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Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers.

The kid asks: why?

Well, it's an axiom. It's called commutativity (which is not even true for most groups).

How do I "prove" the axioms?

I can say, look, there are $3$ pebbles in my right hand and $4$ pebbles in my left hand. It's pretty intuitive that the total is $7$ whether I added the left hand first or the right hand first.

Well, I answer that on any exam and I'll get an F for sure.

There is something about axioms. They can't be proven and yet they are more true than conjectures or even theorems.

In what sense are axioms true then?

Is this just intuition? We simply define natural numbers as things that fit these axioms. If it's not true then well, they're not natural numbers. That may make sense. What do mathematicians think? Is the fact that the number of pebbles in my hand follows the rules of natural numbers "science" instead of "math"? Looks like it's more obvious than that.

It looks to me, truth for axioms, theorems, and science are all truth in a different sense, isn't it? We use just one word to describe them, true. I feel like I am missing something here.

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    $\begingroup$ You're right that axioms cannot be proven - they are propositions that we assume are true. Commutativity of addition of natural numbers is not an axiom. It is proved from the definition of addition, see en.wikipedia.org/wiki/…. $\endgroup$
    – in_mathematica_we_trust
    Apr 2, 2012 at 8:26
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    $\begingroup$ In every rigorous formulation of the natural numbers I've seen, $A+B=B+A$ is not an axiom. It can be formally proven, i.e. from the Peano axioms. $\endgroup$
    – Alex Becker
    Apr 2, 2012 at 8:26
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    $\begingroup$ I like to think of axioms as definitions. If you point to a car and say "that's a car", it would be absurd for someone to reply "you can't prove what a car is". The same goes for, say Zermelo-Franklin set theory. You define sets using axioms, and it is understood when you derive things about them that sets are objects satisfying these axioms. This may seem trivial, but it has been the center of serious debates in the past. For example, most mathematicians define set in a way which excludes the "set of all sets". $\endgroup$
    – Alex Becker
    Apr 2, 2012 at 8:29
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    $\begingroup$ @Jim Thio: in the modern viewpoint, any statement can be taken as an axiom. So the statement of commutativity might be assumed as an axiom in some settings, proved as a conclusion in others, and neither assumed nor proved in others. $\endgroup$
    – Carl Mummert
    Apr 2, 2012 at 11:56
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    $\begingroup$ I don't believe that you need to think of axioms as true. Instead, you can just take axioms as initial points in a system. Or to make a comparison with say Sudoku (pick a logical game) the axioms come as something like the initial numerals. I'd just point this out to the child, and say that the axioms don't have to qualify as true. This perspective comes close to the "axioms as definitions" perspective. The problem with the "axioms as definitions" perspective lies in that were axioms definitions, a single axiom would define. But, sets of axioms define, so axioms aren't definitions. $\endgroup$
    – Doug Spoonwood
    Apr 2, 2012 at 13:29

7 Answers 7

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You only need to "prove" an axiom when using it to model a real-world problem. In general, mathematicians just say "these are my assumptions (axioms), this is what I can prove with them" - they often don't care whether it models a real-world problem or not.

When using math to model real-world problems, it's up to you to show that the axioms actually hold. The idea is that, if the axioms are true for the real-world problem, and all the logical steps taken are sound, then the conclusions (theorems etc.) should also be true in your real-world problem.

In your case, I think your example is actually a convincing "proof" that your axiom (commutativity of addition over natural numbers) holds for your real-world problem of counting stones: if I pick up any number of stones in my left- and right-hands, it doesn't matter whether I count the left or right first, I'll get the same result either way. You can verify this experimentally, or use your intuition. As long as you agree that the axioms of the model fit your problem, you should agree with the conclusions as well (assuming you agree with the proofs, of course).

Of course, this is not a proof of the axioms, and it's entirely possible for someone to disagree. In that case, they don't believe that the natural numbers are valid model for counting stones, and they'll have to look for a different model instead.

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    $\begingroup$ Good answer. I would also add that there are many cases where mathematical theories are based on axioms that are known to be false in the real world - but this doesn't mean they aren't "mathematically true" - only that they aren't relevant to our physical world. $\endgroup$
    – Ophir Yoktan
    Apr 2, 2012 at 17:35
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    $\begingroup$ @Ophir: Right; the Banach-Tarski paradox (which relies on the axiom of choice) comes immediately to mind. $\endgroup$
    – BlueRaja - Danny Pflughoeft
    Apr 2, 2012 at 17:55
  • $\begingroup$ @OphirYoktan I'd think that if you looked hard enough, all axioms will come out false in the real world for something. $\endgroup$
    – Doug Spoonwood
    Apr 2, 2012 at 20:06
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    $\begingroup$ The other answers are good too (and hence +1). But it seems to be the best. Math theorems live in "Math land". It's true in Math land. For it to be true in "material land" you need to show that the rules in Math Lands are isomorphic to the rules in real land. Looks like the number of pebles on my hand are isomorphic to natural number. So far it works fine in computing the number of pebles. So it sort of work. That part is science, and not Math. Somehow I think it's more obvious than science. $\endgroup$
    – user4951
    Apr 16, 2012 at 5:28
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    $\begingroup$ In this particular example (as with so many types of objects that we "count"), the commutativity itself derives from an assumption of some kind of homogeneity, so that, when we evaluate 3+4 and 4+3 we get 7 pebbles in each case. In more sophisticated terms, although free groups are not, in general, abelian, the free group on one letter IS. We actually lose this commutativity when we count "apples and oranges", see? $\endgroup$
    – David Wheeler
    Jun 11, 2012 at 18:03
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The problem of what is means for something to be "true" is a general problem in philosophy which has received a lot of attention, so it is impossible for anyone to give a short and complete answer here. The Stanford Encyclopedia has a nice article on truth. Mathematics is a useful test case for philosophers so a lot has been written on mathematical truth.

