After Godel, Mechanism Reason and Realism

Philosophia Mathematica (III) 14 (2006), 229–254. doi:10.1093/philmat/nkj008Advance Access publication January 9, 2006

After Godel: Mechanism, Reason, and Realism in the

Philosophy of Mathematics†

RICHARD TIESZEN*

In his 1951 Gibbs Lecture Godel formulates the central implication ofthe incompleteness theorems as a disjunction: either the human mindinfinitely surpasses the powers of any finite machine or there existabsolutely unsolvable diophantine problems (of a certain type). In hislater writings in particular Godel favors the view that the human minddoes infinitely surpass the powers of any finite machine and there are noabsolutely unsolvable diophantine problems. I consider how one mightdefend such a view in light of Godel’s remark that one can turn to ideasin Husserlian transcendental phenomenology to show that the humanmind ‘contains an element totally different from a finite combinatorialmechanism’.

. . . one of the things that attract us most when we apply ourselves to amathematical problem is precisely that within us we always hear thecall: here is the problem, search for the solution; you can find it by purethought, for in mathematics there is no ignorabimus.

David Hilbert, 1926

Godel’s incompleteness theorems and some of his related results on(un)decidability, the existence of speed-up theorems, and the consistencyof formal systems mark a turning point in the history of mathematicallogic and the foundations of mathematics. Do they also mark a turningpoint in how we should think philosophically about mathematics? Withthe publication of the Collected Works of Kurt Godel, especially VolumesIII, IV, and V, we have a much better understanding of Godel’s ownassessment of the mathematical and philosophical implications of hisincompleteness theorems. One of his most extensive discussions of thesematters is to be found in his 1951 Gibbs lecture ‘Some basic theorems onthe foundations of mathematics and their implications’ [Godel, *1951].The philosophical picture is filled out in more detail in the drafts of his

† Parts of this paper were read at the American Philosophical Association (PacificDivision) meeting back in 1998. I thank Zlatan Damnjanovic, Paul Benacerraf,C. Anthony Anderson and other audience members for comments. Other parts of thepaper are mostly elaborations, based on material that has now been published in Godel’sCollected Works, of the material presented at the APA meeting.

* Department of Philosophy, San Jose State University, San Jose, California 95192-0096 U. S. A. [email protected]

Philosophia Mathematica (III), Vol. 14 No. 2 � The Author [2006]. Published by Oxford University Press.All rights reserved. For Permissions, please e-mail: [email protected]

paper on Carnap [Godel, *1953/9], in the transcribed document ‘Themodern development of the foundations of mathematics in the lightof philosophy’ [Godel, *1961/?], and in letters and other notes that havenow been published. All of this material raises the question whether weneed to revise some of our philosophical thinking about mathematics. Inparticular, do we need to adjust our views of human reason and realism(platonism) in mathematics and logic? I address this question in somedetail in this paper.

In [Godel, *1951] the central implication of the incompletenesstheorems is formulated as a disjunction:1

Either mathematics is incompleteable in this sense, that itsevident axioms can never be comprised in a finite rule, thatis to say, the human mind (even within the realm of puremathematics) infinitely surpasses the powers of any finitemachine, or else there exist absolutely unsolvable diophantineproblems [of a certain type] . . .

This disjunction is described as a ‘mathematically established fact’ andGodel says that it is of ‘great philosophical interest’. The disjunction ispresumably a mathematically established fact because Godel is thinking ofit, in effect, as a reformulation of his incompleteness theorems. Suppose thatthe human mind is a finite machine (in some sense) and there are for it noabsolutely undecidable diophantine problems. As is well known, well-defined or effectively given formal systems can be viewed as Turingmachines, and Turing machines can be viewed as well-defined formalsystems (see also Godel’s remark on this in the 1964 Postscript to [Godel,1934, pp. 369–370]2). Given this relationship between formal systems andTuring machines we can substitute the notion of Turing machine (TM) for‘finite machine’ in this supposition. Thenwhat the incompleteness theoremsshow is that the supposition could not be true. It could not be true that

(1) the human mind is a finite machine (a TM) and there are for it noabsolutely undecidable diophantine problems.

The incompleteness theorems show that if we think of the human mind asa TM then there is for each TM some ‘absolutely’ undecidablediophantine problem. The denial of the conjunction (1) is, in so manywords, Godel’s disjunction. In formulating the negation of (1) Godel saysthat the human mind ‘infinitely surpasses the powers of any finitemachine’. One reason for using such language, I suppose, is that there are

1 A disjunction like this is also stated in other places in Godel’s writings. See, forexample, Volume V of the Collected Works, pp. 80 and 160.

