Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-14 Thread Irving
 Method liegt der
eigentliche Zauber der
Infinitesimalrechnung. Theodor Ziehen defined logicism in his Lehrbuch
der Logik auf
postivischer Grundlage mit Berücksichtigung der Geschichte der Logik
[1920, p. 173] to
mean that there is an objective realm of ideal entities, studied by logic and
mathematics, and he numbered on that account Lotze, Windelband,
Husserl, and Rickert
among those adhering to logicism.

Having said that: as I wrote in the FOM back in May 2011,

I recall that, many years ago (probably some time in the early or
mid-1980s), Reuben
Hersh gave a colloquium talk in the mathematics  department at the
University of Iowa. I
don't recall the specifics of  that talk, but in its general tenor it
went along the
lines that, in  their workaday world. most mathematicians are
Platonists, working as
though the mathematical structures with which they are working and
which are the subject of theorems exist, whereas, on weekends, they
deny the real existence of mathematical entities.


In the description for Reuben Hersh's What Is Mathematics Really?
(Oxford U. Press,
1997), Hersh's position is described (in part) as  follows:

Platonism is the most pervasive philosophy of mathematics. Indeed, it
can be argued
that an inarticulate, half-conscious Platonism is nearly  universal
among mathematicians.
The basic idea is that mathematical entities exist outside space and
time, outside
thought and matter, in an abstract realm. ...In What is Mathematics,
Really?, renowned
mathematician Reuben Hersh takes these eloquent words and this
pervasive philosophy to
task, in a subversive attack on traditional  philosophies of
mathematics, most notably,
Platonism and formalism.  Virtually all philosophers of mathematics
treat it as isolated,
timeless, ahistorical, inhuman. Hersh argues the contrary, that
mathematics must be understood as a human activity, a social
phenomenon, part of human
culture, historically evolved, and
intelligible only in a social context. Mathematical objects are created
by humans, not arbitrarily, but from activity with existing
mathematical objects, and from the needs of science and daily life.
Hersh pulls the screen back to reveal mathematics as seen by
professionals, debunking many mathematical myths, and demonstrating how
the humanist idea of the nature of mathematics more closely resembles
how mathematicians actually work. At the heart of the book is a
fascinating historical account of the mainstream of philosophy--ranging
from Pythagoras, Plato, Descartes, Spinoza, and Kant, to Bertrand
Russell, David Hilbert, Rudolph Carnap, and Willard V.O.
Quine--followed by the mavericks who saw mathematics as a human
artifact, including Aristotle, Locke, Hume, Mill, Peirce, Dewey, and
Lakatos. ...
- Message from eugene.w.halto...@nd.edu -
   Date: Tue, 13 Mar 2012 17:09:42 -0400
   From: Eugene Halton eugene.w.halto...@nd.edu
Reply-To: Eugene Halton eugene.w.halto...@nd.edu
Subject: RE: [peirce-l] Mathematical terminology, was, review of
Moore's Peirce edition
 To: PEIRCE-L@LISTSERV.IUPUI.EDU PEIRCE-L@LISTSERV.IUPUI.EDU



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

Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-13 Thread Irving
...@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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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

Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-13 Thread Eugene Halton
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

Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-13 Thread Benjamin Udell
Irving, all,

In my previous post I said that I would include the full Peirce quotes, but 
for the first Peirce quote I included only the portion included in the Commens 
Dictionary. For the full quote (CP 4.233), go here: 
http://books.google.com/books?id=3JJgOkGmnjECpg=RA1-PA193lpg=RA1-PA193dq=%22Mathematics+is+the+study+of+what+is+true+of+hypothetical+states+of+things%22

- Original Message - 
From: Benjamin Udell 
To: PEIRCE-L@LISTSERV.IUPUI.EDU 
Sent: Tuesday, March 13, 2012 6:11 PM
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 
edition

Irving, Gary, Malgosia, list,

Irving, I'm sorry that I gave you the impression that I think that a lemma is 
something helpful but unproven inserted into a proof. I mean a theorem placed 
in among the premisses to help prove the thesis. Its proof may be offered then 
and there, or it may be a theorem from (and already proven in) another branch 
of mathematics, to which the reader is referred. At any rate it is as Peirce 
puts it a demonstrable proposition about something outside the subject of 
inquiry. 


