Franklin, lists,
I think that Frederik is largely assuming Peirce's terminology. Peirce
uses the words 'schema' and 'diagram' pretty much interchangeably.
Here are some key quotes on which Frederik is basing his discussion of
the theormatic-corollarial distinction.
http://www.commens.org/dictionary/term/corollarial-reasoning
I once did a summary (footnoted with online links) of key points (at
least as they seemed to me at the time); here it is with a few
adjustments of the links:
Peirce argued that, while finally all deduction depends in one way
or another on mental experimentation on schemata or diagrams,^*[1]*
still in corollarial deduction "it is only necessary to imagine any
case in which the premisses are true in order to perceive
immediately that the conclusion holds in that case", whereas
theorematic deduction "is deduction in which it is necessary to
experiment in the imagination upon the image of the premiss in order
from the result of such experiment to make corollarial deductions to
the truth of the conclusion."^*[2]* He held that corollarial
deduction matches Aristotle's conception of direct demonstration,
which Aristotle regarded as the only thoroughly satisfactory
demonstration, while theorematic deduction (A) is the kind more
prized by mathematicians, (B) is peculiar to mathematics,^*[1]* and
(C) involves in its course the introduction of a lemma or at least a
definition uncontemplated in the thesis (the proposition that is to
be proved); in remarkable cases that definition is of an abstraction
that "ought to be supported by a proper postulate."^*[3]*
[1] Peirce, C. S., from section dated 1902 by editors in the "Minute
Logic" manuscript, Collected Papers v. 4, paragraph 233, quoted only
in part
http://www.commens.org/dictionary/entry/quote-minute-logic-chapter-iii-simplest-mathematics
in "Corollarial Reasoning" in the Commens Dictionary of Peirce's
Terms, 2003–present, Mats Bergman and Sami Paavola, editors,
University of Helsinki. FULL QUOTE:
https://archive.org/stream/TheWorldOfMathematicsVolume3/Newman-TheWorldOfMathematicsVolume3#page/n366/mode/1up
in The World of Mathematics, Vol. 3, p. 1776.
[2] Peirce, C. S., the 1902 Carnegie Application, published in The
New Elements of Mathematics, Carolyn Eisele, editor, quoted in
"Corollarial Reasoning"
http://www.commens.org/dictionary/entry/quote-carnegie-institution-correspondence-4
in the Commens Dictionary of Peirce's Terms, also transcribed by
Joseph M. Ransdell, see "From Draft A - MS L75.35-39" in Memoir 19
http://www.iupui.edu/~arisbe/menu/library/bycsp/l75/ver1/l75v1-06.htm#m19
<http://www.iupui.edu/%7Earisbe/menu/library/bycsp/l75/ver1/l75v1-06.htm#m19>
(once there, scroll down).
[3] Peirce, C. S., 1901 manuscript "On the Logic of Drawing History
from Ancient Documents, Especially from Testimonies', The Essential
Peirce v. 2, see p. 96. See quote
http://www.commens.org/dictionary/entry/quote-logic-drawing-history-ancient-documents-especially-testimonies-logic-histor-5
in "Corollarial Reasoning" in the Commens Dictionary of Peirce's Terms.
The introduction of an idea beyond the explicit conditions of a problem
and not contemplated in the thesis to be proved is precisely a
'complexifying' step. One might think of it as a leveraging of
imagination to deepen understanding, by which vague remark I'm trying to
get at the idea that such complexity is very different from the tedious
complication of hundreds or thousands of trivial computations,
computations that need to be done sometimes even in pure mathematics,
where it is known as 'brute force'. Tedious computation used to be done
by people called 'computers' up until computing machines came into use;
part of Peirce's burden at the Coast Survey was that there came a time
when he had to do his own tedious, lengthy computations and, worse, he
found that his computing power was no longer what it was when he was
younger; errors crept in.
In CP 4.233 (again
https://archive.org/stream/TheWorldOfMathematicsVolume3/Newman-TheWorldOfMathematicsVolume3#page/n366/mode/1up)
in "The Essence of Mathematics", Peirce says,
[....] Just now, I wish to point out that after the schema has been
constructed according to the precept virtually contained in the
thesis, the assertion of the theorem is not evidently true, even for
the individual schema; nor will any amount of hard thinking of the
philosophers' corollarial kind ever render it evident. Thinking in
general terms is not enough. It is necessary that something should
be DONE. In geometry, subsidiary lines are drawn. In algebra
permissible transformations are made. Thereupon, the faculty of
observation is called into play. Some relation between the parts of
the schema is remarked. But would this relation subsist in every
possible case? Mere corollarial reasoning will sometimes assure us
of this. But, generally speaking, it may be necessary to draw
distinct schemata to represent alternative possibilities. [....]
