Minds and Machines

, Volume 20, Issue 3, pp 423–439

Are Turing Machines Platonists? Inferentialism and the Computational Theory of Mind

Authors

    • Department of PhilosophyLouisiana State University
  • Jason Megil
Article

DOI: 10.1007/s11023-010-9203-1

Cite this article as:
Cogburn, J. & Megil, J. Minds & Machines (2010) 20: 423. doi:10.1007/s11023-010-9203-1

Abstract

We first discuss Michael Dummett’s philosophy of mathematics and Robert Brandom’s philosophy of language to demonstrate that inferentialism entails the falsity of Church’s Thesis and, as a consequence, the Computational Theory of Mind. This amounts to an entirely novel critique of mechanism in the philosophy of mind, one we show to have tremendous advantages over the traditional Lucas-Penrose argument.

Keywords

MechanismChurch’s thesisComputational theory of mindDualismInferentialismPlatonismLucas-Penrose argumentBrandomDummettDetlefsenWright

The primary aim of this paper is to demonstrate that inferentialism entails the falsity of Church’s Thesis, and as a consequence undermines the Computational Theory of Mind. This yields an entirely novel anti-mechanist argument, one that has tremendous advantages over extant critiques in the philosophy of mind. As we show, unlike the traditional Lucas-Penrose argument, the inferentialist case is structured such that Gödel’s First Incompleteness Theorem need not be applied to the human mind. Consequently, the argument never has to face the debilitating criticisms that most philosophers take to undermine Lucas-Penrose.

Oxford Inferentialism and Church’s Thesis

We first demonstrate that Dummettian inferentialism entails the falsity of Church’s Thesis. Since Dummett’s is perhaps the canonical inferentialist philosophy of mathematics, one might suspect that the issue is sui generis with respect to the philosophy of mathematics, and hence peripheral to inferentialism more broadly construed. Thus, after initially presenting our argument in the context of Dummett’s inferentialist anti-realism about mathematics, we go onto show that Brandom’s recent presentation of inferentialism as a broader tendency in the philosophy of language is also inconsistent with Church’s Thesis.

a.—Dummettian inferentialists must hold that the human mind can effectively enumerate the set of truths of elementary arithmetic. To establish this, we need only appeal to the three central notions of Dummettian anti-realism: verificationism, molecularism, and the manifestation requirement. Verificationism is simply Dummett’s Heyting-esque equation of truth with idealized provability.

The manifestation requirement is Dummett’s view that our ability to grasp the meaning of a mathematical claim is equivalent to our ability to recognize a proof of it. In The Taming of the True Neil Tennant states the requirement in this manner.

(wpM) For all Φ that the speaker understands: if the condition for the truth of Φ does obtain, then the speaker should be able, if given the opportunity to inspect any truth-maker for Φ, to recognize that the condition for Φ’s truth obtains, or at least be able to get himself into a position where he can so recognize; but if the condition for the truth of Φ does not obtain, then the speaker should be able, if given the opportunity to inspect any truth-maker for Φ, to recognize that the condition for Φ’s truth does not obtain, or at least be able to get himself into a position where he can so recognize (Tenant 1997, 202).

Verificationism equates the truth of mathematical claims with provability, and the manifestation requirement equates our understanding of mathematical claims with the ability to recognize such proofs. Verificationism and the manifestation requirement are intertwined insofar as it is the manifestation requirement that motivates verificationism for Dummettians.1
‘Molecularism’ designates the anti-holistic bent to Dummett’s program. For Dummettians, this anti-holism has fairly strong consequences. For example, when discussing the canonical proofs mentioned in the statement of the manifestation requirement, Tennant writes:

As our first technical corollary giving precise expression to the foregoing, we have the so-called subformula requirement: that every sentence featuring in a normal proof of Q from P1, …, Pn should be a sub-sentence of Q or of one of P1, …, Pn. As our second technical corollary we have the conservative extension requirement: that the rules for the operators actually occurring in the argument ‘P1, …, Pn, so Q’ should suffice for its proof, should the argument be valid (Tenant 1997, 318).

For the purpose of our argument, this conservative extension requirement is the important consequence of molecularism. Its justification as a general inferentialist principle will be addressed in the course of our discussion.

Now, with these three notions in mind, consider the following procedure. Enumerate all possible finite sequences of formulas of first-order arithmetic: S1, S2, S3…. It is a trivial result of computability theory that this can be done for any language with a decidable syntax. Then, take any finite sequence ‘Sn’ and call the last formula of SnPn’. Note that it may just so happen that Sn is a proof of Pn, or it may not be the case that Sn is a proof of Pn. Start at S1 and move down the enumeration in the following manner: if Sn is not a proof of Pn then move to Sn+1. Say that—for the first time—we come to an Sn that happens to be a proof of Pn; call this Pne1’. It is a direct result of the manifestation requirement that one who understood number theory could determine for any finite Sn in the enumeration, whether it is a proof of Pn. So, continue until the next such occurrence: call this Pne2’. Each time a new ‘e’ comes along, label it ‘ei+1’ where ‘i’ is the subscript of the last e found. All the while, enumerate these e’s: e1, e2, e3, … Since the only Pn’s that make it onto the ‘e-list’ are arithmetic truths (as these are the Pn’s such that Sn is a proof of Pn, and for the anti-realist, truth is provability), this procedure will ultimately yield an effective enumeration of the set of truths of elementary arithmetic. That is, the Dummettian inferentialist,2 committed as she is to the manifestation requirement, must hold that the human mind can effectively enumerate the set of truths of elementary arithmetic.

