[FOM] Are Proofs in mathematics based on sufficient evidence

[Irving]

I would add an historical sidelight to Prof. Davis's note on Frege's 
contribution; namely that Charles Peirce also sought to understand the 
nature and concept of proof in mathematics. One may refer in particular 
to two closely related manuscripts (Robin catalog ##16, 17) of 1895, 
described in

Annotated Catalogue of the Papers of
CHARLES S. PEIRCE

BY RICHARD S. ROBIN
THE UNIVERSITY OF MASSACHUSETTS PRESS 1967


as follows:

16. On the Logic of Quantity, and especially of Infinity (Logic of Quantity)
A. MS, n.p., [c.1895], pp. 1, 5-9, 7-18, 18-20.
Several definitions of "mathematics," including Aristotle's and CSP's. 
Mathematical proof and probable reasoning; the system and scale of 
quantity; the importance of quantity for mathematics. But to grasp the 
nature of mathematics is to grasp the three elements, which, with 
regard to consciousness, are feeling, consciousness of opposition, and 
consciousness of the clustering of ideas into sets. Recognition of the 
three elements in the three kinds of signs logicians employ. An 
analysis of the syllogism.

17. On the Logic of Quantity (Logic of Quantity)
A. MS., n.p., [c.1895], pp. 1-9; 7-10 of another draft.
This manuscript should be compared with MS. 16, to which it bears a 
special similarity. See also MS. 250 where CSP defines "mathematics" as 
"the tracing out of the consequences of an hypothesis." Five 
definitions of "mathematics." Benjamin Peirce's definition found 
acceptable with modification. "Science" defined in terms of the 
activity of scientists, not in terms of its content or "truths." 
Probable inference and certain features of mathematical proof (pp. 
7-10).


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> Message: 2
> Date: Thu, 08 Jul 2010 17:17:41 -0700
> From: Martin Davis <eipye at pacbell.net>
> Subject: Re: [FOM] Are proofs in mathematics based on sufficient
> 	evidence?
> To: fom at cs.nyu.edu
> Message-ID: <173028.75684.qm at smtp127.sbc.mail.sp1.yahoo.com>
> Content-Type: text/plain; charset="us-ascii"; format=flowed
>
> I am astonished that on a list devoted to foundations of mathematics,
> none of the comments on Vaughan Pratt's post has even mentioned the
> valuable insights obtained in more than a century of  work in this field.
>
> Gottlob Frege discovered the simple logical rules of inference that
> underlie mathematical proofs. Published proofs are presented in a
> level of detail thought to be appropriate for the intended audience.
> What is appropriate for a first course introducing students to
> rigorous proofs about limits and compactness is very different from
> what is appropriate in an advanced textbook on real analysis, and
> this in turn is much more elaborate than what would appear in a
> professional journal. Carnap proposed the useful metaphor of a
> microscope: if you don't understand how a certain step in a proof
> follows from the previous steps, you turn up the power in Carnap's
> microscope and the gap is filled by more detail. Just as a conceptual
> microscope applied to matter will eventually reach down to the
> molecular level, then to atoms, and then to the atomic constituents,
> so Carnap's microscope will eventually reach down to steps belonging
> to pure logic. Of course no mathematician will want to, or be able
> to, read a proof of anything serious in which all of the steps have
> been filled in. That's where computers come usefully into play as
> much current research shows (despite the prognostications of R.A. de
> Millo, R.J. Lipton and A.J. Perlis).
>
> Already in 1933 G"odel outlined a foundation for mathematics based on
> axioms for sets of higher and higher type (or rank as we would now
> say) stretching into the transfinite. (See Feferman et al eds,
> Collected Works, vol III, pp. 45-53.) The more or less canonical
> Zermelo-Fraenkel axioms encapsulate this process up to a level
> sufficient for almost all ordinary mathematics. G"odel pointed out
> that his construction of an undecidable proposition carried out for
> the set-theoretic hierarchy at a given level becomes provable at the
> next level. There are those who find proofs in terms of the lower
> levels of the hierarchy more convincing than those at higher levels,
> thus providing better evidence for the truth of what is proved. Going
> beyond what can be proved from the full Zermelo-Fraenkel axioms are
> propositions that depend on assumptions about so-called large
> cardinals. Harvey Friedman has been submitting many posts to FOM that
> provide examples.
>
> In addition to this mainstream development, are proposals to restrict
> mathematics to what are thought to be more reliable methods, in
> particular, either by requiring proofs of existence to provide
> constructions (in a suitable sense) of the objects whose existence is
> claimed, or to refuse to permit so-called impredicative definitions.
> Researchers have proposed formal systems intended to encapsulate
> these foundational stances, and much interesting work has shown how
> these systems relate to one another and to mainstream foundations.
>
> Martin Davis
>
>



Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info