Abstract
What understanding of mathematical objectivity is promoted by Peirce’s pragmatism? Can Peirce’s theory help us to further comprehend the role of intersubjectivity in mathematics? This paper aims to answer such questions, with special reference to recent debates on mathematical practice, where Peirce is often quoted, although without a detailed scrutiny of his theses. In particular, the paper investigates the role of intersubjectivity in the constitution of mathematical objects according to Peirce. Generally speaking, this represents one of the key issues for the philosophy of mathematical practice, whereas – with regard to Peirce – these two aspects (intersubjectivity and objectivity) have been studied for a long time, but mostly as unrelated topics. To reconstruct the connection between Peirce’s reflection on intersubjectivity and the objectivity of mathematical theories, the paper is divided into three parts: (1) the first part introduces Peirce’s view of mathematics; (2) the second analyzes his peculiar “pragmatist realism,” which overcomes common dichotomies in the philosophy of mathematics; (3) the third illustrates the role that Peirce attributes to intersubjectivity in science, and investigates to what extent the intersubjective dimension is also essential in mathematics.
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10 January 2023
A Correction to this paper has been published: https://doi.org/10.1007/s11245-022-09880-4
Notes
In this regard, the author specifies: “he [Benjamin Peirce] was extremely attentive to my training when I was a child, and especially insisted upon my being taught mathematics according to his directions” (Peirce 1968, hereafter R 905, CSP 7, n.d.). “R” stands for Peirce’s manuscripts as catalogued by Robin in 1968, followed by the number of the manuscript.See “References” for the abbreviations of Peirce’s works.
It is worth noticing that, if we set aside the series of articles that Peirce published in the Popular Science Monthly in 1877-78, and the so-called “cosmological” series he published in The Monist at the beginning of the 1890s, in this span of time his most important works were devoted to the development of the algebra of logic and published in the American Journal of Mathematics. With reference to mathematics, Peirce’s later research must also be recalled, particularly his concept of continuum and his system of notation known as “existential graphs” (cf. Peirce 2019, hereafter LoF1, Zalamea 2012a, 2022).
This title inspired Carolyn Eisele’s four massive volumes, which collect a great many of Peirce’s writings on mathematics, making numerous manuscripts accessible that would otherwise be difficult to read (see Peirce 1976, hereafter NEM).
To provide a quick overview of Peirce’s “architectonic” of the sciences of discovery, see the following table, based on An Outline Classification of the Sciences (cfr. EP2, pp. 258–262, 1903): (1) MATHEMATICS, divided into (i) Mathematics of Logic, (ii) Mathematics of Discrete Series, (iii) Mathematics of Continua and Pseudo-Continua; (2) PHILOSOPHY (CENOSCOPY), divided into (i) Phenomenology (or Phaneroscopy), (ii) Normative Sciences (divided into ii.a. Esthetics, ii.b. Ethics and ii.c. Logic), (iii) Metaphysics (divided into iii.a. General Metaphysics (or Ontology), iii.b. Psychical (or Religious) Metaphysics, iii.c. Physical Metaphysics; 3) SPECIAL SCIENCES (IDIOSCOPY), divided into (i) Physical Sciences (divided into i.a. Nomological Physics (or General Physics), e.g., Dynamics, Electrics, etc., i.b. Classificatory Physics, e.g., Chemistry, Biology, etc.; i.c. Descriptive Physics, e.g., Geognosy, Astronomy, etc.; (ii) Psychical (or Human) Sciences, divided into ii.a. Nomological Psychics (or Psychology), e.g., Introspectional, Experimental, etc., iii.b. Classificatory Psychics.
e.g., Linguistics, Ethnology, etc., iii.c. Descriptive Psychics (or History), e.g., History, Biography.
Peirce specifies: “Mathematics studies what is and what is not logically possible, without making itself responsible for its actual existence. Philosophy is positive science, in the sense of discovering what really is true; but it limits itself to so much of truth as can be inferred from common experience” (EP2, p. 259, 1903). And, with reference to logic, he clarifies that “it [logic] seeks to know what is, as a fact, the nature of reasoning; while mathematics is a study of the matter of hypotheses” (LoF1, p. 446, 1902).
For the sake of clarity, it must be acknowledged that Peirce’s overall standpoint on mathematics is quite problematic since he sometimes contradicts himself. Moreover, many of Peirce’s reflections come from rather scattered papers, most of which are unpublished manuscripts. For these reasons, there has been a long debate among Peirce scholars about his general view of mathematics (publications discussing this issue range from Haack 1979 and Rosenthal 1984 to Moore 2010, just to mention some of the classical works on the subject). For reasons of space, I will not jump into this debate but will limit myself to an exposition of Peirce’s ideas in light of those articles.
The author further stresses this point: “The logician does not care particularly about this or that hypothesis or its consequences, except so far as these things may throw a light upon the nature of reasoning. The mathematician is intensely interested in efficient methods of reasoning, with a view to their possible extension to new problems; but he does not, quâ mathematician, trouble himself minutely to dissect those parts of this method whose correctness is a matter of course” (LoF1, p. 445, 1902).
