Does mathematics require axioms?

Question

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf
which is also discussed over here: Infinite sets don't exist!?

However, the paragraph which I found most interesting is not really discussed there. I think this paragraph illustrates where most (read: close to all) mathematicians fundementally disagree with Professor NJ Wildberger. I must admit that I'm a first year student mathematics, and I really don't know close to enough to take sides here. Could somebody explain me here why his arguments are/aren't correct?

These edits are made after the answer from Asaf Karagila.
Edit $\;$ I've shortened the quote a bit, I hope this question can be reopened ! The full paragraph can be read at the link above.
Edit $\;$ I've listed the quotes from his article, I find most intresting:

• The job [of a pure mathematician] is to investigate the mathematical reality of the world in which we live.
• To Euclid, an Axiom was a fact that was sufficiently obvious to not require a proof.

And from a discussion with the author on the internet:

You are sharing with us the common modern assumption that mathematics is built up from "axioms". It is not a position that Newton, Euler or Gauss would have had a lot of sympathy with, in my opinion. In this course we will slowly come to appreciate that clear and careful definitions are a much preferable beginning to the study of mathematics.

Which leads me to the following question: Is it true that with modern mathematics it is becoming less important for an axiom to be self-evident? It sounds to me that ancient mathematics was much more closely related to physics then it is today. Is this true ?

Does mathematics require axioms?

Mathematics does not require "Axioms". The job of a pure mathematician is not to build some elaborate castle in the sky, and to proclaim that it stands up on the strength of some arbitrarily chosen assumptions. The job is to investigate the mathematical reality of the world in which we live. For this, no assumptions are necessary. Careful observation is necessary, clear definitions are necessary, and correct use of language and logic are necessary. But at no point does one need to start invoking the existence of objects or procedures that we cannot see, specify, or implement.

People use the term "Axiom" when often they really mean definition. Thus the "axioms" of group theory are in fact just definitions. We say exactly what we mean by a group, that's all. There are no assumptions anywhere. At no point do we or should we say, "Now that we have defined an abstract group, let's assume they exist".

Euclid may have called certain of his initial statements Axioms, but he had something else in mind. Euclid had a lot of geometrical facts which he wanted to organize as best as he could into a logical framework. Many decisions had to be made as to a convenient order of presentation. He rightfully decided that simpler and more basic facts should appear before complicated and difficult ones. So he contrived to organize things in a linear way, with most Propositions following from previous ones by logical reasoning alone, with the exception of certain initial statements that were taken to be self-evident. To Euclid, an Axiom was a fact that was sufficiently obvious to not require a proof. This is a quite different meaning to the use of the term today. Those formalists who claim that they are following in Euclid's illustrious footsteps by casting mathematics as a game played with symbols which are not given meaning are misrepresenting the situation.

And yes, all right, the Continuum hypothesis doesn't really need to be true or false, but is allowed to hover in some no-man's land, falling one way or the other depending on what you believe. Cohen's proof of the independence of the Continuum hypothesis from the "Axioms" should have been the long overdue wake-up call.

Whenever discussions about the foundations of mathematics arise, we pay lip service to the "Axioms" of Zermelo-Fraenkel, but do we ever use them? Hardly ever. With the notable exception of the "Axiom of Choice", I bet that fewer than 5% of mathematicians have ever employed even one of these "Axioms" explicitly in their published work. The average mathematician probably can't even remember the "Axioms". I think I am typical-in two weeks time I'll have retired them to their usual spot in some distant ballpark of my memory, mostly beyond recall.

In practise, working mathematicians are quite aware of the lurking contradictions with "infinite set theory". We have learnt to keep the demons at bay, not by relying on "Axioms" but rather by developing conventions and intuition that allow us to seemingly avoid the most obvious traps. Whenever it smells like there may be an "infinite set" around that is problematic, we quickly use the term "class". For example: A topology is an "equivalence class of atlases". Of course most of us could not spell out exactly what does and what does not constitute a "class", and we learn to not bring up such questions in company.

Answer 1

Is it true that with modern mathematics it is becoming less important for an axiom to be self-evident?

Yes and no.

Yes

in the sense that we now realize that all proofs, in the end, come down to the axioms and logical deduction rules that were assumed in writing the proof. For every statement, there are systems in which the statement is provable, including specifically the systems that assume the statement as an axiom. Thus no statement is "unprovable" in the broadest sense - it can only be unprovable relative to a specific set of axioms.

