Irving, list,
Thank you for your response, erudite and to the point as always.
I agree, it's hard even to imagine a mathematician simultaneously abjuring
abstraction and not abjuring mathematics itself. The main kind of abstraction
that I've read that mathematicians traditonally abjured in earlier centuries
was the abstraction not made to solve an already standing problem (e.g.,
imaginaries are needed for some roots of polynomials). In that narrower sense,
in his Britannica article Dieudonné called "abstractionists" the mathematicians
who abstract freely and exploratively.
How did I go so wrong in my previous post? Well, I believed a sentence (quoted
below) that has long been in the Wikipedia Peirce article. It had references
that I was in a poor position to check. You're saying in effect that the
article is wrong about van Heijenoort's opinion. So it may be wrong about the
two others' opinions as well. Is there an easy way to revise it without adding
much to its length? Will it be okay if I just get rid of the word
"semanticists"? Replace it with "particularists" (a word that I just made up)?
Jean Van Heijenoort (1967),[85] Jaakko Hintikka (1997),[86] and Geraldine
Brady (2000)[79] divide those who study formal (and natural) languages into two
camps: the model-theorists / semanticists, and the proof theorists /
universalists. Hintikka and Brady view Peirce as a pioneer model theorist.
79. a b Brady, Geraldine (2000), From Peirce to Skolem: A Neglected Chapter
in the History of Logic, North-Holland/Elsevier Science BV, Amsterdam,
Netherlands.
85. ^ van Heijenoort (1967), "Logic as Language and Logic as Calculus" in
Synthese 17: 324-30.
86. ^ Hintikka (1997), "The Place of C. S. Peirce in the History of Logical
Theory" in Brunning and Forster (1997), The Rule of Reason: The Philosophy of
C. S. Peirce, U. of Toronto.
If you can help me with that sentence, I'd much appreciate it.
You wrote,
> Setting aside, therefore, the issue of abstraction, the more complex issue
under consideration is that regarding the perceived distinction between model
theorists and semanticists on the one hand and proof theorists on the other.
This is an erroneous distinction insofar as the historical and philosophical
literature, from van Heijenoort forward, distinguishes between two types of
semantics
[SEMANTICS, with some added formatting] Model-theoretic (or intensional)
semantics.
(Actually, van Heijenoort's terminology is itself at first somewhat
misleading, insofar as he initially associated the limited universes of
discourses of the algebraic logicians with the set-theoretic, and not with the
course-of-values of Frege and the set theory of Russell; although he then
immediately corrected himself by associating the Russello-Fregean extensional
semantics with the set theoretical.) Set-theoretic (or extensional, which would
also include Frege's course-of-values, or Werthverlauf) semantics
If I've got it right, you're saying below that the model-theoretic approach
implies logic-as-calculus but not vice versa.
> Having said that, there is, for van Heijenoort and those who came after
him, a complex of dichotomies that are bound together to distinguish
[LOGICS, with some added formatting and futzing] Algebraic logic of De
Morgan, Boole, Peirce, and Schröder Quantification-theoretic - or more
properly, despite van Heijenoort - function-theoretic and set-theoretic logic
of Frege, Peano, and Russell
Logicae utentes, which are logic as calculus only, extensional, but with
restricted universe(s) of discourse, relativism/particularity, and for some,
model-theoretic (possibly with an intensional, rather than extensional,
semantic)
The classical Boole-Schröder calculus. Logica magna, which is logic as
language preeminently, but also as calculus, extensional semantic,
absolutism/universality.
Systems such as Frege's.
Van Heijenoort would agree that it was the incorporation of the
"model-theoretic" or logic as calculus approach of the "Booleans" or algebraic
logicians, by Löwenheim, Skolem, and Herbrand, [continued next right] ...into
the pure lingusitic approach of the Fregeans, that gave "modern" mathematical
logic its character as first-order functional (or predicate) logic and enabled
them and their successors, Gödel preeminently among them, the possibility of
tying the model-theoretic conception of satisfiability to the proof-theoretic
conception of validity, and enabled them to explore the model-theoretic and
proof-theoretic properties of systems such as Hilbert's and the Principia.
And Hilbert, somewhere in between, according to van Heijenoort.
