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I tried to explain to my kids what is an axiom in very nontechnical terminology (also known as layman's term) but I could not find anything that they could relate to easily.

Do anyone have a good explanation?

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    $\begingroup$ Axioms are the truths you cannot deduce from other axioms :) $\endgroup$ – Yves Daoust May 3 '16 at 10:57
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    $\begingroup$ @emory: "axioms are true" is an axiom. $\endgroup$ – Yves Daoust May 3 '16 at 12:04
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    $\begingroup$ @emory: What Yves says is completely false. The axioms of a first-order theory need not be sentences that cannot be deduced from other axioms, and furthermore the definition of axioms has nothing to do with truth. See Ittay's answer for an explicit description of what I am talking about. $\endgroup$ – user21820 May 3 '16 at 12:32
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    $\begingroup$ @user21820: ok, in the same vein, I am adding "axioms are false" as a second axiom, hoping no contradiction. $\endgroup$ – Yves Daoust May 3 '16 at 14:25
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    $\begingroup$ "axioms are true" is an axiom, is an axiom $\endgroup$ – Matt May 3 '16 at 14:44

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In mathematics, every result known descends from something else: it is proven to be true from other facts.

The one exception is axioms: these things we choose to accept without proving them.

We have to choose some axioms, since we cannot prove anything with nothing, but we try and make them as simple and obvious as possible.

For example, Euclidean geometry rests on five axioms, the first of which is "given two points on a plane, it is always possible to construct a straight line passing through these two points". Another states that it is possible to draw a circle with any center and radius.

Using these simple statements, Euclid then proceeds to prove more complex properties of figures on the plane.

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    $\begingroup$ Hi @riccardo-orlando, thank you for your answer. You have the material to be a great teacher and a great researcher. All the best for the future. $\endgroup$ – Andy K May 3 '16 at 7:04
  • $\begingroup$ @riccardo-orlando so does that mean a conjecture is an axiom ? I assume that this definition of conjecture "Conjecture is something that's true but isn't proved(Only conjecture I know is Goldbach's Conjecture , I framed the definition keeping it in mind) " is true ? $\endgroup$ – brainst May 3 '16 at 16:07
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    $\begingroup$ Goldbach's conjecture is just a conjecture. It looks pretty reasonable to assume it's true, but we don't assume anything. However, the instant you write "Assume Goldbach's conjecture is true..." now you have an axiom $\endgroup$ – Cort Ammon May 4 '16 at 3:25
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    $\begingroup$ @brainst In many situations, we assume the laws of thermodynamics to hold true. I think what makes the definitions difficult is that we use the word "law." Really, the "laws of physics" are merely statements which are assumed axiomatically true so often that they look at you really funny when you claim they might not be true. As it so happens, the laws of thermodynamics are not actually true all the time. There are "non-equilibrium thermodynamics" problems for which the laws of equilibrium thermodynamics don't 100% apply (for example, it can be hard to define entropy in those systems) $\endgroup$ – Cort Ammon May 4 '16 at 20:19
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    $\begingroup$ And the laws of thermodynamics are currently believed to have been violated at least once. We don't really have a thermodynamic model of the first instants big bang, because the energy/mass has to come from somewhere. Much of modern cosmology is trying to stitch together a potential birth of our universe which does follow the laws of thermodynamics as we know them, but it's not easy. $\endgroup$ – Cort Ammon May 4 '16 at 20:21
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In former times, axioms were considered to be statements that are so simple and "obviously true" that they cannot be proved (or any attempt to prove them would need to be based on more complicated things - and why bother proving it at all if it is obviously true?).

In today's understanding, an axiom is a statement that is, for the sake of developing a specific theory, taken for granted. For example, the axioms of Euklidean geometry (there is exactly one line passing through two given distinct points" and so on) can be used to rigorously prove all theorems of Euklidean geometry. There is but no inherent "truth" to the theory as such. However, if we verify that the axioms of Euklidean geometry hold for some things or phenomena (e.g., for points and lines and drawn on the Earth's surface) then we can be sure that also all theorems (such as Pythagoras) apply to these. But as soon as we notice that "straight lines" on the Earth surface such as meridians are not as parallel as they should be, the Euklidean theory is no longer fully applicable to these phenomena.

