This question gets asked a lot, in particular by math students, and I think that the answer that is usually given, which is the same as many answers in this page, is overly simplistic: we don't care about ontological truth of our axioms (and, by the way, of the deduction scheme), we just define a set of axioms and define true as provable. This is the formalist view of mathematics, which was the dominant view at the beginning of the 20th century.
This reduces math to a meaningless mind game, as acknowledged by Hilbert himself, one of the major proponents of this view: "Mathematics is a game played according to certain simple rules with meaningless marks on paper".
Unfortunately, this sweeps under the rug important questions, such as "if I prove the correctness of a an algorithm, how does this increase my confidence that the algorithm will actually work?", or "if I compute the hypotenuse of a triangle with Pythagoras theorem, then cut a piece of wood and measure it, will they match?"
This is why ultimately there is some belief involved. From Wikipedia [1]: "In practice, most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system" (emphasis mine).
To work with with a set of axioms, we must believe that that set is both logically consistent, and that in some way its interpretation is consistent with reality. The first question cannot be answered for the interesting sets of axioms, thanks to Godel, and the second is not even well-defined.
For this and many other reasons, philosophy of mathematics has mostly moved on from the logicist/formalist view [2], but as most philosophical problems, there probably will never be a conclusive answer.
To provide small insight into the work of people who actually do doubt ZFC I'd like to source two interesting books.
First is Peter Aczel's work with the Anti-foundation Axiom which allows you to create sets with "infinite regress". Instead of blanket disallowing these kinds of set like ZFC does, he enlarges ZFC by allowing any set which is a unique solution to a group of equations that may be self-referential.
The important point is that the Anti-Foundation Axiom lets us model infinite, streaming structures similar to those modeled in computer programs. It also motivates the somewhat unpopular idea of bisimulation which is very necessary for creating a kind of equality in an AFA world... and also for proving the equality of streaming algorithms.
Second is Lawvere's introduction to thinking about the foundations of mathematics from a Category Theoretic perspective. Lawvere is a proponent of thinking of Set theory as simply one, somewhat interesting Category which can be generated by a more foundational theory and set of axioms. I'm not personally anywhere nearly well-knowing enough to say whether that works, but his book Conceptual Mathematics (http://fef.ogu.edu.tr/matbil/eilgaz/kategori.pdf) gives a very interesting POV on how to work from the Category Theoretical basis to answer some normal Set theoretic questions.
I think that the "infinite, streaming structures" that you are referring to can be modelled in ZFC or HOL using Tarski's fixed-point theorem. See for example the paper by L. C. Paulson, "A fixedpoint approach to implementing (co)inductive definitions".
So the point of this anti-foundation axiom probably lies somewhere else.
Coalgebras are defined as an isomorphism between "sets" X==P(A*X) which allow us to unfold our seed state over non-determistic updates. Such a "set" obviously cannot exist in ZFC directly as it violates set cardinality. It can be a set that satisfies the AFA, though—in fact, that equation I just wrote is sufficient to define it I think and then AFA guarantees uniqueness.
I don't think that either ZFC or ZFC/AFA dominates the other, just that the AFA bakes a framework for infinite streaming structures into the heart of your mathematics. You immediately lose extensional equality in favor of bisimulation, for instance. The point is more to see what centipede mathematics feels like than to claim greater power or expressiveness.
"To work with a set of axioms, we must believe that that set is both logically consistent ... The first question cannot be answered for the interesting sets of axioms"
Consistency of a theory may be proven or disproven in a different set of axioms. For instance, peano or presburger arithmetic are consistent in ZF.
In the field of program verification for instance, people use all kinds of theories that they prove to be consistent.
But I agree that at some point, one has to believe in an axiomatic system.
It's quite common for mathematicians to consider the impact of axioms on a theorem. Mathematicians often consider whether a proof satisfies constructivist constraints, requires the axiom of choice, and consider different geometric foundations (e.g. einstein's spacetime is non-euclidean). This doesn't cause religious disputes, and indeed mathematicians are happy to jump from one set of constraints to another.
