Re: R.J. Lipton • Mathematical Tricks
A formal duality points to a higher unity — a calculus of forms whose expressions can be read in two different ways by switching the meanings assigned to a pair of primitive terms.
I just ran across an old post of mine on the FOM List where I touched on this theme — I’ll copy that here until I get a chance to comment further.
Re: Projective Geometry and Mathematical Logic • Rupert McCallum
C.S. Peirce explored a variety of De Morgan type dualities in logic which he treated on analogy with the dualities in projective geometry. This gave rise to formal systems where the initial constants — and thus their geometric and graph-theoretic representations — had no uniquely fixed meanings but could be given dual interpretations in logic.
It was in this context that Peirce’s systems of logical graphs developed, issuing in dual interpretations of the same formal axioms which Peirce referred to as entitative graphs and existential graphs, respectively. He developed only the existential interpretation to any great extent, since the extension from propositional to relational calculus appeared easier to visualize in that case, but whether there is some truly logical reason for the symmetry to break at that point is not yet known to me.
In exploring how Peirce’s way of doing things might be extended to “differential logic” I’ve run into many themes analogous to differential geometry over GF(2). Naturally, there are many surprises.
Thank you for this interesting post! Could you cite the paper(s?) where he writes about projective / constructive / synthetic geometry, in connection to the original design of graphical logic? I only have the Buchler pink book and the Wiener blue book popular collections. I recall there is some more voluminous series with the logical work — could you recommend a print work that I should have in order to bridge the metaphysical essays to the logical graphs? B
I believe I first noticed this in Volume 4 of Peirce’s Collected Papers. I will try to look it up later today. —Jon
Thank you! By the way, there is much extant literature regarding the Existential Graphs’ isomorphy with Laws of Form (e.g. A whole edition of C&HK of Peirce & Spencer-Brown). In the same spirit of movement from the metaphysical Peirce (the architectonic trichotomism, logic as semiotic, etc.) to Logical Graphs Peirce, can we use the Peirce-Spencer-Brown bridge of iconic logic isomorphy to tease out the metaphysical implications of Spencer-Brown’s thought? Form and indication are an obvious topic of concern to both, and so is distinction and the nature of mathematics and inference. Spencer-Brown is not as prolific an author, but the one great work of Laws of Form plumbs deeper than anything I’m familiar with. I’m taking a class on pragmatism, and working on a mathesis universalis thesis, so I’m trying to combine these two in more ways than one. Thanks!
I happened on Peirce and Spencer Brown during the late 60s and eventually wrote my senior thesis, Complications of the Simplest Mathematics (1976), on a few puzzles posed by their way of doing logic. That work was instigated by Chapter 3 of Peirce’s Minute Logic, titled The Simplest Mathematics, published in his Collected Papers (CP 4.227–323). The adoption of a parallel column paradigm from projective geometry is introduced at CP 4.277 and that is probably where I first saw it.
Thanks! I have to get the Collected Papers volumes. I was a friend of Spencer-Brown, in his last years. I’m working on the applications of their mathematical thought (plus the ancient origins of mathematical original intuition) in pure phenomenology, particularly the area of the mathesis universalis and constructive synthesis in the constitution of lived experience, i.e. how the manifold of apperception becomes the lived world. Fichte’s “original geometry”, the ancient lost “sphaerics”, and the reconstructions of Neoplatonic and Pythagorean systems of the Monas / Peras (the Monad, “Horizoned” or relative and absolute or Aoristos) and Aoristos Dyas (Indefinite and horizonless Dyad : these are both the Origin / Apeiron, and the Odd / Even) dominate my thought in this area. I imagine there is a lot of Peirce’s “simplest” mathematics that is relevant.
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