# Axiomatic Foundations

I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if mathematicians choose the axioms then we are inventing our own maths, surely that is what it is but as it has been practiced and built on for so long then it is too late to change all this so we have had to adapt and create new rules?? I'm not sure how clear what I am asking is or whether it is even understandable but would appreciate any answers or comments, thanks.

• Well, IMHO, perhaps first there was simply some intuitive knowledge, based on practice. Then some clever guy (for example Euclid, though as it turned out, his axiomatic system for geometry was not, well, that clear from our viewpoint) would come and try to build that knowledge into a system (what follows from what)...? And those most familiar systems of axioms that we know, they appeared in the process of systematization, generalization, rigorization, didn't they? (for example the set-theoretic axioms - weren't they a response to paradoxes connected with the naive notion of a set?)
– W_D
Commented Jul 22, 2013 at 8:27
• More questions from my side though than answers, sorry, that's just the vague ideas I've got, not to be considered anything like a fully correct answer...
– W_D
Commented Jul 22, 2013 at 8:28
• Thanks for the help, but am I right in what our saying is that mathematicians create axioms to 'fix' paradoxi (if thats the plural of paradox), then this just says that mathematics is an invention to turn illogical statements that the mind cannot comprehend into comprehendible statements. Commented Jul 22, 2013 at 8:31
• You're asking how we know that Settlers of Catan is the best game. We don't. Some people like to play Settlers of Catan; some like to play Bang!. That's fine. People play the games they like. Commented Jul 22, 2013 at 8:53
• @dfeuer I have discovered a truly marvelous demonstration of the proposition that Settlers of Catan is the best game. Unfortunately, this <strike>margin</strike> comment is too small to contain it.
– joeA
Commented Jul 25, 2013 at 3:05

Mathematics does not exist in a vacuum. It is strongly related, via applications, to the world around us. Mathematicians choose axioms according to what works well when we try to use the insights and results flowing from these axioms to better understand problems (usually from science) that we care about.

To draw an analogy with painting, a painter can surely mix colours in endless combinations and spread paint on canvas in equally endless possibilities. But, artists don't just randomly spread paint on canvas. The reason is that their art does not exist in a vacuum. It is strongly related to human culture, the physical world around us, and the predispositions of the human brain. These dictate what is considered good art, and so guide the artist in the creation of a good painting.

• So if this is true then where does the link between pure and applied maths come? And therefore maths is no more complete than physics or some other imaginary built system? Commented Jul 22, 2013 at 10:06
• @genty4321 Very simplistically, the problems that drive pure math research are intimately tied to the problems of applied maths, which in turn are intimately related to the world around us. And, completely siding with Lakatos on this issue, no, there is no essential difference between mathematics (pure or applied) and physics. Mathematics is not the study of that which results from some axioms. Rather it's the subtle art of cleverly adjusting the axioms when that which results from them does not suit us. Commented Jul 22, 2013 at 10:24
• John Steel once told me that he does not find anything surprising about the applicability of mathematics to the real world, essentially on the grounds you mention. Commented Jul 25, 2013 at 1:08
• @AndresCaicedo I completely agree. There is nothing surprising about the applicability of something that is designed, refined, and refined again with the sole purpose of being applicable. This is opposed to some mathematicians whom I've heard claim that the applicability of mathematics is, for them, proof of the existence of god. Commented Jul 25, 2013 at 1:34

There are (at least) four types of sources for axiomatic systems. Here are the scenarios that I have in mind:

(1) Some mathematical structure, like the plane in geometry or the system of natural numbers, has been recognized as useful for applications and has therefore been studied extensively. So many facts are known about it. In this situation, one might want to organize those facts in a logical system, showing which facts are consequences of which other facts. Of course, to avoid circularity, some facts have to be taken as basic, and then other facts are shown to be consequences of these. The basic facts are called axioms or postulates, and it is desirable to make them as simple and as few as possible, so that one is not assuming things that could rather be proved. Among the axiom systems that arose in this way are Euclid's axioms for geometry (and, in a more rigorous age, Hilbert's axioms for geometry) and Peano's axioms for arithmetic.

(2) Questions have arisen about the legitimacy of some arguments, so it becomes necessary to say exactly what the assumptions are that underlie those arguments. The clearest example of this is Zermelo's (1908) axiomatization of set theory. The immediate problem facing Zermelo was the axiom of choice. It had been used as if obvious, for example in the proof that the union of countably many countable sets is countable. But, when Zermelo pointed it out as an explicit statement and used it in his proof (1904) that all sets can be well-ordered, he got a lot of flak. There were also other points in need of clarification, such as Cantor's distinction between consistent multiplicities (sets) and inconsistent ones. So Zermelo set up a system of axioms on which to base not only the proof of his well-ordering theorem but also the other set-theoretic arguments of the time. (Nowadays, we can view Zermelo's axioms, as well as later extensions by Fraenkel and others, as falling under scenario (1) above, as systematizations of the known facts about the cumulative hierarchy of sets. But, as far as I know, the cumulative hierarchy is not mentioned in Zermelo's writings until 1930. So I regard their introduction in 1908 as a different scenario.)