I do agree that some axioms are grounded in observed reality (e.g. arithmetic, euclidean geometry) but discovering that axioms fail in unfamiliar situations is no generally a cause for soul searching or existential angst.
"This doesn't cause religious disputes, and indeed mathematicians are happy to jump from one set of constraints to another."
For me, that is the difference between an axiom and a belief. People with beliefs think their beliefs to be better than those of others. With axioms, that is not so. In fact, mathematicians like to doubt their axioms "do we really need the parallel postulate" was, for centuries, a subject of thought"
Actually for a while the parallel postulate was a matter of religious zeal. The studies of consistent geometries that had different behaviors for "parallel" lines was, for some reason, profoundly disturbing to some christians who tried to prove that the parallel postulate was not needed (i.e. That non-euclidean geometry was inconsistent). This was a matter of amusement for mathematicians, even though many mathematicians were a bit disturbed by non-euclidean geometry.
Lewis Carrol, who was a mathematician and some kind of Anglican, couldn't stomach complex numbers. So religion can become confused with axioms, but it tends to start with religion, not axioms.
Completeness is overrated. Why does it matter that a system can't prove every possible statement you can write in it? It's not going to give you a wrong answer, it's just going to fail. Why is that particular lack of capability more important than the millions of statements you can't formalize in the first place? Take consistency and build something with it.
P.S. Using a theory to prove its own consistency would be way more suspicious than using a more powerful theory to prove it. "The Bible is true because the Bible says so."
>P.S. Using a theory to prove its own consistency would be way more suspicious than using a more powerful theory to prove it. "The Bible is true because the Bible says so."
Which eventually leaves us either with some top-level assumptions we treat as completely axiomatic, or with infinite regress to higher and higher systems of proof logic.
On the other hand, consistency alone isn't really useful!
Let suppose I design a new proof system to prove arithmetic propositions. Unfortunately, my system is such that it can't produce any proposition. That makes it consistent but useless.
That is interesting, but probably both views have their merits. Belief involves emotional commitment which can create intellectual inertia that can be both good and bad, while formalism places a safe detaching distance from the subject.
Axioms are not beliefs. Axioms are a priori and you formally derive stuff from them using particular logic rules.
Generally speaking, nobody claims that the axioms of set theory are true in any ordinary sense of the word true. Does it makes sense in the real world to assert the existence of an infinite set, and have different sizes of infinities as a consequence? It is irrelevant, you assert the existence of an infinite set because it is practical from a pure mathematical point of view, because you want to model infinite sets like the natural numbers (an abstraction, do they exist? it doesn't matter to mathematicians generally speaking).
Mathematics no longer pretends to describe what is true, what holds in our reality. That idea was abandoned some time ago. A canonical example were non-euclidean geometries, that were studied in the XIXth century for the sake of it. They had applications later, but the motivation for their study and changing Euclid's axioms was formal.
> Generally speaking, nobody claims that the axioms of set theory are true in any ordinary sense of the word true.
Mathematical platonism is a pretty widely held belief, at least amongst pure mathematicians. From Hardy's A Mathematician's Apology,
"I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply the notes of our observations."
A lot of set theorists will speculate about the truth of axioms beyond the standard axioms of ZFC, e.g. large cardinal hypotheses or projective determinacy.
>Does it makes sense in the real world to assert the existence of an infinite set, and have different sizes of infinities as a consequence?
Of course it does. It's very easy for me to write down a finite-sized inductive definition of the natural numbers, which immediately gives me a finite description of all countable infinities and their size.
> Does it makes sense in the real world to assert the existence of an infinite set, and have different sizes of infinities as a consequence?
Yes it does. But I might be biased because I'm a programmer and one of my jobs is adding constraints to unsolvable problems so they become solvable in the infinite set my clients actually care about.
It's surprising how easy it is for somebody to specify a problem that is unsolvable, only for it to become solvable once you ask them what they really want.
There's an infinite variety of axiomatic systems, most of them not much different from, say, a set of beliefs in Santa Claus and magical elves living at the North Pole.
However the axiomatic systems we happen to use most have one important property: they are useful for acquiring new knowledge via scientific method. I.e. they serve as a foundation for theories that make correct predictions as verified by experiment.