The idea that theorematic reasoning often involves a lemma comes not from me 
but from Peirce. Theorematic reasoning, in Peirce's view, involves 
experimentation on a diagram, which may consist in a geometrical form, an array 
of algebraic expressions, a form such as All __ is __, etc.  I don't recall 
his saying anything to suggest that theorematic reasoning is particularly 
mechanical.  I summarized Peirce's views in a paragraph in my first post on 
these questions, and I'll reproduce it, this time with the full quotes from 
Peirce. He discusses lemmas in the third quote.
Peirce held that the most important division of kinds of deductive reasoning is 
that between corollarial and theorematic. He 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 a b Peirce, C. S., from section dated 1902 by editors in the Minute Logic 
manuscript, Collected Papers v. 4, paragraph 233, quoted in part in 
Corollarial Reasoning in the Commens Dictionary of Peirce's Terms, 
2003-present, Mats Bergman and Sami Paavola, editors, University of Helsinki.: 

  How it can be that, although the reasoning is based upon the study of an 
individual schema, it is nevertheless necessary, that is, applicable, to all 
possible cases, is one of the questions we shall have to consider. 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. Theorematic reasoning 
invariably depends upon experimentation with individual schemata. We shall find 
that, in the last analysis, the same thing is true of the corollarial 
reasoning, too; even the Aristotelian demonstration why. Only in this case, 
the very words serve as schemata. Accordingly, we may say that corollarial, or 
philosophical reasoning is reasoning with words; while theorematic, or 
mathematical reasoning proper, is reasoning with specially constructed 
schemata. (' Minute Logic', CP 4.233, c. 1902)

2. Peirce, C. S., the 1902 Carnegie Application, published in The New Elements 
of Mathematics, Carolyn Eisele, editor, also transcribed by Joseph M. Ransdell, 
see From Draft A - MS L75.35-39

Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-12 Thread Benjamin Udell
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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Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-12 Thread Benjamin Udell
Jason, all,

If I had bothered to search on computational mathematics I would have found 
that the potential ambiguity that worried me is already actual, as you clearly 
show.  Do you think that the phrase computative mathematics is too close to 
the phrase computational mathematics for comfort?  I hope not, but please say 
so if it is.

Problem is, the applied in applied mathematics is used in various ways 
that, as Dieudonné of the Bourbaki group pointed out in his Britannica article 
(15th edition I think), jumbles trivial and nontrivial areas of math together, 
and has all too many, umm, applications. One area of pure math X may be 
_applied_ in another area of math Y, whih is to say that Y is the guiding 
research interest. If on the other hand Y is applied in X, then that's to say 
that X is the guiding research interest. And both X and Y remain areas of 
'pure' math. Then there are areas of so-called 'applied' but often nontrivial 
math like probability theory. Then there are applications in statistics and in 
the special sciences. Then there applications in practical/productive 
sciences/arts. And of course, sometimes theoretical or 'pure' math is developed 
specifically for a particular application. (All in all, we won't be able to get 
rid of the term applied, but in some cases we may be find an alternate term 
with the same denotation in the given context).

Best, Ben

- Original Message - 
From: Khadimir 
To: Benjamin Udell 
Cc: PEIRCE-L@listserv.iupui.edu 
Sent: Monday, March 12, 2012 2:14 PM
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 
edition

This latest post caught my attention.

Since my first degree was a B.S. in computational mathematics, I thought that 
I would weigh-in.  

One can make the distinctions as follows, beginning with pure vs. applied 
mathematics.  I will give a negative definition, since I am not so skilled with 
the Peircean terminology used so far; applied mathematics is the use of 
mathematics as a formal, ideal system to specific problems of existence.  For 
instance, consider the use of statistical confidence intervals to solve 
problems in manufactoring relating to the rate of production of defective vs. 
non-defective goods.  Pure mathematics is not bound by existent conditions, but 
pure becomes applied when used in that context.  Hence, I am treated 
applied mathematics as an informal, existential constraint that alters the 
purpose and use of pure mathematics.

Computational mathematics is for the most part a subset of applied mathematics, 
which focuses on how to adapt computational formulas so that they may be run or 
run more efficiently on a given computation system, e.g., a binary computer.  
Computational mathematics, then, is primarily focused on formulas and 
computation of said formulas, which is to be more specific about the limits 
that make it an applied mathematic.

I offer this as a different viewpoint, one coming from where the distinction 
has practical effects.

