Clark, List: CG: I agree that this definitely tends to make nominalism self-refuting which I see as a problem rather than a strength.
A problem for nominalism or for realism? Is it legitimate for a nominalist to deny that holding everything real to be singular is self-contradictory, on the grounds that singularity is not a property? (I am having that very argument with a self-professed nominalist in another context right now.) CG: I assume he’s somewhat platonic about mathematical objects. That is more akin to Godel than the logicists or the constructivists. Yet honestly if someone told me he was a logicist or a constructivist I’d not be at all shocked either. I am not that familiar with the alternatives, but Christopher Hookway, Matthew Moore, and others seem to think that his views--especially his emphasis on diagrammatic reasoning--are closest to mathematical structuralism. As with other sciences, he was more interested in the *methods *of mathematicians than the *objects *of their investigations. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Professional Engineer, Amateur Philosopher, Lutheran Layman www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Mon, Jan 9, 2017 at 3:56 PM, Clark Goble <cl...@lextek.com> wrote: > > On Jan 7, 2017, at 6:52 PM, Jon Alan Schmidt <jonalanschm...@gmail.com> > wrote: > > With reference to individuals, I shall only remark that there are certain > general terms whose objects can only be in one place at one time, and these > are called individuals. They are generals that is, not singulars, because > these latter occupy neither time nor space, but can only be at one point > and can only be at one date. (W2:180-181; 1868) > > Peirce noted here that "the character of singularity" is itself a general, > which seems to render nominalism--the view that everything real is > singular, so nothing real is general--effectively self-refuting. He > defined an individual as a collection of singulars joined across places and > times, which is thus general when taken as a whole. Furthermore, *absolute > *singulars are "mere ideals," such that (ironically) an individual is > really a *continuum *as Peirce came to understand that concept decades > later. Consequently, anything that we cognize *about *individuals is > *necessarily > *general, rather than singular. > > I agree that this definitely tends to make nominalism self-refuting which > I see as a problem rather than a strength. > > While I’m not quite sure how to deal with this issue, I suspect that this > arises out of Peirce’s conception of infinity as opposed to say Cantor’s. > Peirce thinks through it by division while Cantor tends to think through it > in terms of sets of individuals. Since for Peirce any ‘individual’ is > formed from two cuts, that implies a line that can itself be further cut. > It’s really not set theory. > > I’m not sure how Peirce viewed number theory or even how much he knew of > it given how much is from the 20th century. Certainly his father Benjamin > Peirce had worked on the roots of number theory. If we think of number > theory to arithmetic of integers in terms of sets that would appear to lead > to thinking of individuals not as a collection of singulars. > > I did some Googling, since this is an area of Peirce’s thought I’m > ignorant on. He did write on number theory in the paper “Logical Studies of > the Theory of Numbers” around 1890. That paper seems to be somewhat similar > to what Hilbert later did (his 10th problem). That is he was looking for an > algorithm that would tell us if there are proofs. He thought we should do > this by reducing equations to boolean algebra but that appears to merely be > a hypothesis of what one might be able to do. > > I couldn’t find an online copy of that paper. The closest was this > discussion of the paper by Irving Anellis. > > http://www.iupui.edu/~arisbe/menu/library/aboutcsp/ANELLIS/csp&hilbert.pdf > > I’ll confess right up that I just am not at all sure how Peirce viewed > mathematics. Given my own background in mathematics this is pretty > embarrassing as you’d think I’d know something about this. I assume he’s > somewhat platonic about mathematical objects. That is more akin to Godel > than the logicists or the constructivists. Yet honestly if someone told me > he was a logicist or a constructivist I’d not be at all shocked either. > Those seem just as likely a way to conceive of in his philosophy, although > he’d probably then argue that the structures of constructure or logic are > themselves real independent of human thought as possibilities. > > Going back to infinity along with Cantor and Dedekind, Peirce asserted > that Dedekind’s cut actually came from Peirce. Apparently before publishing > on that Peirce had sent Dedekind a paper on such approaches. In contrast to > their approaches Peirce saw a problem that needed to be solved. (Relating > to the other thread, this suggests that he was thinking along metaphysical > lines in what we’d today call modal realism) That is Peirce saw the issue > tied to the logic of possibility. Peirce saw their approach as “inchoate” > which brings to mind that quote on metaphysics we’ve been discussing in the > other thread. > > Generality is, indeed, an indispensable ingredient of reality; for mere > individual existence or actuality without any regularity whatever is a > nullity. Chaos is pure nothing. (“What Pragmatism Is,” CP 5.431 1905) > > The continuum is a General. It is a General of a relation. Every General > is a continuum vaguely defined. (“Letter to E. H. Moore,” NEM 3.925 1902) > > Continuity, as generality, is inherent in potentiality, which is > essentially general. (...) The original potentiality is essentially > continuous, or general. > (“Detached Ideas on Vitally Important Topics,” CP 6.204-5 1908) > > The possible is general, and continuity and generality are two names for > the same absence of distinction of individuals. (“Multitude and Number,” CP > 4.172 1897) > > A perfect continuum belongs to the genus, of a whole all whose parts > without any exception whatsoever conform to one general law to which same > law conform likewise all the parts of each single part. Continuity is thus > a special kind of generality, or conformity to one Idea. More specifically, > it is a homogeneity, or generality among all of a certain kind of parts of > one whole. Still more specifically, the characters which are he same in all > the parts are a certain kind of relationship of each part to all the > coordinate parts; that is, it is a regularity. (“Some Amazing Mazes,” CP > 7.535 note 6 1908) > > The key to Peirce’s metaphysics which seems wrapped up in these > investigation of continuity and the metaphysics of individualism is the > idea that *to be* requires a repetition which demands that it not be > singular in a strong sense. >
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