b.—We now pause to consider some possible objections. Perhaps one could charge that the human mind’s enumeration of arithmetic truths is incomplete; what if there are correct proofs not listed on our enumeration of finite strings S1, S2, S3…? If that were the case, then the list of arithmetic truths generated by the human mind would be a subset of the set of all arithmetic truths, and there would be no reason to conclude that the human mind possesses a procedure by which all of the truths of elementary arithmetic can be enumerated.

While this objection seems plausible, it is not one open to the inferentialist. Assume that we have P, a true sentence not listed as the conclusion of a valid proof occurring in the enumeration of finite sequences S1, S2, S3…. There are only two possible reasons for why P lacks such a proof, the first being that P is a true yet un-provable statement. But, as noted, a core tenet of Dummett’s inferentialism is the verificationist identification of truth with provability. For the Dummettian, one is a Platonist precisely to the extent that one countenances unprovable truths. For whatever it is that makes such truths about infinite totalities true is going to be in some sense beyond the finite and imperfect world humans perceive.

The second possibility for why P might lack a proof is that all proofs of P involved resources outside of elementary arithmetic, and thus sentences not in the enumerated strings. This tack would, however, require accepting an extremely implausible form of holism. Suppose that all of these proofs of P involve state-of-the-art topology. Since, for Dummett, the ability to recognize a proof of P just is an understanding of the meaning of P, we’d now have a situation where a number theorist can’t come to an understanding of what P—a claim in elementary arithmetic—means without being able to recognize an arcane proof in topology, an area in which our number theorist has never worked. It is because of this implausible consequence that the manifestation requirement requires a molecularism strong enough to entail the conservative extension requirement. To imagine that a number theoretic proof is only provable in topology is to imagine that the language of topology does not conservatively extend the language of number theory. But this, when coupled with the manifestation requirement, entails that merely understanding a sentence of number theory requires understanding arcane areas of topology. This is extremely implausible, potentially entailing that nobody understands anything.

Further, such holism is deeply antithetical to the entire Dummettian project. For one can argue that holistic account of grasp of meaning is the only way to justify the use of the traditional truth conditional semantics over Heyting-style proof conditional semantics (Dummett 1991). That is, Dummett holds that the only way one can affirm the manifestation requirement without being a verificationist is by accepting an implausibly strong form of holism.

One might consider weakening the manifestation requirement itself, such that a violation of molecularism wouldn’t entail such strong holism. Perhaps the inferentialist could hold that one only needs to understand a subset of important or canonical proofs (i.e. the ones with conclusions that make their way onto e) in order to understand all true sentences of elementary arithmetic. Since, on this line, the human generated list of truths e would differ from the complete list of arithmetic truths, it would not follow that number theory is effectively enumerable.

This too fails to be a viable option for the inferentialist, however; such a weakening of the manifestation requirement does too much damage. The core tenet of inferentialism is the identification of truth with some form of provability; weakening the manifestation requirement so that understanding of mathematical claims only requires the ability to recognize a subset of proofs of those claims splinters truth and provability apart to such a degree the Dummettian is left with no reason to equate truth with provability. In fact, we conjecture that this is why Dummett sees holism as the only salvation for the non-inferentialist.

In short, the procedure for listing truths of arithmetic delineated above, when combined with three key Dummettian premises (verificationism, molecularism, and the manifestation requirement) leads to the following conclusion: the Dummettian inferentialist must hold that the human mind can effectively enumerate the set of truths of elementary arithmetic.

c.—Now, let us leave the human mind for a moment and turn our attention to Turing Machines and three results in logic: Craig’s Result, Gödel’s first Incompleteness Theorem, and Church’s Thesis.3 Making sense of these requires first explaining some relevant concepts. A theory—or set of sentences—T is axiomatizable if and only if there is a decidable subset of T whose consequences (in the language of T) are just the theorems of T. A set is recursive if, and only if, its characteristic function is recursive, i.e. given any x, a value of 0 or 1 can be assigned to x depending upon whether or not x does (1) or does not (0) belong to said set. A set is recursively enumerable if it is the range of some recursive function. That is, if there is a recursive method through which the members of the set (and perhaps not the objects that do not belong to that set) can be listed; or if one can develop a computer program through which the members of the set can be listed, then the set is recursively enumerable.

Now, Craig proved that if any arbitrary set of sentences is recursively enumerable, then it is axiomatizable. Where our quantifiers range over sets of sentences, we can represent this formally as ‘∀(x)(r.e.(x) → a.(x))’.

Next, note that Gödel’s First Incompleteness Theorem entails that the set of truths of elementary arithmetic is not axiomatizable (∼a.(n)). The set of true statements of elementary arithmetic cannot be captured by any one (consistent) set of axioms, as there will always be sentences—paradigmatically including the relevant Gödel sentence for the axiomatization in question—that are true yet unprovable from the purposed axiomatization.

