Re: Epistemology 201: The Science of Science

From: Jason (jasonstevensNOSPAM_at_free.net.nz)
Date: 02/04/05 

Date: Sat, 05 Feb 2005 11:09:35 +1300

>>Do you see a material implication or logical deduction on the page?
>
> Sure.

So the implication, a formula which is part of the "language of ZFC", is
used in the proof. It is just not written down explicitly or even
formally invoked. It is such a simple thing that everyone is expected
to follow what is going on.

> Your difficulty, then, in seeing any difference between formal
> derivations in ZFC and ordinary mathematical proofs, boils down to the
> observation that they both have something to do with logic?

Not at all. I would say that mathematics includes set theory, which is
an extension of logic. When proofs in set theory are performed, they
are not at odds with its formal system. I'm arguing that all valid
mathematical proofs are in accordance with one formal system or another,
whether the mathematician is aware of it or not. That is, it is
possible to couch any proof in a formal system after the fact, and in
the case of mathematics, a formal system of this kind has been around
for some time. It is possible to reduce all mathematical proofs to
axiom derivations, which is the common ground. The big on in maths is
ZFC. The mathematician doesn't need to know the axioms explicitly,
because there are an infinite number of axiom systems to choose from.
But provided it is a working knowledge that is in accord with ZFC or
explicitly stated system, the proof is fine.

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... >>Do you see a material implication or logical deduction on the page? ... So the implication, a formula which is part of the "language of ZFC", is ... I would say that mathematics includes set theory, ... are not at odds with its formal system. ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... >>Do you see a material implication or logical deduction on the page? ... So the implication, a formula which is part of the "language of ZFC", is ... I would say that mathematics includes set theory, ... are not at odds with its formal system. ...
    (sci.physics)
  • Re: (Not quite) Cantors diagonal proof
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  • Re: (Not quite) Cantors diagonal proof
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    (sci.logic)
  • Re: An uncountable countable set
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