What is the role of definitions in an axiomatic theory?

[C0nfused]

Hi everybody,
Mathematical theories are always based on some axioms. What else makes up an axiomatic theory? I mean , except from the axioms, we need some logical rules to draw conclusions and some definitions. What exactly are these definitions? (define definition!) I mean, can we use these axioms and define whatever we want? Are these "objects" we have defined part of the axiomatic theory? When is a definition considered correct or not ( or just correct for a specific axiomatic theory)? Can we always add new definitions to a theory? And finally, can a theory without definitions be useful ?
Thanks

 

[mathwonk]

why don't you look at soem examples, like maybe hilbert's or birkhoff's axiom systems for geometry?

it seems to me primarily there are "undefined terms" and "axioms" describing properties of these undefined terms.

 

[Hurkyl]

An oft overlooked, yet vitally important piece is the language. You have to have a language even before you can have an axiomatic system!

Usually one starts from a language of logic and the rules of deduction, then states the axioms as statements within this language.

The language consists of things called atomic formulae (things like "x is a real number" or "y is even"), connectives (like "or" or "and"), and logical quantifiers (like "there exists an x such that").

 

[C0nfused]

Thanks for your answers.
I think the "definitions" part of my question wasn't answered, and actually this is mostly my question about. I would really appreciate your help

What exactly are these definitions? (define definition!) I mean, can we use these axioms and define whatever we want? Are these "objects" we have defined part of the axiomatic theory? When is a definition considered correct or not ( or just correct for a specific axiomatic theory)? Can we always add new definitions to a theory? And finally, can a theory without definitions be useful ?

 

[HallsofIvy]

Since you rattle my cage on this one: another important part of the "axiomatic method" is undefined terms. That's what creates the great generality of mathematics. To apply a form of mathematics to some problem, you try to find "interpretations" of the undefined terms so that the axioms are (at least approximately) true. Once you know that, you know that all theorems proved from the axioms are true.

As for definitions- yes, you are pretty much free to define things as you wish- as long as the definitions themselves don't contradict any of the axioms.

If you add new definitions to a theory, as long as you don't add new axioms (and the axioms and old definitions are not violated by those new definitions) you just have a different way of expressing the theory. If you add new axioms related to those new definitions or cause a "redefinition" of things that had already been defined, then you have a new theory.

 

[matt grime]

The mathematical definition of something is (perhaps) the list of properties that the object satisifes. If you want to philosophize about how one defines the word definition without being self referential then try a "philosophy of language" forum.

 

[MathematicalPhysicist]

As for definitions- yes, you are pretty much free to define things as you wish- as long as the definitions themselves don't contradict any of the axioms.

.

i believe they call it "self-consistency", and if they don't call it that so here's another definition for you.

 

[C0nfused]

Thanks for your help. I think I have quite cleared out these things. One last thing: do you have any book suggestions about this topic. I would prefer books that don't specialise only in Geometry, but generally in any mathematical system

 

[honestrosewater]

Thanks for your help. I think I have quite cleared out these things. One last thing: do you have any book suggestions about this topic. I would prefer books that don't specialise only in Geometry, but generally in any mathematical system

You can read the first several pages of by Joseph Shoenfield online (actually, you can search the whole book too). I just came across this book, and it's amazing.
I think it would also help you to know all the different kinds of definitions, especially denotative (or extensional) and connotative (or intentional). Here's a decent explanation. A better explanation can be found in Copi & Cohen's "Introduction to Logic"; There's a good chance your library has it.

 

[mathwonk]

i don't knjow about the formal version of definitions, but in practical mathematics, definitions just isolate concepts that have proven to be useful.

for instance if you know what functions are, then you also know what injective and surjective functions are, but you may not have thought of giving them a special name.

once you realize how useful it is to think in terms of say whether a function is surjective, (because it makes precise the notion that a certain problem f(x) = y always has a solution), then you may stop and make a precise definition of this term.

so in prcatice definitions just stake out certain special situations that are already visible in the landscape you are surveying, and serve to call attention to them.

In my opinion certain definitions like e.g. "relations" are not particularly important, although special types of them, like equivalence relations are very important. for instance defining functions as special types of relations is to me silly, since functions are so much more important than general relations.

of course having said this one can easily think of counterexamples, like the crucial relations of incidence, or ordering, or divides, or whatever...

still it hinders my ability to use a book to teach functions if i also have to teach relations first, when functions are hard enough to teach.

so not all definitions are created equal, a fact that may escape beginning students, who haplessly try to master every one they see.

clearly some definitions are more important than others, like isomorphisms, homomorphisms, continuity, linearity, tangency, convergence, divisibility, group, derivative, deformation, sheaf cohomology group (just kidding).