Dear Irving,
Thanks for your reply. I realize my post was a diversion, not really
addressing the thread. I was simply playing with the idea that the two
mathematicians expressed, to see what would happen if art was considered. As I
said in original post, I agree with what they said for science and math, but
your last post also draws out other arguments on math as a human practice that
are interesting. But when considering art, I do consider its connection to
reality can be more direct, and will try to sound out an argument why. And I'm
trying to mean this in a Peircean sense.
I think claiming that art is fictive and therefore not real because a
literary character such as Helen of Troy has "no real-world counterpart" misses
the point: that claim is a literalization of the real as an existent thing, and
surely the real is more than an existent. The character or the work of art is
the reality embodied, iconically, not referentially. Perhaps it is like the
rainbow: you can't go somewhere else and touch it, its reality involves your
perspective right where you are.
Art does not only appeal to the senses, in my view, but also to
passionate reason. Sound, for example, "appeals" to the senses to determine an
interpretation, but music does that and more. It appeals to the soul, to the
whole passionate self.
Perhaps another way to put it is that authentic art is a primary
interpretant of reality, where authentic science is a secondary interpretant of
reality. Admittedly, art does not deal with many of the questions the sciences
precisely consider. But the real is not limited to what science considers.
Art not only depicts reality, as mathematics can do diagrammatically,
it enters into the creation of it. A work of music can be taken in a scientific
sense as revealing sonic laws, which could reveal a valid component of the
work. But as music it is also an aspect of the musical mind, bodying into
being. A work of art, in this sense, can be an act of creation, of spontaneous
reasonableness, in Peirce's sense that, "The creation of the universe, which
did not take place during a certain busy week, in the year 4004 B.C., but is
going on today and never will be done, is this very development of Reason."
Ongoing creation means that creation issues. How do you measure that reality,
the reality of the immeasurable creation issuing forth? You don't measure it,
you issue it, and in the process you are participant in the reality of creation
issuing forth as well as the iconic interpretant of it. This is the direct
connection to reality I am claiming for living art.
Gene Halton
-----Original Message-----
From: C S Peirce discussion list [mailto:PEIRCE-L@LISTSERV.IUPUI.EDU] On Behalf
Of Irving
Sent: Wednesday, March 14, 2012 9:01 AM
To: PEIRCE-L@LISTSERV.IUPUI.EDU
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce
edition
Not being an aesthetician, I would only say that whereas mathematics
and science are
expected to appeal to reason, the arts are expected to appeal
primarily, if not
necessarily exclusively, to the senses.
That being said, my short and simplistic answer to your question would
be that, as
regards art, and taking literature as an example, no one would claim
that the events or
characters depicted have real components outside of the work of art;
that is to say, a
Helen of Troy may inhabit the pages or the theater stage of a Homer or
a Goethe, but
accepting their "existence" amounts to a suspension of belief for the
sake of the
experience of the work of art. This is tantamount to Russell's appeal
to the theory of
descriptions to render meaningful propositions about the present bald
king of France, or
to Quine's pegasizing, without committing to some Meinongian level of
"being" for the
bald French king or Pegasus. The entities of art are granted to be
fictive, with no
real-world counterpart. Not necessarily so the entities of mathematics
or of the physical
universe, whether, say, triangles or photons.
Not for the more detailed attempt to tackle the question.
The question of notion of mathematical objects as fictive was spurred
for the end of the
nineteenth and early twentieth century philosophy of mathematics by
Hans Vaighinger's
Philosophie des 'Als-ob' (1911), in which mathematical concepts, and
theoretical thought
generally, are understood as fictions or based on fictions, that is, on
deliberately
false assumptions, and taking its cue from Moritz Pasch's still
forthcoming philosophical
study Mathematik am Ursprung (1927). A detailed study of Fiktionen in
der Mathematik
(1926) was undertaken by analytic geometer Christian Betsch (1888-1934)
starting from
Vaihinger's work, and concluding that mathematics does not, after all,
operate from
fictions, certainly not in Vaihinger's sense, although one's
philosophical position,
Betsch argued, determines whether one is willing or unwilling to
countenance fictions, as
do, for example, Millian empiricists. His understanding of the work of
Cantor, Frege,
Peano, and Russell is seen against the background of the role of
fictional entities in
mathematics. In dealing with Frege, for example, Betsch asserts [1926,
p. 224] that his
goal is the "Anförderung der Mathematik" by way of logic, while he
understands that, for
Russell [Betsch 1926, 96] fictions appear in the construction of
mathematical objects.
Against Vaihinger's argument that mathematicis is composed of competing
useful fictions
that essentially nullify each one the effect of the other, Betsch gives
a careful,
detailed, and subtle analysis of mathematics, in particular the
paradoxes that Russell
was instrumental in making widely known; and he concludes that, if with
further diligent
work mathematicians are unable to resolve or altogether banish the
paradoxes, then
mathematics itself must be a magnificent -- and worthwhile -- fiction.