Church’s Thesis has numerous equivalent formulations. We will be concerned with two: (1) if any arbitrary set of sentences is effectively enumerable, then it is recursively enumerable (∀(x)(e.e.(x) → r.e.(x)), and (2) if something is intuitively computable, then it is Turing Machine computable.

Recall that the three Dummettian notions discussed above entail that the human mind can enumerate the set of truths of elementary arithmetic (e.e.(n)). Assume Church’s Thesis is true. Then, if a set of sentences is effectively enumerable, it is recursively enumerable (∀(x)(e.e.(x) → r.e.(x)). So, it follows that the set of truths of elementary arithmetic is recursively enumerable (from ∀(x)(e.e.(x) → r.e.(x) and e.e.(n), obtain r.e.(n) by ∀ and → elimination). Further, by Craig’s Result (∀(x)(r.e.(x) → a.(x)) it follows that the set of truths of elementary arithmetic is axiomatizable (from ∀(x)(r.e.(x) → a.(x) and r.e.(n), obtain a.(n), again by ∀ and → elimination). Gödel’s First Incompleteness Theorem entails that the set of truths of elementary arithmetic is not axiomatizable (∼a.(n)). A contradiction, therefore one of our assumptions is false. Since Gödel’s First Incompleteness Theorem and Craig’s Result are theorems of logic, we must conclude that Church’s Thesis is the culprit. Thus, through negation introduction we reach the following conclusion: for the Dummettian inferentialist, Church’s Thesis is false.4

Pittsburgh Inferentialism

We now wish to step back and examine inferentialism as a broader tendency in the philosophy of language. This involves a brief explication of a recent influential statements of inferentialism (Brandom 2000).5 We are able to show that adherents of this school must accept the three Dummettian notions utilized above, and as a consequence, must also reject Church’s Thesis.

The easiest way to introduce Brandom’s inferentialism is via appeal to Frege, or rather, the two Freges (early and late) that Brandom delineates. According to Brandom, the early Frege emphasized the notion of inference in an important way; specifically, Frege held that the inferences a claim enters into are what fix the conceptual content or meaning of the claim, insofar as “two claims have the same conceptual content if and only if they have the same inferential role” (Brandom 2000, 50). In contrast, the later Frege—and, following him, the mainstream tradition in logic—privileged the notions of reference, truth, and representation over inference. Brandom wishes to follow the early Frege in holding that a claim’s inferential role is what fixes its content.

Brandom holds that speakers often endorse the inferences that fix conceptual content implicitly, while logic is merely a tool that can be used to make these implicit inferential commitments explicit. He gives as an example the claim “Leo is a lion.” When a speaker makes this claim, she implicitly commits herself to a number of other claims, such as “Leo is a mammal.” If the speaker’s language contains logical vocabulary such as the conditional, then this logical vocabulary can be used to make this implicitly endorsed inference explicit. Then the speaker can explicitly state, “If Leo is a lion, then Leo is a mammal,” thereby bringing what was an implicit inference to the surface.

It is important to note that the inferences that fix conceptual content are not equated with logically valid inferences (Brandom 2000, 52–55). Rather, the inferences that fix conceptual content are termed, following Sellars, ‘material inferences.’ As an example of a material inference, Brandom gives the inference from “Pittsburgh is to the west of Princeton” to “Princeton is to the east of Pittsburgh.” Brandom claims that material inferences, if they are recognized to exist at all, are often seen as derivative of logically valid inferences. Brandom wishes to reverse this practice: he first defines the notion of material inference, and then goes on to define the notion of a logically valid inference in terms of material inferences. For Brandom, a logically valid inference is seen as a good material inference that cannot be turned into a bad material inference through the substitution of non-logical vocabulary.

Pittsburgh inferentialism has a number of other broad theoretical commitments: it is a rationalistic as opposed to empiricist doctrine, it is ‘pragmatic’ as opposed to Platonic (it begins with the notion of concept use and then explains conceptual content from there, as opposed to the other way around), it is holistic [“to have any concepts one must have many concepts” (Brandom 2000, 15)], and so on. Many of these issues can be safely left to the side for our purposes here, while a couple of them will be discussed in more detail throughout the remainder of this paper.

What does concern us here is the degree to which this form of inferentialism endorses the three Dummettian notions discussed above: verificationism, molecularism, and the manifestation requirement. If we can show that Brandom’s inferentialism is in fact committed to all three notions, then it follows that our argument is not a mere consequence of Dummett’s philosophy of mathematics.

A—Brandom makes no secret of the degree to which his inferentialism is influenced by Dummett, whose work, along with Frege and Sellars, he cites as one of his principle influences (Brandom 2000, 45). This influence manifests itself in Brandom’s commitment to close variants of the three key Dummettian notions.

Again, Brandom holds that what imbues a claim with content is its ‘inferential role,’ but he understands this term to encompass a claim’s role as both premise and conclusion in inferences. He writes:

Understanding or grasping a propositional content is here presented not as the turning on of a Cartesian light, but as a practical mastery of a certain kind of inferentially articulated doing: responding differentially according to the circumstances of proper application of a concept, and distinguishing the proper inferential consequences of such application (Brandom 2000, 63–64).

