Jerry LRC and Gary R, The point I'm making is true of every branch of experimental science and engineering practice. That includes chemistry, which is the first branch that Peirce studied in detail. JFS> Whenever logicians talk about anything x that exists in any universe of discourse, they write ∃ in their logic. JLRC> This assertion is not true for the chemical symbol system. The logicians are the ones who use the symbol ∃. Every mathematical theory, including arithmetic, uses its own notation and terminology. But every item and pattern of items in their theories have an implicit ∃ in their theoretical foundations. Short summary: Every scientific theory and every engineering plan belongs to the universe of possibilities. The science is a fallible approximation to the universe of actuality, and the engineering is an approximation to a scientific theory about something that might exist in some future state of the universe of actuality. Longer discussion: Peirce insisted that every scientific hypothesis or theory of any kind is fallible. Scientists themselves know that their best theories are reliable only on the domains for which they have been thoroughly tested. They know implicitly -- and often sttate explicitly -- that the most fundamental things in their theories (fields, particles, quarks, strings, ...) are hypothetical entities that exist in their mathematical models. But the evidence for their existence is based on long chains of inference from fallible theories about observable things. Engineers go even further: they admit that "Every model is wrong, but some are useful." They admit that every theory they use for their plans and designs is an approximation. The math for the best available physics is extremely complex, and the engineers often use more computable theories that are known to be false in detail. And Gary, you have mentioned that metaphysics does not require complex mathematics. That is true about the theories of philosophers who don't know much math -- they're the metaphysicians that Peirce denegrates. But those who write about the philosphy of science (Peirce, Galileo, Aristotle, for examples) study as much math as the scientists they are talking about. Many of them (same example) are or were outstanding scientists in their own right. Aristotle, for example, wrote a huge amount about biology, and he had his students carry out experimental science. His writings on embryology, based on those experiments, were the state of the art in that subject for many centuries. Galileo also used experiments and mathematics to interpret those experiments. The metaphysics used in science is as precise and mathematical as the sciences themselves. It's true that the metaphysics required for everyday experience does not require as much math. But "everyday experience" depends very strongly on the culture, education, and technology of the people who have the experience. For Peirce, that is a very high standard. Please read or reread R602.htm. I uploaded it to http://jfsowa.com/peirce/r602.htm . Note the last line of page 13: "phaneroscopic research requires a previous study of mathematics." For Peirce, that includes everything in NEM. John
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