Cute story, but don't mistake it for a second for any sort of casual indictment of the scientific process. ...the story does not even feature the scientific process in the first place (which goes a long way to explain why we find it silly).
There are beliefs which are useful for acquiring new knowledge via scientific method. Like that the laws of physics are universal (the same everywhere in the universe).
This is pure Platonism. I reject Platonism in modern mathematics.
Mathematics should, first, be used to describe the real world. (That is what scientists like Newton were using it for.)
To do that, you don't need an axioms. Nor do you need beliefs. You only need observation of reality. It starts with the concept of a "unit," which can be added or subtracted from other "units." There is, of course, some prerequisite philosophy. ("Reality, you say?" "Why, yes, actually...").
Secondly, mathematics can deal with axioms and "games" and "universes" (to take words from the StackExchange post). But all of mathematics should not be subordinated to such a framework.
I realize my argument is not "obvious." Fully establishing it would take a lot of work, and even then it would be widely rejected by the status quo Platonic mathematics establishment. I'm not claiming to do that in this comment. I'm just putting this alternative perspective out there for anyone who is interested.
Mathematics should, first, be used to describe the real world.
If by "real world" you mean the things and events we can directly witness, that seems to be deliberately crippling mathematics from the start and limiting it to things we already know about.
Depends on what you mean by "directly witness," but that's neither here nor there.
What I am proposing would not cripple mathematics, it would simply replace "axioms" with _facts_ (such as, for a simple one, that adding one unit to another results in having two units). You can then get arithematic, algebra, calculus, and on and on.
If people want to go off and define axioms for made-up worlds that do not necessarily correspond to reality and then go do math with them, that's fine. In fact, I would encourage it.
But we should not be approaching _all of mathematics_ in that way, and that is my point.
Math has the same basis in perceptual concretes (i.e. "real things") as physics, biology, chemistry, etc.
The ZFC axiom set [1] doesn't define a made-up world. Actually, once you understand the axioms (they're somewhat difficult in notation and definitions, but the ideas behind are understandable) they make sense. There are some axioms made to avoid paradoxes (notably ZFC3 [2]), but the majority describe how we understand the world. It'd be difficult to imagine a world where two sets aren't equal even when they have the same elements, or where you couldn't construct a set that contains all elements of two other sets.
How's this different to a set of facts as you say, like one unit + one unit is two units? It's not. I'm not sure of this, but I'd say you could construct a set of axioms based on those facts that is compatible with the ZFC axiom set.
So, why choose ZFC? Because it's simple to define (how'd you formally define what is a unit, and what does the + operation mean?) and because it's more or less consistent with our world. We can extrapolate the things we find while doing mathematics to the real world. And, sometimes, we can discover things of the real world based on findings on things that don't exist. Complex numbers, for example: they don't exist, but they're really useful to define and explain electrical experiments [3].
We approach all of mathematics like this because its easier to study everything within the same system. And it's important that we keep a consistent underlying system so we can be sure that everything we discover based on the set of axioms and rules we've defined is true in a mathematical sense; and then those discoveries could describe our world. Wouldn't we have a consistent system, we could end up with contradictions. If we extracted mathematical concepts from the real world, we wouldn't be sure if a discovery we make from those concepts can be applied to the real world.
TL;DR: Our set of axioms describes properly the real world, is simple and consistent enough and that enables us to infer new ideas that can be applicable to the real world too.
Complex numbers, for example: they don't exist, but they're really useful to define and explain electrical experiments
At risk of turning this into a discussion of what "exist" means, I reckon they do exist :p If they didn't exist, how could we possibly use them, and use them so effectively and consistently?
Of course, if a number "existing" means "I can see this many number of apples" then sure, it doesn't "exist". Yes, this is definitely becoming a discussion of what the word "exist" means applied to number. I'll stop. :)
"_facts_ (such as, for a simple one, that adding one unit to another results in having two units)"
That is not a fact, that is just a consequence of certain sets of axioms, but not of others. It is patently wrong for addition in GF2, where adding one unit to another results in having zero units.
Nope, not wrong in GF(2) at all. In GF(2), adding one unit to another gives you two units. It just happens that in GF(2) having two units is the same thing as having zero units.
