Can Mathematical Axioms Be Arbitrary?

[Svensken]

Hello everyone. I was wondering wether people think that mathematical conclusions can be provisional?

I know about conjectures, but are they really part of maths?

Finally, my definition of an axiom is: self evident and accepted

Do you people think that there are better definitions than mine? (please make them short and nice).

Fun to hear what you think!
-Svensken

 

[arildno]

Axioms:
Stated, arbitrarily chosen "rules of the game".
Many such rules can be picked, each set defining ONE mathematics, just like Gin Rummy and Poker are two different card games due to different sets of rules.

A set of rules is satisfactory if it is not inconsistent (meaning that there does not exist two valid sequences of rule applications leading to contradictory results), and it is USEFUL if it does something more than whiling away the time spent upon playing it.

 

[JSuarez]

I was wondering wether people think that mathematical conclusions can be provisional?

It's not impossible, but unlikely. What happens is this: the steps mathematical proofs are grounded on logical rules, and these, contrary to the widespread opinion, are not given; we do believe that the rules we use are sound, but it's not impossible that they could be, eventually, revised. Historically, the opposite happened more: for example, reductio ad absurdum proofs were only accepted as sound in mathematics in the XIXth century, and today this is regarded, by most people, as a sound logical principle; the general trend has been, so far, mainly one of extension, rather than restriction but, in the future, it may happen otherwise.

I know about conjectures, but are they really part of maths?

Yes. In fact, there are results that are accepted today as most like true, but they have the form of implications: they would be true if we knew for certain that $P \neq NP$, that the Riemman Hypothesis holds, etc.

Finally, my definition of an axiom is: self evident and accepted

That's close to the historical definition of axiom; they were tought as propositions not only accepted, but true, in the strongest sense of the word. In fact, there was a distinction between axioms and postulates: the former could be eventually revised, while you had to be pretty damn convincing to persuade people that the latter need revision.
Today, the notion of axiom is more flexible and pragmatic: they may not be self-evident, but if they are adequate, they may be accepted; they may also be rejected if they appear, at first, self-evident, but later are shown to lead to inconsistencies.

 

[Studiot]

Why do axioms need to be self evident?

Are any of the postulates of Euclid 'self evident' ?

And if they are so self evident why can we drop one for Rieman?

 

[arildno]

Why do axioms need to be self evident?

They don't have to be.

However, axioms ought to be chosen so that "most" questions posed are decidable by the axioms.

Are any of the postulates of Euclid 'self evident' ?

Some would say so, certainly Euclid, with one major exception:
The parallell postulate.

He disliked it so much that he tried to do without it.

 

[wisvuze]

the "rules of the game" is my favourite explanation of axioms.. like Feynman said, there will always have to be something you must assume or take for granted - or else, you'll have a problem of infinite regression

 

[Rasalhague]

Why do axioms need to be self evident?

They don't. As Arildno put it, they're just the rules of the game, and there is more than one game in town ;-) All they have to be is self-consistent; that is, they mustn't lead to contradictions. Apparent self-evidence might be a justification for using a particular axiom in some mathematical theory that purports to model the natural world, as is other kinds of evidence. But the subjective judgment that an axiom is not self-evident isn't grounds for getting rid of it once the game is in play; only if a supposed axiom turns out to be provable from other axioms can it be dispensed with.

Are any of the postulates of Euclid 'self evident' ?

Yes, all of them (with the proviso that the 5th may have been considered a bit less self-evident than the others), Euclid's axioms formalise our inborn intuitions about space. (I'm using axioms as synonymous with postulates here, as I think are you.) Like the axioms of algebra that give us the field of real numbers with addition and multiplication as usually defined, they were traditionally considered self-evident, the starting point from which all proofs must come, and on which proofs could reliably be based. But in modern mathematics, this is treated as just one of many consistent algebraic systems, and Euclidian geometry one of many consistent geometric systems.

And if they are so self evident why can we drop one for Rieman[n]?

Because Riemannian geometry is a different game, played by a somewhat different set of rules.

 

[Werg22]

Why do axioms need to be self evident?

Are any of the postulates of Euclid 'self evident' ?

And if they are so self evident why can we drop one for Rieman?

Euclid's Elements is an axiomatization of geometry. That is, it aims to study geometry by mean of deduction, starting from a few statements taken to be elementary and self-evident.

The concept of a set of axioms as arbitrary "rules of the game" is much more modern. When mathematicians negated Euclid's 5th postulate, they realized that they would get new sets of axioms that were just as logically consistent as Euclid's. It took a long time for mathematicians to first dare this: after all, points and lines were specific kind of objects, and Euclid's axioms were indisputable truths on their matter. A system of axioms incompatible with Euclid's would be nonsense; this, until mathematicians realized that by reinterpreting terms like points and lines, such a system could be made meaningful.

This marked a new chapter in mathematical thought: mathematicians began studying systems of axioms not necessarily attached to any particular interpretation of the terms that appear in them. This is where the idea of axioms as rules for a game arises: we are simply pursuing the logical consequences of some statements that we lay down, without regard to what these axioms could represent, if anything at all. Contrast this with Euclid's Elements, where points, lines and circles were taken to mean particular things.