Quoted from Peirce’s “The Logic of Mathematics in Relation to Education,” published in Educational Review 1898; cf. CP 3.553–562.
However, Peirce still proposed it as the general definition when he wrote the entry ‘mathematics’ for the Century Dictionary (1889–1901).
Peirce states that mathematics deals almost exclusively with “hypotheses which present only systems of relationship that are perfectly regular or as nearly so as the nature of things allows” (EP2, p. 173, 1903). One of the most relevant implications of such a definition is that, with regards to mathematics, Peirce rejects one of Kant’s most important distinction, that between synthetic and analytical judgments. He writes: “Plato, with Aristotle after him, recognizes that mathematics deals wholly with hypotheses, and asserts no matter of fact; and that this is the reason of the necessity of its conclusions. Such is the view of all modern mathematicians. Kant regards mathematical propositions as synthetical judgments a priori, which is perhaps not in such irreconcilable conflict with the mathematicians’ view as it has been assumed to be. Whether a mathematical theorem is ‘synthetical’ or ‘analytical,’ in Kant’s sense, can hardly be answered: it is neither” (LoF1, p. 443, 1902). Accordingly, for Peirce Kant’s distinction between analytical and synthetical judgments fatally overlooks this hypothetical nature of mathematics.
In a similar vein, Peirce underlines that his existential graphs (EG) – the “luckiest find” he made (cf. R 280: 22, 1905) – represent “a moving picture of the action of the mind in thought” (R 298: 1, c. 1905, italics mine; cf. also R 296 CSP 6, c. 1907).
He clarifies this point by comparing mathematics to chemistry: “How to perform a given chemical experiment may be the study of a year; and the performance itself may occupy another. But the mathematician can perform an experiment upon his diagram in a second. The natural result is, that all that element of uncertainty which consists in insufficiency of experiments becomes practically eliminated from mathematics. Everybody knows that an error may occur in adding up a column of figures; and what is possible once is possible always. It is, therefore, possible that everybody has always made a mistake in adding 2 to 2. So that, strictly speaking, that 2 + 2 = 4 is not absolutely certain. But the experiment has been made so many times that practically that possibility of errors is far less than others, such as that the reader has gone crazy (…)” (SWS, p. 51, 1896).
Peirce explains: “Now mathematical phenomena are just as unexpected as any results of chemical experiments. They do not startle the senses, but they involve new ideas. Although mathematics deals with ideas and not with the world of sensible experience, its discoveries are not arbitrary dreams but something to which our minds are forced and which were unforeseen” (Peirce 2010, hereafter SWM, p. 41, 1894). As a forerunner of the idea of mathematics as a historical and creative practice based on signs, see also the pioneering work by De Lorenzo (1971). Particularly relevant to the discussions on the creative aspects of mathematics is Peirce’s distinction between corollary and theoremic/theorematic deduction; cf. Levy 1997, Pietarinen/Bellucci 2014. Peirce writes: “Some relation between the parts of the schema is remarked. But would this relation subsist in every possible case? Mere corollarial reasoning will sometimes assure us of this. But, generally speaking, it may be necessary to draw distinct schemata to represent alternative possibilities. Theorematic reasoning invariably depends upon experimentation with individual schemata” (CP 4.233, 1902).
This expression recalls to Peirce’s famous pragmatic maxim, that in its standard formulation sounds as follows: “Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object” (Peirce 1982-present, hereafter W, followed by the volume number, W3, p. 266).
Further confirmation of this general view may be found in his metaphysics. See Brioschi 2020, part II, ch. 3.
A standpoint which led him to support, in cosmology, a perspective called by him “tychism” (from the Greek tyche), that holds that pure chance is a component of the evolution of the universe.
In this regard, it must be clarified that Peirce’s continuistic standpoint cannot be construed as a kind of reductionism, which collapses the potential (and the ideal) on the actual and experiential. For him an absolute, ideal world, completely detached from experience, does not exist, but distinctions can still be drawn between these modes of reality. As practices evolve, so do ideas and theories. From another perspective, mathematics cannot be understood apart from its practice, but still it is possible to study separately the evolution of mathematical theories, and their consequences for various practices.
With reference to Plato’s conception of the universal forms, Peirce remarks his distance from the former by stating: “In the first place, there is the question concerning the Platonic forms. But putting Platonism aside as at least incapable of proof, and as a self-contradictory opinion if the archetypes are supposed to be strictly universal, there is the celebrated dispute among Aristotelians as to whether the universal is really in things or only derives its existence from the mind. Universality is a relation of a predicate to the subjects of which it is predicated” (W2, p. 471, 1871, emphasis added).
As Peirce clearly states: “The meaning of any speech, writing, or other sign is its translation into a sign more convenient for the purposes of thought; for all thinking is in signs. The meaning of a mathematical term or sign is its expression in the kind of signs in the imaginary or other manifestation of which the mathematical reasoning consists. For geometry, this [expression] is [in] a geometrical diagram” (NEM2, p. 251).
