From: Thomas Walker Lynch Date: Sat, 4 Jul 2026 07:08:27 +0000 (+0000) Subject: call this a first draft of the Computational Naturalism chapter X-Git-Url: https://git.reasoningtechnology.com/machine%20fig.png?a=commitdiff_plain;h=59510979a250f92c0fd3f0dce2c1c55f243d1255;p=TM-2026 call this a first draft of the Computational Naturalism chapter --- diff --git a/document/TM-2026.html b/document/TM-2026.html index fca2f9d..5821baa 100644 --- a/document/TM-2026.html +++ b/document/TM-2026.html @@ -80,6 +80,10 @@ A person might suggest defining S as the set of all definable mathematical objects, forming a universal set. However, if such a universal set S existed, the Axiom of Separation could be applied using the previous predicate to isolate \dot{R}. Because \dot{R} is a valid, definable set, it must reside within S by the very definition of a universal set. But notice that the logic evaluated earlier proved definitively that \dot{R} cannot be a member of S. Yet the existence of a definable set \dot{R} that sits strictly outside of S contradicts the premise that S contains everything. Therefore, within any system governed by the Axiom of Separation, a universal set cannot exist.

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+ The authority to remove the Russell's Paradox set formulation comes from the set S. If we know its definition, then the authority comes through that definition. However, if we merely stipulate that S must be defined, then we are expressing our authority through S by declaring, "Undefined sets are not allowed." In the explanation above, it is only after discovering a set is undefined that we conclude it is not a member of S. I sometimes wonder how mathematics might have evolved had Frege simply taken that approach. We take this question up again in the chapter Computational Naturalism, and discover there is a deeper issue. +

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Stepping back from the mechanics of set definition, a person can observe two competing approaches to establishing mathematical foundations. The first approach is constructive, building complex systems by assembling them upward from fundamental primitives. The second approach relies on islands of meaning, carving out valid spaces from the abstract void using precise rules and axioms, exactly as Zermelo did. Because both methodologies rely entirely upon a rigorous framework of deduction to function, logic itself serves as the essential substrate. Consequently, a complete foundational study requires the examination of three distinct subjects: the primitives used for construction, the rules that bound the theoretical islands, and the underlying logic that evaluates them both.

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- We then define the symbol in computational terms, as done in the prior section. This begins with memory addresses, represented as charge vectors, acting as primitive symbols, and extends to the symbol factory, copy operations, and instance comparison. + We then define the symbol in computational terms, as done in the prior section. This begins with memory addresses, represented as charge arrays, acting as primitive symbols, and extends to the symbol factory, copy operations, and instance comparison.

@@ -1028,20 +1032,32 @@ Where Gödel reduced logic to natural numbers, we go the other direction to expand upon logic from natural numbers.

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+ An axiomatic proof is then a decider that is built up from subroutine calls to the axioms. We might then quantify over all possible compositions of our subroutines in analysis and ask if it is possible that a contradiction decider would return Y or N. +

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Frege's set theory is then the analysis of a logic program against an enumeration of inputs to choose if a proposed symbol is in a set. Perhaps executing such a program is left to first order analysis, or perhaps execution is not practical, and evaluation is left to second order analysis.

- Russell's Paradox is then defined as a Turing Machine that can be analyzed in the second order, but cannot be analyzed in the first order. + Russell's Paradox will then be expressed as a Turing Machine that can be analyzed in the second order, but cannot be analyzed in the first order. That is, the paradox exists merely in the first order as it will never halt when run. However, it is not a paradox in the second order. It is, of course, through second order analysis that we are able to describe why Russell's Set description does not resolve in the first order. +

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+ With the language of Computational Naturalism it is possible to restate every statement ever made by any mathematician; we might say that the mere fact that a mathematician was able to state something qualifies its membership into Zermelo's S set. +

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+ Given our knowledge that a universal halting problem analyzer that decides if a Turing Machine program halts does not exist, it is desirable to have rules that guide our writing only Turing Machine programs that are known to halt. This can be done through construction, as proposed by Russell, or through axioms of separation, mapping, and choice, as proposed by Zermelo and others. But then we know from the completeness and correctness theorems, that when such guiding rules are applied, there will exist Turing Machine programs that do halt, but whose definition cannot be constructed, nor surmised using the said axioms.

- With this language it is possible to restate every statement ever made by any mathematician; the mere fact that a mathematician was able to state it qualifies its membership into Zermelo's S set. + This brings us back to the reference from the Zermelo discussion in the introduction. Specifically, the question posed is if our finding through second order analysis that Russell's paradoxical set formulation will not run in the first order and define a set, does this mean that we are merely using S to state that we are tossing out sets that cannot be defined? The answer is it is not quite this simple. Had Frege said, 'we merely dismiss such sets', we would not have the formalization for the second order analysis. Throwing out Russell's paradox simply because we ran it and it never halted is not a practical approach. Instead, we omit it specifically from S because analyzing R reveals that it fails to define a first order halting machine, and S, by definition, only holds first order halting machines. Zermelo's language is precise and formal. However, it is this bothersome nuance, now articulated here, which caused us to scratch our heads when reading Zermelo the first time.

- Of course, those among us who believe the human brain to be a computational device, or at least the subset of it involved in working on mathematics, are already computational naturalists. + Russell's set formulation, R, can be analyzed to determine that it would not halt if it were run. We still keep it as a second order object, and have even given it a name, R. However, this begs the question, are there machines that cannot even be analyzed in the second order to ascertain if they would return a result in first order analysis, i.e., when they are run? If Gödel has a say here, a person would wager that such machines exist. But then, is there an option for analysis in the third order? Is Turing's halting proof a third order analysis as it reasons about running the second order analyzer? Or is it merely a recursive application of second order analysis?

Addresses and cells