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Jerry Dozoretz

Originally published in Peirce Studies 1 (Institute for Studies in Pragmaticism at Texas Tech University, Lubbock,Texas), 1979, pp. 77-87, this paper is being made available here for purposes of critical feedback and possible revision. Since webpages are not conveniently numbered for scholarly reference, page numbers of the original have been interpolated into the text just before the content for the page so numbered.e.g. "|77|" marks the beginning of page 77.


     It should be noted initially that the thesis I advance in this paper is part of a broader thesis concerning the general nature of inquiry and a host of surrounding epistemological issues. In the broader context I concern myself with questions such as whether indubitability-in-principle plays any role within Peircean epistemic methodology; whether there are, for Peirce, a variety of epistemically admissible kinds of indubitability; what 'the relation is between truth, indubitability and inquiry; and other related questions as well. Here, however, I limit myself to the specific problem concerning the relationship between indubitability, internal reality and the domain of the fictitious; in particular I try to show the importance, for Peirce, of the distinction between the internally real and the fictitious.

     We may begin by noticing that if one cannot be wrong in the judgments one makes -- if one's judgments are indubitable-in-principle -- then the object of such a judgment is precisely what one thinks it to be, no matter what one thinks it to be. Such judgments are, in effect, constitutive of that of which they are the judgment, which makes the nature of the object wholly dependent upon how one views it. Thus if we refer to something as wholly dependent upon how we think it to be, this is to refer to it as though it were a fiction (or figment of our imagination). For what better definition of an imaginary object (or fiction) could there be than that it is an object wholly constituted by what we think it to be? And this is precisely the Peircean conception of a fictional object: "If an object is whatever I or any man or men will have it be, or imagine it, then it is a fiction."1 Or again: "If the attributes of or possible true assertions about an object |78| could vary according to the way in which you or I or any man or actual body of single men, living at any time or times, might think about that object, then that object is what I call a figment.2 Thus when we regard a judgment as indubitable,3 we are, according to Peirce, committing ourselves to holding that the object is (and can be) no more nor less than we think it to be. If one's judgment about the object changed, the object itself would also change: "if the world were a dream, we could just dream otherwise."4 The object would be no different from a creation of the imagination; for there would be no way to distinguish it from one (without, in the attempted distinction-drawing process, begging the question).

     Now Peirce should not be understood to be maintaining that dreams and hallucinations, for example, are in no sense real. On the contrary, he sharply distinguishes between the occurrence of the dream or hallucination and the substance thereof, granting reality to the former and denying it to the latter:

Suppose that I experienced fifty years ago a visual hallucination. In that case it would be true concerning the hallucination that it consisted in my seeing something that was not at the time in the place where I saw it; and this would be true though I had never mentioned it to anybody, or though it had never entered any person's head to suspect such a thing, and though I myself had totally forgotten it. The hallucination would have been real, provided we mean by the hallucination a certain psychical experience of mine. But the substance of the hallucination, that which I say, would be unreal, since everything that would be true of it would be true only in the sense that I, a determinate person, thought, or imagined (a species of thinking) that I saw what I say.5
Moreover, there are imaginable circumstances which would be conducive to viewing the substance or subject matter of a dream (that is, that which was dreamt) as real, the most obvious being an occasion when one dreams something that actually happens:

Suppose that I once dreamed that some describable event took place. Then that event will be 'what' I dreamt and the event of my so dreaming will be the fact 'that' I dreamed so. What I dreamt must be called a Figment, in my terminology, unless, indeed, the dream 'came true' in part, in which case it was, in that part real, however accidentally.6

     Having begun to glimpse the connection between indubitability and fictitiousness we must take great care, for the mere fact that something is a creation of reason or ens rationis7 does not, according to Peirce, entail that it is a figment:  |79|

It may naturally be supposed that things made by thought are necessarily figments, and thus not real. But insist, my reader, upon attaching definite significations to your words, and that done carefully attend to those definite significations, -- without which no genuine science of philosophy is possible -- and you will perceive that a thing made by thought, an ens rationis, is not necessarily a figment, in the sense of something arbitrarily imagined, and therefore, unreal.8