When we look at things in complete generality, in this way, there is no reason to think that the "axioms" for every system will be self-evident. There has been a parallel shift in the study of logic away from the traditional viewpoint that there should be a single "correct" logic, towards the modern viewpoint that there are multiple logics which, though incompatible, are each of interest in certain situations.

No

in the sense that mathematicians spend their time where it interests them, and few people are interested in studying systems which they feel have implausible or meaningless axioms. Thus some motivation is needed to interest others. The fact that an axiom seems self-evident is one form that motivation can take.

In the case of ZFC, there is a well-known argument that purports to show how the axioms are, in fact, self evident (with the exception of the axiom of replacement), by showing that the axioms all hold in a pre-formal conception of the cumulative hierarchy. This argument is presented, for example, in the article by Shoenfield in the Handbook of Mathematical Logic.

Another in-depth analysis of the state of axiomatics in contemporary foundations of mathematics is "Does Mathematics Need New Axioms?" by Solomon Feferman, Harvey M. Friedman, Penelope Maddy and John R. Steel, Bulletin of Symbolic Logic, 2000.

Answer 2

Disclaimer: I didn't read the entire original quote in details, the question had since been edited and the quote was shortened. My answer is based on the title, the introduction, and a few paragraphs from the [original] quote.

Mathematics, modern mathematics focuses a lot of resources on rigor. After several millenniums where mathematics was based on intuition, and that got some results, we reached a point where rigor was needed.

Once rigor is needed one cannot just "do things". One has to obey a particular set of rules which define what constitutes as a legitimate proof. True, we don't write all proof in a fully rigorous way, and we do make mistakes from time to time due to neglecting the details.

However we need a rigid framework which tells us what is rigor. Axioms are the direct result of this framework, because axioms are really just assumptions that we are not going to argue with (for the time being anyway). It's a word which we use to distinguish some assumptions from other assumptions, and thus giving them some status of "assumptions we do not wish to change very often".

I should add two points, as well.

1. I am not living in a mathematical world. The last I checked I had arms and legs, and not mathematical objects. I ate dinner and not some derived functor. And I am using a computer to write this answer. All these things are not mathematical objects, these are physical objects.

   Seeing how I am not living in the mathematical world, but rather in the physical world, I see no need whatsoever to insist that mathematics will describe the world I am in. I prefer to talk about mathematics in a framework where I have rules which help me decide whether or not something is a reasonable deduction or not.

   Of course, if I were to discuss how many keyboards I have on my desk, or how many speakers are attached to my computer right now -- then of course I wouldn't have any problem in dropping rigor. But unfortunately a lot of the things in modern mathematics deal with infinite and very general objects. These objects defy all intuition and when not working rigorously mistakes pop up more often then they should, as history taught us.

   So one has to decide: either do mathematics about the objects on my desk, or in my kitchen cabinets; or stick to rigor and axioms. I think that the latter is a better choice.

2. I spoke with more than one Ph.D. student in computer science that did their M.Sc. in mathematics (and some folks that only study a part of their undergrad in mathematics, and the rest in computer science), and everyone agreed on one thing: computer science lacks the definition of proof and rigor, and it gets really difficult to follow some results.

   For example, one of them told me he listened to a series of lectures by someone who has a world renowned expertise in a particular topic, and that person made a horrible mistake in the proof of a most trivial lemma. Of course the lemma was correct (and that friend of mine sat to write a proof down), but can we really allow negligence like that? In computer science a lot of the results are later applied into code and put into tests. Of course that doesn't prove their correctness, but it gives a "good enough" feel to it.

   How are we, in mathematics, supposed to test our proofs about intangible objects? When we write an inductive argument. How are we even supposed to begin testing it? Here is an example: all the decimal expansions of integers are shorter than $2000^{1000}$ decimal digits. I defy someone to write an integer which is larger than $10^{2000^{1000}}$ explicitly. It can't be done in the physical world! Does that mean this preposterous claim is correct? No, it does not. Why? Because our intuition about integers tells us that they are infinite, and that all of them have decimal expansions. It would be absurd to assume otherwise.

It is important to realize that axioms are not just the axioms of logic and $\sf ZFC$. Axioms are all around us. These are the definitions of mathematical objects. We have axioms of a topological space, and axioms for a category and axioms of groups, semigroups and cohomologies.