Nevertheless, a "good" axiom system is one that allows us to apply it to interesting things. As among those interesting things are the theory of natural numbers, geometry, and set theory, certain axiom systems (Peano, Euklid, Zermelo-Frenkel) relating to these theories have become somewhat standard.

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    $\begingroup$ is the consistent spelling 'Euklid' intentional? $\endgroup$ – Ittay Weiss May 3 '16 at 6:39
  • $\begingroup$ Hi @Hagen-von-eitzen, many thanks for your answer. It is a good one for the kids. I'll have a better ammo when I'll explain it to them. $\endgroup$ – Andy K May 3 '16 at 6:59
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    $\begingroup$ I guess Hagen is German, like me; we spell Euklid this way. $\endgroup$ – Guntram Blohm May 3 '16 at 7:10
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    $\begingroup$ @IttayWeiss Guntram's guess is right. I can never remember to adjust the spelling of Εὐκλείδης to English habits. Sorry for the .. inkonvenience $\endgroup$ – Hagen von Eitzen May 3 '16 at 14:01
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    $\begingroup$ it's quite $\omicron \kappa$ @HagenvonEitzen $\endgroup$ – Ittay Weiss May 3 '16 at 20:43
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To a mathematician, or a philosopher, in the pre-modern era (and to some crackpots of the modern era) an axiom refers to a truth so basic that it cannot be proved or argued about from other truths. A basic elementary fact to 'obviously' true that the only way to argue about it is to tautologically proclaim "it simply is true!!!". For instance, the fact that addition of natural numbers is commutative, namely that $m+n=n+m$ for all natural numbers $m,n$, might be taken as an example of an axiom in that sense about the natural numbers.

In the modern era, we are no longer obsessed about what things really are (whatever that means) and thus we no longer treat axioms as described above. Instead, we recognise that mathematics is a formal system consisting of a language in which to express yourself, a logical system one uses to manipulate statements, and an initial collection of statements we take to be true since we wish to see what the consequences of the properties expressed by these axioms are. Thus, any statement at all can be taken as one of many axioms. For instance, one may wish to study the natural numbers axiomatically by choosing no axioms at all. Well, without breaking eggs, you can't make an omelette, so all you'll get are tautologies. On the other extreme, you may take as axioms all the statements true about natural numbers. That will be a terrific achievement, but, of course, it is not realistic you can ever list your axioms in this case. The golden path lies in finding a clever, efficient, formulation of properties (of the natural numbers) that you deem to be characteristic, and state those as your axioms. So, in the modern context, the axioms of a system are simply an axiomatic description of some properties of whatever you wanna study (rather than an attempt to describe what things are, we describe what you can do with those things). For the natural numbers, the Peano axioms do the job.

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    $\begingroup$ You might recognise. I, on the other hand, recognize! :-) $\endgroup$ – Asaf Karagila May 3 '16 at 6:40
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    $\begingroup$ as long as we agree that there is no recognishion involved. $\endgroup$ – Ittay Weiss May 3 '16 at 6:43
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    $\begingroup$ Hi @IttayWeiss, many thanks for the historical explanation of an axiom. It'll prove handy. It is a funny one. Younger, you were learning and you only need to understand the things you've learned for yourself. And now all of the sudden, you are learning with someone's else e.g my kids in this instance, and you see how much pedagogy it is needed to break something big (and sometime not so big like here but not that evident) into manageable parts. Thanks again $\endgroup$ – Andy K May 3 '16 at 7:08
  • $\begingroup$ you're very welcome @AndyK enjoy! $\endgroup$ – Ittay Weiss May 3 '16 at 8:06
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From a mathematical point of view, axioms are the fundamental self-imposed rules of the games which mathematicians play. For example, a considerable part of mathematics deals with structures (think of numbers) for which the rule holds that you can replace $a·b$ with $b·a$. From these fundamental rules, all other rules can be deduced – actually that’s already the actual game: deducing rules from existing rules. For example if you have the rules that $3·4$ can be replaced with $12$ and that $12−7$ can be replaced with $5$, you can deduce the rule that $3·4−7$ can be replaced with $5$.

