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 >> >> --------------------------------------------------------------------------------- >> You are receiving this message because you are subscribed to the >> PEIRCE-L listserv. To remove yourself from this list, send a message >> to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in >> the body of the message. To post a message to the list, send it to >> PEIRCE-L@LISTSERV.IUPUI.EDU >> > > > ----- 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 > > --------------------------------------------------------------------------------- > You are receiving this message because you are subscribed to the > PEIRCE-L listserv. To remove yourself from this list, send a message > to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in > the body of the message. To post a message to the list, send it to > PEIRCE-L@LISTSERV.IUPUI.EDU > > --------------------------------------------------------------------------------- > You are receiving this message because you are subscribed to the > PEIRCE-L listserv. To remove yourself from this list, send a message > to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in > the body of the message. To post a message to the list, send it to > PEIRCE-L@LISTSERV.IUPUI.EDU > ----- 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 --------------------------------------------------------------------------------- You are receiving this message because you are subscribed to the PEIRCE-L listserv. To remove yourself from this list, send a message to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in the body of the message. To post a message to the list, send it to PEIRCE-L@LISTSERV.IUPUI.EDU --------------------------------------------------------------------------------- You are receiving this message because you are subscribed to the PEIRCE-L listserv. To remove yourself from this list, send a message to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in the body of the message. To post a message to the list, send it to PEIRCE-L@LISTSERV.IUPUI.EDU