2 Page numbers in Godel references are to the Collected Works.

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denumerably many different Turing machines and for each of them thereis some absolutely unsolvable diophantine problem of the type Godelmentions. So Godel’s disjunction, understood in this manner, ispresumably a mathematically established fact. It is not possible to rejectboth disjuncts.

Three possibilities are left open by the disjunction:

(2) the human mind does not infinitely surpass the powers of any finitemachine (TM) and there are absolutely unsolvable diophantineproblems. That is, the human mind is a finite machine and there arefor it absolutely unsolvable diophantine problems.

(3) the human mind infinitely surpasses the powers of any finitemachine and there are absolutely unsolvable diophantine problems.

(4) the human mind infinitely surpasses the powers of any finitemachine and there are no absolutely unsolvable diophantineproblems.

Godel has some interesting things to say about each of these possibilities butit is clear from some of his later writings in particular that he favors option(4). In this paper I want to consider what it could mean to favor this optionand how one might defend such a choice. I think that in order to defend thechoice we need to reassess the role of human reason and of realism inmathematics. In discussions I had with Hao Wang in the mid-nineteeneighties Wang told me that Godel hoped to use ideas in Husserl’sphenomenology to show, among other things, that the human mind is nota machine. A passage of a draft letter found recently in the Godel Nachlassconfirms Wang’s comments (see [van Atten and Kennedy, 2003, p. 460]).My discussion belowwill be structured by this suggestion (see also Part II of[Tieszen, 2005]). In the final section of the paper I will include a few ideasabout the prospects for solving openmathematical problems that are impliedby or at least compatible with the non-mechanist view of the mind I present.

1. Some Remarks of Godel on Minds and Machines

I begin by noting a few remarks of Godel that reflect his anti-mechanistreading of the incompleteness theorems. These remarks form thebackdrop for much of what I will say below. One such remark alreadyappears in an early paper:

The generalized undecidability results do not establish anybounds for the powers of human reason, but rather for thepotentialities of pure formalism in mathematics . . .Turing’sanalysis of mechanically computable functions is independentof the question whether there exist finite non-mechanicalprocedures . . . such as involve the use of abstract terms on thebasis of their meaning. [Godel, 1934, p. 370]

AFTER GODEL: MECHANISM, REASON, AND REALISM 231

Note that while the procedures Godel speaks of here are finite they areevidently not mechanical because they involve the use of ‘abstract termson the basis of their meaning’. Human reason may be able to use abstractterms on the basis of their meaning whereas this is not a possibility inthe case of pure formalism or mechanism. In the discussion below I willhighlight this idea. A similar theme is sounded in the following passage,although Godel adds that human reason is capable of constantlydeveloping its understanding of the abstract terms:

Turing in his 1937 . . . gives an argument which is supposed toshow that mental procedures cannot go beyond mechanicalprocedures. However, this argument is inconclusive. WhatTuring disregards completely is the fact that mind, in its use,is not static, but constantly developing, i.e., that we understandabstract terms more and more precisely as we go on usingthem, and that more and more abstract terms enter the sphereof our understanding. [Godel, 1972a, p. 306]

There is evidently supposed to be some sense in which minds, but notmachines, are or could be ‘constantly developing’ their understanding of‘abstract terms’. In yet another paper [*1961/?, p. 385] Godel says thatwhat a machine cannot imitate is the activity in which more and more newaxioms become evident to us on the basis of our grasp of the (abstract)meaning of the primitive notions of mathematics.

A letter that appears in Volume V of the Collected Works presents anice summation of his view:

My theorems only show that the mechanization of mathe-matics, i.e., the elimination of the mind and of abstractentities, is impossible, if one wants to have a satisfactoryfoundation and system of mathematics.

I have not proved that there are mathematical questionsundecidable for the human mind, but only that there is nomachine (or blind formalism) that can decide all number-theoretic questions (even of a certain very special kind).

Likewise it does not follow from my theorems that thereare no convincing consistency proofs for the usual mathe-matical formalisms, notwithstanding that such proofs mustuse modes of reasoning not contained in those formalisms.What is practically certain is that there are, for the classicalformalisms, no conclusive combinatorial consistency proofs(such as Hilbert expected to give), i.e., no consistency proofsthat use only concepts referring to finite combinations of

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symbols and not referring to any infinite totality of suchcombinations.