The above is an example of why I keep talking about complexity in the
sense of nontriviality. A theorem in the old sense, that is, as opposed
to a corollary, is a proposition whose proof requires, at least as a
practical matter, some 'complexifying', active new-idea-adding
experimentation of theorematic reasoning. Such reasoning does not just
add steps and operations, but incorporates ideas in ways that enrich the
understanding, make 'new gestalts', to borrow some lingo that may sound
hokey today. The mathematical theorem's nontriviality is its character
of being a mathematical theorem in the sense of not being a mathematical
corollary; it's such a theorem's non-corollarity. The theorem's
nontiviality reflects, is, in a sense, the needed theorematicity of its
proof, and for that very reason it reflects also the prospect of its
occasioning in turn further theorematic proofs of further theorems,
whatever they might be, as opposed to mere corollaries; its all about
deepened understandings, as opposed to merely additional tidbits, soever
multitudinous, of information. It takes nontrivia to make nontrivia.
It's true that I bring in a methodology-of-inquiry perspective in
addition to the critique-of-arguments perspective taken by Peirce in
analyzing theorematic and corollarial reasonings. But I think that it
does matter in understanding the role of theorematic reasoning in
mathematics, and in relating the ideas of theorematic and corollarial
reasonings to the common parlance (at least what I've been told of it)
of mathematicians, where 'nontriviality', 'depth', 'fecundity' are
prized characters of proven propositions.
It's not that the theorematic deduction brings something to light while
the corollarial deduction brings nothing to light. It's a matter of
degree as you say; indeed what seems theorematic to a schoolchild may
well seem corollarial to a mathematician. Peirce generally discusses
reasoning and inquiry in the context of discovery rather than in the
context of justification, as Frederik pointed out; and we never entirely
depart the context of discovery even when we're focused on
justification. Anyway, corollarial reasoning that is not manifestly
redundant (redundant like '/pq/, ergo /p/') does provide some jot of
novelty or nontriviality; the categorical syllogisms (such as All A is
B, all B is C, ergo all A is C) are deductive forms designed to assure
some modicum of novelty in corollarial conclusions; and massive,
brute-force corollarial computation may bring things to light that we
couldn't find otherwise (it still plays a big role in the proof of the
four-color theorem). What Peirce says is that sometimes corollarial
deduction won't suffice, and that then theorematic deduction is needed
in order to bring something to light.
Whew. I'm not sure I've addressed all in your post, but I'll let it
stand for now and retract who knows what tomorrow.
Best, Ben
On 4/19/2015 5:12 PM, Franklin Ransom wrote:
---------- Forwarded message ----------
From: *Franklin Ransom* <pragmaticist.lo...@gmail.com
<mailto:pragmaticist.lo...@gmail.com> >
Date: Sun, Apr 19, 2015 at 5:11 PM
Subject: Re: [biosemiotics:8342] Re: [PEIRCE-L] Natural Propositions,
Ch. 10: Corollarial and Theorematic Experiments with Diagrams
To: biosemiot...@lists.ut.ee <mailto:biosemiot...@lists.ut.ee>
Ben, lists,
Thank you, Ben, for a post that is (clearly) on topic.
Frederik notes, in the fourth definition of theorematic reasoning,
that it involves schemata rather than words. Actually, he qualifies
this claim, noticing that Peirce says even words are schemata, but
rather simple schemata. Theorematic reasoning typically involves then
complicated schemata. It is really a matter of degree or gradation
though, as corollarial reasoning typically involves simpler schemata
and theorematic reasoning typically involves complicated schemata,
relative to each other. In the text, p.276-7, Frederik seems to
associate schemata with diagrams, so that corollarial reasoning makes
less use of diagrams and theorematic reasoning makes greater use of
diagrams.
If I recall correctly, this is all that is really mentioned about
complexity or complication. Otherwise, there is the discussion in the
chapter regarding the possibility that some theorematic reasoning,
using a different logic system (by this, meaning a different set of
axioms and rules), may be reworked as corollarial reasoning, because
not needing to include something new or foreign to the premises and
conclusion as the other logic system would have required. I believe
that is in p.280-3.
As I understand it, what Frederik takes to be most essential is the
introduction of something new or foreign to the reasoning, and not so
much the relative simplicity or complexity of the reasoning. This is
probably due to the flexibility of some reasonings as being capable of
classification under either head, depending upon the logic system at work.
With respect to nontriviality or depth, this isn't really discussed in
the chapter. The point of the chapter is less about the value of
theorems than it is about explaining what theorematic diagrammatic
reasoning is and what its significance is. In fact, the significance
seems to be less about the importance of theorematic reasoning in
mathematics and more about the importance of theorematic reasoning for
epistemology, i.e. for knowledge whether of the scientific sort or of
the everyday sort.
