And the definition of axiom is ?

[Alan1000]

And the definition of "axiom" is...?

There are (broadly speaking) two basic definitions of the word "axiom".

The classical definition, with which Plato, Euclid, and Aristotle wrestled, is roughly this:

"An axiom is a proposition for or against which no evidence can be adduced, but the truth of which, it appears intuitively impossible to deny".

The modern definition is:

"An axiom is a proposition which forms the starting point for a chain of deductive argument."

An example of the former might be, "the whole is greater than the part" (vide Euclid). An example of the latter might be, "Einstein's Theory of General Relativity".

With special reference to mathematics in the 21st Century, what are the relative merits and demerits of these two definitions? Discuss.

 

[lurflurf]

The modern definition is more practical. The earlier definition is naive. There is much value in axioms that seem wrong, axioms that are wrong, compeating sets of axioms, axioms that can be deduced from other axioms, axioms that can be used to deduce other axioms, axioms that seem like they might follow from others but probably do not, and so on.It is better to not be so limited.

 

[Hurkyl]

In formal logic, an axiom is simply a statement chosen to be called as such. As lurflurf said, it's a rather practical approach -- you state axioms so you can talk about the theory they generate.

The logical/mathematical content of the classical version is the same; it is just coupled with some rhetoric. With the modern practice of basing work on hypotheses, or conjecturing certain statements are true, I don't think there is any value left -- all that remains is the more negative rhetorical aspects, such as dogmatic assertions of "truth" or the implied belittling of anyone who would look at the speaker's presumptions with a critical eye.

 

[Goongyae]

In mathematics, axioms are simply starting points assumed to be true. Derived theorems are tautologies in the sense that they are a restatement of the axioms. Whether any of the axioms or theorems correspond to any aspect of reality is up to the practicioner to find out, not necessarily the mathematician.

In the Austrian School of economics, a field which employs a lot of verbal logic and uses little mathematical calculation, the axiom "people value leisure" is considered to be empirical; it's based on watching reality.

However, the axiom that "people (conciously) act in order to remove felt uneasiness" is often considered to be a priori in the sense that considering the opposite would be absurd. In fact, one who formulates an argument against this axiom only helps reinforce it. This "a priori" kind of axiom more closely matches your first definition for axiom.

EDIT: BTW I think many of the axioms of metamathematics, i.e. basic logic, are generally taken to be a priori. I don't know of any case where non-conventional higher-order logic axioms are used. However, some mathematicians, the intuitionists specifically, will reject some things like the law of the excluded middle. For them, it's not correct to say that something which is not false is necessarily true.

 

[Goongyae]

With the modern practice of basing work on hypotheses, or conjecturing certain statements are true, I don't think there is any value left [in a priori axioms]

From Ludwig von Mises' The Ultimate Foundation of Economic Science http://mises.org/books/ultimate.pdf:

"the treatment that positivism accords... implies a theorem for which no experimental justification whatever can be provided, viz., the theorem that all observable phenomena are liable to a reduction to physical and chemical principles. Whence do the positivists derive this theorem? It would be certainly wrong to qualify it as an a priori assumption. A characteristic mark of an a priori category is that any different assumption with regard to the topic concerned appears to the human mind as unthinkable and self-contradictory. But this is certainly not the case with the positivist dogma we are dealing with. The ideas taught by certain religious and metaphysical systems are neither unthinkable nor self-contradictory. There is nothing in their logical structure that would force any reasonable man to reject them for the same reasons he would, e.g., have to reject the thesis that there is no difference and distinction betwen A and non-A.

The gulf that in epistemology separates the events in the field investigated by the natural sciences from those in the field of thinking and acting has not been made narrower by any of the findings and achievements of the natural sciences. All we know about the mutual relation and interdependence of these two realms of reality is metaphysics. The positivist doctrine that denies the legitimacy of any metaphysical doctrine is no less metaphysical than many other doctrines at variance with it."

 

[JSuarez]

With special reference to mathematics in the 21st Century, what are the relative merits and demerits of these two definitions? Discuss.

And what looks like homework, smells like homework and tastes like homework is...