Robert, List: For Peirce, mathematics is indeed a science of *discovery*, but it is not a *positive *science.
CSP: Mathematics is the most abstract of all the sciences. For it makes no external observations, nor asserts anything as a real fact. When the mathematician deals with facts, they become for him mere "hypotheses"; for with their truth he refuses to concern himself. The whole science of mathematics is a science of hypotheses; so that nothing could be more completely abstracted from concrete reality. ... Mathematics is not a positive science; for the mathematician holds himself free to say that *A* is *B* or that *A* is not *B*, the only obligation upon him being, that as long as he says *A* is *B*, he is to hold to it, consistently. (CP 3.428, 1896) CSP: Perhaps you will ask me whether it is possible to conceive of a science which should not aim to declare that something is positively or categorically true. I reply that it is not only possible to conceive of such a science, but that such science exists and flourishes, and Phenomenology, which does not depend upon any other *positive science*, nevertheless must, if it is to be properly grounded, be made to depend upon the Conditional or Hypothetical Science of *Pure Mathematics*, whose only aim is to discover not how things actually are, but how they might be supposed to be, if not in our universe, then in some other. (CP 5.40, EP 2:144, 1903) CSP: Pure mathematics differs from the positive sciences in not making any categorical assertions, but only saying what would be true in case certain hypotheses were true, and not undertaking to be in the least responsible for there being anything in nature corresponding to its hypotheses, whether exactly or approximately. The positive sciences do undertake to assert what the characters of the experiential facts are. (EP 2:146, 1903) CSP: Science of Discovery is either, I. Mathematics; II. Philosophy; or III. Idioscopy. 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. Idioscopy embraces all the special sciences, which are principally occupied with the accumulation of new facts. (CP 1.183-184, EP 2:259, 1903) No one is disputing that in Peirce's classification, mathematics is the science on which all other sciences depend for their principles, nor that every other science has its mathematical part accordingly. Nevertheless, when we employ mathematics *within *phaneroscopy, we are still practicing phaneroscopy, not mathematics; and when we employ mathematics or phaneroscopy *within *semeiotic, we are still practicing semeiotic, not mathematics or phaneroscopy. Peirce consistently defines mathematics *per se* as the science that draws necessary conclusions about hypothetical states of things. CSP: Mathematics is, therefore, the study of the substance of hypotheses, or mental creations, with a view to the drawing of necessary conclusions. (NEM 4:268, c. 1895) CSP: The most abstract of all the sciences is mathematics. That this is so, has been made manifest in our day; because all mathematicians now see clearly that mathematics is only busied about *purely hypothetical questions*. As for what the truth of existence may be the mathematician does not (qua mathematician) care a straw. (CP 1.53, 1896) CSP: Of late decades philosophical mathematicians have come to a pretty just understanding of the nature of their own pursuit. I do not know that anybody struck the true note before Benjamin Peirce, who, in 1870, declared mathematics to be "the science which draws necessary conclusions," adding that it must be defined "subjectively" and not "objectively." A view substantially in accord with his, though needlessly complicated, is given in the article "Mathematics," in the ninth edition of the *Encyclopædia Britannica*. The author, Professor George Chrystal, holds that the essence of mathematics lies in its making pure hypotheses, and in the character of the hypotheses which it makes. What the mathematicians mean by a "hypothesis" is a proposition imagined to be strictly true of an ideal state of things. In this sense, it is only about hypotheses that necessary reasoning has any application; for, in regard to the real world, we have no right to presume that any given intelligible proposition is true in absolute strictness. On the other hand, probable reasoning deals with the ordinary course of experience; now, nothing like *a course of experience* exists for ideal hypotheses. Hence to say that mathematics busies itself in drawing necessary conclusions, and to say that it busies itself with hypotheses, are two statements which the logician perceives come to the same thing. (CP 3.558, 1898) CSP: The third elementary way of reasoning is *deduction*, of which the warrant is that the facts presented in the premisses could not under any imaginable circumstances be true without involving the truth of the conclusion, which is therefore accepted with necessary modality. But though it be necessary in its modality, it does not by any means follow that the conclusion is certainly true. When we are reasoning about purely hypothetical states of things, as in mathematics, and can make it one of our hypotheses that what is true shall depend only on a certain kind of condition ... we can be certain of our