To ignore that fact is to bury your head in the sand and insist that axioms are only for logicians and set theorists.

Answer 3

It seems that many people regard the author's view as naive or un(der)informed. I disagree.

There is a well-known phrase attributed to Kronecker (presumably originally stated in German, and perhaps I am slightly misquoting the English translation as well) that "God created the natural numbers, and all else is the work of man". This is (in my view) an essentially anti-axiomatic declaration, which aligns fairly closely with the point of view in the essay under consideration, namely that mathematics is the investigation of certain "god-given" objects, such as the natural numbers, or the Lie group $G_2$ (to take an example from the essay).

This view is partly Platonist (in the sense that that term is generally used in these sorts of discussions, referring to a belief in a non-formal mathematical reality) and partly constructivist. It is one that I'm personally sympathetic to, and I don't think I'm alone in that. I regard ZFC as a convenient framework for doing mathematics in, but not as the actual basis underlying the mathematics I do; the natural numbers and the investigation of their properties are (in my view) much more fundamental than ZFC or other axiomatic systems that might encode them --- and the same goes for $G_2$ (again in my view)!

My view might be a minority one among working mathematicians (I don't really know), but I know that I'm not the only one who holds it. I also know others who genuinely believe that everything they do rests on ZFC, and that this is of crucial importance.

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Another thing: it is often said that even though many mathematicians don't explicitly invoke the axioms of ZFC in their work, they are implicitly resting on those foundations. Personally, I don't find this convincing; I think it is often the case that those who do believe that everything necessarily rests on ZFC find it easy to construe what others are doing as (implicitly) resting on those foundations. But those who don't believe this also won't accept the claims that their work implicitly relies on those foundations.

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Just to be clear, by the way: my comments here are not meant to apply to things like theorems in group theory, or commutative algebra, or Lie theory, where one derives consequences from the axioms that a structure satisfies (although they might apply in certain contexts where set-theoretic issues potentially intervene); obviously there axioms play a role, although, as the author writes, in these contexts axioms might be better construed as definitions. Rather, they apply to the basic objects of mathematics like the natural numbers, Diophantine equations, and so on.

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It also seems worth mentioning something here which I also made a comment about on another answer:

It doesn't seem to currently be known whether FLT is proved in PA, or only in some more sophisticated axiomization of the natural numbers. On the other hand, there is no doubt among number theorists that the proof is correct. How is such a situation possible? In my view, it's because people ultimately verify the proof not by checking that it is consistent with some specified list of axioms, but by checking that it accords with their basic intuition of the situation, an intiuition which exists prior to any axiomization.

In the end, it will presumably be possible to isolate precisely those properties of the natural numbers that are used in the proof, whether it is the axioms of PA or something stronger, but my point is that the proof is known to be correct although what precise properties of $\mathbb N$ are being used is not yet known! This is because we can argue about $\mathbb N$ based on our intrinsic understanding of it, without having to encode all the aspects of that understanding that we use in precise axiomatic form.

Answer 4

[The] common modern assumption that mathematics is built up from "axioms" ... is not a position that Newton, Euler or Gauss would have had a lot of sympathy with, in my opinion. ... [C]lear and careful definitions are a much preferable beginning to the study of mathematics.

But the very reasons for objecting to a crude "first lay down some axioms and see what follows" model of mathematical knowledge apply equally to a "first fix the definitions" model. Definitions are not laid down at the outset, once and for all, "carved in stone", but often have to be tweaked as we explore successful and unsuccessful proofs. What definitions it is fruitful to use is something mathematicians discover by exploration, trial and error.

There's a famous and wonderfully thought-provoking discussion of the way mathematical knowledge grows, and the way that our axioms and definitions get refined together as we go along, in Imre Lakatos's Proofs and Refutations (1976), which any maths student should sometime read.

Answer 5

For most purposes, axiom, definition, theorem, postulate, lemma, corollary, proposition, and all other similar terms are simply pedagogy, and there is essentially no mathematical content in the distinction between them. (although "axiom" and "theorem" have a precise technical meaning in the setting of formal logic. But the usual caveats about mixing up formal and informal meanings apply)

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I am one of those formalists the author decries. I am a formalist because I recognize the following.

Arguments involve hypotheses and rules of inference. In regards to hypotheses, we have two basic approaches:

• One can state hypotheses up front,
• One can make them up on the fly.

In regards to inferences, we have two basic approaches:

• One can state acceptable rules of inference up front,
• One can make them up on the fly.

