Conventionalism and axioms are certain stages of development of science. That is, these statements should not become a dogma. This kind of assumption is the hypothesis that allows us to develop the submission. For example, the interesting idea that the Earth is flat and rests on three pillars. After some time, this axiom has been rejected.

The value for a set of axioms is just that of a definition.  The point of such definitions is that they allow you to prove theorems, preferably for cases in a more concrete context. Often they provide some intuitive feel e.g. using a geometric language (e.g. in topology) or generalising some well known facts to a much wider context (e.g. the definition of K-groups is a very wide generalisation of the construction of the integers from the natural numbers). However, don't take intuitive too serious. It often takes years of honing one's intuition to find axioms/definitions intuitive. It also takes years of training to not get too carried away by intuition and "prove" wrong theorems. 

Much the same is true in physics. Axioms have no place in physics, but standard models certainly have and they serve pretty much the same purpose as axioms in mathematics. The model of space as three dimensional Euclidean geometry is very much a formalisation of our everyday experience. Turns out, our experience gives us the wrong idea as we would recognise pretty much immediately if we were used to zipping around at 99% of light speed through the universe.  Physicists, of course, know this and many have spend many years honing their intuition to think intuitively of particles zapping through Lorentzian (non Euclidean) space-time or the even weirder non-local effects of quantum physics.  And again, intuition (or "common knowledge") is fallible and may lead to even experts believing wrong things. 

I don't really understand your point about value.  Good axioms are good because of their usefulness.

Dear Dr. Rogier Brussee,

thank you for your most valuable contribution.

It came to my mind the following observations.

Considering a classic fundamental text as the Euclid's Elements, in which the "common notions" (koinai ennoiai), which are the axioms of the text, have mostly a universal value, in the sense that you could put them between the premises of any discipline,it is stated, for example, that two things equal to the same thing are equal to each other, that by adding and subtracting the same things to equal things are still obtained equal things, etc.

The only possible exception is – maybe -constituted by the fourth of these common notions, which says that things "coincident" are equal to each other. Today this "coincidence" is understood as congruence of geometrical figures, so that is specific of geometry to affirm that two geometric figures are equal (or equivalent) if congruent, that is, if they can be superimposed on each other by means of transformations that do not the "deform" them (isometric). From this point of view, the fourth of the common notions of Euclid should be regarded as specific to geometry, but it seems that Euclid meant it in more generic terms, and not by chance today the concept of congruence has spread to almost all areas of mathematics, where denotes generally a kind of equivalence. As the postulates (aitemata) of Euclid, they are all specific to geometry.

While the approach described above is moreover quite compatible with Euclid's Elements, it is not at all compatible with most modern terminological choices, such as those that they have  imposed from the Newton's Principia and up one of the basic texts of modern deductive disciplines, as are the Foundations of Geometry (1899) by Hilbert.

Newton, in fact, defines axioms his famous three laws of motion, and for the rest they appear between the premises only of definitions. In Newton, therefore, apart from maintaining the claim to define the terms, all the axioms are "specific" to physics.

As for Hilbert, he is the founder of that modern approach which intends to give up the claim to define the terms explicitly, putting together a sufficient number of assumptions, such as to allow to be able to deduce theorems even without having any mental representation of what to which the terms refer. Thus Hilbert eliminates all definitions widely increasing the number of premises from which to take the moves, after which all these premises call them indifferently axioms, and speaks of axiomatic system to indicate any deductive system, while the inherent discipline is generically defined axiomatic .

As Newton and especially Hilbert employ this terminology it seems to emerge the desire to define axiom any premise on which it can be made a deductive system, giving up altogether to speak of postulates.

This is partly true, but we must also observe that in the introduction of these texts appear only those that the modern theory of formal systems would define formulas, or expressions that give certain properties to the terms or they make certain relationships exist between two or more terms. Instead, they never appear among those conditions, the rules of inference, that is, those premises that establish under what conditions you may deduct certain other formulas. The inference rules are in fact those commonly understood as "basic principles" which underpin the deductive process, and are not included in the introduction of an "axiomatic system" as Hilbert’s.

Thus, in spite Hilbert’s choice of words of Hilbert and others seem to indicate a desire to bring every premise to an axiom, a closer analysis of the structure of these systems seems to indicate the need to reserve that term only for the formulae, excluding the inference rules.

Having done that, if you decide to define postulates the inference rules, having the latter mostly a "logical character, you would have a total reversal of the first approach outlined previously, that is more or less compatible with the lexicon of Euclid's Elements. To avoid such a contrast between the various options, and also because there is no valid reason to define postulates only the inference rules, it is preferred to extend the concept of postulate to all the premises of a deductive system, whether logical or specific, be formulae or rules of inference, after which the postulates that are not inference rules are defined axioms.

Thanks for reading and have my best regards.

Gianrocco Tucci

Dear mr. Tucci,

I am afraid I don't quite understand. In mathematics the axioms are the premises. As I heard a mathematician say "I can define what I want".  The rules of inference are part of them, even if they are not always explicitly stated. Of course, you can define what you want, but you can not prove what you want once you have stated the rules. It is not so common to change the rules of logic (although people do tend to worry a bit about using the axiom of choice), but in the end logic can be considered as just another branch of algebra, and just as in other branches of algebra you can change or relax the rules. E.g. the classic Aristotilean logic allowing proof by contradiction can be replaced by other forms of logic like intuitionistic logic (which tends to be not so intuitive), which then makes some things unprovable although that includes the existence of nasty counterexamples! Computer scientists have used a subtly different form of logic for automatic theorem provers like the coq system and they have used it both for checking the correctness of computer processors and proving the Feith Thomson theorem (a highly non trivial theorem in group theory) from basic axioms. In fact, there is a rather interesting program by Voevodsky for new "Grundlagen" based on these ideas and ideas in algebraic topology. 

In Hilberts time the "Grundlagen" discussions were  raging. You can argue that social, philosophical and economic developments made the time ripe for such discussions, but I believe the most important input in these discussions was the fact that mathematical apparatus had progressed to the point where you run into problems without a formal apparatus. Something as basic and in everyday mathematical use as Lebesgue integration and measure theory simply does not work if you don't know that the cardinality of the real numbers is strictly larger than that of the integers, i.e. the naive and intuitive notion of "infinity" is just not sufficient. Without formalisations like set theory you cannot even express or think about different notions of infinity.  On the other hand once you have the proper formalisation (in this case set and measure theory) and have learned to say things like "the countable union of measurable sets is measurable" without so much as blinking an eye (i.e you have reached the level of a second year undergraduate math student), you can forget about the finer points of set theory and the whole machinery of measure theory just makes it easier to work with integrals and partial differential equations.

So to conclude, the formalisation that the Goettinger school was certainly a leading proponent of, turned out to be a necessity to make progress.  Having said that, one cannot deny that a lot of formalisation has taken place because generalisation and formalisation has become to be viewed as valuable in its own right, sometimes forgetting  (free after Einstein)  "as formal as necessary, but not more formal".

There is a nice discussion (without any formulas) of the issue of rigor and intuition in mathematics by some of the best living mathematicians in 

http://arxiv.org/abs/math/9404229 (reacting to http://arxiv.org/abs/math/9307227 by Jaffe and Quinn). 

Especially noteworthy are the contributions by Michael Atiyah, Renee Thom and Morris Hirsch.

Dear Rogier,

have my sincere thanks for your excellent exposition of (if I may say) the "state of the art". I'll certainly read the reference you suggested with a great pleasure as it is written without formulae.

I hope to hear from you again sinceI have learned with profit your interventions in RG.

Best regards,

Gianrocco