In short, for Brandom, what gives a mathematical claim its content is both (1) what other claims makes the claim true, and (2) what other claims the original claim makes true. Further, our grasp of the meaning of a claim is equated with our ability to recognize both when we are justified in asserting a claim and what follows from the assertion. This view translates into mathematics in the following manner: we grasp the meaning of a mathematical claim when we are able to correctly recognize a proof of the claim, and when we can correctly use the claim as a premise in proofs of other statements.

Recall Dummett’s manifestation requirement: the grasp of the meaning of a claim is equated with our ability to recognize a proof of the claim. So clearly, insofar as Brandom equates our grasp of the meaning of a mathematical claim with our ability to recognize a proof, Brandom endorses the manifestation requirement. However, Brandom adds the additional proviso that we must also be able to use the claim as a premise in other proofs.6

Now, just as our infinite sequence of finite strings S1, S2, S3… contains all possible proofs of a claim, it also includes all possible instances where the claim will act as a premise in a proof. The only change is that such strings of sentences are also relevant to the meaning of the premises, since for Brandom what follows from a claim is also determinative of that claim’s meaning. But this is of no consequence for our argument. As regards the manifestation requirement, it applies to Brandom’s view as well.

B—Likewise, Brandom’s inferentialism is committed to some form of verificationism. He claims that the standard, Platonistic order of explanation was such that one began by assuming “that one has a prior grip on the notion of truth,” and then went onto use that notion of truth “to explain what good inference consists in.” Brandom wishes to reverse this; he starts “with a practical distinction between good and bad inferences,” and then goes onto discuss “truth as talk about what is preserved by good moves” (Brandom 2000, 12). A good move is a good inference, and in mathematics, a good inference is a good proof. So, truth is a derivative of proof.

To approach this from another angle, Brandom states that “understanding … a content is grasping the conditions that are necessary and sufficient for its truth” (158). For Brandom, the content of a claim arises through the inferential relations the claim enters into; thus, to grasp the inferential relations of a claim is to grasp the conditions that are necessary and sufficient for its truth.7 In mathematics, an inferential relation is a proof relation, so, to understand a proof of a claim is to grasp the conditions that are necessary and sufficient for the claim’s truth. Proof is truth; Brandom is committed to Dummettian verificationism.

C—It more difficult to show why Brandom must defend molecularism with regards to logical and mathematical vocabulary. Succeeding here requires digressing into an examination of the tonk logical operator problematic.

The tonk operator (Prior 1960; Belnap 1962) is an operator that has the introduction rule corresponding to ‘or’ and the elimination rule of ‘and’. When tonk is added to a language it creates ‘a runabout inference ticket;’ it can be used to prove any Q from any P. Starting with P, we can use tonk/or introduction to infer P tonk Q, then we can use tonk/and elimination to infer Q.

The inferential role for tonk is well defined, so, the question now arises, “what, exactly, is wrong with tonk?” Prior held that the problem with tonk is that it allows one to deduce false conclusions from true premises; this answer, however, is clearly not open to the inferentialist, as it renders truth prior to inference. Prior’s view implies that to see what the correct inference rules are we have to appeal to the model-theoretic account of logical consequence to ensure that there is no row on the truth table where all of the sentences in the premises are true and the conclusion false. This is obviously not an inferentialistically acceptable option.

So, inferentialists now face the following challenge: show what is wrong with tonk in wholly inferentialist terms. To answer this challenge, inferentialists have adopted the conservative extension requirement, which (again) holds that

when logical vocabulary is being introduced, one must constrain such definitions by the condition that the rule not license any inferences involving only old vocabulary that were not already licensed before the logical vocabulary was introduced, that is, that the new rules provide an inferentially conservative extension of the original field of inferences (Brandom 2000, 68).

Another way to put this is, when one adds a new logical constant to a language, one must be sure that this addition does not lead to any new inferences not involving the new constant, otherwise, one would not have “a conservative extension of the original field of inferences” (Brandom 2000, 68). Tonk violates the conservative extension requirement, so, in adding the requirement, inferentialists have defeated tonk on their own terms.

Here molecularism enters, since the conservative extension requirement implies that a logical truth should just depend upon the meanings/inferential rules of the logical operators in the sentence expressing that logical truth.8 Remember that Brandom endorses the conservative extension requirement with regard to logical vocabulary to avoid the problem with tonk; so he accepts a molecularist philosophy of logic.

So, clearly Brandom must accept molecularism for logic, but need he for math? This is a complicated issue. Intuitively he does, as inferentialism seeks to characterize math without recourse to demonstrative inferences (without non-linguistic evidence such as a red blotch justifying, “This is red”). It is in virtue of this avoidance that inferentialism manages to avoid mathematical Platonism. For the inferentialist, there are no objects in Plato’s heaven by which we could secure demonstrative inference (i.e. the soul pointing and saying, “This is omega”). But then strictures about logical truths should (as they do for Dummettians) apply to mathematical truths.

If one finds that stricture uncompelling, then that is O.K. as our master argument applies just as well to the set of second-order logical truths, which are also provably non-axiomatizable. Instead of starting with an enumeration of sequences of sentences of first-order number theory, we start with sequences of sentences of second-order logic. Then everything proceeds as before.