From a practical point of view, reality also follows certain sets of rules. For example, if we tie two pieces of string together, the length of the resulting string is the same, no matter in which order we tie them together. These sets of rules happen to be the same as those which mathematicians like to play with. In the string example, it corresponds to the rule $a+b=b+a$. Hence, if mathematicians deduce a new rule from the axioms (the fundamental rules), this rule also applies to reality and allows us to make statements about reality. In fact, mathematicians only came up with certain sets of rules because they reflect reality (and they wanted to eat). Sidenote: What makes mathematics so damn useful is that a mathematical set of rules can often be identified with many real sets of rules.

It should be noted that when I was saying “reality follows certain sets of rules”, I should have said: “reality seems to follow certain sets of rules”. We do not actually know that the rules that we assume to be followed by reality actually are followed by reality all the time. We have very good reason to assume so, because we have tested most of them for thousands of years and have not found any counter-examples, but that’s as far as it goes. Mathematicians tend not to worry about this but just assume that the axioms are true. There are two reasons for this:

  • Mathematicians do not want to worry about what rules hold for reality. They leave that job to physicists, chemists and other scientists (who are much better at this anyway).

  • As already mentioned, a single mathematical axiom can often be applied to many real-life objects, problems, and similar. Thus, whenever mathematicians find a new rule to deduce from axioms, it can be applied to many real-life problems. If mathematicians worried about whether axioms are true in each possible application, they would never get to do any actual mathematics.

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  • $\begingroup$ Hi @Wrzlprmft your answer is very neat and detailed. But how do I explain your answer to a 14-years old and 11-years old respectively...? $\endgroup$ – Andy K May 3 '16 at 13:02
  • $\begingroup$ @AndyK: Rules of games, elementary arithmetics and pieces of string should be something that children can relate to. And you can probably directly relate the rules-of-games aspect to their current mathematics education. And to be honest, only the first paragraph is the actual answer to the question, the rest is only to answer the “but, why?”. $\endgroup$ – Wrzlprmft May 3 '16 at 13:09
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    $\begingroup$ I understand your explanation. It is an interesting one. +1 for the effort. $\endgroup$ – Andy K May 3 '16 at 13:11
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layman's terms

Axioms are things you have to assume to get started thinking about something.

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  • $\begingroup$ Hi @user1717828 the only answer's I've started thinking is the following mmmmhh... (2 ing, you beat me) $\endgroup$ – Andy K May 3 '16 at 9:29
  • $\begingroup$ I have to use axioms to find my car keys? I see where you're getting at, but that is stretching the definition a bit. $\endgroup$ – Neil May 3 '16 at 14:50
  • $\begingroup$ Perhaps 'deducing' instead of 'thinking about'. $\endgroup$ – Dennis Jaheruddin May 3 '16 at 21:42
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    $\begingroup$ @Neil: If you modeled "find car keys" mathematically then yeah, there would be axioms somewhere in there. $\endgroup$ – GManNickG May 3 '16 at 22:47
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    $\begingroup$ @Neil: Nobody here is suggesting that when you can't find car keys, you got your axioms wrong (whatever that means). If you actually took the time to model the process of finding car keys, there will be axioms. E.g., "Keys exist" might be an axiom; if not that, then "Some objects exist, and there is an object I call my keys"; if not that, then "The universe exists, in it are objects, etc....". At some point, you assume some fact as true because you need to do make any further deductions. This doesn't mean you're always consciously modelling things mathematically, of course. $\endgroup$ – GManNickG May 4 '16 at 19:46
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When you want to build a house, you have to put in a foundation first, no matter whether this house is a mansion or a prefab hut.
But what do you put the foundations on? There has to be a building site - some ground that is marked out and fenced off and leveled.
This house is our theorem or lemma; we add wing extensions as a corollary. The foundation is our set of assumptions, the things we need to be true in order for the house to stand up. And the site that the foundations go into/onto, is defined by the axioms we use.