There is a separate problem that the word "true" is used to mean several things in mathematics: it can mean just "true", or it can mean "true in a particular structure". For example, the latter meaning is intended when we say that the axiom of commutativity is true in some groups and not in others. The notion of truth in a structure is well studied in mathematical logic. But I think the question above is about plain "truth" not about "truth in a structure".

The easiest way to define what plain "truth" means is to believe that there is some "real" mathematical structure, consisting of mathematical objects that actually exist. This viewpoint is called mathematical Platonism or mathematical realism. Then a statement is "true" if it is true when interpreted as a statement about these real mathematical objects. For example, from this viewpoint the statement "Addition of natural numbers is commutative" is true because the actual addition operation on the actual set of natural numbers is commutative.

There are other "anti-realist" theories of truth that do not presuppose that there are independently existing mathematical objects that can be used to test the truth of a statement. (One problem with the realist versions is that it is far from clear how we would be able to tell whether mathematical objects have various properties using our five senses.) Some go as far as replacing truth with provability; for example, this is one way to understand the motivations for intuitionism. But most mathematicians maintain that there is a difference between truth and provability.

There is a separate issue that most of the time the word "true" is used in mathematical proofs, it is just a turn of phrase that could be omitted. For example, instead of saying "We know that $A \to B$ is true, and we have proved $A$, so $B$ is true", we can say "We have assumed $A \to B$, and we have proved $A$, so we may conclude $B$". This shows up when the proofs are formalized: formal proofs in most theories (e.g. the theory of groups, ZFC set theory) do not have any way to refer to "truth", they simply manipulate formulas. The idea, of course, is that if the assumptions are true then the conclusion is true. But the formal proof itself will not make reference to plain "truth".

The question goes on to ask how we would know (in a realist theory, for example) that addition of natural numbers is commutative. Someone could say "you prove it from other postulates" but then the problem would be how to know that those postulates are true. In the end, the question is how to know that any postulate about the actual natural numbers is true. This is a major issue for mathematical realism, as I mentioned above. The most common answer is that humans have some form of insight which allows us to determine the truth of some (but not all) mathematical propositions directly, without having a formal proof of those propositions. The commutativity of natural number addition is one of those propositions: by thinking about addition and natural numbers, we are drawn to conclude that the addition is commutative. In the end this is how we justify all postulates in geometry, set theory, arithmetic, etc. The realist positions is that although we cannot prove them formally, we can come to believe they are true by thinking about the objects they describe.

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Tarski's undefinability theorem states (roughly) that one cannot define the truth of arithmetic statements entirely within the confines of the language of formal arithmetic. This marks a sharp boundary between two things that one might naively have called truth: on the one hand, the formal proof systems we use as mathematicians; on the other, truth as we understand it in the world.

The latter, of course, takes some work to understand- and would-be logicians are often liable to tie themselves in knots when they first try to get their head around Tarski's elegant summary that:

'Snow is white' is true if and only if snow is white

In short, though, the idea is a simple one- a proposition, including an axiom, gets its truth all and only from correspondence with the world.

The trouble being in practice that we do not have an a priori picture of the world to hand, so instead we build correspondences with abstract set theoretic constructions, built to keep step with the formal systems themselves, called models (check out these notes by marker for a solid introduction). The philosophical rationale here is that sets are a good way of describing generalised objects and their constituent parts, and therefore are a fair description of possible worlds.

That there exist models of your arithmetic axioms, both in set theory (see eg. Von Neumann Ordinals) and out in the world are the closest thing we can come to proving the truth of arithmetical statements, since truth and proof are seperate entities...

Addendum: You may also enjoy this account of Tarskian semantic truth as scientific truth, an all time favourite of mine.

