# Are axioms/postulates always so self-evident that they don't need any proofs

## Main Question or Discussion Point

Hi

I'm a math layman so please be simple and straightforward. Thanks.

Are axioms/postulates in any field always so self-evident that they don't need any proofs? I could say '1+1=2' is an axiom, am I allowed to say this, or are there some requirements for an axiom to qualify as one? Could an axiom be a nonsensical truth? I suppose it could be. It depends on who is interpreting it and level of their knowledge. There is an axiom in theory of relativity that light always travels at the same speed no matter what. Well, for me it just flies over the head because I'm dumb in these things. Could you please help me with the queries on the axioms/postulates? Thanks a lot.

PS: I'm not really sure if it's really a math related stuff, if it isn't then please move it to the appropriate section. Thanks.

D H
Staff Emeritus
Axioms never need proofs. Google the term.

gb7nash
Homework Helper
All it is, is a logical starting point.

epenguin
Homework Helper
Gold Member
'Axiom' is really a term of mathematics not physics. You quoted something in relativity, which is physics.

However parts of physics may be so well established, or such useful models that they can be treated pretty much as pure mathematics and developed as formal axiomatic system. As in mathematics they don't even have to be true for that, though if they are not true enough to be useful they will not be popular. I am not a physicist, but I think some physicists go for that, more regard it with wry detachment, but we will soon hear.

For math proper, yes originally in the first example still regarded as a sort of model or prototype, Euclidean geometry, they were thought to be self-evident. And a statement about the real world. Later it emerged that other less self-evident sets of axioms worked as mathematics, and that being about the real world is not a mathematical fact. Ordinary arithmetic too has axioms which appear self-evident and about reality, but much the same could be said as about geometry.

No, they do not have to be self-evident. You will soon find some that are not self-evident because you cannot make head or tail of what they mean. For purists they don't mean anything. It is not obvious why those axioms have been created.

As well as axioms to do anything with them you need definitions. You have axioms about points and lines for example, but then you need to define circles and triangles to use the axioms on.

It is often said axioms can be anything we find amusing. I would like to invite criticisms of that statement: either it is not true or there must be some criterion of what we find amusing.

Anyway there are some limitations. A system is of no interest if it is too artificial, limited in scope or application. Questions about Roman numerals for instance are not interesting. Then a system of axioms might be uninteresting because it is just an example of a more general one. On the other hand sometimes that is of interest. Sometimes it is of interest to strip down a system and find which of its result (and concepts) come from the more limited system. Then it has to be consistent. Well, not obviously self-inconsistent to start with. But could it be non-obviously inconsistent? I expect we shall hear more about that. I would like to know if they have ever gone on developing a system for any long time and then found it inconsistent. Then there is a criterion of usefulness perhaps not be be thought of in too limited and literal a way, though there is the necessary-for-electrical engineering kind of usefulness.

The bit above about the stripped-down axioms applies in science outside math and is sometimes overlooked and mistaken. You have a theory, which involves a number of assumptions, and you say the experiments agree with it so back it up. But unless you know what you can strip out of the assumptions and still predict those results you don't actually know, and can illude yourself, what the experiments have told you.

chiro
Hi

I'm a math layman so please be simple and straightforward. Thanks.

Are axioms/postulates in any field always so self-evident that they don't need any proofs? I could say '1+1=2' is an axiom, am I allowed to say this, or are there some requirements for an axiom to qualify as one? Could an axiom be a nonsensical truth? I suppose it could be. It depends on who is interpreting it and level of their knowledge. There is an axiom in theory of relativity that light always travels at the same speed no matter what. Well, for me it just flies over the head because I'm dumb in these things. Could you please help me with the queries on the axioms/postulates? Thanks a lot.

PS: I'm not really sure if it's really a math related stuff, if it isn't then please move it to the appropriate section. Thanks.
Axioms are basically a set of minimal assumptions that mathematicians make and build on them to prove properties about particular systems or representations.

Usually you don't want to have any axiom that is unnecessary: ie if one axiom can be proven from other axioms, then you discard that axiom from your baseline axioms.

Most 'objects' in mathematics whether they be groups, sets with special properties, vector spaces and so on have a minimal amount of axioms which mathematicians study and prove a variety of results that basically start from the axioms.

They do this because a lot of these "things" like vector spaces are commonly used in different areas including the physical sciences. So if there is a result that someone needs about some type of object (example vector space), then someone can take that result and use it for whatever purpose they had in mind.

You can think of axioms as rules that define, rather than rules that describe. They define the rules. As they define the rules, they have no logical precedent, and thus no need for proof.

