[FOM] FOM: What is a proof?
Ron Rood
ron.rood at planet.nl
Fri Jan 23 07:20:20 EST 2009
Q1. Frege seems to me a typcial case of someone accepting Aristotle's
traditional point of view in virtually all respects. And I think that
Frege thought so for mathematics as well as for any other field of
science. For Frege, axioms are true and known propositions; theorems
are logically derived from axioms and are hence also known
propositions (or so Frege believed). I do not have any of Frege's
works on my desk now, but I think that finding a number of good quotes
is not that hard.
Q2. For Hilbert, a set of axioms serves to provide an implicit
definition of a collection of models. I think that Hilbert hence at
least thought that axioms are not true with respect to some fixed
domain of objects. See the first pages of Grundlagen der Geometrie. I
don't know whether Hilbert has said something about where models come
from. I find it hard to say whether Hilbert believed that axioms are
known and in what sense. I do recall a paper from Bernays where he
says that Hilbert held that knowledge is something that applies to a
system of propositions and is not, or not primarily, a "propositional
attitude". I am not sure whether Hilbert accepted the view that a
proof in mathematics is a logical deduction.
Q2. Kant rejected the view that a proof in mathematics is a logical
deduction. According to Kant, a proof in mathematics is a
construction. A construction in mathematics is the a priori exhibition
of an object (Critique of Pure Reason, A713/B741). Kant did accept the
view that axioms in mathematics are true and known and I think also
for other fields of science.
Ron
Op 22 jan 2009, om 17:29 heeft John Corcoran het volgende geschreven:
> FOM: What is a proof?
> Aristotle's general TRUTH-AND-CONSEQUENCE CONCEPTION OF PROOF was
> meant to
> apply to all demonstrations. According to him, a PROOF is an extended
> argumentation that begins with premises known to be TRUTHS and that
> involves
> a chain of reasoning showing by deductively evident steps that its
> conclusion is a CONSEQUENCE of its premises; a proof is a [logical]
> DEDUCTION whose premises are known to be [material] TRUTHS. Starting
> with
> premises they know to be true, knowers demonstrate a conclusion by
> DEDUCING
> it from the premises. It is essential to this conception that a proof
> provides KNOWLEDGE of the truth of its conclusion to anyone for whom
> it is a
> proof. Every proof produces (or confirms) KNOWLEDGE of (the truth
> of) its
> conclusion for every person who comprehends the demonstration.
> Persuasion
> merely produces OPINION.
>
> Q1. Who are the prominent figures in foundations of mathematics that
> ACCEPT
> this view, perhaps with qualifications? Please give quotes if
> possible, or
> at least give traceable references.
> Q2. Who are the prominent figures in foundations of mathematics that
> REJECT
> this view, perhaps with qualifications? Please give quotes if
> possible, or
> at least give traceable references.
> Q3. What are the qualifications required by prominent figures in
> foundations
> of mathematics that ACCEPT this view with qualifications?
>
> Q4. What are the objections by prominent figures in foundations of
> mathematics that REJECT this view?
>
>
> John Corcoran, PhD, DHC
> Professor of Philosophy
> University at Buffalo
> Buffalo, NY 14260-4150
> corcoran at buffalo.edu
> http://philosophy.buffalo.edu/people/faculty/corcoran/
> http://www.acsu.buffalo.edu/~corcoran/
>
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