In both cases, one approach is far more convincing than the other. :)

When a person says things like

a fact that was sufficiently obvious to not require a proof

the only meaningful content is the statement "I will assume this statement"; everything else is either purely rhetorical, and only holds weight if you buy into the rhetoric.

(Assuming, of course, that you don't consider "Wildberger thinks Euclid thought something was obvious" to be a logically valid argument for some conclusion. And even if you do think something like that, such a rule of inference can be very tricky to apply correctly)

It doesn't matter whether we truly believe mathematics is a meaningless game or something that tells us about the "reality of the world in which we live"; either way, there is going to be some statements we accept, some rules of inference we accept, and other statements we deduce from these. And if we do a good job putting all of the hypotheses up front and eliminating extraneous window dressing, you can't even tell the difference between the two philosophies.

Answer 6

One of the pleasant properties of the "game played with symbols" is that it doesn't matter why you're playing it, everyone still gets the same answers out. You can play it because you think it describes some "real" but abstract thing, or because you think the purpose of mathematics is to predict the universe and that by manipulating symbols you can do that, or you can play it because you like symbols. Nobody cares, they can use your results anyway. The same is not true of intuitive reasoning.

There's more than one way to provide a foundation for mathematics. The most widely-referenced at the moment is axiomatic set theory, but 2000 years of valuable results in mathematics were obtained without it, and with only the occasional mishap. The clever (and perhaps surprising) thing about axiomatic set theory is that it could be "slid in" underneath all of that reasoning, in a way that avoided fundamentally changing what mathematicians accept as a proof in most fields.

The metamath project seeks to compile proofs from ZF(C) of everything. It's interesting that even where such elementary proofs don't already exist, because mathematicians simply haven't written out the full detail of every proof in predicate calculus, nobody expects the project to fail to produce them. Mathematicians "can tell" that they're making arguments that formalise even without formalising them, obviously with some small scope for error.

As such, it doesn't matter that Euclid wasn't reasoning about a set that models a particular theory, because somebody who does that, or in general who reasons from axioms, can get the same results.

Sometimes people who care about axioms don't get the same results. In Euclid's scheme, Pasch's Axiom doesn't follow, which Euclid didn't notice. AFAIK that is not because he was stating true facts in a sensible order and it would not have been sensible to state this. He just overlooked it, it was so self-evident that he didn't even notice it evidencing itself. I think it's fairly clear that Pasch improved on Euclid's work by driving out such details. Euclid intended his list of axioms to include everything that he was taking for granted, so it's useful to reason just from the axioms you've identified instead of from anything that's self-evident, and thereby identify any mistakes Euclid may have made in declaring the list complete.

Or take the "axiom" that Euclid himself was troubled by, the parallel postulate. When considering non-Euclidean geometries in general, part of their value is that they have some things in common with Euclidean ones, and some things different. How is the difference characterised? By different axioms. Now, if Euclid felt that an axiom was something inherently true then that's fine as far as it goes, but if he held to his opinion that the parallel postulate is true then that would have rendered him incapable of considering a non-Euclidean geometry in light of his other axioms. That's a limitation of refusing to consider axioms to be negotiable. I never met Euclid, but I find it hard to believe that a great mind would be inherently limited in that way. He got a certain distance in the time available to him, but did not discover everything interesting about his procedure for reasoning. Discovering more interesting things caused modern mathematicians to start viewing axioms differently, and to view what mathematicians had been doing for 2000 years differently.

I also agree with axioms-as-definitions. You can by all means write down your axioms and rules of procedure, and use them on the basis that they're worthwhile in themselves, or that any foundation that provides a model for them will do, and you don't care to address the philosophical question of what that foundation might be. I don't think these parts of what the author says are controversial, the bit that's tricky is to reject the formal foundations entirely. I don't know what the author means by "beginning to the study of mathematics", but if he's talking about the training of a student then I doubt anybody would argue that children should be taught ZF before learning to count. As such it follows that ZF does not come first, if any formalism does it's PA.

I bet that fewer than 5% of mathematicians have ever employed even one of these "Axioms" explicitly in their published work

This sounds like a point that, if you want to make it seriously, you can investigate by statistical sampling. It's an interesting point, and let's assume it's true, but ultimately if you write $x \notin x$ you are not explicitly appealing to the axiom of regularity but you are appealing to a result that you have seen proved (with a very short proof) from ZF, and anyone likely to read your paper has seen this proof too. And so on upwards to results with much longer proofs. As metamath shows, there's no fixed boundary between results that can be formalised and results that can't.