Jason H.

On Mon, Mar 12, 2012 at 12:47 PM, Benjamin Udell bud...@nyc.rr.com wrote:

  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

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Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-11 Thread Irving

Ben Udell asked:


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?


Given that Peirce wrote at MS L75:35-39 that:

Deduction is only of value in tracing out the consequences of
hypotheses, which it regards as pure, or unfounded, hypotheses.
Deduction is divisible into sub-classes in various ways, of which the
most important is into corollarial and theorematic. Corollarial
deduction is where 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. Ordinary syllogisms and some deductions
in the logic of relatives belong to this class. 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. The subdivisions of theorematic deduction are of very high
theoretical importance. But I cannot go into them in this statement.

the answer to the question would appear to be: no.

Whereas Peirce's characterization of theorematic and corrolarial
deduction would seem, on the basis of this quote, to have to do with
whether the presumption that the premises of a deductive argument or
proof are true versus whether they require to be established to be
true, and seems more akin, at least peripherally, to the
categorical/hypothetical status of the premises, the distinctions
theoretical - computational which I suggested and likewise Pratt's
creator - consumer are not at all about the deriving theorems or the
what is assumed about the truth of the premises. Rather the distinction
between creator-theoretician vs. consumer-practitioner is a distinction
in which the former is concerned (in the main) to develop new
mathematics on the basis of the mathematics that has already been
established, whereas the consumer practitioner borrows and utilizes
already established mathematics for purposes other than establishing
new mathematical results. The example which I cited, of Riemann and
Minkowski vs. Einstein is applicable here. Riemann expanded known
mathematical results regarding three-dimensional geometries to
n-dimensional geometries (Riemann manifolds) and contributed to the
development of non-Euclidean geometries, and Minkowski starting from
non-Euclidean geometries, in particular parabolic and hyperbolic,
arrived at his saddle-shaped space, and Minkowski taught Einstein the
mathematics of Riemannin and Minkowski geometry, who used it to work
out the details of relativity, but, unlike Riemann or Minkowski, did
not create any new mathematics, just utilized the already given
mathematics of Riemann and Minkowski to mathematically solve a
particular problem in physics. I think most would agree with the
proposition that Einstein was a physicist, rather than a mathematician,
albeit unassailably a mathematical physicist, who employed already
established mathematics and mathematical equations to advance physics,
and along those same lines, I think most would likewise agree with the
proposition that Einstein was not a mathematician. This does not, of
course, take away from his status as a physicist.

By the same token, Newton can be credited as both a mathematician, for
his fluxional caculus as well as a physicist, although his invention --
and I would not want to get into the Newton-Leibniz battle here -- of
the calculus was developed in large measure for the purpose of doing
physics. But the fact that Newton (although he used geometry rather
than the calculus in the mathematics of the Principia) obtained the
fluxional calculus in part to advance mathematics (a major advance over
Cavalieri's ponderous method of indivisibles, and in part to work out
and express mathematically the laws of gravity and of terrestrial and
celestial mechanics, illustrates that a theoretical/ applied
distinction is somewhat artificial as compared with the theoretical -
computational distinction and creator - consumer distinction.


- Message from bud...@nyc.rr.com -
   Date: Wed, 7 Mar 2012 14:41:08 -0500
   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



Irving,

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? That identification seems not without
issues but still pretty appealing to me, but maybe I've missed
something. (For readers unfamiliar with Peirce's way of
distinguishing theormatic from corollarial, see further below where
I've copied my Wikipedia summary with reference links

Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-11 Thread malgosia askanas
Irving wrote, quoting Peirce MS L75:35-39:

Deduction is only of value in tracing out the consequences of
hypotheses, which it regards as pure, or unfounded, hypotheses.
Deduction is divisible into sub-classes in various ways, of which the
most important is into corollarial and theorematic. Corollarial
deduction is where 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. Ordinary syllogisms and some deductions
in the logic of relatives belong to this class. 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. The subdivisions of theorematic deduction are of very high
theoretical importance. But I cannot go into them in this statement.


[...] Peirce's characterization of theorematic and corrolarial
deduction would seem, on the basis of this quote, to have to do with
whether the presumption that the premises of a deductive argument or
proof are true versus whether they require to be established to be
true [...]