So, Brandom is committed to the manifestation requirement, verificationism, and molecularism for a non-axiomatizable discourse (minimally, second order logic; arguably mathematics as well). It follows that one can then combine these considerations with the effective procedure delineated above, and then simply re-run the argument we used to falsify Church’s Thesis on Dummett’s anti-realism. In short, both Dummett’s anti-realism and Brandom’s inferentialism entail the falsity of Church’s Thesis.

Inferentialism and the Computational Theory of Mind

Above, we demonstrated that inferentialists must reject Church’s Thesis. Here, we show that as a consequence they also must hold that the mind can in-principle do things that no computer can.

Recall that we established that inferentialists must hold that the human mind can effectively enumerate the set of truths of elementary arithmetic (e.e.(n)). This set is provably not-recursively enumerable. There is no contradiction here as long as Church’s Thesis is false.

Now note that by standard theorems of logic, the form of computability revelant to the Computational Theory of Mind (von Neumann computability) is equivalent to a host of other concepts such as Turing machine computability, Lambda definability, and recursivity. As a consequence of the equivalence there is no von Neumann computer program to enumerate the members of a non-recursively enumerable set, such that any arbitrary member of that set will at some point show up on the enumeration.

As a consequence of inferentialism, the human mind can enumerate the set of truths of first order arithmetic. But, again, a consequence of Gödel’s First Incompleteness Theorem, computers cannot do this very thing. Therefore, the failure of Church’s Thesis for the inferentialist is exactly the failure of the Computational Theory of Mind.

We should note now that we take this to be “friendly ammendment” to the package of views one gets from Dummett and Brandom. The quotes which open this paper make clear Dummett’s discomfort both with Church’s Thesis and the Computational Theory of Mind, and we conjecture that something like the above proofs moved him. Brandom’s anti-naturalism is notorious (Dennett 2009). While he has not suggested that the failure of Church’s Thesis protects human dignity from the creeping depredations of Naturwissenschaft, the results of this paper show how he could.

Lucas-Penrose, Putnam, and Benacerraf

Our argument that inferentialists must reject both Church’s Thesis and the Computational Theory of Mind is entirely novel. To show just how strong it is, we must contrast it with extant “Gödelian” anti-mechanist arguments. It is not difficult to establish that none of the debilitating problems faced by those arguments are faced by ours.

On the face of it, the Lucas-Penrose argument seems unproblematic. In The Emperor’s New Mind he states:

whatever…algorithm a mathematician might use to establish mathematical truth…there will always be mathematical propositions, such as the explicit Gödel proposition (for the formal system associated with that algorithm), that his algorithm cannot provide an answer for. If the workings of the mathematician’s mind are entirely algorithmic, then the algorithm…that he actually uses to form his judgments is not capable of dealing with the (Gödelian) proposition…constructed from his personal algorithm. Nevertheless, we can (in principle) see that (that proposition) is true. That would seem to provide him with a contradiction, since he ought to be able to see that also. Perhaps this indicates that the mathematician was not using an algorithm at all (Penrose 1989, 538–539).

The basic idea is clear: if we were Turing machines, we’d have the same abilities and limitations of Turing machines. We don’t. So we’re not! Mechanism is false, and cognitive science, symbolic artificial intelligence, the philosophy of mind, and the relevant areas of psychology need to find a new foundation. Of course, there are severe problems with the Lucas-Penrose version of this argument.

The first canonical criticism (Putnam 1960) of this is simple and elegant. As is well known, Gödel’s Incompleteness Theorem is only applicable to consistent formal systems. It follows that if we want to apply Gödel’s Incompleteness Theorem to the formal system that the human mind is computing, then we must establish that the human mind is consistent. Unfortunately for the anti-mechanist though, Gödel’s Second Incompleteness Theorem demonstrates that the consistency of a formal system cannot be established from within the system itself, so, if we are Turing machines implementing a formal system, we can never establish our own consistency.

It is generally recognized that Putnam’s criticism of Lucas-Penrose is devastating, and much of the literature surrounding Lucas-Penrose centers on the consistency issue. As we show below, Lucas tried to overcome this criticism (which even he calls ‘serious’) on several occasions, while Putnam has reiterated the criticism (most recently in his 1995 review of Penrose’s Shadows of the Mind). Others who utilize, remark on the severity of, or try to defeat Putnam’s criticism include Mortensen, Landau, Irvine, Bowie, and Hutton.

The second canonical criticism (Benacerraf 1967) is severe as well. The construction of a Gödel sentence is a rather complicated affair; obviously, if one is going to infiltrate a formal system to construct an undecidable sentence one needs to have a firm grasp of the system in question. Benacerraf points out that the formal system the human mind implements might be so complex that we just simply may never obtain insight into its specific character to the degree needed to construct our version of the Gödel sentence. We created formal systems such as the one used in the Principia, so we can understand it to the degree needed to construct its Gödel sentence, and we can see the truth of its Gödel sentence, but even so, it doesn’t follow that we can see the truth of, let alone construct, our own Gödel sentence. That is, the best Lucas-Penrose can muster is a disjunction:

either no (formal system) encodes all human arithmetical capacity—the Lucas-Penrose thought—or any system which does has no axiomatic specification which human beings can comprehend (Wright 1995, 87).9

Clearly, this disjunction does not defeat mechanism.