The lack of explicit appeal doesn't prove whether or not the axioms are fundamental to the work. However, any given paper relies on some set of results, and if you replaced ZFC with something else that produces those same results then you wouldn't need to change the paper. That's what those playing with foundations are up to. It's perfectly reasonable to state discontent with foundations, but the difficult and enlightening task would be to provide an alternative. A naive notion of classes in place of things "too big to be sets" may or may not do the job. The author asserts that it does (by way of an example, the full list of tricks to form his foundation presumably is longer).

So, I think more than lip service is paid to foundations, but as against that results are accepted whose proof could in fact be more rigorous in the sense that they're not yet computer-verifiable in symbolic logic but could be made so in the opinion of both author and readers. Take from that what you will as to whether the formal work and/or the opinion that the formalization could be done, are "necessary". In the mean time, the author's main point is true that most mathematicians don't spend a lot of time worrying about foundations, and seem to do all right.

Answer 7

Back to the original question: Does Mathematics Require Axioms?

The best answer I can think of is: Not at all - until they do.

In actual practice, working mathematicians go about developing new mathematics using tools of ordinary human thinking and speaking - they model abstract objects as pictures (in the head or on the board); they 'look at the objects' to 'see' what features they have; they use previous results in arguments very informally; they 'handwave' arguments; etc. When they appeal to some property to justify an inference in chats with colleagues, they don't bother justifying the appeal as long as the colleague accepts it. Even when they finally write up their results and have to be more technically precise, they still use a lot of informal language and almost never refer back to specific axioms for justification, because they expect their readers will know what they mean.

And it is a historical fact that all of the major mathematics that has been created was well-developed quite a bit before anyone felt a need to introduce axioms. The creation and explosive develoment of Calculus/Analysis proceeded for over two hundred years before people felt a need to axiomatize it (or rather the Real Numbers on which it is based). The basic results of Geometry were well known before Euclid wrote the Elements. Heck, people were doing arithmetic, and later number theory, thousands of years before anyone thought to create axioms for the natural numbers.

Judging from history, it seems mathematicians resort to axiomatization in two circumstances: (a) They need to teach a subject to ordinary students rather than dedicated mathematicians, and the old 'handwaving' won't do; (b) The old 'handwaving' unexpectedly leads to contradictions or other bogus results. Geometry is the prototype for (a) - Euclid was a teacher, and needed a textbook to organize the subject for his pupils. Set Theory is a classic case of (b) - Cantor's own reasoning produced overt paradoxes, which were eliminated by axiomatization [Zermelo, Russell, etc]. Calculus was a combination of both - axiomatization began because mathematicians like Bolzano and Weirstrauss had to teach it to ordinary students, but found all the usual arguments to be both logical gibberish [infinitesimals??] and pedagogical disasters.

Answer 8

First of all, as far as I know, no one really knows anything about Euclid, much less what was going through his mind when formulating his "axioms". Be that as it may, axioms exist for a reason, not just to giddy formalists and logicians. It is true that most mathematicians never explicitly make use of any axioms, and as you say most can't even recall a single one of them. But the fact of the matter is they serve a precise purpose in mathematics, as mathematics is and should be independent of any sort of real world measurements (through real world measurements may indeed guide our intuition in mathematics). The main problem is the age old scenario of the iterated question "why?". Upon sufficient iteration of the question "why?" (which is a legitimate question), you will always end up in a land where the only way out is to answer with "axioms", there's just no other way if you want to stay within the realm of pure mathematics. And even though I never think about axioms and have never once used them, I understand that they serve a purpose, which to me is an obvious one that people should embrace if they are to truly understand the nature of mathematics and it's distinction from science, which is intrinsically empirical.

Answer 9

I can understand the writer's frustration with the ZF axioms. I myself found them so counter-intuitive that I have had to develop my own simplified versions. (OK, maybe I'm just not that clever!)

The one area where you absolutely cannot avoid dealing with each of the axioms of set theory (and logic) is in the development of automated theorem provers and proof checkers. But there is no reason to be so spooked by the notion of an infinite set. They can be handled quite easily and safely. I think this terror of the infinite must have been some kind of over-reaction to the well known inconsistencies of naive set theory.