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From Peirce 1885 unpublished till _Collected Papers_, "An American Plato" - Review: Josiah Royce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
On 12/3/2025 5:58 PM, Jon Alan Schmidt wrote:
_ _ _ _ _ _ _ _ _ _ ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . ► UNSUBSCRIBE FROM PEIRCE-L . But, if your subscribed email account is not your default email account, then go to https://list.iu.edu/sympa/signoff/peirce-l . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.Ben, List:
I changed the subject line to match the topics that your post addresses.
BU: I think Peirce seldom if ever wrote about the result of "infinite" inquiry. He said that inquiry pushed far enough or for long enough will reach the truth - sooner or later - but still inevitably.
We are using different terms but seem to be saying essentially the same thing. The pragmaticistic definition of truth as what an infinite community *would* affirm after infinite investigation is derived from Peirce's well-known statement, "The opinion which is fated to be ultimately agreed to by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real" (CP 5.407, EP 1:139, 1878). In my own words, truth is the final interpretant of every sign whose dynamical object is a reality. Accordingly, what I am discussing is a real but potential or ideal infinity, not an actual infinity; again, a regulative principle and an intellectual hope--what Peirce sometimes calls a "would-be." He says so himself in the subsequent paragraph.
CSP: Our perversity and that of others may indefinitely postpone the settlement of opinion; it might even conceivably cause an arbitrary proposition to be universally accepted as long as the human race should last. Yet even that would not change the nature of the belief, which alone could be the result of investigation carried sufficiently far; and if, after the extinction of our race, another should arise with faculties and disposition for investigation, that true opinion must be the one which they *would ultimately* come to. "Truth crushed to earth shall rise again," and the opinion which *would finally* result from investigation does not depend on how anybody may actually think. (CP 5.408, EP 1:139, 1878; bold added)
Moreover, in his very next published article, he refers to "an *unlimited* community" and "a hope, or calm and cheerful wish, that the community may last *beyond any assignable date*," thus facilitating "the *unlimited *continuance of intellectual activity" (CP 2.654-5, EP 1:150, 1878; bold added). His further definitions of truth after the turn of the century reflect his even stronger embrace of scholastic realism, as well as his development of semeiotic. "Truth is that concordance of an abstract statement with the *ideal limit* towards which *endless investigation would tend* to bring scientific belief" (CP 5.565, 1902; bold added). "Now thought is of the nature of a sign. In that case, then, if we can find out the right method of thinking and can follow it out,--the right method of transforming signs,--then truth can be nothing more nor less than the last result to which the following out of this method *would ultimately* carry us" (CP 5.553, EP 2:380, 1906; old added).
BU: As I recall, Peirce had doubts about the reality of things in mathematics, but he thought that some of those things imposed themselves on the mind with a forcefulness very like that of the real.
These might be the remarks that you have in mind.
CSP: The pure mathematician deals exclusively with hypotheses. Whether or not there is any corresponding real thing, he does not care. His hypotheses are creatures of his own imagination; but he discovers in them relations which surprise him sometimes. A metaphysician may hold that this very forcing upon the mathematician's acceptance of propositions for which he was not prepared, proves, or even constitutes, a mode of being independent of the mathematician's thought, and so a *reality*. But whether there is any reality or not, the truth of the pure mathematical proposition is constituted by the impossibility of ever finding a case in which it fails. (CP 5.567, 1902)
The realities that pure mathematicians study are not actualities (2ns) with which they react, but logical possibilities (1ns) that they imagine, along with necessary consequences (3ns) that they draw from them--some of which can be far from obvious when they initially formulate their hypotheses, and are thus surprising whenever they are discovered. Peirce's distinction between corollarial and theorematic (or theoric) reasoning comes into play here, even though both are deductive (e.g., see CP 7.204-5, EP 2:96, 1901; NEM 4:1-12, 1901; CP 4.612-6, 1908; NEM 3:602, 1908). As a result, "Mathematics is purely hypothetical: it produces nothing but conditional propositions. Logic, on the contrary, is categorical in its assertions" (CP 4.240, 1902).
Regards,
Jon Alan Schmidt - Olathe, Kansas, USA
Structural Engineer, Synechist Philosopher, Lutheran Christian
www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidtOn Tue, Dec 2, 2025 at 1:33 PM Benjamin Udell <[email protected]> wrote: >
Jon, list,
I dip in for a moment, then vanish. I wanted to reply to posts by Edwina, Robert, and Ulysses but got busy as I do these days. I hope I'll get to those.
Jon, you wrote,
What an infinite community *would* affirm after infinite investigation is precisely how Peirce explicates the meaning of *truth* in practical terms--those beliefs whose corresponding habits of conduct *would* never be confounded by any *possible* future experience.
I think Peirce seldom if ever wrote about the result of "infinite" inquiry. He said that inquiry pushed far enough or for long enough will reach the truth - sooner or later - but still inevitably. The inquiry that continues indefinitely, by an indefinite community of inquirers, will attain, sooner or later, definite increase of knowledge. Each increase in actual knowledge occurs, as I understand it, at a finite remove from the inquiry's beginning, while you sound like you're discussing an actual infinity - e.g., an infinity of years or an infinity of one year's achieved subdivisions (sounds like it would get infinitely hot) - after which the truth is reached. I remember over 10 or 15 years ago discussing on peirce-l with Clark Gobel the idea of an inquiry into the full meaning of one's wife, not just one's wife as a sign of this or that or the weather today, but as one's wife per se, as representing everything that one's wife may represent. I thought that such an inquiry was so open-ended that maybe it _would_ require an eternity of inquiry, like the final entelechy of the universe (or whatever Peirce called it) maybe because a real example of "full meaning" is somehow too 2nd-order semiosic, to be dealt with finitely. Well, Clark seemed not to like that idea, while I was thinking vaguely (indeed as I'm no expert) of Turing oracles and the like.
I ought to note that, as to the reality of undiscovered legisigns, Peirce himself seemed reluctant to assert the reality of things in pure mathematics - discovered or undiscovered. I've long much leaned in favor of it - maths as discovered, not invented. The mathematician Kronecker split the difference, saying that God created the integers, all the rest is the work of man. As I recall, Peirce had doubts about the reality of things in mathematics, but he thought that some of those things imposed themselves on the mind with a forcefulness very like that of the real. Unfortunately I lost the email drafts where I kept the quotes. Maybe one will need to allow of "grades" of realness. I have no idea how to do that in a non-handwaving way.
Best, Ben
_ _ _ _ _ _ _ _ _ _ ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . ► <a href="mailto:[email protected]";>UNSUBSCRIBE FROM PEIRCE-L</a> . But, if your subscribed email account is not your default email account, then go to https://list.iu.edu/sympa/signoff/peirce-l . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