My concern about corollarial reasoning is that, since corollarial
reasoning does involve experimentation, what should be the point of
experimentation if nothing unnoticed or hidden ever appeared as a
result? I don't doubt that theorematic reasoning is better for the
purpose, I just don't think that it's a hard-and-fast line to be drawn
between theorematic and corollarial reasoning. Perhaps my concern
would be better answered though if it were made clearer what the role
of these reasonings is in the context of scientific method, which
would allow for a clearer account of the Holm example.
-- Franklin
On Sun, Apr 19, 2015 at 2:05 PM, Benjamin Udell <bud...@nyc.rr.com
<mailto:bud...@nyc.rr.com> > wrote:
Franklin, lists,
I agree with Jon, thanks for your excellent starting post.
You wrote,
[....] Why can't corollarial reasoning, which involves
observation and experimentation, reveal unnoticed and hidden
relations? After all, on p.285-6, Frederik mentions the work
of police detective Jorn "Old Man" Holm and his computer
program, which Frederik describes as a "practical example of
corollarial map reasoning" (p.285). In this example, Holm uses
the corollarial reasoning to reveal information about the
whereabouts of suspects. Doesn't the comparison of the map
reasoning with suspects' testimony end up revealing unnoticed
and hidden relations?
There's a distinction that some make between complexity and mere
complication. Corollarial reasonings may accumulate mere
complications until the result becomes hard to see, although it
involves little if any complexity in, more or less, the sense of
depth or nontriviality.
I don't know whether there's a theorematic approach to Jørn Holm's
diagrammatization that would show its result in a nontrivial
aspect, and anyway its diagrammatic, pictorial presentation
already leaves one in no doubt that a pattern is revealed. A good
example involving alternate proofs that seem corollarial and
theorematic is the Monty Hall problem, a popular puzzle based in
probability theory. I remember reading an essentially corollarial
proof of the answer, and seeing a round diagram that showed how
alternatives lead inevitably to the conclusion in the diagram's
center. The answer to the Monty Hall problem remains, however,
notoriously counter-intuitive to people; the essentially
corollarial but multi-step proof - in words, even with the round
diagram - often leaves people with nagging vague doubts. They get
that it must be true but they feel that they don't fully get the
problem, they keep re-examining the problem, wondering whether it
was well disambiguated, etc. (it describes an actual standard
scenario on a popular TV game show). But the problem's answer has
also a proof that deserves to be called theorematic (even if it is
not very much so) since it involves varying the conditions of the
problem, adding things not contemplated in the thesis, going a
little deeper into the mathematical possibilities. One increases
the number of doors in the scenario from 3 to 10. With 10 doors,
the basically the same solution makes obvious sense, then one
reduces the number doors from 10 to 9 to 8, etc. down to 3, and
sees that the basic solution does not change at all; people get
satisfied (for whatever that's worth). It has become hard to avoid
running into that proof if one searches the Internet for "Monty
Hall problem". I also vaguely remember a geometric problem
involving the fitting of circles, shown to me by a roommate during
college; he was dissatisfied with a particular usual proof, he
wanted a proof that gave more satisfactory understanding, and it
turned out to be more imaginative and, as I'd call it now,
theorematic.
Nontriviality or depth of a result should not be confused with
mere complication and lengthiness of a proof; take the Pythagorean
theorem, which is considered both deep and not very hard to prove.
The nontriviality or depth of a theorem consists not in the
difficult complication of proving it but in its favorability as a
bridge to further nontrivial lessons or, to put it less
recursively, its favorability for use as a basis for further
proofs almost as if it were another postulate even though it is
entailed by the postulates and axioms already granted. It's a
place where one can come to rest for a while and set up camp; if I
were to coin a word dedicated to expressing it I'd say
'basatility'. Likewise the nontriviality or depth (apart from mere
complication as distinguished from complexity) of a proof of a
theorem is properly its favorability as a basis for further
lessons. (I'm not sure that there is much difference between
'depth' and 'power' of a theorem or a proof.) The nontrivial or
deep is more or less _/difficult/ _ (which is a usual connotation
especially of 'nontrivial') since, of course, it requires some
corresponding depth or or nontriviality of understanding and
perspective.
(One should distinguish such depth, complexity, etc., of theorems
and proofs also from the logical complexity that a fact or datum,
as a relation or complexus of relations, possesses; I mean such
'complexity' as quantified and characterized by valence,
transitivity or intransitivity, etc. This is likewise as one
distinguishes the novelty or new aspect of a deductive conclusion
from Shannonesque quantity of information or 'newsiness'.)
Best, Ben
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