What is the difference between an axiom, a hypothesis and a postulate? I'm a student of physics and not know much of philosophy. All I know is that axioms (for example, the axioms of Euclidean geometry) cannot be mathematically proved. But so does a postulate and a hypothesis.

  • It is a common misconception that 'axioms cannot be proven', this is not the case. Axioms are assumed to be true without a proof. Some axioms in certain contexts are redundant, meaning they can be proven form the rest of the axioms, for example the axiom of pairing is redundant over the rest of ZF set theory. That doesn't mean the axiom of pairing isn't an axiom. Axioms are assumed to be true without proof, not always unprovable. Postulate is 99% of the time synonymous with axiom.
    – Not_Here
    Jun 25 '18 at 14:38
  • 1
    In fact, one of the inference rules you're allowed to use during a proof is to just restate the axioms. This is in a round about way, a proof of the axiom itself. Obviously this isn't a convincing proof because it starts with the assumption that A is true and ends with A being true, but it's still in the technical sense of the word a 'proof'. So it's not that axioms cannot be proved, it's that they're trivial to prove since we already assume them to be true. Sometimes other axioms we assume are strong enough to prove them on their own, which leads to redundancies.
    – Not_Here
    Jun 25 '18 at 14:46
  • Axiom and postulate : NO difference. Jun 25 '18 at 14:57
  • @MauroALLEGRANZA, "Postulate" can be used as a verb. "Axiom," not so much. Jun 25 '18 at 15:05

Axiom is from the Greek ἀξίωμα :

that which is thought fit, a requisite; that which a pupil is required to know beforehand; a self-evident principle.

Postulate is from the Latin postulo :

I ask, request, desire, demand. I pretend, claim.

See : Anthony Lo Bello, Origins of mathematical words, John Hopkins UP (2013), page 41 :

Euclid did not use the word axiom, instead calling his geometrical assumptions αἴτηματα, demands, postulates, and the nongeometrical ones koinai ennoiai, common notions. It was Proclus (fifth century A.D.), in his Commentary on the First Book of Euclid's Elements, who first called the latter statements axioms, and his decision has prevailed to this day.

The medieval Latin translators rendered αἴτηματα by petitiones, requests, and koinai ennoiai by scientia universaliter communis, knowledge common to everyone.

αἴτημα is from αἰτέω : I ask, request, demand.

Thus, it has the same meaning of the Latin : postulo, that is the correct Latinn translation of aitēma.

See also Aristotle, Post.An., I :

[76a32] I call principles in each genus those which it is not possible to prove to be.

[76a38] Of the things they use in the demonstrative sciences some are proper to each science and others common — but common by analogy, since things are useful in so far as they bear on the genus under the science. Proper: e.g. that a line is such and such, and straight so and so; common: e.g. that if equals are taken from equals, the remainders are equal.

Thus, we can sey that Euclid organizes his Elements according to Aristotle's view :

there are five common notions (later called: axioms), i.e. self-evident truths of a general nature, and five postulates (αἴτηματα), principles specific of the geometrical science.

Hypothesis is from Greek ὑπόθεσις from hupotíthēmi, “I set before, suggest”.

See also : Arpad Szabò, The Beginnings of Greek Mathematics, Reidel (1978), Ch.3.26 The difference between postulates and axioms, page 302 :

One of the facts which has emerged from our discussion [...] is that aitemata and axiomata were synonymous. Furthermore, the manner in which they were used by every ancient mathematician except Euclid suggests that these terms were interchangeable with a nuber of other words having to do with the foundations of mathematics.

For example, Book I of Archimedes' On the Sphere and Cylinder begins with two lists of such principles. Tho assertions in the first list are called axiomata and are obviously definitions, whereas those in tho second list are called lambanomena and seems to be related to Euclid's postulates.


Axioms and postulates are very similar.

Axioms are taken to be self evidently true (usually) and tools for further reasoning.

A postulate is some assumption which you consider true simply for the sake of argument. It may not be true.

A hypothesis is a proposed answer to some question or some general truth claim. Usually this refers to a truth claim made for empirical reasons such as to explain some set of observed facts.

I also have a background in physics so I'll give you an example:

The laws of logic are taken as axioms. Mathematics also has certain logical axioms (though it's a topic of debate as to whether or not you can fully reduce mathematics to classical logic). They are taken as self evident truths. E.g. the law of non-contradiction. It seems evidently true that something cannot simultaneously be and not be.

When Einstein develop special relativity (SR), he famously applied his 'two' postulates (even though there are actually more required to derive SR but that doesn't matter).

One of the postulates was that the speed of light 'c' in a vacuum is constant. He used this as a postulate since Maxwell derived the speed of electromagnetic wave propagation in 'free space' and (as far as I remember) it turned out that his value was very very similar to the value of the speed of light which had already been measured. Einstein decided to assume this was the case and reason from that to understand the consequences, even though this was not conclusive and could have been false.

Using these postulates, he used the axioms of logic and mathematics to derive SR which contains some set of hypotheses about how the world should work. Then these hypotheses are subject to empirical test.

Since the empirical successes of SR, it is considered 'true' so has been kind of elevated to the status of an accepted theory though, fundamentally, it is a set of hypotheses.

That's what they all mean and how they are all related.

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