For in addition to figments (that which is arbitrarily imagined and wholly constituted by what somebody thinks it to be), which are unreal, and the real (which is independent of what you or I or any number of men may think about it), there is for Peirce a kind of middle ground, as it were, which is neither fictional and thereby unreal nor yet real in the aforementioned sense. In what follows I shall endeavor to show that in order to isolate and describe the "middle ground" mentioned we shall have to make use of Peirce's "internal"/"external" reality distinction, and that the distinction between positive truth and mathematical truth may be best understood in this context.

     The mere fact that something is a creation of reason or ens rationis does not, as we have seen, entail the conclusion that it must be unreal or fictional.9 Since, according to Peirce, "the mathematician is merely reasoning about his own mental creations"10 or, to state the point somewhat differently, since mathematical knowledge "is merely knowledge of the creations of our own imagination,"11 and inasmuch as the objects which the mathematician studies "are creations of his own conscious volition, so that no part of them can be hidden from him,"12 these mental creations would be clearly fictional if indubitability were present in this context as well, though otherwise not. Now, David Savan, for example, argues that the "reasonings of pure mathematics" are indubitable in the sense in which they "cannot conceivably be doubted."13 If it is true that Peirce does indeed opt for indubitability here, that would provide us with an indirect way of showing that the subject matter of such reasonings is unreal, that is, fictional. What, then, are Peirce's views concerning the nature of "the reasonings of pure mathematics"?

     I take it that an instance of this kind of reasoning is expressed by the proposition that two times two equals four, considered as a conclusion. Does Peirce allow for absolute certainty here? On the contrary, he argues as follows:

That twice two is four is often used as a type of perfect certainty. The basis of our confidence in the proposition lies in the extreme ease with which we can at any time make the experiment of duplicating an imaginary object, then duplicating the resulting pair, and comparing the final result with that of adjoining to the first imaginary object a second, a third, and a fourth like it. But this is only an |80| experiment. -- It may have been performed with the same result a billion times; still mistakes in addition do occur, according to all experience; and what happens once may happen any number of times, and on every one of the first billion trials. It is therefore not absolutely impossible, that every one of the experiments hitherto made upon the result of multiplying 2 by 2 have been in error in the same way; so that it is as nearly certain as anything can well be that the proposition 2 X 2 = 4 is not absolutely certain. Of course, it would be silly to doubt it. But it is one thing to hold to that practical maxim, and quite another to hold it to be beyond the bare possibility of error.14

It is quite clear from this, and numerous other similar passages,15 that Peirce does indeed allow for practical certainty16 in a mathematical setting, but not for absolute certainty. That is, he does allow for that kind of certainty in which, for practical purposes, something is treated as though it were immune from error; but such a practical treatment or "maxim" is not to be identified with that kind of certainty which would, from a logical point of view, preclude the "bare possibility of error." Such reasonings are "certain" but not indubitable; consequently, the reasonings of pure mathematics are best considered as undoubted and not as indubitable in character.

     This is consistent with other things Peirce has to say about the nature of mathematics. He is not unhappy with talk of the subject matter of mathematical reasoning being real so long as we understand that the kind of reality which obtains here is not properly identifiable with that reality which is the supposed subject matter of empirical judgments. The reality of the latter (empirical) is what Peirce calls "external reality" while that of the former (mathematical) is "internal reality":17

Of realities, some are internal, some are external. An external reality is an object whose characters are not only independent of the thoughts of me or you about this object but are also independent of our thoughts about any other object; while an internal reality though not dependent on our opinions about it does depend upon some thought.18