The association of logica utens with algebraic logic and "calculus only" was a
bit surprising to me; I thought that logica utens was logic used in practice
rather than acquired by theoretical study. I guess the idea is that their
algebraic logic was concerned with formalizing and rendering theoretically
explicit the kinds of logic that people 'automatically' use or might use
(which connects with particularity, relation to context, etc.).
I end with a Latin digression: Since the phrases _logica utens_ and _logica
docens_ have long seemed a bit odd to me in view of their seeming Latin
meanings, what with the present (and active) participles, I took the
opportunity to look them up in the Century Dictionary. Under "logic":
Abstract logic, the general theory of logic (also called _logica docens_,
_general_ and _theoretical logic_): opposed to _concrete logic_, or logic as an
element of active thought in the prosecution of science (also called _logica
utens_, _special_ and _practical logic_). The terms _logica utens_ and _docens_
are derived from _logicus utens_, he who draws conclusions, and _logicus
docens_, he who frames demontrations. But the corresponding distinction of
branches of science is not very clear, and the terms are often used vaguely and
incorrectly.
Now it makes sense. It comes from the idea of the _reasoner_ teaching (docens)
/ using (utens). From logicus to logica for logic itself, one might have
expected _logica docta_ and _logica usa_ but obviously that didn't happen.
(Note to any Latin students: _utens_ is from _utor_, a deponent verb, i.e.,
conjugated in the passive voice with the sense of active voice, but a deponent
verb's present (active) participle does not follow the "reverse voice" rule,
and it has active, not passive, sense. I'm not expert enough by any means, but
I've seen the deponent past (and passive) participle use in active sense when
part of a verb, and in passive sense when used as an adjective.) End of Latin
digression.
Best, Ben
----- Original Message -----
From: "Irving"
To: PEIRCE-L@LISTSERV.IUPUI.EDU
Sent: Saturday, December 03, 2011 5:07 PM
Subject: Re: [peirce-l] SLOW READ: On the Paradigm of Experience Appropriate
for Semiotic
Ben Udell wrote [begin quote]:
> Gary F., list,
> ...
> You wrote:
>> Abstraction (in the sense above) obviously has its uses in the process
of learning from experience, but not to the degree that it can *replace*
experience. My guess is that this is the same issue that Irving and others have
been dealing with in this thread with regard to ?formalism?, but not being a
mathematician, i don't always follow their idiom.
> I'm not a mathematician either, and Irving can correct me if he wants to
plow through my prose, but I agree that the issue is related. There's a related
issue of model theorists and semanticists, versus proof theorists, who are more
like formalists. Model theorists and semanticists see formal languages as being
_about_ subject matters which are 'models' for the formalism. Somebody once
told me that when I say that, in a deduction, the premisses validly imply the
conclusions, that's proof-theoretic in perspective, but when I say that, in a
deduction, if the premisses are true then the conclusion is true, that's
model-theoretic in perspective.
> Peirce is usually classed on the model theorist/semanticist side, and
Goedel's aim is said to have been to show that mathematics can't be regarded as
pure formalism, a show about nothing. Proof theorists and formalists are more
inclined to see math as formal calculi, systems of marks transformable
according to rules, not as language _about_ things. Now, calculation, as far
as I can tell, is (deductive mathematical) reasoning with terms. E.g.,
(trivially) "5 ergo 5"*, instead of "there is a horse ergo there is a horse".
I can kind of see how propositions (a.k.a. zero-place terms) versus (other)
terms, would align with facts, real objects, etc., versus marks. If you look at
propositions as marks, then they're like term-inviting clumpish things (as
opposed to proposition-inviting facts or states of affairs.)
> But it's an alignment by some sort of affinity or correlation, not
identity. Semantics is concerned not just with reference by propositions but
with reference by terms to things; the terms are not ideally non-referring
marks in semantics. For a formalist, the marks _are_ the things.
[end quote]
If I understand aright, one of the issues being raised by Ben and Gary is the
link between abstraction and formalism, and whether there is a connection as
well between model theorists and semanticists on the one hand, and proof
theorists on the other, where the latter are close to formalists as being
abstractionist.
