How is a system of axioms different from a system of beliefs?

Question

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?

Answer 1

To paraphrase Robert Mastragostino's comment, a system of axioms doesn't make any assertions that you can accept or reject as true or false; it only specifies the rules of a certain kind of game to play.

It's worth clarifying that a modern mathematician's attitude towards mathematical words is very different from that of a non-mathematician's attitude towards ordinary words (and possibly also very different from a classical mathematician's attitude towards mathematical words). A mathematical word with a precise definition means precisely what it was defined to mean. It's not possible to claim that such a definition is wrong; at best, you can only claim that a definition doesn't capture what it was intended to capture.

Thus a modern interpretation of, say, Euclid's axioms is that they describe the rules of a certain kind of game. Some of the pieces that we play with are called points, some of the pieces are called lines, and so forth, and the pieces obey certain rules. Euclid's axioms are not, from this point of view, asserting anything about the geometry of the world in which we actually live, so one can't accept or reject them on that basis. One can, at best, claim that they don't capture the geometry of the world in which we actually live. But people play unrealistic games all the time.

I think this is an important point which is not communicated well to non-mathematicians about how mathematics works. For a non-mathematician it is easy to say things like "but $i$ can't possibly be a number" or "but $\infty$ can't possibly be a number," and to a mathematician what those statements actually mean is that $i$ and $\infty$ aren't parts of the game Real Numbers, but there are all sorts of other wonderful games we can play using these new pieces, like Complex Numbers and Projective Geometry...

I want to emphasize that I am not using the word "game" in support of a purely formalist viewpoint on mathematics, but I think some formalism is an appropriate answer to this question as a way of clarifying what exactly it is that a mathematical axiom is asserting. Some people use the word "game" in this context to emphasize that mathematics is "meaningless". The word "meaningless" here has to be interpreted carefully; it is not meant in the colloquial sense (or at least I would not mean it this way). It means that the syntax of mathematics can be separated from its semantics, and that it is often less confusing to do so. But anyone who believes that games are meaningless in the colloquial sense has clearly never played a game...

Answer 2

The role of axioms is to describe a mathematical universe. Some settings, some objects. Axioms are there only to tell us what we know on such universe.

Indeed we have to believe that ZFC is consistent (or assume an even stronger theory, and believe that one is consistent, or assume... wait, I'm getting recursive here). But the role of ZFC is just to tell us how sets are behaved in a certain mathematical settings.

The grand beauty of mathematics is that we are able to extract so much merely from these rules which describe what properties sets should have.

Whether or not you should accept an axiomatic theory or not is up to you. The usual test is to see whether or not the properties described by the axioms make sense and seem to describe the idea behind the object in a reasonable manner.

We want to know that if a set exists, then its power set exists. Therefore the axiom of power set is reasonable. We want to know that two sets are equal if and only if they have the same elements, which is a very very reasonable requirement from sets, membership and equality. Therefore the axiom of extensionality makes sense.

What you should do when you attempt to decide whether or not you accept some axioms is to try and understand the idea these axioms try to formalize. If they convince you that the formalization is "good enough" then you should believe that the axioms are consistent and use them. Otherwise you should look for an alternative.

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Related:

1. I talked about the difference between an idea and its mathematical implementation in this answer of mine which might be relevant to this discussion as well (to some extent).

Answer 3

While I agree that faith is not required to accept the rules of certain game in order to play that game, I am pretty sure that, if the rules of the game, and the consequences of playing by those rules, were entirely arbitrary, a lot of mathematicians wouldn't be able to earn a living by doing mathematics.

For ex. according to some recent stats, mathematician is one of the most employable occupations in the U.S., mostly in finance industry. If that doesn't indicate that the rules have some validity in the real world, I don't know what does?!

Would physicists use mathematics if it were completely incapable to describe what they are theorizing and measuring?

There is evidence that axioms may be true, a lot of evidence.

EDIT: I have found the following article by R.W.Hamming very illuminating: The Unreasonable Effectiveness of Mathematics

Some quotations:

The Postulates of Mathematics Were Not on the Stone Tablets that Moses Brought Down from Mt. Sinai.

also

The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way. For over thirty years I have been making the remark that if you came into my office and showed me a proof that Cauchy's theorem was false I would be very interested, but I believe that in the final analysis we would alter the assumptions until the theorem was true.