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My favourite axiom demo is Peano's first which reads Zero is a number.

It is clear and obvious to be true but clearly difficult (pointless?) to prove.

There's also the Reflexive Property which states:

For any quantity a, a = a.

Although this may not (technically) be an axiom it has all the properties

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  • $\begingroup$ Hi @OldCurmudgeon, yeah! I loved that one. I was going on a long rant with water as a liquid to the kids. But your example beats mine in simplicity and elegance. $\endgroup$ – Andy K May 3 '16 at 13:25
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    $\begingroup$ @AndyK - I also recall from my university days (a looong time ago now) that 1 = 1 is also an axiom of a kind but I cannot find a reference to post. $\endgroup$ – OldCurmudgeon May 3 '16 at 13:35
  • $\begingroup$ I loved the a looong time ago now and now we are teaching our kids. #Timeisflying $\endgroup$ – Andy K May 3 '16 at 13:45
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Things whose existence is taken for granted either due to common past experience and practice or a given thing handed down to us and unquestioningly adopted. When questioned, one cannot give a previous basis (blinks, scratches head etc.)

Often the axiom is at a foundational or fundamental level as a sort of primordial premise for consequential if-then common logic of syllogism. When such an axiom is changed and fresh attachments of logic are made with new axioms strange surprising and unexpected experiences and results ensue.

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  • $\begingroup$ Hi @Narasimham, not sure why your answer did not garner more vote. It is a nice one. $\endgroup$ – Andy K May 4 '16 at 9:55
  • $\begingroup$ Thanks for your sentiment., @ Andy K. Just wishing to share in and being able to express ... that is sufficient satisfaction already. In this case we know how non-euclidean geometry took 18 centuries to be born after Euclid. The fifth postulate I am referring to.Even the big Gauss was averse to revealing his mind afraid of what the commoners may think or reject.. Pavlov showed how common perception is conditioned.. you join the bandwagon and all would be well !! $\endgroup$ – Narasimham May 4 '16 at 10:32
  • $\begingroup$ Don't worry @narasimham, you're a good mathematician. To be a great mathematician takes time, loads of rebuke and the will to get-up again after a each rebuttal. If you have the will, you will do better. If you look at Marie Curie, how many rebuttal did she has to endure...? ;) $\endgroup$ – Andy K May 6 '16 at 7:58
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Axioms are rules, and the game is to show what's allowed by the rules.

Just like rules, axioms are true because they we say they are true. Also just like rules, we can impose whatever axioms we want!

In the real world, the rules we follow are generally the ones everyone agrees upon (like not stealing) rather than some ones people make up arbitrarily. Similarly, while mathematicians can impose whatever axioms they want, they often focus on axioms that are agreed upon to be interesting or important in some way, rather than ones that are made up arbitrarily.

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In simple terms, an axiom is a statement which is widely recognized to be true. In mathematics this statement is something like the fundamental theorem of algebra, whose validity is is hardly worth questioning.

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  • $\begingroup$ -1 Simple but completely wrong. And the fundamental theorem of Algebra's validity is hardly worth questioning? That is a joke, isn't it? $\endgroup$ – miracle173 Jan 26 '17 at 6:05
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Mathematicians occupy themselves with questions like 'If fact A is true, what else turns out to be true?'. Once you get started, one fact leads to another and another. However, you have to get started somewhere.

Axioms are the things we assume to be true at the start of a thought process so that we can proceed to explore the consequences of those assumptions.

Examples are:

  • Assume that there is an operation called addition such that a + b = b + a for all values of a and b

  • Assume that there is a special number called 0 such that a + 0 = a for all values of a

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