As Peirce clarifies, “the most important consequence of it [pragmatism], by far, (…) is that under that conception of reality we must abandon nominalism” (CP 8.258, 1904). For Peirce what nominalists deny is the reality of anything general (cf. also R 717, CSP 9, c. 1895), whereas he strongly advocates the reality of laws, generals (that is, universals) and regularity.
See for instance CP 6.604, where Peirce explicitly rejects the charge of “being what he [Paul Carus, editor-in-chief of The Monist] calls a ‘constructionist’,” underling that his method “has neither been in theory purely empirical, nor in practice mere brain-spinning; and that, in short, my friend Dr. Carus’s account of it has been as incorrect as can be.”
In a similar vein, some years later he confessed his doubts about the exhaustiveness of his father’s definition: “In one respect, however, I am in doubt whether my father’s definition is adequate or not. It seems to make the deduction of the consequences of hypotheses the sole business of the mathematician. Now it cannot be denied that immense mathematical genius goes into the framing of those hypotheses. (…) Yet, regarded from the point of view of pure mathematics, which utterly disregards actual fact, the mere creation of imaginary states of things is less science than poetry. It is only when a hypothesis is considered as an approximate rationalization of those phenomena of nature that suggested it, that it can be reckoned as a contribution to science” (LoF1, p. 446, 1902). But this could be misleading, considering that Peirce several times reiterates that mathematics is not the science of what is (actuality), but of what can be and would be (possibility and potentiality). As a consequence, the fact that mathematical constructs might represent “an approximate rationalization” is secondary with regard to the nature of mathematics; it does not represent per se the business of mathematician.
This is not to say that Peirce thinks that poetry is purely arbitrary or that it is constrained by conventions. For him, good poetry has also an important, cognitive import. As the author remarks in 1903: “I hear you say: ‘All that is not fact; it is poetry.’ Nonsense! Bad poetry is false, I grant; but nothing is truer than true poetry. And let me tell the scientific men that the artists are much finer and more accurate observers than they are, except of the special minutiae that the scientific man is looking for. I hear you say: ‘This smacks too much of an anthropomorphic conception.’ I reply that every scientific explanation of a natural phenomenon is a hypothesis that there is something in nature to which the human reason is analogous; and that it really is so, all the successes of science in its applications to human convenience are witnesses” (EP2, p. 193).
See next section for a more in-depth analysis of the intersubjective side of the matter.
SWM, p. 83, 1906: “It is true that what must be is not to be learned by simple inspection of anything. But when we talk of deductive reasoning being necessary, we do not mean, of course, that it is infallible. But precisely what we do mean is that the conclusion follows from the form of the relations set forth in the premiss.”
In this regard, see Campos 2010.
This element nonetheless testifies to the inalienable subjective, and fallible, aspect of mathematics according to Peirce.
Indeed, in this regard Peirce states: “The person divides himself into two parties which endeavor to persuade each other. From this and sundry other strong reasons, it appears that all cognitive thought is of the nature of a sign or communication from an uttering mind to an interpreting mind” (R 498, c. 1906).
With reference to the existential graphs, the dialogue is between what Peirce calls “the graphist” and the “grapheus,” or the interpreter. Cf. for instance R 450, CSP 190, 1905: “The surface … shall be called our sheet of assent, and every proposition the regular expression of which the imaginary graphist, with the concurrence of the interpreter shall at any time place upon that sheet of assent, shall be understood to be mutually agreed to, by them both, as representing the truth upon the universe of discourse. … It renders the graphist responsible for the truth of the proposition he writes.”
The relevance of intersubjectivity is such that Peirce goes so far as to call his own doctrine “conditional idealism” in his philosophical discourse, in order to emphasize how something can be called ‘objective’ if in the long run the entire scientific community acknowledges it as a result to which all research must lead (cf. EP2, p. 419, 1907).
As he clearly states in the same excerpt, “the progress of investigation carries them [different minds with the most antagonistic views] by a force outside of themselves to one and the same conclusion. This activity of thought by which we are carried, not where we wish, but to a foreordained goal, is like the operation of destiny” (W3, p. 273, 1878). For a more detailed analysis of the connections between Peirce’s theory of inquiry and metaphysics, see Brioschi 2022.
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Acknowledgements
I am deeply grateful to the anonymous reviewers for their precious comments and critiques, which helped me detect and correct the imprecisions and ambiguities present in the previous draft. This research was funded by the Department of Philosophy “Piero Martinetti” of the University of Milan under the Project “Department of Excellence 2018–2022” awarded by the Ministry of Education, University and Research (MIUR).
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Brioschi, M.R. C.S. Peirce on Mathematical Practice: Objectivity and the Community of Inquirers. Topoi 42, 221–233 (2023). https://doi.org/10.1007/s11245-022-09857-3
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DOI: https://doi.org/10.1007/s11245-022-09857-3