I would disagree with this reading of the Peirce passage.  It seems
to me that the distinction he is making is, rather, between (1) the case
where the conclusion can be seen to follow from the premisses
by virtue of the logical form alone, as in A function which is continuous
on a closed interval is continuous on any subinterval of that interval
(whose truth is obvious without requiring us to imagine any continuous
function or any interval), and (2) the case where the deduction of the
conclusions from the premisses requires turning one's imagination
upon, and experimenting with, the actual mathematical objects
of which the theorem speaks, as in A function which is continuous
on a closed interval is bounded on that interval.

-malgosia

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Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-07 Thread Irving

About two and a half weeks ago, Garry Richmond wrote (among other
things), in reply to one of my previous posts:


You remarked concerning an older, artificial, and somewhat inaccurate
terminological distinction between practical or applied on the one hand
and pure or abstract on the other. In this context one finds Peirce
using pure, abstract and theoretical pretty much interchangeably,
while I agree that theoretical is certainly newer and I can see why
you think it is less artificial and inaccurate than the other two. But
on the other side of the distinction, while practical seems a bit
antiquated, applied appears to me quite accurate and legitimate. My
question then is simply this: what is the terminology used today in
consideration of this distinction? Is it, as I'm assuming, theoretical
and applied? Further, are there other important distinctions which
aren't aspects or sub-divisions of these two terms? Where, for example,
would you place Peirce's mathematics of logic, which he characterizes
as the simplest mathematics including a kind of mathematical valency
theory (to use Ken Ketner's language of monadic, dyadic, and triadic
relations retrospectively analyzed as tricategorial). A more
fundamental question: is there a place for this kind of 'valental'
(Ketner) thinking in contemporary mathematics or logic?


The characterization which I propounded obviously mirrors to a
considerable extent the medieval distinction between logica utens and
logica docens. The reason that I regard such distinctions between the
older, artificial, and somewhat inaccurate terminological distinction
between practical or applied on the one hand and pure or abstract on
the other is that the history of mathematics demonstrates that much of
what we think of as applied mathematics was not particularly created
for practical purposes, but turned out in any case to have
applications, whether in one or more of the mathematical sciences or
for other uses, but from intellectual curiosity, that is, for the sake
of illuminating or extending some aspect of a mathematical system or
set of mathematical objects, just to see where [else] they might lead,
what other new properties can be discovered; and as many examples in
the history of mathematics in the other direction, that new fields of
mathematics were developed for the sake of solving a particular problem
or set of problems in, say physics or astronomy, that led to the
development of abstract or theoretical systems. One might point to
numerous particular aspects of work, e.g., in real analysis that grew
out of dissatisfaction with Newton's fluxions or Leibniz's
infinitesimals in their ability to deal with problems in terrestrial
mechanics or in celestial mechanics. As a separate mathematical
problem, there is the issue of functions which are everywhere
continuous but nowhere differentiable, which lead Weierstrass to his
work in formalizing the theory of limits in terms of the epsilon-delta
notation. And Cantor's work in set theory emerged specifically as an
attempt to provide a mathematical foundation for Weierstrass's real
analysis. The “peculiarly behaving” functions of Jacobi and Weierstrass
turned out also to be applicable; the motion of a planar pendulum
(Jacobi), the motion of a force-free asymmetric top (Jacobi), the
motion of a spherical pendulum (Weierstrass), and the motion of a heavy
symmetric top with one fixed point (Weierstrass). The problem of the
planar pendulum, in fact, can be used to construct the general
connection between the Jacobi and Weierstrass elliptic functions.
Another example: group theory, as a branch of algebra, was used by
Felix Klein as a way of organizing geometries according to their
rotation properties; but group theory itself arose from the work of
Abel, Cayley, and others, to deal with generalizations of algebra, in
particular in their efforts to solve Fermat's Last Theorem and to
determine whether quintic equations have unique roots. The application
by Heisenberg and Weyl of group theory to quantum mechanics, makes
group theory, in this respect at least, applicable, as well as pure.
This is why I suggest that a more useful distinction is between
theoretical and computational rather than pure and applied.