Theorems, on the other hand, describe truths derived from the rules defined by axioms. They describe implications of the rules set for as axioms, and thus require proof of the validity of those implications.

disregardthat
It is a common misconception that accepting axioms is a type of suspension of disbelief. It's not about being self-evident, not even evident, because they do not say anything that for which evidence could be given in a meaningful way. E.g. the different geometries does not contradict each other. It is analogous to accepting the rules of chess as the true rules of chess, meaningless to say the least. Might they not be false, might not all chess players be playing the wrong set of rules for chess? Of course not.

The self-evidence of axioms only seem to make sense when the mathematics that evolved on this basis is utilized in, say, physics; but then of course it's about real evidence, not self-evidence.

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Thank you, everyone.

I understand it a bit more now. Jarle, you say it reall well: "It is a common misconception that accepting axioms is a type of suspension of disbelief." So, are the 'basic' rules of any game such as chess also axioms? I don't want you to take me into discussions of time because I won't understand much and it would be waste of your energy and time. They say that light travels at a constant speed is an axiom/postulate in theory of relativity. Is it really? I would say it should be something like a fact because it's about something 'physical'. I wouldn't say a horse always runs at a constant speed because it's false. If I had said a horse in the fastest running animal known to humans, even then it would be a fact not an axiom. Doesn't axiom give room to something like 'artistic license' where you can play around with things? Please guide me. Thanks a lot.

chiro
Thank you, everyone.

I understand it a bit more now. Jarle, you say it reall well: "It is a common misconception that accepting axioms is a type of suspension of disbelief." So, are the 'basic' rules of any game such as chess also axioms? I don't want you to take me into discussions of time because I won't understand much and it would be waste of your energy and time. They say that light travels at a constant speed is an axiom/postulate in theory of relativity. Is it really? I would say it should be something like a fact because it's about something 'physical'. I wouldn't say a horse always runs at a constant speed because it's false. If I had said a horse in the fastest running animal known to humans, even then it would be a fact not an axiom. Doesn't axiom give room to something like 'artistic license' where you can play around with things? Please guide me. Thanks a lot.
What happens in the physical sciences is that if some theory stands the test of time and the test of many experiments its usually upgraded to a law. Examples include Newtons Laws, Laws of Thermodynamics and so on.

I know I'm going to get flamed for this, but most experiments simply test very specific results and the results are very constrained to some particular subset of the science. I don't know if anecdote is the right word but in the context of science as a whole it makes sense to refer to it in this way.

What happens generally is that scientists find anomalies and then try to analyze and explain them. When the results don't fit in current theory, then it has to be changed and that often means generalizing the assumptions in some manner.

Chiro, thanks a lot. If you don't mind, would you please directly answer some of the questions in my previous post because this way I would learn more and could easily clear my misconceptions. Anyway, I do appreciate the help I have received from you. Thanks.

Thank you, everyone.

I understand it a bit more now. Jarle, you say it reall well: "It is a common misconception that accepting axioms is a type of suspension of disbelief." So, are the 'basic' rules of any game such as chess also axioms? I don't want you to take me into discussions of time because I won't understand much and it would be waste of your energy and time. They say that light travels at a constant speed is an axiom/postulate in theory of relativity. Is it really? I would say it should be something like a fact because it's about something 'physical'. I wouldn't say a horse always runs at a constant speed because it's false. If I had said a horse in the fastest running animal known to humans, even then it would be a fact not an axiom. Doesn't axiom give room to something like 'artistic license' where you can play around with things? Please guide me. Thanks a lot.
Best wishes
Jackson

Would someone please step in to help me?

Hi
I'm a math layman so please be simple and straightforward. Thanks.
It's hard for us to be simple and straightforward -- it goes against our nature. ;) However, I think I can do it just this once. Have a look at this: http://mathworld.wolfram.com/Axiom.html

If you still have concerns, reply back and we can get back to the lofty, nonlinear discussions that we all love so much.

There is an axiom in theory of relativity that light always travels at the same speed no matter what. Well, for me it just flies over the head because I'm dumb in these things. Could you please help me with the queries on the axioms/postulates?
In Special Relativity, it is a postulate that the speed of light is constant and the theory is built on this "assumption". If it can be shown that the speed of light in a vacuum is not in fact constant, then the theory is invalidated. Postulates may be thought of as "reasonable assumptions offered without proof". That does not mean postulates are obvious, self-evident facts. For example is I have postulate 1: "All dogs are bigger than cats" and postulate 2: "All cats are bigger than horses", then I can construct a theory that "All dogs are bigger than horses". This is reasonable theory as long as the postulates are sound. However, if it can be shown that in reality, some horses are in fact bigger than cats, then postulate 2 is false and the theory is false. So while you have "artistic licence" to pick whatever postulates you like, any theory based on them can be invalidated if your choice of postulates is not sound.