Peirce was truly a mathematical philosopher. He was himself a first-rate mathematician, the most gifted son of Benjamin Peirce, the foremost American mathematician of the time; he often made his philosophical points by means of mathematical arguments and examples; and he believed that philosophy must begin with logic, which rests in turn upon mathematics. Moreover, many of his central concepts—most notably that of continuity—are as much mathematical as they are philosophical. For all of these reasons we cannot fully understand Peirce's philosophy unless we come to grips with the mathematical dimensions of his thought.
Kuhn asserted that during times of revolutionary science, anomalies refuting the accepted theory have built up to such a point that the old theory is broken down and a new one is built to take its place in a so-called "paradigm shift."
If you're interested in learning more about the philosophy of science, you might want to begin your investigation with some of the big names in the field:Aristotle (384-322 BC) Arguably the founder of both science and philosophy of science.
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The notion of essential predication is connected to what aretraditionally called the categories(katêgoriai). In a word, Aristotle is famous for havingheld a “doctrine of categories”. Just what that doctrinewas, and indeed just what a category is, are considerably more vexingquestions. They also quickly take us outside his logic and into hismetaphysics. Here, I will try to give a very general overview,beginning with the somewhat simpler question “What categoriesare there?”
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In the last century, Aristotle’s reputation as a logician hasundergone two remarkable reversals. The rise of modern formal logicfollowing the work of Frege and Russell brought with it a recognitionof the many serious limitations of Aristotle’s logic; today,very few would try to maintain that it is adequate as a basis forunderstanding science, mathematics, or even everyday reasoning. At thesame time, scholars trained in modern formal techniques have come toview Aristotle with new respect, not so much for the correctness ofhis results as for the remarkable similarity in spirit between much ofhis work and modern logic. As Jonathan Lear has put it,“Aristotle shares with modern logicians a fundamental interestin metatheory”: his primary goal is not to offer a practicalguide to argumentation but to study the properties of inferentialsystems themselves.