It was, I think Vaughn Pratt who very recently (in a post to FOM)
proposed that the distinction between pure and applied be replaced
by a more reliable and compelling characterization in terms of the
consumers of mathematics; between those who create mathematics and
those who do not create, but make use of, mathematics. Given this
fluidity between theory and practice -- and one can find numerous
examples of mathematicians who were also physicists, e.g. Laplace, even
Euler, I think it would be beneficial to adopt Pratt's creator and
consumer distinction. A notable example of the latter would be
Einstein, who, with the help of Minkowski, applied the Riemannian
geometry to classical mechanics to provide the mathematical tools that
allowed 

Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-07 Thread Benjamin Udell
, University of Helsinki. 
2.. 2 Peirce, C. S., the 1902 Carnegie Application, published in The New 
Elements of Mathematics, Carolyn Eisele, editor, also transcribed by Joseph M. 
Ransdell, see From Draft A - MS L75.35-39 in Memoir 19 (once there, scroll 
down). 
3.. 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 in Corollarial Reasoning in the Commens Dictionary of 
Peirce's Terms.
- Original Message -  
From: Irving 
To: PEIRCE-L@LISTSERV.IUPUI.EDU 
Sent: Wednesday, March 07, 2012 8:32 AM 
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 
edition
About two and a half weeks ago, Garry Richmond wrote (among other things), in 
reply to one of my previous posts:

   You remarked concerning an older, artificial, and somewhat inaccurate 
terminological distinction between practical or applied on the one hand and 
pure or abstract on the other. In this context one finds Peirce using pure, 
abstract and theoretical pretty much interchangeably, while I agree that 
theoretical is certainly newer and I can see why you think it is less 
artificial and inaccurate than the other two. But on the other side of the 
distinction, while practical seems a bit antiquated, applied appears to me 
quite accurate and legitimate. My question then is simply this: what is the 
terminology used today in consideration of this distinction? Is it, as I'm 
assuming, theoretical and applied? Further, are there other important 
distinctions which aren't aspects or sub-divisions of these two terms? Where, 
for example, would you place Peirce's mathematics of logic, which he 
characterizes as the simplest mathematics including a kind of mathematical 
valency theory (to use Ken Ketner's language of monadic, dyadic, and triadic 
relations retrospectively analyzed as tricategorial). A more fundamental 
question: is there a place for this kind of 'valental' (Ketner) thinking in 
contemporary mathematics or logic?
The characterization which I propounded obviously mirrors to a considerable 
extent the medieval distinction between logica utens and logica docens. The 
reason that I regard such distinctions between the older, artificial, and 
somewhat inaccurate terminological distinction between practical or applied 
on the one hand and pure or abstract on the other is that the history of 
mathematics demonstrates that much of what we think of as applied mathematics 
was not particularly created for practical purposes, but turned out in any case 
to have applications, whether in one or more of the mathematical sciences or 
for other uses, but from intellectual curiosity, that is, for the sake of 
illuminating or extending some aspect of a mathematical system or set of 
mathematical objects, just to see where [else] they might lead, what other new 
properties can be discovered; and as many examples in the history of 
mathematics in the other direction, that new fields of mathematics were 
developed for the sake of solving a particular problem or set of problems in, 
say physics or astronomy, that led to the development of abstract or 
theoretical systems. One might point to numerous particular aspects of work, 
e.g., in real analysis that grew out of dissatisfaction with Newton's fluxions 
or Leibniz's infinitesimals in their ability to deal with problems in 
terrestrial mechanics or in celestial mechanics. As a separate mathematical 
problem, there is the issue of functions which are everywhere continuous but 
nowhere differentiable, which lead Weierstrass to his work in formalizing the 
theory of limits in terms of the epsilon-delta notation. And Cantor's work in 
set theory emerged specifically as an attempt to provide a mathematical 
foundation for Weierstrass's real analysis. The peculiarly behaving functions 
of Jacobi and Weierstrass turned out also to be applicable; the motion of a 
planar pendulum (Jacobi), the motion of a force-free asymmetric top (Jacobi), 
the motion of a spherical pendulum (Weierstrass), and the motion of a heavy 
symmetric top with one fixed point (Weierstrass). The problem of the planar 
pendulum, in fact, can be used to construct the general connection between the 
Jacobi and Weierstrass elliptic functions. Another example: group theory, as a 
branch of algebra, was used by Felix Klein as a way of organizing geometries 
according to their rotation properties; but group theory itself arose from the 
work of Abel, Cayley, and others, to deal with generalizations of algebra, in 
particular in their efforts to solve Fermat's Last Theorem and to determine 
whether quintic equations have unique roots. The application by Heisenberg and 
Weyl of group theory to quantum mechanics, makes group theory, in this respect 
at least, applicable, as well as pure. This is why I suggest that a more 
useful distinction is between theoretical and computational rather than pure

Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-02 Thread Catherine Legg
Thanks for the thoughtful reply, Gary!

