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 ---------
   Date: Mon, 12 Mar 2012 13:47:10 -0400
   From: Benjamin Udell <>
Reply-To: Benjamin Udell <>
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

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

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
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
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

[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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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

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