Amusingly, he warns in the introduction:

I am well aware that much of what I say, especially about the nature of mathematics, will annoy many mathematicians. My experimental approach is quite foreign to their mentality and preconceived beliefs

Answer 4

One can believe a set of clearly contradictory statements (indeed, people often do). If it can be proven that a set of statements entails a contradiction, though, it certainly cannot be taken as an axiomatic system.

Answer 5

There are similarities about how people obtain beliefs on different matters. However it is hardly a blind faith.

There are rules that guide mathematicians in choosing axioms. There has always been discussions about whether an axiom is really true or not. For example, not long ago, mathematicians were discussing whether the axiom of choice is reasonable or not. The unexpected consequences of the axiom like the well ordering principle caused many to think it is not true.

Same applies to axioms that are discussed today among set theorists. Set theoretical statements which are independent of the ZFC. There are various views regarding these but they are not based on blind belief. One nice paper to have a look at is Saharon Shelah's Logical Dreams. (This is only one of the views regarding which axioms we should adopt for mathematics, another interesting point of views is the one held by Godel which can be found in his collected works.)

I think a major reason for accepting the consistency of mathematical systems like ZFC is that this statement is refutable (to refute the statement one just needs to come up with a proof of contradiction in ZFC) but no such proof has been found. In a sense, it can be considered to be similar to physics: as long as the theory is describing what we see correctly and doesn't lead to strange things mathematicians will continue to use it. If at some point we notice that it is not so (this happened in the last century in naive Cantorian set theory, see Russell's Paradox) we will fix the axioms to solve those issues.

There has been several discussion on the FOM mailing list that you can read if you are interested.

In short, adoption of axioms for mathematics is not based on "blind faith".

Answer 6

The best example is the Euclidean geometry and non-Eucliedean geometry. The first four postulates are common to these geometries and are rather elegant

1. A straight line segment can be drawn joining any two points
2. Any straight line segment can be extended indefinitely in a straight line
3. Given any straight line segment, a circle can be drawn having the segment as radius and one end point as center.
4. All right angles are congruent.

The fifth one did not share their grace

(5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

This postulate is not used in the first twenty-eight proposition of Euclid's Elements and it is a rather ugly duckling.

Euclid thought of this postulate to be inferior to the other four and certainly Euclid would have find it far preferable to prove this ugly duckling rather than to have to assume it. But he found no proof and therefore adopted it.

Over the centuries untold numbers of people gave untold years of their lives in attempting to prove that the fifth postulate was itself part of the four-postulate geometry.

Saccheri assumed the fifth postulate is opposite (he hoped he will find a contradiction) and worked with that as his fifth postulate and he got the hyperbolic geometry.

Non-Euclidean geometry was discovered in 1823 simultaneously, in one of those inexplicable coincidences by Janos Bolyai and Nikolay Lobachevskiy.

The meanings of "point", "line" and so on can be determined by the sets of theorems in which they occur and there are many geometries that have the first four axioms and they are all internally consistent but only Euclidean geometry is consistent with the external world.

This is from chapter IV, Consistency, Completeness and Geometry of Godel, Escher, Bach: an Eternal Golden Braid by Douglas H. Hofstadter

Answer 7

I would submit the following: first, axioms in mathematics do not say anything about the Natural world, whereas beliefs usually do, I think. Second, for axioms, we know full well they are arbitrary and can be changed. I wouldn't say that we are "convinced" any axioms are "true". We know that axioms are an arbitrary set of givens we have chosen to derive some theorems, and it is part of the game to alter the axioms or consider a new set of axioms to derive other theorems. I think it is fairly different for beliefs where people would tend to be convinced the beliefs are "true"?

Answer 8

Is anyone forcing us to use math always and only for reasoning about the actual world?

Counting seems practical. Certainly we spend lots of time learning about counting because it is known to be useful. Yet, those of us who are not particle physicists, where do we find anything in the actual world that can really be counted, that really has integrity in the manner of the integers (the counting numbers and their negatives)? Perhaps I'm counting watermelons. I'm not a plant biologist. Who am I to say that a watermelon is really one melon and not two melons, if it grows so constricted in the middle that the constriction is only full of rind? Even if no one has a definitive answer whether it is one melon or two melons, we need not conclude that learning to count was a waste of time. We could simply say that counting has many uses, just not this particular use.