An internal reality is, as Peirce puts it, dependent upon some thought, because "the mathematical work, proper, begins with a hypothesis already assumed."19 We are, as Peirce well realizes, treading dangerously close to collapsing mathematical reasoning into purely fictional discourse, in which case, of course, talk of the possibility of being mistaken (a methodological requirement for the subject matter of a discourse being real) or of reality (in any sense) would be stretched beyond recognition. Yet Peirce is careful here; for although he admits that there is a sense to saying that these "assumed hypotheses" are "arbitrary creations,"20 |81| this is not the sense in which the nature of the subject matter of such reasoning is open to change at the whim or fancy of the inquirer (as is the case with fictions, properly so called). After asserting that mathematics deals with the "internal world" Peirce continues in a way designed to distinguish it from the "not real" or "fictitious":

If our arbitrary creations, had, when once created, no insistency at all, no power of resisting the will, we could not reason about them at all. They would not have the consistency required to permit a necessary proposition to be woven out of them. Mathematical truth is at least as irresistible as empirical truth; but of the substance of the latter truth we have a perception; it is the mighty universe that compels our thought, while the former is an invisible giant that holds us in his grip.21

Although this quotation only provides us with, at best, a very weak ground for distinguishing mathematics from fictions, we shall shortly pursue further Peirce's general rationale for this approach; but let us forestall consideration of this rationale until we have gained an understanding of precisely how the possibility of error is generated in a mathematical setting. This understanding will, in turn, aid us in our attempt to get clear on the grounds provided by Peirce for this distinction.

     I argue elsewhere (in connection with my more general thesis) that with respect to perceptual judgments the possibility of being mistaken springs from the nature of the subject matter of such judgments itself -- that is, from the fact that such judgments are about "external reality." Inasmuch, then, as mathematics deals with assumed hypotheses and is responsible for the truth of its claims insofar as they "would be true in case those initial statements [assumed hypothesis] should be exactly true,"22 it would seem that, in contradistinction to that kind of reasoning which deals with what we take to be the real, external world,23 the possibility of being mistaken in mathematics would be purely a matter of blunders in our applications of the leading principles of (deductive) inference. Goodness or badness of reasoning here could well be, for Peirce, a question of thought and not of fact and perhaps Peirce would allow for defining it "subjectively," though only in the sense in which truth could in this case be a matter of coherence with those assumed hypotheses (and, in that way, be "hypothetical" in character). Since Peirce tells us that mathematical propositions and geometrical propositions "are all of the same character"24 and since he describes mathematics and pure geometry in similar ways, the following quotation lends considerable plausibility to our speculation:

The truth of pure geometry is the truth of an arbitrary hypothesis, while the truth of geometrical optics is the truth as to what the eye would see if its perceptions were sufficiently fine and if light had not |82| those characters which slightly falsify geometrical optics. Or we may conceive of geometrical optics as itself taking account of those circumstances. The truth of logic is not the truth of an arbitrary hypothesis.25

The contention that a kind of coherence theory of truth may be involved here stands in need of further examination, though. Although there are, within Peirce's writing's, numerous passages supportive of this contention as described above, notice must nonetheless be taken of his discussions of what he calls "theorematic deduction," in particular.26 For Peirce recognizes a kind of deduction wherein something "new" (in the sense of not being contained either explicitly or implicitly in the premises) emerges as a product of, or component in, some deductively generated progression. Were all deduction the mere tracing out of the consequences of the hypotheses, theses, axioms, and the like (functioning as the starting points for any given piece of mathematical reasoning), mistakes would surely be due to blunders in our applications of the leading principles of deductive inference. That is, were all mathematical deduction characterizable as what Peirce calls "corrolarial deduction" this account would hold. Yet since Peirce tells us that theorematic deduction "requires the invention of an idea not at all forced upon us by the terms of the thesis"27 it would seem that a mistake here could not be accounted for along the aforementioned lines. To gain a clearer understanding of just what is involved in what Peirce calls theorematic deduction it would be helpful to note that the idea "invented" takes the form of a "postulate", which for Peirce is the "affirmation of a possibility."28 On the other hand, in merely tracing out consequences (corrolarial deduction) what we are doing is "carefully taking account of the definitions of the thesis to be proved."29 Now since definitions, axioms and the like differ from postulates in that they consist in the "denial of a possibility,"3O that is, serve to limit the scope and domain of the reasoning, the following example may prove helpful in coming to understand theorematic deduction, move us toward an understanding of what being mistaken in theorematic deduction would be like, and give us a more complete understanding of truth in a mathematical setting.