The first part of my reply in this case is that neither intuitionists (such as
Brouwer) or logicists (such as Frege or Russell) abjure abstraction any more
than formalists. Indeed, Piaget formulated his genetic "constructive
epistemology" for his developmental psychologist Jean Piaget, describing
abstract reasoning as the final stage of cognitive development by referring
directly to Brouwer. The expression "constructivist epistemology" was first
used by Piaget in 1967, in the article "Logique et Connaissance scientifique"
in the Encyclopédie de la Pléiade. Piaget refers directly to the Brouwer and
his radical constructivism. (See, e.g., my "La psicologia di Piaget, la
matematica costruttivista e l'interpretazione semantica della verita secondo la
teoria degli insiemi" (Nominazione: Rivista Internazionale di Logica 2 (1981),
174-188) on how Piaget's psychology describes the epistemology of number and
set theory.
Setting aside, therefore, the issue of abstraction, the more complex issue
under consideration is that regarding the perceived distinction between "model
theorists and semanticists on the one hand and proof theorists on the other.
This is an erroneous distinction insofar as the historical and philosophical
literature, from van Heijenoort forward, distinguishes between two types of
semantics, namely the set-theoretic (or extensional, which would also include
Frege's course-of-values, or Werthverlauf, semantics) and the model-theoretic
(or intensional). (Actually, van Heijenoort's terminology is itself at first
somewhat misleading, insofar as he initially associated the limited universes
of discourses of the algebraic logicians with the set-theoretic, and not with
the course-of-values of Frege and the set theory of Russell; although he then
immediately corrected himself by associating the Russello-Fregean extensional
semantics with the set theoretical.)
Having said that, there is, for van Heijenoort and those who came after him, a
complex of dichotomies that are bound together to distinguish the algebraic
logic of De Morgan, Boole, Peirce, and Schröder on the one hand from the
"quantification-theoretic -- or more properly, despite van Heijenoort,
function-theoretic and set-theoretic logic of Frege, Peano, and Russell. All of
the elements of this complex are to be brought together in my forthcoming "Van
Heijenoort's Conception of Modern Logic, in Historical Perspective" for the
special issue of Logica Universalis commenmorating the van Heijenoort
centenary. These are on the one side: logica magna, which is logic as language
preeminently, but also as calculus, extensional semantic,
absolutism/universality; logicae utenses, which are logic as calculus only,
extensional, but with restricted universe(s) of discourse,
relativism/particularity, and for some, model-theoretic (possibly with an
intensional, rather than extensional, semantic). The former characterizes such
systems as Frege's, the latter, the classical Boole-Schroder calculus. Van
Heijenoort would agree that it was the incorporation of the "model-theoretic"
or logic as calculus approach of the "Booleans" or algebraic logicians, by
Löwenheim, Skolem, and Herbrand, into the pure lingusitic approach of the
Fregeans, that gave "modern" mathematical logic its character as first-order
functional (or predicate) logic and enabled them and their successors, Gödel
preeminently among them, the possibility of tying the model-theoretic
conception of satisfiability to the proof-theoretic conception of validity, and
enabled them to explore the model-theoretic and proof-theoretic properties of
systems such as Hilbert's and the Principia. So: Where does Hilbert stand in
all of this? For van Heijenoort, Hilbert stood somewhere between these two
camps. For example, it was Herbrand, examining Hilbert's conception of "being a
proof" that allowed Herbrand to treat valdity as satisfiability in every model
-- which he did, of course, through "Herbrand expansion", which is just
Peirce-Mitchell-Schröder defining quantified formulas as logical sums and
products for a k-ary universe, of every arity k. Likewise, it was Löwenheim,
adopting the Peirce-Mitchell-Schröder definition of quantifiers as logical sums
and products, that made the Löwenheim-Skolem Theorem [LST] possible. (But
Peirce already had a finite version of the LST in 1886 (the annotation on p,
464 for "The Logic of Relatives: Qualitative and Quantitative", W5:376, ll.
31-36, is mine.)
The basic formative texts for this historico-philosophical approach are: van
Heijenoort, "Set-theoretic Semantics", "Logic as Calculus and Logic as
Language", "Relativism and Absolutism in Logic" and "Système et métasystème
chez Russell", and Hans Sluga, "Frege Against the Booleans".
Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info
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