conclusions, *provided no blunders have been committed*. ... It is to ideal states of things alone--or to real states of things as ideally conceived, always more or less departing from the reality--that deduction applies. (CP 2.778, 1902) CSP: It was Benjamin Peirce, whose son I boast myself, that in 1870 first defined mathematics as "the science which draws necessary conclusions." This was a hard saying at the time; but today, students of the philosophy of mathematics generally acknowledge its substantial correctness. ... For all modern mathematicians agree with Plato and Aristotle that mathematics deals exclusively with hypothetical states of things, and asserts no matter of fact whatever; and further, that it is thus alone that the necessity of its conclusions is to be explained. This is the true essence of mathematics; and my father's definition is in so far correct that it is impossible to reason necessarily concerning anything else than a pure hypothesis. ... Mathematics is the study of what is true of hypothetical states of things. That is its essence and definition. Everything in it, therefore, beyond the first precepts for the construction of the hypotheses, has to be of the nature of apodictic inference. No doubt, we may reason imperfectly and jump at a conclusion; still, the conclusion so guessed at is, after all, that in a certain supposed state of things something would necessarily be true. Conversely, too, every apodictic inference is, strictly speaking, mathematics. ... It is difficult to decide between the two definitions of mathematics; the one by its method, that of drawing necessary conclusions; the other by its aim and subject matter, as the study of hypothetical states of things. (CP 4.229&233-234&238, 1902) In short, according to Peirce, mathematics is *strictly deductive* in its method and *strictly hypothetical* in its subject matter. Claiming otherwise is revisionism. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Tue, Jul 13, 2021 at 1:49 PM robert marty <robert.mart...@gmail.com> wrote: > Jon, List, > To write that Mathematics is a* strictly* hypothetical science is not > only to totally misunderstand mathematical activity and the place of > Mathematics in the Sciences of Discovery but also to show revisionism > towards Peirce's philosophy of science. > Here is an excerpt from my current work: > > (Beginning) > > *"1- The classifications of Sciences according to C.S. Peirce.* > > Peirce provided several classifications, more or less detailed, in > constant reference to Auguste Comte with whom he shares the principle > according to which each level depends for its construction on the level > that precedes it (except for the first one of course). He proposes the > image of the "Well of Truth". The simplest classification includes only > three divisions (MS 1345), but it envelops all the others which will be > expansions of it. > > > *Every systematic philosopher must provide himself a classification of the > sciences. Comte first proposed to arrange the sciences in a series of > steps, each leading another. This general idea may be adopted, and we may > adapt our phraseology to the image of the well of truth with flights of > stairs leading down into it.* > > (Peirce, MS 1345, undated, transcription 1976: NEM, III.2, 1122) [emphasize > mine] > > How I consider the Tommi Vehkavaara's compilation of all the Peirce > classifications of sciences from 1902 to 1911. I view to examine the > relation of dependencies between Mathematics and Semiotics. In the > Sciences of Discovery, I extract the part of the path that goes down from > the entrance of the well to the Semiotics. This path is a sequence that > alternates labels and dependencies. The terms in italics are labels whose > strict utility is classificatory; they do not imply any dependency. On the > other hand, the terms in bold are disciplinary fields. The signs "|" > indicate membership in a discipline class. The signs "↓" indicate > dependence between the disciplinary fields concerned. > > > > *Theoretical science** (pure scientific inquiry)* > | > > > *A) Science of Discovery (Science of Research) | **AI -Mathematics * > ↓ > *AII **Cenoscopy-Philosophia Prima** (positive science)* > > > * | (Epistêmy, 1902) "necessary philosophy" *| > *AIIa Phenomenology-Phaneroscopy* > ↓ > *AIIb Normative Sciences* > | > *AIIbiii Logic-Science of general laws of signs-Formal Semiotic-Semiotic* > > > > *Table 1: the sequence of labels and dependencies.* > > > > If I retain only the effective dependencies the path is reduced to this > one: > > *AI -Mathematics (**Theoretical science, Science of Discovery )* > ↓ > *AIIa Phenomenology-Phaneroscopy *(positive science)* (* > *Cenoscopy, necessary philosophy) *(which rests upon familiar, general > experience) > ↓ > > *AIIbiii Logic-Science of general laws of signs-Formal Semiotic-Semiotic *- > study of Universal Categories (all present in any phenomenon): Firstness, > Secondness, Thirdness > > *Table 2: The sequence of dependencies (**the labels concerned are > indicated in brackets**).