John, Gary F., List: JFS: Intuitionistic logic is a restriction on the permissible rules of inference.
I suppose that is one way to view the omission of the (so-called) law of excluded middle, but it is precisely equivalent to saying that non-Euclidean geometry is a restriction on the permissible rules of inference because it omits the parallel postulate. JFS: That makes it impossible to use many widely accepted theories of mathematics ... Likewise, omitting the parallel postulate makes it impossible to use many widely accepted theories of geometry. Should we reject all non-Euclidean systems on that basis? On the contrary, "Mathematics is the study of what is true of hypothetical states of things" (CP 4.233, 1902), so it is perfectly appropriate (and potentially enlightening) to examine the results obtained from adopting a *different *set of hypotheses as the starting point. JFS: And it's the foundation for his theory of continuity -- which Abraham Robinson proved was consistent in 1960. Most mathematicians familiar with Peirce's thought recognize that Robinson's non-standard analysis is *not *the rigorous mathematical treatment of continuity that most closely conforms to Peirce's conception, pointing instead to synthetic differential geometry and smooth infinitesimal analysis. And what is the logic of these category-theoretic systems? Intuitionistic logic, because it is *not *the case that an infinitesimal is either zero or non-zero. Please see my *Transactions* paper about this, "Peirce's Topical Continuum: A 'Thicker' Theory" ( https://www.jstor.org/stable/10.2979/trancharpeirsoc.56.1.04). JFS: In applications to science and engineering, especially computer science, nobody uses intuitionistic logic. That is quite a sweeping claim. How could anyone know for sure that *absolutely nobody* in these fields uses intuitionistic logic? JFS: The reason why is that it "blocks the way" of using the most convenient, efficient, and flexible methods of reasoning. As I have said before, and as Gary F. basically said below, so what? Inquiry is not the same thing as "using the most convenient, efficient, and flexible methods of reasoning." If the latter is one's purpose, then indeed, intuitionistic logic is likely the wrong choice. However, Peirce stated over and over that this was not *his *purpose in studying logic, or at least not his *primary *purpose; and it is not mine, either. If one's purpose is instead to explore the effects of omitting excluded middle from classical logic without positing additional truth values, then intuitionistic logic is the *perfect *choice. Someone who tries to prevent such an investigation, or even just diminish its legitimacy, is the one who is blocking the way of *inquiry*. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt On Wed, May 19, 2021 at 7:51 AM <[email protected]> wrote: > John, I appreciate the clarification in your post: when you wrote that > intuitionist logic “blocks the way of inquiry”, what you really meant was > that *it "blocks the way" of using the most convenient, efficient, and > flexible methods of reasoning.* Peirce’s idea of *inquiry*, and > specifically of the inquiry he was developing by means of Existential > Graphs, was very different from that, especially in his one presentation of > EGs to a mixed audience (meaning an audience not composed entirely of > mathematicians). He made this clear at the very beginning of his second > Lowell lecture <https://gnusystems.ca/Lowell2.htm> of 1903: > > [[ Before beginning, let us distinctly recognize the purpose which this > system of expression is designed to fulfil. It is intended to enable us to > separate reasoning into its smallest steps so that each one may be examined > by itself. Observe, then, that it is not the purpose of this system of > expression to facilitate reasoning and to enable one to reach his > conclusions in the speediest manner. Were that our object, we should seek a > system of expression which should reduce many steps to one; while our > object is to subdivide one step into as many as possible. Our system is > intended to facilitate the *study* of reasoning but not to facilitate > reasoning itself. Its character is quite contrary to that purpose.]] > > > > Gary f. > > > > *From:* [email protected] <[email protected]> *On > Behalf Of *John F. Sowa > *Sent:* 19-May-21 00:59 > *To:* [email protected] > *Subject:* Re: [PEIRCE-L] Intuitionistic logic > > Gary R, > > I'm glad you asked. > > GR> Please explain how this "blocks the way of inquiry" for folk like me > who are apparently radically deficient in mathematics and logic so simply > can't see it as such. > > > Intuitionistic logic is a restriction on the permissible rules of > inference. That makes it impossible to use many widely accepted theories of > mathematics -- among them, the theory that there are hierarchies of > infinities. > > Peirce was one of the mathematicians who discovered a proof of that point > independently of Georg Cantor. And it's the foundation for his theory of > continuity -- which Abraham Robinson proved was consistent in 1960. > > In applications to science and engineering, especially computer science, > nobody uses intuitionistic logic. The reason why is that it "blocks the > way" of using the most convenient, efficient, and flexible methods of > reasoning. > > The mainstream mathematicians don't stop intuitionists from developing > their own pet theories. They just ignore them. > > John >
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