The issue you raise about how deduction and induction should be
categorised is an interesting one. I had always thought of deduction
as falling clearly under secondness, due to the compulsion involved.
But you are right to note that in theorematic deduction the mind is
not passive but active, and that this form of reasoning was very
important to Peirce.

I don't see how one might interpret induction as secondness though.
Though a *misplaced* induction may well lead to the secondness of
surprise due to error. H...

Cheers, Cathy

On Thu, Feb 23, 2012 at 1:49 PM, Gary Richmond gary.richm...@gmail.com wrote:
 Cathy, list,

 When I first read your remark suggesting that the birth, growth and
 development of new hypostatic abstractions should be in the position
 of 3ns rather than argumentative proof of the validity of the
 mathematics as I had earlier abduced, I thought this might be another
 case of the kind of difficulty in assigning the terms of 2ns and 3ns
 in genuine triadic relations which had Peirce, albeit for a very short
 time in his career, associating 3ns with induction (while before and
 after that time he put deduction in the place of 3ns as necessary
 reasoning--I have discussed this several times before on the list and
 so will now only refer those interested to the passage, deleted from
 the 1903 Harvard Lectures--276-7 in Patricia Turrisi's edition--where
 Peirce discusses that categorial matter).

 I think his revision of his revision to his original position may have
 been brought about by the clarification resulting from thinking of
 abduction/deduction/induction beyond critical logic (where they are
 first analyzed as distinct patterns of inference), then in methodeutic
 where a complete inquiry--in which  hypothesis formation is 1ns, the
 deduction of the implications of the hypothesis for testing is 3ns,
 and, finally, the actual inductive testing is 2ns--provides a kind of
 whetstone for categorial thinking about these three. (Yet, even in
 that 1903 passage he remarks that he will leave the question open.)

 Be that as it may, I am beginning to think that you are clearly on to
 something and that that transforming of a predicate into a relation
 which we call hypostatic abstraction certainly ought to be in the
 place of 3ns. Re-reading parts of Jay Zeman's famous and fine article
 on hypostatic abstraction further strengthened that opinion. See:
 http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm  Zeman
 writes:

 It is hypostatic or subjectal abstraction that Peirce is interested
 in; a hint as to why he is interested in it is given in his allusions
 in these passages to mathematical reasoning [. . .] Jaakko Hintikka
 has done us the great service of bringing to our attention and tying
 to contemporary experience one of Peirce's central observations about
 necessary—which is to say mathematical—reasoning: this is that
 nontrivial deductive reasoning, even in areas where explicit
 postulates are employed, always considers something not implied in the
 conceptions so far gained [in the particular course of reasoning in
 question], which neither the definition of the object of research nor
 anything yet known about could of themselves suggest, although they
 give room for it.

 As is well known, Peirce calls this kind of reasoning theorematic
 (in contrast to corollarial reasoning) because it introduces novel
 elements into the reasoning process in the form of icons, which are
 then 'experimented upon in imagination.' 

 Zeman quotes Hintikka to the effect that Peirce himself seems to have
 considered a vindication of the concept of abstraction as the most
 important application of his discovery [of the theorematic/corollarial
 distinction] and then remarks that Peirce would indeed have agreed
 that the light shed on necessary reasoning by this distinction helps
 greatly to illuminate the role of abstraction. . .

 See, also: EP2:394  where Peirce comments that it is hypostatic
 abstraction that leads to the generalizality of a predicate and, of
 course, what is general is 3ns. In short, I think you are quite right
 Cathy to have suggested that correction of my categorial assignments.
 As Peirce notes near the end of the Additament to the Neglected
 Argument, hypothetic abstraction concerns itself with that which
 necessarily would be *if* certain conditions were established
 (EP2:450).

 Best,

 Gary

 On 2/21/12, Catherine Legg cl...@waikato.ac.nz wrote:
 Gary wrote:


 For the moment I am seeing these
 three as forming a genuine tricategorial relationship, which I'd diagram
 in my trikonic way, thus:

 Theoretical mathematics:

 (1ns) mathematical hypothesis formation (creative abduction--that piece
 of mathematics)
 | (3ns) argumentative proof (of the validity of the  mathematics)
 (2ns) the mathematics itself

 [...]

