**REFUTATION OF THE GOLEM**

__The Golem__

A

*golem*is in legend a clay figure brought to life by magic. In contemporary times the urge to make a golem finds expression in the claim of strong AI (in philosophy called

*functionalism*) that a computer can be made that is indistinguishable as to behaviour from human life, and hence must

*be alive*and conscious.

__General__

1. The only way to

*prove*that a machine behaves functionally exactly like a human being is to construct it. No such machine has ever been made. “It shall be made!” is an article of faith in materialism. No one is obliged to believe this. The first retort to strong AI is:

*put up or shut up!*Where is your machine?

2. The mind renders experience intelligible to its own understanding by imposing order upon it. Human experience, concept laden, is structured by universals. But the material faith in the

*golem*is artificially buttressed against this observation by a doctrine imposed by

*fiat*of nominalism: that there are no universals, and that all human understanding arises from sensation alone (empiricism). Since this is a fiat, no one is obliged to believe it. Furthermore, time as we know it is subjective time. The mind imposes subjective time upon its experience and subjective time is nothing akin to the objective time of theoretical science.

3. There is a rationalist philosophy of mathematics. There is a synthetic logic.

__Technical__

1. A computer is a machine that operates on binary data, 1s and 0s, and as such can only be described by a mathematical structure that deploys at most one primitive concept, such as a binary function, set-membership or in category theory, the arrow. A computer is bound to an abstract surface (space) known as a Boolean algebra and is constrained by a logic known as first-order logic. But more than one primitive is deployed in mathematics. For example, the primitive concepts of infinity, potential and actual, cannot be derived from a Boolean function.

2. In analysis (the theory of the continuum and limits) the axiom of completeness is irreducibly second-order; it cannot be expressed in the language of first-order logic. The equivalences between the various forms of the completeness axiom (Dedekind, Heine-Borel, Cantor, Cauchy etc.) cannot be computable.

3. Number theory deploys the concept of the potential infinite in its fundamental argument of mathematical induction, which is synthetic, and is not a principle of first-order logic. Number theory entails the Archimedean property that the natural numbers are not bounded above, but in set theory a supremum of the natural numbers is defined. Therefore, number theory cannot be embedded in set theory. Since the Archimedean property is a consequence of the axiom of completeness, analysis is also not set theoretical.

4. The Dirichlet pigeon-hole principle (used in fixed-point theorems) is a synthetic principle of human reason that cannot be expressed in first-order logic or by Boolean functions. In group theory, Lagrange’s theorem that the order of a subgroup divides into the order of the group, cannot be formalized in first-order logic. The expressive power of mathematics transcends that of first-order logic.

5. Close examination of the structure of a Boolean algebra reveals that it alone is insufficient to model even the analytic logic of first-order inferences. Analytic logic is an instance of synthetic logic, and not vice-versa.

__Further__

These observations above are sufficient to refute the existence of the golem. However, there are three additional and untested claims on this website that, if accepted as valid, categorically rule out the golem.

1. Analysis requires an Axiom of Indestructibility of Extension: Every proper part of an extended portion of space is extended. Or in the words of Kant: “Space … consists solely of spaces, time solely of times. Points and instants are only limits, that is, mere positions which limit space and time.” This axiom has always been implicit in analysis, but for historical reasons has hitherto not been recognized. Then the continuum is known to require two primitives – points and extensions – and no such structure could possibly be computable. (But analysis requires other primitives – at least both the potential and actual infinite. The continuum should already be known to be a non-computable ideal structure of mathematics.)

2. It has long been disputed whether Godel’s incompleteness theorem supplies a counter-argument to the golem. It is a claim here that (a) there is a universal form of Gödel’s theorem; (b) if this universal theorem is assumed to be a first-order theorem, then it does entail a formal contradiction. This shows that Gödel’s theorem in its universal form is a counter-argument to the golem. There is also a huge amount of work on the semantics of Gödel’s theorem posted, which demonstrates that a structure becomes essentially complete when it is modeled by the continuum. There is a deep relationship between the semantics of Gödel’s theorem and analysis that does demonstrate that Gödel’s theorem is a counter-example to the golem.

3. It is a claim here that the Halting problem can be solved by mathematical induction. Since the Halting problem is non-computable, this would be a categorical counter-example to the golem.