Jon AS, Gary F, List,
JAS: the question is simply whether Peirce ever explicitly states that "mathematics is the universe of all possibilities." If not, then nobody can definitively claim that this is what he intended. The assumption that mathematics specifies all of what Peirce called "real possibilities" is the working hypothesis of all scientists from the ancient Greeks to the present -- independently of what words they use to express their opinions. Peirce's many years of work and writings about science and engineering show that he was fully convinced of the power of mathematics. His many criticisms of "metaphysicians" who don't use mathematics show his belief in the importance of mathematics for every branch of philosophy. JAS: As I have pointed out before, Peirce's "three Universes" do not consist of possibilities, actualities, and necessities; they consist of Ideas, Brute Actuality, and Signs (CP 6.455, EP 2:435, 1908) or Possibles, Existents, and Necessitants (EP 2:478-479, 1908). Moreover, Peirce does not claim that all possibilities are real, only that some possibilities are real (CP 5.453, EP 2:354, 1905; CP 4.581, 1906); and pure mathematics is not concerned at all with whether any given possibility is real, only with what would follow necessarily from it. The first sentence is inconsistent with the quotations cited in the middle of that paragraph. The last sentence is true, and it shows why pure mathematics is the basis for phaneroscopy. It's the first stage in the identification and classification of the elements of experience. After phaneroscopy, further evaluation depends on the normative sciences. To see the similarities, compare those quotations to my version: Ideas, Brute Actuality, Signs (CP 6.455, EP 2:435, 1908) Possibles, Existents, Necessitants (EP 2:478-479, 1908) Possibilities, Actualities, Necessities (AKA Laws) (JFS) The only difference between my version and Peirce's is in the choice of suffixes. As a compromise, I would be happy with a selection from both of Peirce's versions: Possibles, Actuality, Necessitants. As for the issue about which possibles are real possibles, that depends entirely on the applications. As scientists make new discoveries, different versions of mathematics may represent different possibles that may be considered real. There is no way of knowing which of the infinitely many possible mathematical theories may someday prove to be useful. JFS: {This discussion] shows the power of mathematics to specify anything that exists in the universe and beyond. That includes any imaginable or even unimaginable level of infinities. jas: This is quite a sweeping claim. How could mathematics specify something that is unimaginable? Very easily. Cantor's hierarchies of infinities include infinitely many unimaginable structures and patterns of structures. The mathematics of quantum mechanics has a huge number of surprises. Physicists have learned how to do the math. But after a century of research, they are still stumbling across surprising results that defy intuition. There are fractals and multidimensionl patterns that are important for various applications, but only the simplest examples can be imagined. JFS: Those examples [of birds, bats, moths, and Olympian gymnasts] show the huge amount of processing that takes place in seconds or less. And all of it can be specified in some version of mathematics, even though it may take many years for neuroscientists to discover the proper kind of mathematics to use. JAS: This is a statement of faith, not fact. No. It's a brief summary of the ongoing work in neuroscience, which has made enormous progress in the past 30 years. The artificial neural networks implemented in computer systems have been successful in complex perception, such as recognizing faces in a crowd or driving an automobile. But the neuroscientists are working diligently on the far more complex brains from nematodes to fruit flies to birds to humans. They have learned a lot, and mathematics is key to every stage of the science. But there is much more work to be done. In a previous note, I commented on the following points by Gary, but I'd like to add another remark, based on the above. JFS: Phenomenology/phaneroscopy analyzes experiences in the phaneron in order to classify and determine the elements of experience. But as Peirce said, the same kinds of experiences may come from external sensation, from imagination, or from memories. GF: The phrase kinds of experiences is ambiguous. My specific answer to the question I posed would be that the distinction between the actual world and a world of imagination arises from awareness of the difference between Secondness and Firstness, together with the recognition that these distinct elements of experience are equally elementary. It's not ambiguous. It's sufficiently general to include the others, but I agree that it's important to distinguish them. GF: The question is not about distinctions in general, but about a specific distinction which normative science cannot rely on mathematics to supply. Your general statement glosses over the fact that according to Peirces classification of sciences, normative logic depends for its principles on phenomenology/phaneroscopy (as well as on mathematics, from which it cannot inherit this distinction in principle). To illustrate the issues, consider an extremely vivid dream. The initial stages of phaneroscopy for analyzing the experience would be the same as the analysis of sensations from a similar situation while awake. But the evidence that it is a dream might not come from any particular experience, but from an unusual transition between experiences. An obvious example would be hiking in the woods and suddenly waking up in bed. Other examples may be physically impossible actions or features that are known to be false. People may also have false memories or deja vu experiences that require detailed methodeutic or even clinical therapy to untangle. The initial interpretations would occur in phaneroscopy. But more complex analyses would require normative logic and semeiotic. John
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