 Wouldn't argumentative proof be the 2ness, and the 3ness would be
 something like the birth, growth and 

Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-02 Thread Stephen C. Rose
1. Hypothesis (Abduction)

2. Induction  3.   Deduction
But isn't it also the case that we can mix firsts, seconds and thirds if we
think it appropriate. As in Terms Propositions Symbols.
Best, S
*ShortFormContent at Blogger* http://shortformcontent.blogspot.com/



On Fri, Mar 2, 2012 at 12:25 PM, Catherine Legg cl...@waikato.ac.nz wrote:

 Thanks for the thoughtful reply, Gary!

 The issue you raise about how deduction and induction should be
 categorised is an interesting one. I had always thought of deduction
 as falling clearly under secondness, due to the compulsion involved.
 But you are right to note that in theorematic deduction the mind is
 not passive but active, and that this form of reasoning was very
 important to Peirce.

 I don't see how one might interpret induction as secondness though.
 Though a *misplaced* induction may well lead to the secondness of
 surprise due to error. H...

 Cheers, Cathy

 On Thu, Feb 23, 2012 at 1:49 PM, Gary Richmond gary.richm...@gmail.com
 wrote:
  Cathy, list,
 
  When I first read your remark suggesting that the birth, growth and
  development of new hypostatic abstractions should be in the position
  of 3ns rather than argumentative proof of the validity of the
  mathematics as I had earlier abduced, I thought this might be another
  case of the kind of difficulty in assigning the terms of 2ns and 3ns
  in genuine triadic relations which had Peirce, albeit for a very short
  time in his career, associating 3ns with induction (while before and
  after that time he put deduction in the place of 3ns as necessary
  reasoning--I have discussed this several times before on the list and
  so will now only refer those interested to the passage, deleted from
  the 1903 Harvard Lectures--276-7 in Patricia Turrisi's edition--where
  Peirce discusses that categorial matter).
 
  I think his revision of his revision to his original position may have
  been brought about by the clarification resulting from thinking of
  abduction/deduction/induction beyond critical logic (where they are
  first analyzed as distinct patterns of inference), then in methodeutic
  where a complete inquiry--in which  hypothesis formation is 1ns, the
  deduction of the implications of the hypothesis for testing is 3ns,
  and, finally, the actual inductive testing is 2ns--provides a kind of
  whetstone for categorial thinking about these three. (Yet, even in
  that 1903 passage he remarks that he will leave the question open.)
 
  Be that as it may, I am beginning to think that you are clearly on to
  something and that that transforming of a predicate into a relation
  which we call hypostatic abstraction certainly ought to be in the
  place of 3ns. Re-reading parts of Jay Zeman's famous and fine article
  on hypostatic abstraction further strengthened that opinion. See:
  http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm  Zeman
  writes:
 
  It is hypostatic or subjectal abstraction that Peirce is interested
  in; a hint as to why he is interested in it is given in his allusions
  in these passages to mathematical reasoning [. . .] Jaakko Hintikka
  has done us the great service of bringing to our attention and tying
  to contemporary experience one of Peirce's central observations about
  necessary—which is to say mathematical—reasoning: this is that
  nontrivial deductive reasoning, even in areas where explicit
  postulates are employed, always considers something not implied in the
  conceptions so far gained [in the particular course of reasoning in
  question], which neither the definition of the object of research nor
  anything yet known about could of themselves suggest, although they
  give room for it.
 
  As is well known, Peirce calls this kind of reasoning theorematic
  (in contrast to corollarial reasoning) because it introduces novel
  elements into the reasoning process in the form of icons, which are
  then 'experimented upon in imagination.' 
 
  Zeman quotes Hintikka to the effect that Peirce himself seems to have
  considered a vindication of the concept of abstraction as the most
  important application of his discovery [of the theorematic/corollarial
  distinction] and then remarks that Peirce would indeed have agreed
  that the light shed on necessary reasoning by this distinction helps
  greatly to illuminate the role of abstraction. . .
 
  See, also: EP2:394  where Peirce comments that it is hypostatic
  abstraction that leads to the generalizality of a predicate and, of
  course, what is general is 3ns. In short, I think you are quite right
  Cathy to have suggested that correction of my categorial assignments.
  As Peirce notes near the end of the Additament to the Neglected
  Argument, hypothetic abstraction concerns itself with that which
  necessarily would be *if* certain conditions were established
  (EP2:450).
 