     Recently a case was reported in the news about a man who attempted to change his name to '1069'. His request having been denied both by the Internal Revenue Service and by the Social Security Administration, he took his case to court. The judge ruled that there was nothing in the law to preclude his changing his name to '1069', and so the decision of the court was that he be allowed to institute his requested name change. Of interest to us here is the way in which the decision was generated. In Peircean terminology, the laws of relevance denied a variety of possibilities; the judge, however, in setting a precedent affirmed a possibility which the man in question proceeded to actualize; that is, the judge laid down a postulate -- engaged in "theorematic deductiojn." His primary |83| concern was to insure that his decision did not contradict the laws by affirming a possibility which had been excluded by the laws. This he accomplished. Here, had the judge's decision been mistaken, it wouldn't have been grounded in a misapplication of the leading principles of deductive inference (as ordinarily understood). It would rather have been due to a violation of the principle of noncontradiction. In this way we can see that being mistaken in a mathematical setting can occur in one of two basic ways. It could either be a matter of inferential blunders or of generating a contradiction by affirming a possibility which is denied by the axioms, hypotheses, etc., that is, by the starting points of mathematical reasoning. In this way we can see how a coherence theory of truth can still hold within a mathematical setting; for what remains at issue is whether or not the deductively (corrolarial or theorematic) generated movements or consequents cohere, or are consistent, with the assumed hypotheses, and in this way the hypothetical character of mathematical truth is preserved.31

     Let us now return to the mathematics/fiction (internally real/ unreal) distinction. Surely the mere fact that mathematical objects have a kind of "insistency" does not, in and of itself, provide us with a means for distinguishing such objects from fictions -- if for no other reason than that many fictional objects also have this kind of insistency. How, then, are we to draw this distinction? (It is not as though we are seeking criteria or decision procedures, for these I believe, are not in tune with the Peircean methodology; it is rather that we are seeking differences, grounds, and epistemological commitments for approaching this as a crucially important Peircean distinction.) One avenue of approach is provided by Peirce's conception of truth in a mathematical setting. As we have seen Peirce makes the truths of the products of mathematical reasoning (the consequents) dependent upon the truth of the starting points (the antecedents) of that reasoning. That is, our deductively (which, for Peirce, is synonymous with mathematically)32 generated conclusions would be true just in case our hypothetically generated initial premises (axioms, definitions, etc.) turn out to be true of something; and that, in turn, is something which -- as is the case with all "truths" properly so called, for Peirce -- awaits our discovery. Yet with fictions it is clearly different. For in this sort of case, any "deductive movement" and the "consequences" thereof do not await, for their "truth," any discoveries yet to be made concerning their "antecedents." Why? Because, by definition, as wholly constituted by their creator, they cannot be false, and so talking of them in terms of their turning out to be true ( or false) is devoid of sense. That is, the reference to the future so essential to Peirce's treatment of truth is altogether missing in a fictional setting. The opening predicates of a work of fiction are not hypothetical; that is, they do not function as hypotheses. From this it seems clear that by focusing on the hypothetical character of mathematical truth we gain a partial insight into the internal reality/fiction distinction. It is as if, in the case of the fiction, we have |84| what masquerade as categorical truths. That is, the declarative sentences in a work of fiction -- fictitious "assertions" -- have the form of categorical truths; but what keeps them from being genuine categorical truths is that there is no possibility of our being mistaken; there is simply nothing. over and above what the fiction's creator tells us, which awaits our discovery.

