Dear Irving,
A digression, from the perspective of art. You quote probability
theorist William
Taylor and set theorist Martin Dowd as saying:
> "The chief difference between scientists and mathematicians is that
> mathematicians have a much more direct connection to reality."
>> This does not entitle philosophers to characterize mathematical reality
>> as fictional.
Yes, I can see that.
But how about a variant:
The chief difference between scientists, mathematicians, and artists is that
artists have a much more direct connection to reality.
This does not prevent scientists and mathematicians to characterize artistic
reality
as fictional, because it is, and yet, nevertheless, real.
This is because scientist's and mathematician's map is not the territory, yet
the artist's art is both.
Gene Halton
-----Original Message-----
From: C S Peirce discussion list [mailto:PEIRCE-L@LISTSERV.IUPUI.EDU] On Behalf
Of Irving
Sent: Tuesday, March 13, 2012 4:34 PM
To: PEIRCE-L@LISTSERV.IUPUI.EDU
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce
edition
Ben, Gary, Malgosia, list
It would appear from the various responses that. whereas there is a
consensus that Peirce's theorematic/corollarial distinction has
relatively little, if anything, to do with my theoretical/computational
distinction or Pratt's "creator" and "consumer" distinction.
As you might recall, in my initial discussion, I indicated that I found
Pratt's distinction to be somewhat preferable to the
theoretical/computational, since, as we have seen in the responses,
"computational" has several connotations, only one of which I initially
had specifically in mind, of hack grinding out of [usually numerical]
solutions to particular problems, the other generally thought of as
those parts of mathematics taught in catch-all undergrad courses that
frequently go by the name of "Finite Mathematics" and include bits and
pieces of such areas as probability theory, matrix theory and linear
algebra, Venn diagrams, and the like). Pratt's creator/consumer is
closer to what I had in mind, and aligns better, and I think, more
accurately, with the older pure (or abstract or theoretical) vs.
applied distinction.
The attempt to determine whether, and, if so, how well, Peirce's
theorematic/corollarial distinction correlates to the
theoretical/computational or creator/consumer distinction(s) was not
initially an issue for me. It was raised by Ben Udell when he asked me:
"Do you think that your "theoretical - computational" distinction and
likewise Pratt's "creator - consumer" distinction between kinds of
mathematics could be expressed in terms of Peirce's "theorematic -
corollarial" distinction?"
I attempted to reply, based upon a particular quote from Peirce. What I
gather from the responses to that second round is that the primary
issue with my attempted reply was that Peirce's distinction was bound
up, not with the truth of the premises, but rather with the method in
which theorems are arrived at. If I now understand what most of the
responses have attempted to convey, the theorematic has to do with the
mechanical processing of proofs, where a simple inspection of the
argument (or proof) allows us to determine which inference rules to
apply (and when and where) and whether doing so suffices to demonstrate
that the theorem indeed follows from the premises; whereas the
corollarial has to do with intuiting how, or even if, one might get
from the premises to the desired conclusion. In that case, I would
suggest that another way to express the theorematic/corollarial
distinction is that they concern the two stages of creating
mathematics; that the mathematician begins by examining the already
established mathematics and asks what new mathematics might be
Ben Udell also introduces the issue of the presence of a lemma in a
proof as part of the distinction between theorematic and corollarial.
His assumption seems to be that a lemma is inserted into a proof to
help carry it forward, but is itself not proven. But, as Malgosia has
already noted, the lemma could itself have been obtained either
theorematically or corollarially. In fact, most of us think of a lemma
as a minor theorem, proven along the way and subsequently used in the
proof of the theorem that we're after.
I do not think that any of this obviates the main point of the initial
answer that I gave to Ben's question, that neither my
theoretical/computational distinction nor Pratt's "creator" and
"consumer" distinction have anything to do with Peirce's
theorematic/corollarial distinction.
In closing, I would like to present two sets of exchanges; one very
recent (actually today, on FOM, with due apologies to the protagonists,
if I am violating any copyrights) between probability theorist William
Taylor (indicated by '>') and set theorist Martin Dowd (indicated by
'>>'), as follows:
>> More seriously, any freshman philosopher encounters the fact that there are
>> fundamental differences between physical reality and mathematical reality.
> Quite so. And one of these is noted by Hilbert (or maybe Hardy,
> anyone help?) >-
> "The chief difference between scientists and mathematicians is that
> mathematicians have a much more direct connection to reality."
>> This does not entitle philosophers to characterize mathematical reality
>> as fictional.
> Quite so; but philosophers tend to have a powerful sense of entitlement.
the other, in Gauss's famous letter November 1, 1844 to astronomer
Heinrich Schumacher regarding Kant's philosophy of mathematics, that:
"you see the same sort of [mathematical incompetence] in the
contemporary philosophers.... Don't they make your hair stand on end
with their definitions? ...Even with Kant himself it is often not much
better; in my opinion his distinction between analytic and synthetic
propositions is one of those things that either run out in a triviality
or are false."
