[FOM] FOM: What is a proof?

[Ron Rood]

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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