Generally mathematical propositions are literally for the sake of argument. An axiom is a structure, little more than that. Perhaps an axiom gives to some otherwise undefined terms some otherwise meaningless names and claims that they satisfy an otherwise undefined relation. If we then reason out the consequences of this axiom, we do not thereby commit to the truth of any proposition in the actual world except perhaps the value of spending our time in this manner.

When we reach any logical conclusion from the axiom, we have found a proof that the axiom implies the conclusion. A proof is a structure. In finding a proof of a consequence, we have however not identified any actual or imagined objects for which the axiom is true.

When we do identify objects to correspond to the unknowns in an axiom, we have a model. Model theory is something other than proof theory. Even then we have not reduced the axiom to the model. In general there are many models for a particular proposition. A famous example is that Euclid's postulates (that is, axioms) of geometry, minus the one known as the "parallel postulate", provide valid but different models of geometry in the plane, on the sphere, or on a hyperbolic surface such as $z = x^2 - y^2$.

Answer 9

I am going to go ahead and claim that there is in many ways not much of a difference.

As others here have explained very well, mathematical axioms can be seen as a set of rules and statements describing a mathematical system. And in similar terms, religious beliefs can be seen as a set of rules and statements describing the world we live in (or, as some state, don't actually live in).

However, the term "faith" is mostly meaningless (and, I would claim, irrelevant) as far as mathematics are concerned. To have faith that something is true requires that there is uncertainty about that truth. A lack of proof. Axioms do not need proof, they are the basis of proofs. To have faith in an axiom is pointless.

As for the question of accepting or rejecting axioms, I would claim this to be a moot point. There is no need to accept or reject any particular axiom because there is nothing stopping two mathematical systems coexisting, one with the axiom in question and one without.

Of course, for practical purposes one cannot explore an infinite number of systems, so the question becomes one of usefulness. Does this system, with these axioms, allow me to make useful or interesting statements, or does it not? Most systems will not. But some, like for example that of Euclidean geometry, have resulted in a vast number of useful statements.

The accepting and rejecting of axioms is an issue for religions only because religions tend to make statements not about their own system, but about the world. This is, according to me, the one important way in which mathematical and religious axioms differ: mathematical axioms only make statements about their own systems, while religious axioms may (and often do) make statements about anything and everything.

Where mathematical systems can coexist because they each create their own little hypothetical universe, religions often can not and require faith because we only have this one universe which we insist on making statements about.

To see what happens when we instead start making religious-style statements about another universe, have a look at any well written fantasy or science fiction book. Suddenly axioms which may appear very religious if taken out of context become more mathematical in nature; like mathematical systems they define their own universe, and having "faith" in them once again becomes meaningless.

Answer 10

One point that is missing in the other answers and that I believe is important is that mathematicians have adopted ZFC as their primary axiom system because it is useful for reasoning about the real world.

When discussions about this topic arise the main focus (probably rightly so) is to explain that the axioms and the resulting mathematics are "meaningless" (in the sense Qiaochu described) and that mathematics is more like a thought experiment or a game. And that is important, because it allows us to explain why the use of $i$ or the use of uncountable sets is indeed reasonable.

But on the other hand, ZFC excels at building objects that describe the real world. The real numbers are a great approximation of what we have in mind when we talk about length, statistics allow us to abstractly grasp uncertainty and physics as a whole describes reality in an eerily precise way. And even more important, the implications ZFC allows us to make about the behavior of these objects and concepts match our observations of the real world. That is, our logic, our axiom system and our meta-mathematics captures the way the real world "ticks".

Of course that is no accident: ZFC was build to capture as many concepts that mathematicians thought to be self evident as possible. But if we would live in a different universe we would probably adopt a different set of axioms. As such I understand an axiom system as a system of beliefs that may or may not be well grounded in reality. And I believe in ZFC in the way that I believe in physics: Even though it is totally arbitrary that our universe follows the mathematical concepts found by physicists, it has non the less proven to be a very good approximation.