(3) People notice that very similar ideas and proofs are occurring in different areas. The elementary arithmetic of addition of integers, or real numbers, or complex numbers is very similar to the behavior of the operation of composition of permutations of finite sets or of rotations of space. In this situation, it is worthwhile to isolate the basic features common to these different contexts and deduce other common features from the basic ones (axioms) once and for all, rather than treating each context individually. Thus, the examples I just mentioned are all subsumed by the axioms for groups. Notice that here the axioms are intended to apply to many different structures (numbers, permutations, etc.) whereas in (1) (and perhaps also (2)), the axioms are intended to describe one specific structure. In (1), the existence of different models of the axioms is an unintended feature or bug; in (3) it is the main reason for formulating the axioms.

(4) Just plain curiosity. For example, given Euclid's axioms for plane geometry, let's see what happens if we replace the parallel postulate by some contrary assumption. Nowadays, such non-Euclidean geometries are seen as descriptions of interesting structures (like the hyperbolic plane), but when such axioms were first considered, no such structures were known, and in fact these "strange" axioms were expected to be contradictory. In principle, anybody can make up and study whatever axioms (s)he wants. Whether anyone else will pay attention, though, is a more difficult question. Axiomatic systems that don't fit under (1), (2), or (3) above had better come with some serious motivation, or the person who introduces and uses them is likely to be ignored.

There's one important thing that any choice of axioms should at least do, which is to be consistent, i.e. you cannot prove both a theorem and it's inverse, because if you could, you could prove anything at all, which would not give you any interesting theory.

That said, thanks to Gödel, we know that ZFC cannot be proved to be consistent using ZFC (unless it is inconsistent), so we have little way of truly knowing whether the axioms we have chosen 'work', so to say, but intuitively they don't seem to be inconsistent.

I would say that when one creates axioms, one creates them with a specific goal in mind, so while we have an infinite amount of axioms to choose from, we're only interested in the subset that describes some object which behaves in a way which fits with our intuition.

Take the Peano axioms as an example, we want to describe the natural numbers, and we have some intuition on how these should behave. We want a number 0, we want a way to count upwards, we don't want to end up back at 0 if we count for long enough, etc. The axioms is just a way to use logic to explain what we already 'knew'. If I created a different set of axioms for the natural numbers, but where, once I got to 17, I started back at 3, I think we'd all agree that while the mathematical object I've created might be perfectly valid, it is not the natural numbers I've described.

Any other set of axioms is chosen in much the same way. We have a mathematical object we want to describe, we try to create a set of axioms which does the job, and see where the axioms lead us. When we see where the axioms go, we may decide from 'outside' the logic whether this set of axioms really was what we wanted, or if they should be changed to better suit what our intuition tells us should happen.

• It does help thanks, but surely you could disprove something (using a perfectly valid set of axioms) that has already been proved using another set of axioms, but as both sets of axioms used are consistent then you will have shown that maths cannot always be consistent and that you can 'defy' logic by both proving and disproving something? Commented Jul 22, 2013 at 8:44
• @genty4321: I can say "The winner is the team which scores the most goals" and later say that "The winner is the one to control the most territory on the board." These are inconsistent, but that's okay, because one of them is about hockey (or soccer, or ...) and the other is about Go (or Stratego, or ...). Commented Jul 22, 2013 at 8:46
• @genty4321 I suppose that you are not defying logic the way you described. For instance, there are different "geometries" - euclidean, non-euclidean (which contradict the fifth postulate in some way), they are considered separately, not to be mixed together! Or, for instance, you can prove something assuming the axiom of choice, but if you assume its negation, something opposite might hold, but that doesn't disprove logic as the means of conducting your proofs at all. It's the matter of you assumptions...
– W_D
Commented Jul 22, 2013 at 8:51
• @genty4321 A proof is only valid given a set of axioms, so formally you'd have to say 'Given axioms X, then P', and then 'Given axioms Y, then Not P'. This only means that the truth value of P is dependent on whether you follow the axioms of X or Y. Then the choice on whether X or Y is the 'correct' axioms to use would depend on intuition and 'outside' logic. Commented Jul 22, 2013 at 8:54
• @dfeuer Yes but they are for two different games so it does not affect the logic Commented Jul 22, 2013 at 8:55

The axioms of mathematics come, in many cases, from the mathematical objects that are being axiomatized.