Betsch himself,
whose Dr. Phil. from the University of Tübingen, Zur analytischen
Geometrie der dualen
Grössen, was written under the direction of Alexander von Brill
(1847-1935), was, as
[Gödel 1931] noted, prepared to accept fictions in mathematics in the
sense only of
tentative definitions or hypothetical propositions. In a doctoral
thesis for the
University of Erlangen in 1912 and published as a book the following
year [Lapp 1912;
1913], Adolf Lapp (b. 1888) undertook an epistemological analysis of
truth from the
perspectives of Heinrich Rickert, Edmund Husserl, and Vaihinger's Die
Philosophie des Als
Ob. For Vaihinger, who drew his evidence from outdated seventeenth- and
eighteenth-century thinkers and without reference to or knowledge of
the work of Karl
Weierstrass and others, the differential and integral calculus were the
quintessential
examples of this kind of fictional thinking. Vaihinger's justification
of infinitesimals
as "a method of opposite mistakes" [Vaihinger 1911 (2nd, 1913, ed.),
511ff.], as Stephan
Körner noted, preluded it from being taken seriously by
mathematicians. Friedrich
Waismann reported (without citation) that the popular
nineteenth-century author Lübsen
(otherwise unidentified, but indubitably the philosophical
mathematician Heinrich
Borchert Lübsen (1801-1864), author of the extremely popular and
long-lived Ausführliches
Lehrbuch der Analysis [1853], the Einleitung in die Infinitesimal-Rechnung zum
Selbstunterricht: mit Rücksicht auf das Nothwendigste und
Wichtigste [1855], and
similar works, creator of a method for self-instruction in mathematics)
wrote of the
differential calculus as a "mystical method operating with infinitely
small quantities,"
the differential being a mere "breath, a nothing," with George
Berkeley's [1734] comment
on the "ghosts of departed quantities" being quoted. In his textbook
expounding the
infinitesimal calculus, Lübsen (see 5th (1874) ed. [1855, p. 228])
therefore writes that
"In dieser wahrhaft schöpferischen Leibniz'schen 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" and include bits and
> pieces of such areas as probability theory, matrix theory and linear
> algebra, Venn diagrams, and the like). Pratt's creator/consumer is
> closer to what I had in mind, and aligns better, and I think, more
> accurately, with the older pure (or abstract or theoretical) vs.
> applied distinction.
>
> The attempt to determine whether, and, if so, how well, Peirce's
> theorematic/corollarial distinction correlates to the
> theoretical/computational or creator/consumer distinction(s) was not
> initially an issue for me. It was raised by Ben Udell when he asked me:
> "Do you think that your "theoretical - computational" distinction and
> likewise Pratt's "creator - consumer" distinction between kinds of
> mathematics could be expressed in terms of Peirce's "theorematic -
> corollarial" distinction?"
>
> I attempted to reply, based upon a particular quote from Peirce. What I
> gather from the responses to that second round is that the primary
> issue with my attempted reply was that Peirce's distinction was bound
> up, not with the truth of the premises, but rather with the method in
> which theorems are arrived at. If I now understand what most of the
> responses have attempted to convey, the theorematic has to do with the
> mechanical processing of proofs, where a simple inspection of the
> argument (or proof) allows us to determine which inference rules to
> apply (and when and where) and whether doing so suffices to demonstrate
> that the theorem indeed follows from the premises; whereas the
> corollarial has to do with intuiting how, or even if, one might get
> from the premises to the desired conclusion. In that case, I would
> suggest that another way to express the theorematic/corollarial
> distinction is that they concern the two stages of creating
> mathematics; that the mathematician begins by examining the already
> established mathematics and asks what new mathematics might be
>
> Ben Udell also introduces the issue of the presence of a lemma in a
> proof as part of the distinction between theorematic and corollarial.
> His assumption seems to be that a lemma is inserted into a proof to
> help carry it forward, but is itself not proven. But, as Malgosia has
> already noted, the lemma could itself have been obtained either
> theorematically or corollarially. In fact, most of us think of a lemma
> as a minor theorem, proven along the way and subsequently used in the
> proof of the theorem that we're after.
>
> I do not think that any of this obviates the main point of the initial
> answer that I gave to Ben's question, that neither my
> theoretical/computational distinction nor Pratt's "creator" and
> "consumer" distinction have anything to do with Peirce's
> theorematic/corollarial distinction.
>
> In closing, I would like to present two sets of exchanges; one very
> recent (actually today, on FOM, with due apologies to the protagonists,
> if I am violating any copyrights) between probability theorist William
> Taylor (indicated by '>') and set theorist Martin Dowd (indicated by
> '>>'), as follows:
>
>>> More seriously, any freshman philosopher encounters the fact that there are
>>> fundamental differences between physical reality and mathematical reality.
>
>> Quite so. And one of these is noted by Hilbert (or maybe Hardy,
>> anyone help?) >-
>
>> "The chief difference between scientists and mathematicians is that
>> mathematicians have a much more direct connection to reality."
>
>>> This does not entitle philosophers to characterize mathematical reality
>>> as fictional.
>
>> Quite so; but philosophers tend to have a powerful sense of entitlement.
>
> the other, in Gauss's famous letter November 1, 1844 to astronomer
> Heinrich Schumacher regarding Kant's philosophy of mathematics, that:
> "you see the same sort of [mathematical incompetence] in the
> contemporary philosophers.... Don't they make your hair stand on end
> with their definitions? ...Even with Kant himself it is often not much
> better; in my opinion his distinction between analytic and synthetic
> propositions is one of those things that either run out in a triviality
> or are false."
>
> ----- Message from bud...@nyc.rr.com ---------
> Date: Mon, 12 Mar 2012 13:47:10 -0400
> 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
>
>
>> 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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>
>
> ----- End message from bud...@nyc.rr.com -----
>
>
>
> 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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----- End message from eugene.w.halto...@nd.edu -----
Irving H. Anellis, Ph.D.
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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