>From Osher Doctorow [EMAIL PROTECTED], Tues. Dec. 3, 2002 1326 Tim May gives a very detailed account of his ideas on category and topos theories, and I will only comment on a few of his ideas and some of Ben Goertzel because of space and time limitations.
I think that Tim and I, and hopefully Ben, do not differ on the extreme usefulness of being able to generalize concepts across many different fields and subfields. MacLane and Lawvere's category theory TRIED to do that, and the effort is certainly commendable. Perhaps it is more than commendable. As one who seldom receives commendations, I may tend to give them less often to others than I did when I started out in mathematics/science. Nevertheless, I perceive or understand what Ben refers to as a certain lack of deep results in category theory as compared with my Rare Event Theory for example - although Ben understands it relative to his own experiences. The theorems that Tim has cited are one counterexample class to this, but where are the great predictions, where is there anything like the Einstein Field Equation, the Schrodinger Equation, Newton's Laws, Fermat's numerous results, Maxwell's Equations, the Gauss-Bonnet Theorem and its associated equation that ties together geometry and topology, Non-Euclidean/Riemannian Geometry, Euclidean Geometry, the Jacobsen radical, Gauss-Null and related sets in geometric nonlinear functional analysis, Godel's theorems, or even Hoyle's Law or the Central Limit Theorems or the almost incredible theorems of Nonsmooth Analysis and Kalman filters/predictors and Dynamic Programming and the Calculus of Variations and Cantor's cardinals and ordinals and Robinson's infinitesimals and Dirac's equations and Dirac's delta functions and Feynmann's path history integrals and diagrams and the whole new generation of continuum force laws and on and on. Sure, category theory can go in to many fields and find a category and then take credit for the field being essentially a category, and I can go into many fields and find plus and minus and division and multiplication analogs and declare the field as an example of Rare Event Theory [RET] or Fairly Frequent Event Theory [FFT or FET] or Very Frequent Event Theory [VFT or VET] or a plus field or a minus field or a division field or a multiplication field. And both Category and RET-FET-VET theories can show that many of their concepts cross many fields. This is very commendable, although to me it is old hat to notice that something like a generalization of a group crosses many branches of mathematics, whereas RET-FET-VET classes such as GROWTH, CONTROL, EXPANSION-CONTRACTION, KNOWLEDGE-INFORMATION-ENTROPY tend to cross not only branches of mathematics but branches of physics and biology and psychology and astrophysics and on and on. But string and brane theory are suffering from precisely what category theory is suffering from - a paucity of predictions of the Einstein and Schrodinger kind mentioned in the second paragraph back, and a paucity of depth. Now, Tim, you certainly know very very much, but how are you at depth [question-mark - my question mark and several other keys like parentheses are out]. I will give an example. Socrates would rank in my estimation as a Creative Geniuses of Maximum Depth. The world of Athens was very superficial, facially and bodily and publicly oriented but with relatively little depth, and when push came to shove, rather than ask what words meant, it preferred to kill the person making the inquiries. What it was afraid of was going deep, asking what the gods really were, why so-called democracy ended at the boundaries of Athens and even was inapplicable to all people in Athens, what democracy really was, why the individual and the group/humanity were not equally important, when the Golden Mean and the Golden Extreme as I would call it [for example, valuing Knowledge rather than compromising between Knowledge and Ignorance] applied. You mentioned, Tim, that the Holographic Model is still very hypothetical. Are we to understand that G. 't Hooft obtained the Nobel Prize for a very hypothetical idea [question-mark] among others. I have actually generalized the Holographic Principle and it follows from RET-FET-VET Theory. But it happens to be an example of DEPTH of a type that Category Theory does not know how to handle. It says that LOWER DIMENSIONS CONTAIN MORE KNOWLEDGE-INFORMATION THAN HIGHER DIMENSIONS - IN FACT, ALL OF IT, with appropriate qualifications. I will conclude this rather long posting with an explanation of why I think Lawvere and MacLane and incidentally Smolin and Rovelli went in the wrong direction regarding depth. It was because they were ALGEBRAISTS - their specialty and life's work in mathematics was ALGEBRA - very, very advanced ALGEBRA. Now, algebra has a problem with depth because IT HAS TOO MANY ABSTRACT POSSIBILITIES WITH NO [MORE CONCRETE OR NOT] SELECTION CRITERIA AMONG THEM. It is somewhat like the Ocean - if an explorer worships the Ocean, then he will go off in any direction that Ocean seems to be leading him, and at least half of the time he will be wrong in terms of applications and predictions and usefulness and depth and so on. ANALYSIS, on the other hand, uses algebra and geometry and topology and number theory, but its core ideas are LIMITS [ONE- OR TWO-SIDED], MEASURES, CHANGE, and a few other things, and how these relate to geometries and topologies and algebras. Rovelli and Smolin and Ashtekar and Schwarz and Witten are slightly different types of algebraists - they are either ALGEBRAIC GEOMETERS or ALGEBRAIC TOPOLOGISTS, with PHYSICS thrown in of course. Their string and brane and knot and loop theories have about the same depth as those of MacLane and Lawvere in terms of not predicting new things. Oh, they do marvellously at REINVENTING THE WHEEL, at reconstructing QM and QFT, etc. Osher Doctorow ----- Original Message ----- From: "Tim May" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Tuesday, December 03, 2002 11:30 AM Subject: Re: Applied vs. Theoretical