Lucas has formulated a number of responses to Putnam and Benacerraf (Lucas 2002), none of which are particularly convincing. In an effort to sidestep Gödel’s Second Incompleteness Theorem, Lucas points out that Gentzen proved the consistency of arithmetic utilizing meta-mathematical considerations, the moral being that perhaps we can establish our consistency through a similar strategy.10 While Lucas admits that an effort such as Gentzen’s depends “upon the wider set of principles” used being consistent, and that “this assumption can be called into question,” he feels that he has found a ray of hope. However, the fact that the consistency of these meta-mathematical considerations can be called into question is extremely problematic; indeed, this is one of the key morals of Gödel’s Theorems. Of course Gödel did not undermine proofs such as Gentzen’s; rather, he demonstrated that such proofs will fail to serve the foundational purposes to which philosophers might harness them. The consistency of the assumptions used in Gentzen’s proofs is no more secure than that of the systems being proved consistent.

Lucas also argues that if we were inconsistent, we’d assert any Q; we don’t, so we’re consistent. This, too, is unconvincing. A contradiction implies any Q only in classical systems, and it is probable that our actual inferential behavior is best modeled with paraconsistent systems.

Lucas’ reply to Benaceraff is not conclusive either. Lucas argues that the burden of determining what kind of Turing machines can capture the programs that we instantiate lies on the mechanists’ shoulders, not his. Lucas charges that mechanists don’t tell us much about the specific Turing machines that could serve as a representation for the program he instantiates, yet, he “cannot be just an abstract idea of a Turing machine” and “although Benacerraf pleads ignorance, it must in principle be knowable which machine it is that purports to represent me.” In short, Lucas sees nothing that would, in principle, prevent us from obtaining the necessary grasp of our own formal specification to the degree needed for the Gödel sentence construction, and he doesn’t feel that it is his responsibility to provide the details of such a formal specification.

Perhaps Lucas is right: maybe somehow we could not only come to know what specific Turing machines could represent us, but also somehow obtain a strong enough understanding of the machines that we could perform the Gödel sentence construction, recognize its truth, and thereby defeat mechanism. And perhaps Lucas-Penrose could even overcome Putnam’s criticism somehow. However, this is speculative: it would be far better if one could formulate an anti-mechanist argument with fewer ifs.

For our purposes, the most important moral of this discussion is that neither the Putnam nor the Benacerraf criticism applies in any manner to our Gödelian anti-mechanist argument.

There are two senses (not sufficiently distinguished in the literature) in which a computing device can be said to have a Gödel sentence. In the first sense, the program can itself be represented as an axiomatic theory, with inputs being further premises and outputs being conclusions. Since theories canonically representing such programs must be rich enough to encode recursive functions, they have Gödel sentences. So theories encoding programs have Gödel sentences. To the extent that human beings are digital computers, and minds are programs, it will be possible (relative to a method of axiomatically representing the programs for the computer) to construct Gödel sentences for a human mind.

The second, entirely different, sense in which a computing device might be said to have a Gödel sentence is the sense in which that device is capable of enumerating a set of truths axiomatized by a theory. If the theory is of the right sort (and if it is rich enough, consistent, and such that a digital computer can enumerate the full set of truths) then a Gödel sentence can be generated from an axiomatization of a theory for the enumerated sentences. As with the previous sense of “applying Gödel’s theorem to the mind,” there will be different Gödel sentences depending upon which of the syntactically distinct axiomatizations are chosen for the theory the mind can enumerate.

What is essential is that our anti-mechanist argument does not “apply” Gödel’s First Incompleteness Theorem in either of these problematic senses. In our argument, Gödel’s Theorem is only used to get a contradiction when paired with Church’s Thesis and the claim that there is an effective enumeration of first order arithmetic. But this is just a fact of logic. On the other hand, both Putnam and Benacerraf’s criticisms concern the sense in which people or Turing machines can be said to “have” Gödel sentences (Putnam) and whether they could know what they are even if we did (Benacerraf). Though our argument deserves the sobriquet Gödelian, neither of the canonical criticisms of anti-mechanist Gödel arguments apply.

Wright’s Intuitionistic Version of Lucas-Penrose

In a recent paper (Wright 1995) Crispin Wright argues that Benacerraf’s criticism of Lucas-Penrose is misguided, and that an intuitionist can overcome Putnam’s objection. Here we briefly remark on Wright’s first argument before focusing on his second. Wright holds that in the mechanist versus anti-mechanist debate, we can either discuss actual, real world, limited human minds or idealized human minds that can, in principle, do anything an actual human mind could do if the actual human mind’s abilities and efforts could be finitely extended to the necessary degree. Wright claims idealized human minds should be at issue in the debate, as obviously the output of any actual mind will be finite and easily capturable by an algorithm/Turing machine. Further, upon realizing this, Benacerraf’s objection comes to be seen as misguided; it is possible for us, in principle, to finitely extend our capacities to the degree needed to survey and comprehend any axiomatized formal system in a manner needed for the Gödel sentence construction. Consequently, Benacerraf’s second disjunct is not needed. Again, we point out that even so, arguing that we can theoretically, in principle, understand any Turing machine representation of us to the degree needed for the Gödel sentence construction and actually carrying out the construction are two very different things; Lucas-Penrose would be more convincing if it could do the latter, if it depended upon fewer ‘ifs’.11

