Why do we formalize conceptions?

Question

Why do we always try to formalize conceptions? Let's take the naive conception of sets, why do we try to write down a list of axioms? what do we earn in doing so? I'm looking especially for references.

Answer 1

It is so to achieve a flawless system that is connected, pure of assumptions and possible errors. There are many examples, in mathematics especially of "obvious" things that were used for granted and later proved wrong.

Here is a great thread with tons of examples: 'Obvious' theorems that are actually false

But the concept of sets isn't a good example of that. While the theory alone is the basis for the whole modern mathematics, the concept of sets is one of the rare things that in mathematics are defined by itself, or to say so, something taken for granted. When we take in consideration a set of numbers we assume that they are a group but that doesn't give them any properties without a relation. Being a part of a set alone means nothing without a proper relation.

So the short answer to that question would be, strong basis.

Answer 2

I suggest you - not as a real aswer, but as an hint for further reflections - this quotation from George Boolos, from The iterative conception of sets (1971) [reprinted into George Boolos, Logic Logic and Logic (1998), page 13-on, and also into Paul Benacerraf & Hilary Putnam (editors), Philosophy of Mathematics : Selected Readings (2nd ed 1983), page 486-on ] :

It is not to be presumed that the concepts of set and member of can be explained or defined by means of notions that are simpler or conceptually more basic. However, as a theory about sets might itself provide the sort of elucidation about sets and membership that good definitions might be hoped to offer, there is no reason for such a theory to begin with, or even contain, a definition of "set." That we are unable to give informative definitions of not or for some does not and should not prevent the development of quantificational logic, which provides us with significant information about these concepts.

We axiomatize (and formalize) concepts not in order to define (we cannot define all), but in order to "elucidate", i.e. to clarify, to deepen our understanding of them.

Answer 3

Think of formalizing conceptions as a method of resolving disputes in mathematics. Interested parties must come to an agreement on a common notation, common rules of inference and a common, agreed upon list of assumptions about the system under study. Ideally, verifying a proof should be a mechanical procedure with no room for ambiguous interpretations of the rules and assumptions.

From this common ground, it is hoped that that the interested parties will always be able reach a consensus on any mathematical issue. If there is still a lack on consensus on some issue, they must go back and re-examine this supposed common ground for errors and inconsistencies.