John, Jon, Gary F, Edwina, List, I don't believe that it is in any way controversial that not only Peirce, but virtually every serious scholar makes a distinction between theory and practice and, likewise, between pure and applied mathematics.
Here I'd like to comment on the later distinction (i.e., between pure and applied mathematics); but rather than offering abstract definitions of these, to emphasize the legitimate interest of many folk on this list in the importance of real world applications of mathematics and science, I'll instead quote from a college website commenting on how pure and applied mathematics are distinguished in their undergraduate degree programs. These simple summaries are meant to help students decide which major (pure or applied mathematics) to enroll in. Edwin Ding, PhD, an associate professor in the Department of Mathematics, Physics, and Statistics at APU, noted that the mathematics major focuses on pure mathematics. He explained that pure mathematics deals with the theoretical side of math and has a greater concentration on proofs, theorems, and abstract concepts. “The applied math major, on the other hand, focuses more on applying analytical/computational math techniques to solve real-world problems in different fields,” said Ding. These fields can include actuarial science, biology, physics, computer science, and statistics. According to Ding, “Both majors start with the foundational courses, such as calculus sequence and ordinary differential equations, and go into different specializations later on.” https://www.apu.edu/articles/the-difference-between-mathematics-degrees-applied-math-vs-pure-math/ What it seems to me that De Tienne emphasizes in that aspect of his presentation having the purpose of positioning pharneroscopy within Peirce's *Classification of Sciences*, a work, btw, of *Science of Review*, concerned with sciences *qua* scientific *disciplines *as distinct from how the knowledge of each of these will be employed in the actual work of any given scientist or group of sciences. So, if one wants to prepare for work in, for example, biology, a student would most likely want to have and, indeed, need to have some training in several sciences in addition to biology, for example, in mathematics, logic, chemistry, etc. But her more advanced training would *emphasize* the distinctive research characters (methodology) of the science of biology, a science which she may have a burning interest in which may eventually lead to her making important discoveries in it. As I see it, Peirce's *Classification of Sciences* is principally devised to point to the specific *subject matter* of individual sciences as they have already been developed, but equally importantly, may be developed, and will be researched by generations of future scientists with special interests in particular sciences*. Of* course these interests and abilities may not be limited to a single science, may even encompass many sciences (Peirce was himself certainly an authentic polymath). And while preparatory studies for one to become, say, a chemist or a psychologist may overlap to some (small) extent, yet as their individual educations broaden and deepen, their scientific training and, of course, ultimately their scientific practice, *will diverge* considerably. Naturally it is not unheard of for, say, an accomplished botanist to also be an expert anthropologist. But if so, that scientist must make deep and serious studies of both those sciences, and likely much, much more. It is likely that programs in any scientific discipline will most likely require students to take courses in mathematics and logic, perhaps the history of science, etc. But once they are actively working in their special fields, if they continue as research scientists they will likely want to make significant progress in their respective field(s), to "further" them, as the English idiom would have it. For, as Peirce comments: Science is not a fixed, unchangeable body of propositions. After a thousand years the general face of science may be modified past recognition. 1899 | From Comte to Benjamin Kidd | CN 2:214 Thus, science "consists in a *disposition* of living men" and women actively working in various scientific disciplines, genuine science being the work and research of those trained in and engaged in the various scientific disciplines. CSP: Science is research; and research is science, from the first moment when the researcher casts aside all desire to prove his present opinions right, and burns with ardent desire to find out wherein they are wrong. Science thus consists in a disposition of living men. . . Peirce continues in this passage to note that for contemporary, active, *developing* research that "the true divisions of science will be those which divide living men.: 1902 | Minute Logic: Chapter II. Prelogical Notions. Section I. Classification of the Sciences | MS [R] 426:12 Thus, he brings together these two notions of scientific subject matter and the actual scientific practice of scientists working more of less collaboratively in particular sciences, the last in consideration of the so-called "Perennial Classification": CSP: . . .what I mean by a “science,” both for the purpose of this classification & in general, is the life devoted to the pursuit of truth according to the best known methods on the part of a group of men who understand one another’s ideas and works as no outsider can. . . they are pursuing a branch of truth according. . . to the best methods that are known at the time. I do not call the solitary studies of a single man a science. It is only when a group of men, more or less in intercommunication, are aiding and stimulating one another by their understanding of a particular group of studies as outsiders cannot understand them, that I call their life a science. 1905 | Adirondack Summer School Lectures | MS [R] 1334:11-13 So, while scientists interests may not, and most likely *will *not be limited to a particular science -- as in a very strong sense today they *cannot* be so limited -- yet we tend to establish educational and research programs in particular fields: in biology, physics, chemistry, logic, political science, linguistics, etc. and we call those who come to practice these sciences biologists, physicists, chemists, logicians, etc. Therefore, I'm not quite sure what is really controversial here. To identify a science's subject matter (even as Peirce notes that it could develop fairly out of recognition over a long period of time), and to note that some people concentrate their research in particular sciences, as indeed they must, neither of these in any way limits the present and future scope of any science, nor compels any individual researcher to restrict her research to a single science. Indeed, it seems to me that we live in an age of inter- and cross-disciplinary research. I would hope these comments are as reasonable and non-controversial as I believe that they really ought to be. Establishing and/or developing a science is the work of scientists, and Peirce's classification of sciences is meant as a mere tool -- and aid -- to that establishment and development. Best, Gary R “Let everything happen to you Beauty and terror Just keep going No feeling is final” ― Rainer Maria Rilke *Gary Richmond* *Philosophy and Critical Thinking* *Communication Studies* *LaGuardia College of the City University of New York* >
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