In his attempt to overcome Putnam’s criticism, Wright first recounts the traditional intuitionistic account of negation (as first characterized by Heyting and later Dummett): “The negation of A is demonstrated by any construction which demonstrates that a contradiction could be demonstrated if we had a demonstration of A.” If any proof of A would lead to a contradiction, then we can conclude A. Wright then makes some further assumptions about our system (‘S’) before proceeding: (1) S is intuitionistically endorsed in the sense that no proof in S violates the principles of intuitionism, and (2) S is computationally adequate such that S is of sufficient strength to do all elementary arithmetic computations. As Wright points out, if these assumptions didn’t hold in a given system, no intuitionist would want to deal with the system to begin with.

Wright countenances the construction of the Gödel sentence for S. He then argues that one can utilize the intuitionistic account of negation to establish G’s truth without having to assume the consistency of S. Basically, if we assume that we have a proof of G. (“I am not provable”), this leads to a contradiction, so, by the intuitionistic account of negation (whereby we can conclude A if an assumption that we have a proof of A leads to contradiction), it follows that G is true and therefore unprovable. It then follows that an intuitionist supporter of Lucas-Penrose can now evade the need to establish the human mind’s consistency, as at no point did Wright need to appeal to S’s consistency to establish the truth of G.

All this being said, Wright’s argument has been convincingly refuted (Detlefson 1995). Michael Detlefson criticizes Wright for intermixing formal with informal arithmetic in his proof, for being unclear at times, and for making an invalid inference at one point (where Wright concludes that the Gödel sentence is provable in S). A large portion of Detlefsen’s article is devoted to a reformulation of Wright’s argument so that it avoids these problems; we’ll move past this aspect of Detlefsen’s critique to focus on aspects that are more germane to our current effort.

Detlefsen shows that even if Wright’s argument goes through that it would not refute mechanism. Rather, the best Wright’s argument can produce is a “stand-off” between the (intuitionist) mechanist and (intuitionist) anti-mechanist. To show why, Detlefsen first claims that the mechanist has one responsibility as regards intuitionism: to show that there is every body of intuitionistic thought is formalizable.

Detlefsen then goes onto examine the relationship between I (intuitionist arithmetic) and S (the formal system in which the Gödel sentence is produced) in Wright’s argument. S was held to be (by Wright) a formal system with an intuitionist logic, computationally adequate and of sufficient strength. Then, I was characterized in terms of the following three properties (1) every proof of S is an I demonstration, (2) I is intuitionistic, and (3) I can prove S’s Gödel sentence. Wright’s arithmetic system I is an outgrowth of the formal system S, it is based on S. As a consequence, the best Wright’s argument can demonstrate is that any formal system that can purport to represent I cannot be this particularS, and not—as Wright wishes—that there is no formal system can formalizes I. The structure of Wright’s argument is such that the generalization Wright wants to make (i.e. noS can formalize I) simply cannot be had. Each time Wright defeats one particular S, the mechanist can simply produce another S that is complete with respect to I. Then Wright can then demonstrate that the new S is incomplete, the mechanist can again respond, and the process can continue ad infinitum.

The moral here is that, just as our argument never has to face the two most serious criticisms of the original Lucas-Penrose argument, our argument also does not have to face Detlefsen’s criticism of Wright’s argument. Again, this is because our argument does not in any way involve finding actual or idealized peoples’ Gödel sentences. As with Putnam and Benacerraf, Detlefsen’s critique finds no purchase.

Platonistic Turning Machines and Intuitionistic Disembodied Minds

In conclusion, we’d like to note that combining the fact that Dummettian inferentialism entails the falsity of the Computational Theory of Mind with Dummett’s claim that his inferentialism is the only (or even just best) alternative to Platonism leads to bizarre conclusions. So first assume for the sake of argument that Dummett’s inferentialism vs. Platonism dichotomy is legitimate. We have shown above that Dummettian inferentialism entails the falsity of Computational Theory of Mind. (inf. → ∼C.T.M). It follows from this that if the Computational Theory of Mind is true, then inferentialism is false, (C.T.M. → ∼inf). Now, assuming Dummett’s dichotomy (inf. ∨ Platonism), if one wants to hold the Computational Theory of Mind as true, one must become a Platonist in the philosophy of mathematics (this argument instances the schema (R → ∼P), (PQ), R |- Q). So, by conditional introduction, we have C.T.M. → Platonism. Strangely, the Computational Theory of Mind, a theory that supposedly demystified the workings of the human mind, entails the truth of a view that renders mathematical knowledge and reality mysterious—Platonism. Turing machines are Platonists.