     A further motive for maintaining this distinction is provided by Peirce's account of those entia rationis which come under the general heading of "hypostatic abstractions": the procedure whereby we convert, for the purposes of analysis, a predicate into a subject.33 For example, suppose some imaginary political bosses are wondering about the "electability" and "public presence" of some possible candidate (Jones) under their consideration. Suppose further that they have found out that Jones is satisfactory, as regards political stance, but also that he is shy. Wondering what kind of campaigner he would be and what sort of public appeal he might have, they decide to find out all they can about "shyness" and about the electorate's disposition toward it. They convert what had been the predicate portion ("is shy") of their original understanding of Jones into a subject, namely, shyness, for the purposes of analysis. They then proceed to find out all they can, from clinical and experimental psychologists, sociologists, etc., about shyness and what is involved therein, and, on the basis of these findings, predicate of Jones a host of inter-related characters developed out of their analysis of shyness. They finally make their decision on the basis of what they take to be their enhanced understanding of Jones. Now were it the case that all abstractions were fictions, then by moving from Jones' being shy to a consideration of shyness they would have removed themselves from any possibility of discovering any real characteristics of Jones. The results of their analysis would be irrelevant. Such a move could not enhance their understanding of this possible candidate for public office one iota.

     However, Peirce would argue that, on the contrary, the simple fact that they had to resort to an abstraction (which in some cases cannot be avoided) should not be viewed in this way. He feels that were we to so view it, our attempts to analyze things of concern to us would invariably be thwarted. That is, just as we cannot do without language when we engage in inquiries, so we often cannot proceed without the aid of abstractions. It is indeed the case that the truth status of our discoveries concerning the abstracted element of our understanding is dependent upon the truth of categorical statements (for example, "Jones is shy," "Smith is shy," etc.);34 but that is just another way of saying that the existence of shyness consists in the fact that somebody is shy. The upshot of this is that the truths of abstractions are hypothetical in character; that is, abstractions are real in that sense of reality which Peirce refers to as the "internally real." Moreover, there is an essential need, in Peirce, for the reality of abstractions. His basic realism demands it. After all, according to Peirce, a collection is an abstraction and what else is a |85| "natural kind" but a collection in re, a natural grouping or sortal. That is, the reality of natural kinds is part of Peirce's realism; he sees nature as unfolding in a way closely analogous to that process of mind wherein we collect and group things. The nonfictitiousness of collections is thus crucial to Peirce's overall realism. If one were to attempt to treat of abstractions as fictions one could not help thereby having committed oneself to a form of nominalism which Peirce abhorred. It is for all of these reasons that we must maintain a distinction between the internally real and the fictitious.

     What I have argued is that within the Peircean epistemic framework there is an essential need for a well-maintained distinction between the internally real and the fictitious. Moreover, it is clear that Peirce is able to draw and defend this distinction once indubitability-in-principle has been tied to, and reserved for, the domain of the fictitious. In view, then, of the possibility of being mistaken within a mathematical setting, he develops just that distinction which his realism demands; namely, the distinction between the fictitious and the internally real.



1. MS 333, p. 19.

2. Collected Papers, 6.328; see also MS 372, p. 11; MS 609, pp. 7f.; MS 683, pp. 33f. (herein Peirce acknowledges his having borrowed this use and understanding of 'fiction' from Duns Scotus). Notice should also be taken of the fact that Peirce occasionally uses 'fictile' as the contrary of 'real'; see, in this vein, MS 656, p. 36: "as the contrary of 'real' I employ the word fictile . . . ." Attend also to MS 1601, pp. 8f., wherein Peirce discusses the reality of certain entia rationis, and wherein he distinguishes genuineness and reality: "The Figment remains a Figment whether it professes to be Real or acknowledges itself as a Figment -- while the Real, as such, makes no profession at all. Genuineness is one thing; Reality quite another. If it is created by the Creator of the Three Universes [Peirce's modes of Reality -- the possible, the actual, and the general or would-be] it is Real. But if it be an ens rationis, may it not equally be Real? It certainly may; and yet we now seem to be 'warm' as they say in a game of guessing. The figment may be Real from one point of view. A dream, e.g., is a Real dream. Yet it may profess to be just the figment it is, as any professed work of fiction does. Or it may make no more profession of any kind than a Real thing usually makes, that is to say no profession at all. But its fictionsness [sic] consists in its being a phenomenon or other object of Representation which is 'due' to -- or created, or represented by -- merely nothing but somebody's thought."  |86|

3. Unless otherwise indicated, I will use the term 'indubitable' throughout this paper to mean indubitable-in-principle, that is, that relative to which there is no possibility of being mistaken.