* > > The study of dependencies requires elucidation of this rather vague > concept. One thing is certain: we must find mathematics in semiotics, > otherwise, we will at worst betray Peirce's work, at best revise it. Given > the importance of semiotics in his work, including in the pragmatism of > which it is a basis, this must be done step by step with great care. > > *1.1 AI- Mathematics* > > The following selection of quotations provides sufficient information to > accurately understand both the intrinsic nature of each disciplinary field > and nature and function of these dependencies.*"* > > > > *This double assertion, first, that logic ought to draw upon mathematics > for control of disputed principles, and second that ontological philosophy > ought in like manner to draw upon logic, is a case under a general > assertion which was made by Auguste Comte, namely, that the sciences may be > arranged in a series with reference to the abstractness of their objects; > and that each science draws regulating principles from those superior to it > in abstractness, while drawing data for its inductions from the sciences > inferior to it in abstractness. So far as the sciences can be arranged in > such a scale, these relationships must hold good." *(CP 3.427 ; 1896 The > regenerated logic) [emphasize mine] > > > > *"I would classify the sciences upon the general principle set forth by > Auguste Comte, that is, in the order of abstractness of their objects, so > that each science may largely rest for its principles upon those above it > in the scale while drawing its data in part from those below it.23 At their > head I would place Mathematics, for this irrefutable reason, that it is the > only one of the sciences which does not concern." *(EP2: 35 ; 1898 from > MS 437) [emphasize mine] > > > * "Among the theoretical sciences, I distinguish three classes, all > resting upon observation, but being observational in very different senses. > The first is mathematics, which does not undertake to ascertain any matter > of fact whatever, but merely posits hypotheses, and traces out their > consequences. It is observational, in so far as it makes constructions in > the imagination according to abstract precepts, and then observes these > imaginary objects, finding in them relations of parts not specified in the > precept of construction. This is truly observation, yet certainly in a very > peculiar sense; and no other kind of observation would at all answer the > purpose of mathematics".* (1902, Minute Logic: Chapter II. Prelogical > Notions. Section I. Classification of the Sciences (Logic II); CP > 1.239-240) [emphasize mine] > > These texts do not only say that mathematics is superior in abstractness > to any other science, and therefore first of all to Phenomenology. They > also say that it is capable of drawing the consequences of hypotheses > extracted from the data of Phenomenology to which it provides its > principles. It is in this that the relation of dependence of Phenomenology > with respect to Mathematics consists. Finally, it can proceed to the > creation of imaginary objects on which it can make observations of a > particular kind that will perhaps allow it to discover new relations > between the data that motivated the hypotheses. *To say that it is a > negative science is to limit it to its deductive capacities.* But it is > obviously much more than that, as Peirce expresses it in MS 1345: > > > > > *" We divide the whole into three great parts : - mathematics, the study > of ideal constructions without reference to their real existence, > -empirics, the study of phenomena with the purpose of identifying their > forms with those mathematics has studied, - Pragmatics, the study of how > we ought to behave in the light of the truths of empirics." *[C.S. > Peirce, 1976: NEM , vol III.2 1122] > > (End of quote) > > This does not prevent one from looking at Phaneroscopy as a positive > science. It is always possible to refuse to see the dependence on > mathematics and to tinker without using it, but this will be an > epistemological deficiency and it is because of this that I am talking > about revisionism. The unjustifiable use of the term "strictly" expresses a > will. It is your strictly personal contribution No surprise, The terms > "strictly > hypothetical" appear only one time in CP 6.467 and in EP2: 440, without > relation to Mathematics. > > This does not prevent one from looking at Phaneroscopy as a positive > science. It is always possible to refuse to see the dependence on > mathematics and to tinker without using it, but this will be an > epistemological deficiency and it is because of this that I am talking > about revisionism. The unjustifiable use of the term "strictly" expresses a > will to reject Mathematics from the field of Phenomenology where, according > to Peirce, it operates. It is your *strictly* personal contribution. > > So there is still a pending question: So there is still a question > pending: with what forms do you identify the forms you find in the > phenomena? > > > This is what I had strictly to say today about the Classification of > Sciences. I still have a lot to say about Mathematics. > > Robert Marty > Honorary Professor ; PhD Mathematics ; PhD Philosophy > fr.wikipedia.org/wiki/Robert_Marty > *https://martyrobert.academia.edu/ <https://martyrobert.academia.edu/>* >
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