Example 1: Euclidean Geometry Many ancient cultures needed to work with geometric ideas in order to build things. Euclid famously wrote a system of axioms for plane geometry based on his existing intuition about the plane. Much later, in the 1800s, it was realized that Euclid's axioms were problematic (e.g. Euclid used Pasch's axiom implicitly but his axioms cannot prove it). Hilbert developed a different axiomatic system for geometry that did not have the problems that they ten realized Euclid's axioms had.

So how did Hilbert come up with those axioms? He already understood plane geometry! The goal was to isolate some basic properties that would be able to develop the theory he wanted.

Example 2: Topology According to the Wikipedia article on topology:

Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.

Again, these researchers had many examples of mathematical objects at hand, and the goal was to isolate certain properties of those objects and write down axioms describing just those properties.

There are many more examples that can be given, including group theory, Zermelo-Fraenkel set theory, graph theory, matroid theory, computable functions, and more. None of these was invented "out of thin air" as a set of axioms - they each developed over time as more examples were considered.

One way that progress in mathematics happens is that we see a phenomenon that happens in a particular example, and we ask what properties of the example are needed for the phenomenon to happen. If the class of all objects that have those properties turns out to be interesting, then the properties may become axioms for a new class of objects. Studying those axioms may show us new phenomena, leading to new systems of axioms, and so on.

Of course the (presumably) first axiom system was that of Euclid's Geometry. This book though was a systematisation of knowledge at the time, and it seems reasonable to suppose it started as a teaching course. If you are giving a course, you have to decide where you are going to start, and it seems reasonable to start with basic assumptions. Of course we do not know what was the evolution of this book!

Axioms often evolve through practice and trial and error. People carry out a particular kind of argument and then realise it can be carried out under certain abstract assumptions.

The virtues of abstraction are several.

1) To cover several examples, and in this to make analogies.

2) To be available for new examples, and so new analogies.

3) To simplify proofs.

The last advantage may be surprising, but the reason is that the axioms sort out the essentials which are required for the proof, and allow the casting off of excess baggage.

All this is intended to emphasise that axioms arise from lots of study, of examples, of proofs, and of other axiomatic systems.

To illustrate: we all know that 2+3 = 3 +2, and 2 x 3 = 3 x 2. To say these are examples of a commutativity law is to make an analogy between addition and multiplication of numbers.

Things get more exciting when you get an analogy between addition of knots and multiplication of numbers: see this knot exhibition.

• Another question I have is if something is proved for one set of axioms then you can use it for that set of axioms but wouldn't you have to prove it for another set of axioms before you could use if for the second set of axioms. So for instance must you prove $2+2=4$ for every set of axioms before you can use it in any set of axioms, and is this for any piece of maths? Commented Jul 23, 2013 at 11:17
• Aristotle, who died before Euclid was born, developed an deductive axiom system for syllogisms, which I believe was the first axiomatic system in Europe. I do not know about other cultures, but experience shows me not to discount them. Commented Jul 25, 2013 at 0:56

I will speak boldly and assert that the very purpose of an axiomatic system lies in developing a theory deductively. This implies that the axiomatic system will basically help you or your machine generate theorems which require little to no insight on your part (though of course, any insight you have, if correct, will help). Now what properties will axioms prefeably have if this holds true? I suggest the following:

1. Axioms preferably will be few in number. The more axioms you have, the more you take for granted. The point of an axiomatic system lies in the deductions made and having the ability to tell what deductions can get made. This becomes clearer when you have fewer axioms.
2. The axioms preferably will be independent for system S in the sense that using the same rules of inference, for any axiom A will not come as derivable from a system (S-A) with all the other axioms of S, but not having A as an axiom (some authors do NOT seem to realize this as important). In other words, using the rules of inference you've taken for granted, you can't logically deduce any axiom from the other axioms. If you can, then you could eliminate one of the axioms, and you could have deduced more, in some sense, if you had worked with the smaller set of axioms instead of the full set of axioms.
3. The rules of inference for the theory will get kept to a minimum also. In effect, if you have a lot of rules of inference, you take a lot for granted and consequently in some sense deduce less than you could have if you had fewer rules of inference. (this part may come as the most contentious.)
4. The rules of inference will come as sufficiently powerful to deduce a lot of theorems.
5. The axioms preferably will be consistent in the sense you can't derive a proposition and its negation in the theory. If you do have inconsistent axioms, then you may not have deduced anything.
6. The axioms preferably will be at least relatively easy to use, so that you can deduce more. This in part will depend on who or what does the deducing, so this does imply different axiom sets as potentially feasible, since not everyone thinks the same way, and not all machines behave the same way. For this reason short axioms oftentimes will come as preferable to long axioms, though again NOT ALWAYS, because everyone thinks differently to some degree, and every machine differs from other machines.