Sounds like you want to take a subset of axioms that you like, label them "facts", and go from there. Presumably, you'll end up with a subset of mathematics.
Math has the same basis in perceptual concretes (i.e. "real things") as physics, biology, chemistry, etc.
Can you show me what I get when I subtract three apples from two apples? How about showing me however many apples I need to square in order to get negative one apple?
You don't blindly believe axioms, it is just that certain sets of axioms happen to allow us to make useful predictions about the world. While you could have philosophical discussions about weather certain axioms are true (whatever that means), and what their exact relationship with the world is, you don't need to do this to see that axioms are useful to arrive at mathematical conclusions, which then can be used to make predictions about the world.
Why do we believe that if we take 1111 apples and then add 2222 more we will get 3333 of them? Because we have seen from experience that a certain way of applying mathematics to the real world gives great results. The same for conclusions following known axioms, they seem to generally work well for describing our world.
You don't blindly believe axioms, it is just that certain sets of axioms happen to allow us to make useful predictions about the world. While you could have philosophical discussions about weather certain axioms are true (whatever that means), and what their exact relationship with the world is, you don't need to do this to see that axioms are useful to arrive at mathematical conclusions, which then can be used to make predictions about the world.
Why do we believe that if we take 1111 apples and then add 2222 more we will get 3333? Because we have seen from experience that a certain way of applying mathematics to the real world gives great results. The same for conclusions following known axioms, they seem to generally work well for describing our world.
A good post although I suggest rephrasing "useful" to something like "experimentally falsifiable". With a pretty wide definition of "experimentally" and "world", especially for non-applied math.
If you have no way to do a falsifiable test, and can't just substitute in a X and do abstract analysis of X instead of worrying about X, then its just a belief.
I'd be careful categorizing stuff eternally as beliefs not axioms. Blindly only permitting geometry to be Euclidean meant some exciting non-euclidean results couldn't happen until it was "allowed" to be thought about somewhat recently (well, a century or two ago...). Also physicists have this amazing ability to turn pure math into applied math, at least over a long enough historical scale. And physicists have a pretty good ability at coming up with crazy experimental methods to test, look at the last 75 yrs or so of quantum physics... so let me get this straight, you do what with two geiger counters and a truly giant magnet and some radioactive atoms or what? Or shine a light beam or ion beam thru that weird magnet?
There maybe some axioms that are analogues to rules of a game and so are neither true nor false.
But in practice, the bulk axioms are chosen and retained because the inventor thinks they or their consequences have some mapping to reality. This mapping is tested through applying the resultant math to some field and getting predictive success. In this regard the bulk of axioms are the same as any other beliefs except they are not arbitrary beliefs, and they are useful or believed to have the potential of usefulness.
And part of the problem is merely semantic. The word "belief" differs from the term "axiom" in other critical ways: "belief" regularly involves a call to action and includes concept that do not go though any tests for consistency let alone predictive success. If one asks "what is the difference between an assumption and an axiom" there is a lot less excitement.
Most interesting, I think, is the idea that basic axioms should be different than assumptions at all. That they can be definitively decided or the part of reality they map to is real and can be found.
I do not see why we would necessarily have evolved the mental equipment to work out every law of nature. No other animal has that, and while we clearly have better mental capacity I can't see why we would have evolved _every_ mental capacity.
The major difference is that nothing requires that your set of beliefs be consistent: however, if a given set of axioms allows you to prove and disprove some assertion P, you pretty much can't use that set of axioms anymore! Well, you can, but rather than the normal huge castles made of sand that follow from any reasonably strong set of axioms, you will end up with a small puddle of mud that's useless.
In this sense, a simple discovery (of an axiomatic system's inconsistency) would essentially cause that set of axioms never to be used again.
Can you imagine that a simple logical proof would immediately cause the Pope to abandon making certain statements together?
Suppose a religious scholar produces an argument that the universe described by,
"Jesus Christ, the only Son of God" + "who was born of the Virgin Mary" + "his kingdom will have no end" is an inconsistent Universe.