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    $\begingroup$ We can also define black as the color of the snow and it'll be true too. One thing for sure is the color of the snow is the same with the color of cloud and milk and will produce the same light with the same wavelength. $\endgroup$
    – user4951
    Apr 16, 2012 at 5:30
  • $\begingroup$ Truth and Proof are separate entities only to formulations with no (underlying) meaning. ie purely syntactic ones. If you think this is a good educational process then teach also Goedel's theorem that in that conext will prove that what you teach is meaningless. $\endgroup$
    – Nikos M.
    May 28, 2014 at 0:54
  • $\begingroup$ PS. Goedel managed to prove the theorem using underlying meaning interpretations, not purely syntactic manipulation $\endgroup$
    – Nikos M.
    May 28, 2014 at 0:56
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Initially, the study of number is empirical.

As a child, we observe that 5 objects + 2 objects is the same as 2 objects + 5 objects experimentally, and come to hold this as a truth.

There is some relevant child psychology here which I am not familiar enough with to comment on. I believe it is known that children are not born with a sense of commutativity, but develop it quite early.

So, we can say, for a child, commutativity is axiomatic - we observe it so often, that we come to hold it as a basic intuitive truth of living.

Search result: example of relevant child psych research

In the study of philosophical mathematics, we have decided that a much simpler set of axioms is possible. The Peano axioms do not include commutativity as an axiom. However, commutativity follows from the axioms we have decided.

Axioms here are considered to be the "basic/minimal truths of mathematics".

But, in reality, we have chosen these axioms because they fit well with a mathematics that we are interested to study - linking clearly to our earlier intuitive ideas of number.

One of those intuitive ideals is commutativity - the Peano axioms would have been rejected quickly if they were not compatible with commutativity of natural numbers. Commutativity need not be an axiom any more, because the idea is now included within other, simpler axioms.

The axioms of a system can evolve based on the requirements of the system, as they have here. We start with the childish intuitive axiom of commutativity, developing into the 19th Century Peano axioms, and the 20th Century Zermelo-Frankael axioms.

The axioms are "true" in the sense that they explicitly define a mathematical model that fits very well with our understanding of the reality of numbers.

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I like axioms that only formalize what we intuitively believe to be true. Say I intuitively believe that if I combine two groups together, I will end up with a collection that has the same size regardless of which group came first. To proceed with rigorous mathematics, I might make this one axiom of many that define a formal algebraic structure consistent with my understanding of the size of groups of things in the real world. I don't think of the axiom as either true or false in any universal sense; I think of it as consistent with my intuition for the real world and as true by definition when doing formal mathematics. (As noted, commutativity is not usually an axiom for $\mathbb{N}$.)

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  • $\begingroup$ Good answer, deserves more upvotes. $\endgroup$
    – goblin GONE
    Dec 15, 2014 at 5:48
  • $\begingroup$ @alex.jordan So can we say that because we have build our mathematics with axioms derived from nature (inuition) mathematics describe real world so well? Now the question is can they describe it always? Would there be a case where according to mathematics we should get a result A but we get a result B? $\endgroup$
    – user599310
    Aug 5, 2020 at 17:01
  • $\begingroup$ @user599310 Hmm, maybe the Banach-Tarski paradox is of interest. Of course, when "according to mathematics we should get a result A but we get a result B" it tends to make you rethink your axioms. So it's a bit of a self-correcting process. $\endgroup$
    – 2'5 9'2
    Aug 5, 2020 at 17:06
  • $\begingroup$ @alex.jordan It seems so counterintuitive. We can model nature based on maths (irrespective that we can do maths w/o care about nature/physics) and then it is like checking if the axioms of maths are built into nature. Then Banach-Tarski comes up. Maybe our world is one of the worlds that fit very well. I think the fact that we have that feeling that maths are inherent in nature is that whatever we imagine even if it seems impossible according to laws of physics maths can describe it. $\endgroup$
    – user599310
    Aug 5, 2020 at 19:16
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Actually, commutativity of addition applied to the real world is just a special case to an much more general identity: If I give you two things, and give someone else the same two things in opposite order, you both receive the same things.

For example, I give you an apple, then a pear, and I give the other person first a pear, then an apple, I've given you both the same two things: An apple and a pear.

In the same way, if I give you three apples, and then five apples, and I give the other person five apples, and then three apples, I've given both of you the same, namely eight apples. And therefore $3+5=5+3$.

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Say I am explaining to a kid, 𝐴+𝐵 is the same as 𝐵+𝐴 for natural numbers. The kid asks: why?

"If I have 3 apples in one hand and 2 apples in another, how many apples do I have?"

"5."

"Does it matter which hand the apples are in?"

"No."

"That's why."

Well, I answer that on any exam and I'll get an F for sure.

I'd give you an A+. You proved it to me.

They can't be proven

You can prove assertions about axioms from other axioms.

yet they are more true than conjectures

Axioms are assumed true. "Conjectures" are unknown; by definition they lack proof from said axioms.

or even theorems

Thereoms are just as true as the axioms (in theory).

Axioms are assumed to be true. Thereoms depend on axioms and a proof. Thereoms, then, are lengthier than axioms. If you trust your axioms because they've worked over and over, you could trust your axioms more than a particular proof (particularly if your proof is long). (And now you're using meta-axioms to judge axioms more trustworthy than a theorem.)

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