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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.

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  • Are you asking why human reasoning is discursive? Or why we need phantasms ("mental images") to understand?
    – Geremia
    Jun 6, 2014 at 17:19
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    Are you familiar with Russell's Paradox? For a good historical account of the move from naive to axiomatic set theories look at Fraenkel, Bar-Hillel, Levy Foundations of Set Theory, chapter II. Jun 6, 2014 at 17:19

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I'd argue against your thesis. We don't always formalise conception; simply because many concepts are not amenable to formalisation. Kant for example, made a heroic effort in trying to formalise ethics - his axiom being the categorical imperative.

What is true is that many concepts are studied, and their study turns into a scientia - that is a domain of knowledge. I use the word scientia delberately to differentiate this from modern science, which is one form of scientia.

Even in mathematics, where formalisation is important; in fact, so important in the contemporary situation that there is a philosophy of mathematics that is associated with it; formalisation hasn't always been important. For example, arithmetic as opposed to plane geometry wasn't axiomatised in Ancient Greece, but in the late 19C, over two millenia later; an interval of over 2 millenia hardly speaks of always and must.

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  • I don't get the point of your answer. It's like someone asks "Why do we change the oil in the car?" and one, instead of saying "Because otherwise you're going to ruin it" answers "Well, there are a lot of people who don't do it, and they sometimes have fatal crash, but still, you don't have to do it.".
    – Ant
    Jun 7, 2014 at 13:54
  • @Ant: ok, now give me a formalisation of the concept contained in your comment. Jun 7, 2014 at 16:58
  • I'm talking about mathematical concepts. If someone asks why is it that we must formalize them, answering "Really you don't, they didn't for thousands of years! Yes their math was contradictory, but still" sounds weird to me.. I don't mean to be rude, it's just that I don't see it as constructive. Maybe I misinterpreted ?
    – Ant
    Jun 7, 2014 at 17:47
  • @ant: you might be talking about mathematics; but the question asked about conceptions; granted, though - his one example was formalising set theory. During the two millenia that numbers weren't formalised by the Peano Axioms, negative numbers, zero, fraction & complex numbers were invented; these are major achievements which had nothing to do with formalisation per se; and that fact ought to be noted because its important; In what way is this not constructive? Jun 7, 2014 at 18:29
  • note, i'm not saying formalisation isn't important; but that it isn't always necessary;you can also think of formalisation as just another concept - how does one formalise that? Jun 7, 2014 at 18:33
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An easy answer is that the naive conception of sets lead to absurd conclusions.

Let's take for example what is know as Russel's paradox

According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox.

(and there is also the classic xkcd)

So to answer your question, there is no space in mathematics for concepts that are not well defined by a set of axioms, because that would eliminate rigor from demonstrations and lead to wrong results.

(Of course it also possible to choose axioms that lead to contradictions, but once one finds out, we can change the axioms and start again)

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  • The barber's paradox is NOT an instance of Russell's paradox. To see this note that Russell's paradox involves a comprehension principle concerning sets: For every property there is a set containing exactly its instances. But the barber does not involve a principle guaranteeing the existence of barbers. That's why the barber can be 'solved' simply by denying the existence of someone satisfying the paradoxical condition. Indeed, first-order logic itself already rules this out since ∃y∀x(Ryx ↔ ¬Rxx) (R a binary predicate) is not first-order satisfiable. Wish set theory was that easy!
    – sequitur
    Jun 6, 2014 at 21:22
  • @sequitur you're right, thank you! I was imprecise ;-)
    – Ant
    Jun 6, 2014 at 21:29

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