  Best,
 
  Gary
 
  On 2/21/12, Catherine Legg cl...@waikato.ac.nz wrote:
  Gary wrote:
 
 
  For the moment I am 

Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-02-22 Thread Gary Richmond
Cathy, list,

When I first read your remark suggesting that the birth, growth and
development of new hypostatic abstractions should be in the position
of 3ns rather than argumentative proof of the validity of the
mathematics as I had earlier abduced, I thought this might be another
case of the kind of difficulty in assigning the terms of 2ns and 3ns
in genuine triadic relations which had Peirce, albeit for a very short
time in his career, associating 3ns with induction (while before and
after that time he put deduction in the place of 3ns as necessary
reasoning--I have discussed this several times before on the list and
so will now only refer those interested to the passage, deleted from
the 1903 Harvard Lectures--276-7 in Patricia Turrisi's edition--where
Peirce discusses that categorial matter).

I think his revision of his revision to his original position may have
been brought about by the clarification resulting from thinking of
abduction/deduction/induction beyond critical logic (where they are
first analyzed as distinct patterns of inference), then in methodeutic
where a complete inquiry--in which  hypothesis formation is 1ns, the
deduction of the implications of the hypothesis for testing is 3ns,
and, finally, the actual inductive testing is 2ns--provides a kind of
whetstone for categorial thinking about these three. (Yet, even in
that 1903 passage he remarks that he will leave the question open.)

Be that as it may, I am beginning to think that you are clearly on to
something and that that transforming of a predicate into a relation
which we call hypostatic abstraction certainly ought to be in the
place of 3ns. Re-reading parts of Jay Zeman's famous and fine article
on hypostatic abstraction further strengthened that opinion. See:
http://web.clas.ufl.edu/users/jzeman/peirce_on_abstraction.htm  Zeman
writes:

It is hypostatic or subjectal abstraction that Peirce is interested
in; a hint as to why he is interested in it is given in his allusions
in these passages to mathematical reasoning [. . .] Jaakko Hintikka
has done us the great service of bringing to our attention and tying
to contemporary experience one of Peirce's central observations about
necessary—which is to say mathematical—reasoning: this is that
nontrivial deductive reasoning, even in areas where explicit
postulates are employed, always considers something not implied in the
conceptions so far gained [in the particular course of reasoning in
question], which neither the definition of the object of research nor
anything yet known about could of themselves suggest, although they
give room for it.

As is well known, Peirce calls this kind of reasoning theorematic
(in contrast to corollarial reasoning) because it introduces novel
elements into the reasoning process in the form of icons, which are
then 'experimented upon in imagination.' 

Zeman quotes Hintikka to the effect that Peirce himself seems to have
considered a vindication of the concept of abstraction as the most
important application of his discovery [of the theorematic/corollarial
distinction] and then remarks that Peirce would indeed have agreed
that the light shed on necessary reasoning by this distinction helps
greatly to illuminate the role of abstraction. . .

See, also: EP2:394  where Peirce comments that it is hypostatic
abstraction that leads to the generalizality of a predicate and, of
course, what is general is 3ns. In short, I think you are quite right
Cathy to have suggested that correction of my categorial assignments.
As Peirce notes near the end of the Additament to the Neglected
Argument, hypothetic abstraction concerns itself with that which
necessarily would be *if* certain conditions were established
(EP2:450).

Best,

Gary

On 2/21/12, Catherine Legg cl...@waikato.ac.nz wrote:
 Gary wrote:


 For the moment I am seeing these
 three as forming a genuine tricategorial relationship, which I'd diagram
 in my trikonic way, thus:

 Theoretical mathematics:

 (1ns) mathematical hypothesis formation (creative abduction--that piece
 of mathematics)
 | (3ns) argumentative proof (of the validity of the  mathematics)
 (2ns) the mathematics itself

 [...]

 Wouldn't argumentative proof be the 2ness, and the 3ness would be
 something like the birth, growth and development of new hypostatic
 abstractions?

 Cheers, Cathy

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-- 
Gary Richmond
Humanities Department
Philosophy and Critical Thinking
Communication Studies
LaGuardia College--City University of New York

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