Ben, List: BU: Peirce used the word "indefinite" as much closer, in signification, to "vague" than to "infinite".
I agree and did not mean to imply otherwise. However, in the quotations that I provided, he explicitly refers to "an unlimited community" and "endless investigation." I suppose that I could amend "what an infinite community would believe after infinite investigation" to "what an unlimited community would believe after endless investigation" accordingly, but is there a meaningful difference between these two summary formulations? Again, this is not an *actual *"final opinion" that *will *be reached "after infinite time" (your words), it is a real but "*ideal *limit towards which endless investigation *would *tend to bring scientific belief" (Peirce's words, italics mine). As he says earlier in the same paragraph from which you quoted, "The final opinion which would be sure to result from sufficient investigation may possibly, in reference to a given question, never be actually attained, owing to a final extinction of intellectual life or for some other reason. In that sense, this final judgment is not *certain *but only possible" (CP 8.43, 1885). He adds later in that paragraph, "there is nothing to distinguish the unanswerable questions from the answerable ones, so that investigation will have to proceed as if all were answerable ... it really makes no difference whether or not all questions are actually answered, by man or by God, so long as we are satisfied that investigation has a universal tendency toward the settlement of opinion" (ibid.). BU: It's interesting that Peirce, as you point out, did assert that there [is] truth in mathematics, even though he seemed reluctant to go all-in on mathematics harboring the real. Indeed, he was not a mathematical Platonist himself but said "that the typical Pure Mathematician is a sort of Platonist. Only, he is [a] Platonist who corrects the Heraclitan error that the Eternal is not Continuous. The Eternal is for him a world, a cosmos, in which the universe of actual existence is nothing but an arbitrary locus. The end that Pure Mathematics is pursuing is to discover that real potential world" (CP 1.646, RLT 121, 1898). That is why *truth *in pure mathematics properly takes the form of if-then propositions, with real *logical *possibilities (e.g., postulates) as their antecedents and real *conditional *necessities (e.g., theorems) as their consequents. "Thus, the mathematician does two very different things: namely, he first frames a pure hypothesis stripped of all features which do not concern the drawing of consequences from it, and this he does without inquiring or caring whether it agrees with the actual facts or not; and, secondly, he proceeds to draw necessary consequences from that hypothesis" (CP 3.559, 1898). Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Fri, Dec 5, 2025 at 7:29 AM Benjamin Udell <[email protected]> wrote: > Jon, list, > > Peirce used the word "indefinite" as much closer, in signification, to > "vague" than to "infinite". It seems convenient to call indefinitely long > times or distances infinite because of some part-way cognateness between > the words, and because times or distances may seem infinite to us lowly > mortals, and because Ancient Greek _apeiros_ seems to have been used in > both senses "indefinite" and "infinite", and because, if there are any > inquiries that seem interminable, still it seems quite plausible that > intelligent beings will bag their answers after infinite time. > > In F.R.L. (1899) http://www.princeton.edu/~batke/peirce/frl_99.htm , > Peirce said that Auguste Comte said that we earthlings never would be able > to discover the chemical composition of the stars, and that, very soon > afterward, the spectroscope was invented ("discovered," quoth Peirce, very > decidedly, I suspect) which soon enough revealed the chemical composition > of the stars. Peirce usually thought in terms of a definite increase of > knowledge after some actually elapsed definite time, given in advance a > prospect of an indefinite amount of time to play with in the first place. > To say guarantee the final opinion after infinite time seems like unneeded > cheating, anyway confusing to people new to Peirce. Interestingly more > precise would be to say what kind of question _would_ require an infinite > time. The "full meaning" or full final interpretant of one's spouse? > People mention the halting problem as maybe solvable (even deductively) > with infinite time. The difference remains infinite between (A) finite, > soever indefinite and soever prolonged or extended but still finite, and > (B) infinite. > > From Peirce 1885 unpublished till _Collected Papers_, "An American Plato" > - Review: Josiah Royce > CP 8.43, also in Writings 6. Note that Peirce took a somewhat cosmic view > even as he discussed "questions asked," not questions _askable_ (expectably > or imaginably or whatever). > > BEGIN QUOTE > The problem whether a given question will ever get answered or not is not > so simple; the number of questions asked is constantly increasing, and the > capacity for answering them is also on the increase. If the rate of the > latter increase is greater than that of the [former] the probability is > unity that any given question will be answered; otherwise the probability > is zero. > END QUOTE. > > Peirce was discussing definite and finite amounts of time of actual > discovery, and an indefinite amount of time in which to discover. Peirce > didn't promise that which Quine and others later wanted, an observable > normal rate of progress in inquiries. It's a question of contingent mental > evolution, not of partly conditional but still pre-programmed vegetable > growth. Of course you know all that. But the involvement of vague, > indefinite future dates of discovery doesn't morph by itself into the > involvement of infinities of inquiry. Sorry, I'm repeating myself, I guess > it's time for the old man to take a nap. > > Thanks for the Peirce quote that you found, it's exactly the passage that > I was thinking of. > > It's interesting that Peirce, as you point out, did assert that there > truth in mathematics, even though he seemed reluctant to go all-in on > mathematics harboring the real. His usual definitions of truth and the real > lock the two ideas together. Well, the ideas are regulative, not > speculative, but one suspects that Peirce would welcome a stronger argument > for the real in mathematics. (It would be terminologically easier if we > called the real numbers singulions and the complex numbers binions. Maybe > not easier, what would we call the imaginary numbers?) > > Best, Ben >
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