Re: Fwd: [biosemiotics:8342] Re: [PEIRCE-L] Natural Propositions, Ch. 10: Corollarial and Theorematic Experiments with Diagrams

[Benjamin Udell]

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
   
   (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 so

Re: Fwd: [biosemiotics:8342] Re: [PEIRCE-L] Natural Propositions, Ch. 10: Corollarial and Theorematic Experiments with Diagrams

[Franklin Ransom]

Ben, lists,

It looks like Ben's post was sent to Peirce-L, but not the biosemiotics
list-serv. For biosemiotics list members, please see below for the post to
which I am responding.

"I think that Frederik is largely assuming Peirce's terminology. Peirce
uses the words 'schema' and 'diagram' pretty much interchangeably."

Yes, Ben, I would have guessed as much. In connection with this, I wonder
whether Peirce would have said Kant's schematization in the Critic of Pure
Reason is in fact a diagrammatization of some sort.

As to the rest of what you had to say, I don't find myself really in any
disagreement.

I see that Frederik mentioned that whether something is corollarial or
theorematic is not relative to a person's intelligence. I would suppose you
meant that a mathematician would have a much more advanced logic system
available to think about then the (average) schoolchild, in which case your
remark would make sense. While Frederik is right to point out that only
whether something new or foreign is introduced is what makes the reasoning
theorematic (I believe I mentioned in a previous post that this is what is
signficant for Frederik, and not so much the complexity of the schema), it
is also true that what logic system one is using will affect what counts as
corollarial reasoning and what as theorematic; that is, it will affect
whether the something "new" or "foreign" is really new or foreign to the
system, and the typical schoolchild probably has a much simpler logic
system to work with than the typical mathematician.

Actually, it just struck me that I mentioned that the complexity is not so
important for Frederik's account, and you have continued discussing
complexity to show how it is important. I'm sorry, I didn't mean to imply
that non-triviality is unimportant. It does get mentioned in the text at
some point, but does not play a prominent role, not nearly as much as the
point that something new or foreign must be introduced into the reasoning.
Though, I do wonder somewhat whether non-triviality is connected to a
theorem not being easily absorbed into a logic system that could turn what
was originally a theorematic reasoning into a purely corollarial reasoning.
Perhaps the newer or more foreign the idea, the more nontrivial and fecund
it may turn out to be?

-- Franklin

On Mon, Apr 20, 2015 at 4:18 PM, Benjamin Udell  wrote:

>  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://w