4. MS 126, p. 6.

5. MS 852, pp. 11f.; see also MS 680, p. 24: "The substance of a dream is not real, because whatever is true of it is so in that so the dreamer dreamed; and dreaming is only a particular kind of thinking. But that fact of the dream, if it actually took place, is a real fact, because it 'will always remain true that so and so was dreamed, whether the dreamer remembers it or not."

6. MS 683, assorted pages, pp. 24f.

7. Wishing to "restore the word to its original signification" Peirce explains that by "ens" he understands "whatever can be, among other things, named, or mentioned, or to which we can direct our thought.'" (MS 722, p. 1.) See also MS 719, p. 1: "The term ens (plural: entia) a present participle formed from esse, was employed by the schoolmen to denote whatever can be mentioned and whatever the mind can be directed towards, whether it exists or not, or however absurd it may be. In this sense I shall use the word."

8. MS 200, p. 34. For whatever reason, the editors of the Collected Papers did not include this paragraph, which indicates Peirce's intention to show how some entia rationis may be real. The omission (in 6.327) of reality and externality presents us with a discussion from which the reader can in no way infer that it is designed by Peirce to illustrate the point that not all entia rationis are fictions.

9. Peirce, in a very late notebook (MS 659), dated 1910, cites as a typical confusion, with which he has had to deal throughout his lifetime, one in which we approach "being unreal as the same as existing only in a person's mind." (p. 30.)

10. MS 94a, p. 18.

11. MS 860, p. 14.

12. MS 693, p. 300.

13. David Savan, "Decision and Knowledge in Peirce," Transactions of the Charles S. Peirce Society, 1 (1965), p.41.

14. MS 335, pp. 1-4.

15. See, for example, MS 839, assorted pages, pp. 4f., wherein Peirce offers an analysis of "7 X 3 = 21" along these lines. Also see MS 590 pp. 1f.; MS 620, pp. 19f. (esp. p. 20); MS 1601, p. 6; MS 596, pp. 1Of.

16. MS 94a, p. 18; MS 839, pp. 4f.; MS 693, pp. 230-234 and pp. 30Of.

17. See MS 293, assorted pages, pp. 47f; MS 637, assorted pages, pp. 27f.

18. MS 333, p. 19.

19. MS 94a, p. 18 (emphasis added). See also MS 852, pp. 6f. and MS 872, p. 2: "The mathematician's business is to start from a supposed state of things, which is described in general terms."

20. MS 283, assorted pages, p. 53.

21. MS 283.   |87|

22. MS 278d, p. 49.

23. See, for example, MS 283, p. 86: "Logic is the science of truth and falsity. . ."

24. MS 589, p. 9.

25. MS 283, assorted pages, pp. 56f.

26. Collected Papers, 2.267; The New Elements of Mathematics, vol. IV, pp. 1ff. (esp. pp. 8-12), also pp. 38,42, and 49.

27. New Elements of Mathematics, vol. IV, p. 8.

28. New Elements of Mathematics.

29. New Elements of Mathematics.

30. New Elements of Mathematics.

31. See, for example, Collected Papers, 4.240.

32. Collected Papers, 3.353,3.509.

33. See MS 280, assorted pages, p. 42; MS 431, pp. 14f.

34. See, for example, Collected Papers, 3.642.

END OF:  Dozoretz, "The Internally Real, the Fictitious, and the Indubitable"


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