Must one even assume the truth of Dummett’s dichotomy here? Assume that inferentialism and Platonism are merely the most probable depictions of mathematical knowledge and reality; perhaps there is a third alternative, say ‘x?’ Even still, the fact that the Computational Theory of Mind entails the falsity of inferentialism implies that if the Computational Theory of Mind were true, then this would produce further evidence for Platonism, insofar as this would eliminate the inferentialism disjunct, thereby raising the probabilities of ‘Platonism’ and ‘x’.

Finally, the Computational Theory of Mind is usually seen as the most plausible form of materialism in mind. But then one could plausibly posit the following dichotomy: C.T.M. ∨ dualism. Now, if true, inferentialism, since it entails the falsity of Computational Theory of Mind, would provide evidence for dualism.

The conclusions that Turing machines must be Platonists and that intuitionists must be dualists are so extravagant that these arguments in their favor might be viewed as reductio proofs of the dichotomies involved. Nonetheless, almost all philosophical presentations of the relevant debates contrast inferentialism and Platonism on the one hand and either the Computational Theory of Mind or dualism on the other. Therefore, our argument does show that most philosophers should conclude that inferentialism raises the probability of, and hence provides evidence for, Dualism, and equally that Computational Theory of Mind provides evidence for Platonism.

One must now admit that metaphysical naturalism has been struck a severe blow. The paradigm naturalistic philosophy of math and logic has been shown to undermine the paradigm naturalistic philosophy of mind, and vice versa. Mystery mongers everywhere should rejoice.

Footnotes
1

Exactly how this must go is much more complicated, and philosophically interesting, than would first appear (Cogburn 2004).

 
2

It must be noted that this argument is independent of issues concerning intuitionism. While the Dummettian intuitionist must accept the conclusion, so must one who (albeit perhaps mistakenly) believes that verificationism, the manifestation requirement, and molecularism can be consistently held by a defender of the universal validity of strictly classical (non-intuitionist) inferences such as the law of excluded middle or double negation elimination.

 
3

Proofs of the relevant lemmas from this section can be found in any good book on logic and computability theory (Boolos and Jeffrey 1989; Epstein and Carnielli 1999).

 
4

If it weren’t for the role molecularism plays, our argument might be considered to be merely an interpretation of what Dummett seemed to have in mind in the epigram at the beginning of this paper. Dummett seems to be asserting that the manifestation requirement is inconsistent with Church’s Thesis. However, we’ve shown this not to be true. Rather, the manifestation requirement, verificationism, and molecularism are jointly inconsistent with Church’s Thesis. Also note that it is not at all clear to many contemporary intuitionist anti-realists that Church’s Thesis is false. In fact, two noted arguments for intuitionism (Dragalin 1988; Tenant 1997) use Church’s Thesis as a premise! It is a consequence of the current paper that it is no accident that Dummett’s own arguments (Cogburn 2000; Cogburn 2005d) never do.

 
5

Of course, Dummett’s own views apply outside of mathematics, though their original motivation concerned worries about infinity. By focusing on one each of Dummett and Sellars’ intellectual heirs, we hope that our discussion will further rapprochement between these two schools of thought.

 
6

In the case of cashing out distinctively empirical concepts this is a non-trivial and important addition, as Dummettian harmony does fail in important ways once we leave the realm of logic and mathematics. Explicating this issue would bring us too far afield of the current argument though.

 
7

This comes from Sellar’s classic paper, “Meaning as Functional Classification,” which describes the three types of pattern governed linguistic behavior.

(1) Language Entry Transitions: The speaker responds to objects in perceptual situations, and in certain states of himself, with appropriate linguistic activity.

(2) Intra-linguistic Moves: The speaker’s linguistic conceptual episodes tend to occur in patters of valid inferences (theoretical and practical), and tend not to occur in patterns which violate logical principles.

(3) Language Departure Transitions: The speaker responds to such linguistic conceptual episodes as ‘I will now raise my hand’ with an upward motion of the hand, etc. (Sellars 1980, 423–424)

Elsewhere (Cogburn 2005a) one of the authors ties this to both Dummett and Christopher Peacocke’s views about content.

 
8

Incidentally, this shows why Brandom should not find intuitionistically unacceptable principles to be logical principles. Classical negation does not conservatively extend intuitionistic logic. When one adds classical negation rules to the introduction and elimination rules for the other logical operators, one can derive Putnam’s Theorem (P → Q) ∨ (Q → P), which does not contain negation. Intuitionist negation, on the other hand, does conservatively extend the negation free fragment of the logic.

 
9

As is pointed out by Wright, this ‘disjunctive view’ seems to be the view endorsed by Gödel himself in his famous Gibbs lecture.

 
10

The source material for these quotes (Lucas 2002) is a web page, so page numbers are not given.

 
11

One must note that our proof does rely on the performance-competence distinction. The ability to recognize proofs at the heart of mathematical understanding is an idealized procedure instantiated by understanders. However, as has been shown (Cogburn 2005b) the idealization must be pretty tightly tied to inferential behavior of actual people. Unlike the case of proof recognition, it is not clear how the idealized ability to recognize one’s own Gödel sentence is tied to actual inferential behavior. So there is an important distinction between our appeal and Wright’s. However, conceding the point to Wright is not a large concession. It will be clear that Detlefsen’s critique of Wright does not affect our argument.

 

Copyright information

© Springer Science+Business Media B.V. 2010