It won't suddenly be the case that nobody will mention these three things in the same axiomatic set again, just because they're (through some rigorous proof) inconsistent.
That's a huge, huge difference. It's like the difference between chess and Shakespearean criticism!
A couple of important distinctions between axioms and beliefs:
1) axioms are assumed to be logically consistent with each other, at least when the word "axiom" is understood strictly; it's possible that someone may believe two things that are inconsistent with each other
2) belief is an involuntary mental acceptance that a particular claim is true. You can't make yourself believe something that you know to be false
Axioms are definitions you set before you start to derive the rest of the system. If you want to, you can test the axioms and the derivations against reality and you have science.
Beliefs are definitions you think somebody else set that you can't change before you start to derive the rest of the system. In most cases, you are not permitted to test beliefs against reality.
You don't get your feelings hurt when an assumption turns out to be wrong. But if you were to be betrayed by someone you believed (same as assumed they were telling you the truth or had your interests in mind), it would hurt your feelings because you had invested some of your ego with that assumption. Same for those with religious beliefs.
I'm using axiom and assumption interchangeably, as I don't see what the difference between the two is. However, assumption is a much simpler word and means pretty much the same thing to everyone, so it's easier to use. I guess axiom is something like a generally accepted assumption.
A reality, often unacknowledged, is that mathematicians proceed largely on the basis of elegance. As Achilles and the Tortoise shows us, basic logic has built-in points of infinite regress, and the only way past them is to agree that the logic is sound.
Mathematicians who must choose between two sets of axioms will pick the one they consider most elegant, every time. The fact that this is ultimately an aesthetic judgement is clear: Erdős Pál, arguably the finest mathematician of the latter half of the 20th century, described proofs that tickled his sensibilities as "Pages from the Book".
The question is about comparing axioms with belief and religion; so my question is kind of off-topic, but I have to ask: How about philosophy? How does proving a philosophical theorem start? Is it by an axiom or something else? To put it another way, what is the philosophical equivalent of scientific method?
A statement about facts belongs to the natural sciences. A statement about numbers (or dually: sets) belongs to math. A statement about statements, belongs to philosophy. Since the scientific method requires testability against facts, while (in this definition) statements can impossibly be facts in reality, it is not possible to apply the scientific method in philosophy. The same holds true for programming. Any statement about programming languages or programs is either mathematical or philosophical, but never scientific.
In my opinion, if the set of beliefs is orthogonal and irrefutable, it is a set of axioms. Orthogonal: It is not possible to derive one belief from another or a combination of other beliefs. Irrefutable: Counterexamples have not been found or counterexamples can simply not ever be found. It is obvious, however, that orthogonal and irrefutable do not mean "true". On the contrary, "true" means that a statement can be derived from the axioms, while the axioms by definition themselves cannot. Axiomatic systems are simply a form of rigorous and highly systematic beliefs, but not necessarily internally consistent nor necessarily consistent with reality.
In the same sense that an axiom can't be proved, it can neither be refuted. I think you mean they're consistent: that you can't infer a paradox or a contradiction from your set of axioms.
This reduces math to a meaningless mind game, as acknowledged by Hilbert himself, one of the major proponents of this view: "Mathematics is a game played according to certain simple rules with meaningless marks on paper".
Unfortunately, this sweeps under the rug important questions, such as "if I prove the correctness of a an algorithm, how does this increase my confidence that the algorithm will actually work?", or "if I compute the hypotenuse of a triangle with Pythagoras theorem, then cut a piece of wood and measure it, will they match?"
This is why ultimately there is some belief involved. From Wikipedia [1]: "In practice, most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system" (emphasis mine).
To work with with a set of axioms, we must believe that that set is both logically consistent, and that in some way its interpretation is consistent with reality. The first question cannot be answered for the interesting sets of axioms, thanks to Godel, and the second is not even well-defined.
For this and many other reasons, philosophy of mathematics has mostly moved on from the logicist/formalist view [2], but as most philosophical problems, there probably will never be a conclusive answer.
[1] http://en.wikipedia.org/wiki/Foundations_of_mathematics#Part...
[2] http://en.wikipedia.org/wiki/Philosophy_of_mathematics