John S, Robert, Jon A, List,
Here are a few questions about the philosophical import of the mathematics of category theory. I'm hoping that you might have some ideas that would shed some light on the matter. 1. Category theory represents a class of objects as nodes and then connections are made in terms of directed arrows. It has the general aim of exploring structure-preserving morphisms within a mathematical system (e.g., a topology). Similarly, it can be used to explore maps between different systems of objects (e.g., between sets and groups). In these two respects, category theory bears a striking resemblance to the system of representations and resulting analysis developed by A.B Kempe in his A Memoir of the Theory of Mathematical Form. The upshot of both systems is that they enable the user to study the composition and decomposition of relations within and between virtually any mathematical system. Do you find the similarities between contemporary category theory and Kempe's system similarly striking? 2. Peirce was much impressed with the depth of the Kempe's insights into the formal features that are essential for inquiry in any area of mathematics. At this same time, he thought that Kempe's analyses fell short in a number of respects. One shortcoming was the result of treating objects as nodes that are connected by several arrows. This mirrors a central feature of Euler's graph theory. Peirce raises an objection that this does not provide the resources needed to analyze objects that are connected by three or more arrows in Kempe's systems. Peirce thought that the introduction of a branching relation that is inherently triadic in character might help to remedy this shortcoming. Do any of Peirce's objections apply both to Kempe's analysis of mathematical form and to contemporary category theory? 3. One of the striking features of category theory is that it enables the user to compare the structural features of wildly different types of mathematical systems. As such, systems that are infinite and continuous can be compared directly with systems that finite and discrete. Given Peirce's classification of these different kinds of mathematical systems, where does category theory itself fall? In particular, does the category of categories (i.e., the application of the theory to itself) belong to the part of mathematics that studies infinite and continuous systems, or does it belong to the study of systems that are finite and discrete, or does it fall somewhere in the middle? Yours, Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: John F. Sowa <[email protected]> Sent: Friday, May 29, 2020 6:19:48 AM To: Peirce-L Subject: [PEIRCE-L] The sciences: mathematical and classificatory Robert and Jon, I was browsing through and deleting some old email, and I came across the points quoted below. I also remember that Jon claimed that Peirce's word 'classificatory' for normative science made it sound trivial. But there are only three kinds of science: (1) mathematical, (2) classificatory. and (3) some combination of #1 and #2 in various proportions. Physics, after Galileo and Newton, became highly mathematical, but it has always depended on classification for its choice of hypotheses. Chemistry began as the classificatory science of alchemy, but it became more and more mathematical as it used physics to interpret and develop its study of the phenomena. And biology was almost purely classificatory until the late 20th century. As for phaneroscopy, Peirce derived his semeiotic by applying mathematics to the analysis of experiences in the phaneron. (See R602, Photometric Researches, and the sections of CP vol. 1 which the editors labeled 'Phenomenology'.) Pure mathematics (which includes mathematical logic) is the only science that is purely mathematical. But mathematics in practice depends on the classificatory sciences for the choice of various hypotheses to study. As for phaneroscopy, Peirce derived his semeiotic by applying mathematics to the analysis of experiences in the phaneron. (See R602, Photometric Researches, and the sections of CP vol. 1, which the editors labeled 'Phenomenology'.) That is why I used the term "formal semeiotic" for the methods Peirce used to derive his categories. And I agree that mathematical category theory is a powerful method for analyzing the structure. I believe that it makes a strong case for the label "formal". The next step after deriving formal semeiotic is to use it to classify the open-ended variety of phenomena labeled aesthetics, ethics, and rhetoric -- essentially every book on those subjects from Aristotle to the present. All the work on government and legal matters combines ethics and rhetoric with some logic (mostly limited to Aristotle's version). All that work is classificatory, and it's by no means trivial. John --------------- RM> Thank you very much John for these texts I did not ... I could have known the R602 since I had access to microfilmed manuscripts, but at the time, I was mainly interested in sign classifications and I did not think that this ms could contain such assertions . I particularly remember this sentence: "Phaneroscopic research requires a previous study of mathematics. The type of mathematics depends on the application, and there is no limit on the amount and depth of mathematics that may be needed" that I have had the opportunity to show to all those who will tell me that it does not have the ability to access the elementary definitions of category theory ... "Scientific research is a fighting sport". -------------------------- JFS> Your summary of the issues is very good, and I strongly agree with the need for examples. In a search for examples, I went back to _Photometric Researches_, which I believe is essential for understanding the development of Peirce's philosophy. It's not an accident that it was published in 1878, the same year in which "How to make our ideas clear" was published in _Popular Science Monthly_. (Peirce met the editor of that magazine when they were both in Sicily, observing the solar eclipse.) The title of Chapter 1 is "The sensation of light." See http://jfsowa.com/peirce/PRexcerpts.pdf . On page 2, he wrote "Light considered purely as something in the external world may be called _noumenal light_. Light considered as an appearance, and as a function of the sensation, such that it is measured by the convention just mentioned, may be termed _phenomenal light_. Photometry generally concerns phenomenal light..." Just reading a few pages (my PRexperts, for example) is enough to show how deeply Peirce's phaneroscopy is grounded in his scientific and engineering research. It's also important to compare that research with R602, which was written after his 1903 classification of the sciences. See http://jfsowa.com/peirce/r602.htm : p 12> But preliminary to normative science, which is essentially classificatory, — stop to take that well in, I beg you, gentle reader, there should be a nomological science, which shall make out all the different indecomposable elements which enter into everything that is conceivably possible, discriminates them with care, and shows how they can be varied and combined. This science I hesitate to call phenomenology after Hegal, for fear of marring his peculiar conception of it; and therefore, though I think it is essentially the same thing under a somewhat different aspect, I will name phaneroscopy. p 13> It is the science of the different elementary constituents of all ideas. Its material [m13] is, of course, universal experience, — experience I mean of the fanciful and the abstract, as well as of the concrete and real. Yet to suppose that in such experience the elements were to be found already separate would be to suppose the unimaginable and selfcontradictory. They must be separated by a process of thought that cannot be summoned up Hegel-wise on demand. They must be picked out of the fragments that necessary reasonings scatter; and therefore it is that phaneroscopic research requires a previous study of mathematics. Observation from p, 12: normative science is essentially _classificatory_. That implies that logic as semeiotic is a classification of the ways that formal logic is applied to practice. The reasoning methods of logic as semeiotic are derived from mathematical logic. Many people who talk about Peirce's semeiotic ignore the fact that his reasoning methods are based on mathematical logic. Observation from p, 13 of R602 and pp. 10 and 11 of PRexcerpts: Phaneroscopic research requires a previous study of mathematics. The kind of mathematics depends on the application, and there is no limit on the amount and depth of mathematics that may be needed. John
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