----- Message from bud...@nyc.rr.com ---------
Date: Mon, 12 Mar 2012 13:47:10 -0400
From: Benjamin Udell <bud...@nyc.rr.com>
Reply-To: Benjamin Udell <bud...@nyc.rr.com>
Subject: Re: [peirce-l] Mathematical terminology, was, review of
Moore's Peirce edition
To: PEIRCE-L@LISTSERV.IUPUI.EDU
> Malgosia, Irving, Gary, list,
>
> I should add that this whole line of discussion began because I put
> the cart in front of the horse. The adjectives bothered me.
> "Theoretical math" vs. "computational math" - the latter sounds like
> of math about computation. And "creative math" vs. what -
> "consumptive math"? "consumptorial math"? Then I thought of
> theorematic vs. corollarial, thought it was an interesting idea and
> gave it a try. The comparison is interesting and there is some
> likeness between the distinctions. However I now think that trying
> to align it to Irving's and Pratt's distinctions just stretches it
> too far. And it's occurred to me that I'd be happy with the
> adjective "computative" - hence, theoretical math versus computative
> math.
>
> However, I don't think that we've thoroughly replaced the terms
> "pure" and "applied" as affirmed of math areas until we find some way
> to justly distinguish between so-called 'pure' maths as opposed to
> so-called 'applied' yet often (if not absolutely always)
> mathematically nontrivial areas such as maths of optimization (linear
> and nonlinear programming), probability theory, the maths of
> information (with laws of information corresponding to
> group-theoretical principles), etc.
>
> Best, Ben
>
> ----- Original Message -----
> From: Benjamin Udell
> To: PEIRCE-L@LISTSERV.IUPUI.EDU
> Sent: Monday, March 12, 2012 1:14 PM
> Subject: Re: [peirce-l] Mathematical terminology, was, review of
> Moore's Peirce edition
>
> Malgosia, list,
>
> Responses interleaved.
>
> ----- Original Message -----
> From: malgosia askanas
> To: PEIRCE-L@LISTSERV.IUPUI.EDU
> Sent: Monday, March 12, 2012 12:31 PM
> Subject: Re: [peirce-l] Mathematical terminology, was, review of
> Moore's Peirce edition
>
>>> [BU] Yes, the theorematic-vs.-corollarial distinction does not
>>> appear in the Peirce quote to depend on whether the premisses - _up
>>> until some lemma_ - already warrant presumption.
>>> BUT, but, but, the theorematic deduction does involve the
>>> introdution of that lemma, and the lemma needs to be proven (in
>>> terms of some postulate system), or at least include a definition
>>> (in remarkable cases supported by a "proper postulate") in order to
>>> stand as a premiss, and that is what Irving is referring to.
>
>> [MA] OK, but how does this connect to the corollarial/theorematic
>> distinction? On the basis purely of the quote from Peirce that
>> Irving was discussing, the theorem, again, could follow from the
>> lemma either corollarially (by virtue purely of "logical form") or
>> theorematically (requiring additional work with the actual
>> mathematical objects of which the theorem speaks).
>
> [BU] So far, so good.
>
>> [MA] And the lemma, too, could have been obtained either
>> corollarially (a rather needless lemma, in that case)
>
> [BU] Only if it comes from another area of math, otherwise it is
> corollarially drawn from what's already on the table and isn't a
> lemma.
>
>> [MA] or theorematically. Doesn't this particular distinction, in
>> either case, refer to the nature of the _deduction_ that is required
>> in order to pass from the premisses to the conclusion, rather than
>> referring to the warrant (or lack of it) of presuming the premisses?
>
> [BU] It's both, to the extent that the nature of that deduction
> depends on whether the premisses require a lemma, a lemma that either
> gets something from elsewhere (i.e., the lemma must refer to where
> its content is established elsewhere), or needs to be proven on the
> spot. But - in some cases there's no lemma but merely a definition
> that is uncontemplated in the thesis, and is not demanded by the
> premisses or postulates but is still consistent with them, and so
> Irving and I, as it seems to me now, are wrong to say that it's
> _always_ a matter of whether some premiss requires special proof. Not
> always, then, but merely often. In some cases said definition needs
> to be supported by a new postulate, so there the proof-need revives
> but is solved by recognizing the need and "conceding" a new postulate
> to its account.
>
>> [MA] If the premisses are presumed without warrant, that - it seems
>> to me - does not make the deduction more corollarial or more
>> theorematic; it just makes it uncompleted, and perhaps uncompletable.
>
> [BU] That sounds right.
>
> Best, Ben
>
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----- End message from bud...@nyc.rr.com -----
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
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