fourth paragraph, immediately before the section heading “Etymology”.
Your are particularly interested in this part:
to be "true" is an open question[citation needed]
I shall try to reformulate, rearrange and explain the sentence in question:
In the philosophy of mathematics
I.e. we are not talking about mathematics in the sense of setting up theories and solving problems; we are talking about trying to understand the significance and value of such activities.
There is an open question
There is a particular question which is still open, i.e. does not have a generally accepted answer.
The question is “what does it mean to say that a given mathematical statement is true?”
This has at least four layers: (we ask) what it means to assert the truth of a statement.
“Is it meaningful to say that a given mathematical statement is true?”
In particular, this question applies to axioms,
since axioms are mathematical statements, given a special status but of the same nature as other statements like theorems, conjectures and fallacies.
We are asked what (if anything) it means to say that an axiom (like “0 is a natural number” or “adding one to a natural yields another”) or a theorem (like “$2 + 2 = 4$”) is true. We may also ask what is the difference between saying “$2 + 2 = 4$” and saying ‘the statement “$2 + 2 = 4$” is true’.
This is different from the way we normally approach mathematics, where we may (for the above examples) start with some perceptions of collections we can count, spot patterns (a collection may be empty, we can add things to it), find a formalism that seems to correspond to our experience, and then concentrate largely on the formalism. In other cases our starting point may be perceptions of mathematics itself.
To answer this question, we have to expand our scope to include more than the formalism, and ask what it means to talk about things “corresponding to our experience” or being true in some other sense.
Many views of what mathematics is about and what its statements mean can be found in the section on ‘Contemporary schools of thought’ in the Wikipedia article on the philosophy of mathematics. They include: