Ben and lists, At 04:37 PM 9/25/2014, Benjamin Udell wrote:
You snipped a bit too much of what I said. I was talking about mathematics not just in the sense of doing the math, but of the mathematical objects themselves.
HP: I snipped where I did because I have no disagreement there. I am not making my point clear. I understand your belief in "mathematical objects themselves." I have no trouble imagining that, say, logic and prime numbers exist independent of human thought. Many creative mathematicians (e.g., Gödel) and some physicists (e.g., Penrose) explicitly claim to be Platonists. Nevertheless, these are apparently empirically irrefutable (unfalsifiable) beliefs, and therefore so are the opposite beliefs. So are realism and nominalism, etc. Arguments, often acrimonious, for and against each of these beliefs have been going on between brilliant minds for thousands of years with no sign of a "winner." I see this as strong evidence (as do Kant, Hertz, Poincaré, Born, and others) that we cannot logically or empirically decide these epistemic issues. Nevertheless, it is also a fact that passionate believers and non-believers in each case are responsible for creating the greatest mathematical and scientific discoveries. There is a lesson here that Polanyi, for one, has elaborated.
At least as important as this lesson are the revolutionary empirically verified discoveries of the 20th Century. Classical physics thinking really took a hit with quantum mechanics and relativity. As Clark just described it, the confidence of the classical era has been lost. Today, physicists know their theories are incomplete or wrong, Cosmology and particle theories also have major uncertainties. Still, the necessity of the complementarity of models is still generally acknowledged. This is complementarity in Bohr's strong sense that if they were formally combined the models would appear contradictory (e.g., reversible vs. irreversible, deterministic vs. probabilistic, discrete vs. continuous models).
Logic also took a hit in the 20th Century. Quantum theory does not satisfy classical logic (e.g., failure of the distributive law of propositional logic), and proofs of Incompleteness, and Uncomputability by Post, Gödel, Turing, as well as the logically unresolvable issues, like the Church-Turing Thesis and the Continuum Hypothesis showing that complementarity would be a logical as well as an empirical necessity in modeling reality. Turing's question, "Can machines think?" is undecidable because, as Turing understood, the words machine and thought are ambiguous. Given this word ambiguity, the psychologism question: Can matter think? (i.e., brain matter) is just as undecidable as Turing's question.
In this type of undecidable argument, the psychological evidence shows that thinking processes in human brains are so extremely varied that mutual agreement (except to disagree) is often impossible. Of course anecdotal evidence of this fact begins with history, but today there is more objective evidence. This fact of variation in brains is important for understanding cultural differences, especially between philosophy and science. Poincaré's and Russell's "irremediable differences" over logic's foundations is a classic example. That type of difference occurs in our discussions of Peirce here. I don't see that Pierce himself recognized the source of these differences.
In 1880 William James called the source of such irremediable difference, "temperaments," but we now know there are more specific individual differences in brains, but James' point about their role in disputes is still valid: "Yet in the forum he [the disputant] can make no claim, on the bare ground of his temperament, to superior discernment or authority. There arises thus a certain insincerity in our philosophic discussions: the potentest of all our premises is never mentioned."
Howard
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