. . . It is not the structure itself of the deductive systemswhich is being threatened with a breakdown, but only a certaininterpretation of it, namely its interpretation as a blindformalism.

. . . [O]ur knowledge of the abstract mathematical entitiesthemselves (as opposed to the formalisms corresponding tothem) is in a deplorable state. This is not surprising in view ofthe fact that the prevailing bias even denies their existence.(Letter to Rappaport [Godel, 2003b, pp. 176–177])

I will be offering, in effect, an extended commentary on these remarks.Godel’s formulation of his disjunction in his 1951 paper and his remarkson the limitations of mechanization in mathematics are typicallyconnected with arguments in favor of realism (platonism), the existenceof abstract concepts, meaning clarification, and mathematical intuition,and against various reductionistic views of the human mind or of humanreason. Mathematical realism is combined with a view according to whichreason transcends mechanism about the mind. A view of reason thattranscends mechanism is required if we are to avoid the position thatthere are mathematical questions undecidable by the human mind.

2. A Broader View of Godel’s Disjunction

What the incompleteness theorems show, as we said, is that it must befalse that

(1) the human mind is a finite machine (a TM) and there are for it noabsolutely undecidable diophantine problems.

This puts certain views about mathematics in a bind. We cannot bemechanists about the mind or, in the appropriate sense, strict formalists,and yet be unmitigated optimists about mathematical problem solving. Onoption (2) listed above it appears that if the human mind is a finitemachine (a TM) then we must say that there are for it absolutely undecid-able problems. But why would we want to accept the consequent of thisconditional in the absence of conclusive evidence for the antecedent? Itake it that nothing definitive can be said at this time in favor of the truth ofthe antecedent.

The disjunction Godel formulates in the 1951 paper is, in my view,profitably read in connection with what he says in his 1961 paper, ‘Themodern development of the foundations of mathematics in the light ofphilosophy’. This is the paper in which Godel describes the developmentof foundational research in mathematics in terms of skepticism,


materialism, and positivism on one (leftward) side and spiritualism,idealism, theology, and metaphysics on the other (rightward) side.Empiricism and pessimism (as a skepticism about knowledge) belong onthe leftward side of this schema and apriorism belongs on the rightwardside. Godel says that, on the whole, the development of philosophy sincethe Renaissance has gone from right to left. This development has alsomade itself felt in mathematics although mathematics has not yielded asquickly to the empiricist zeitgeist that has evolved since the Renaissance.Hilbert’s proof theory is an example of an attempt to bring mathematics,which seems to display a kind of apriorism, into line with the zeitgeist.It is an attempt to preserve, on the one hand, some of the rightwardfeatures of mathematics and the mathematician’s instincts; namely, that aproof of a mathematical proposition should provide a secure grounding forthe proposition and that every precisely formulated yes-or-no questionin mathematics should have a clear-cut answer. It tries to bring this intoline with the zeitgeist, however, by construing mathematics in terms ofconcrete, purely formal, systems.Mathematics is viewed as something likea game of manipulating finite strings of symbols according to preciselyspecified finite sets of rules. This accords with the zeitgeist since oneneeds to refer only to concrete and finite objects in space—objects that aregiven directly to the senses—and purely mechanical operations on suchobjects. What the incompleteness theorems show, according to Godel, isthat Hilbert’s attempt to combine empiricism or materialism with aspectsof classical mathematics will not work. If we do not want to give up on thecertainty that is possible in mathematical knowledge or the idea that forclear questions posed by reason, reason can also find clear answers, thenwe need to find a new combination of the leftward and rightwarddirections. The incompleteness theorems show that Hilbert’s attempt wastoo primitive and tended too strongly in the leftward direction. What weshould try to do, therefore, is not to secure mathematics by proving certainproperties by a projection onto material systems, that is, by a projectiononto physical symbol systems. Rather, we should try to cultivate ordeepen our knowledge of the abstract concepts that lead to setting up thesemechanical systems and, by the same means, to gain insights into thesolvability of all meaningful mathematical problems. In this case wewould not need to give up on Hilbert’s optimism about mathematicalproblem solving.

In an interesting note from sometime in the nineteen thirties Godel saysthat the limitations established by his incompleteness theorems could beinterpreted as (i) establishing that certain problems are absolutelyundecidable or, on the other hand, (ii) only showing that something waslost in the transition from understanding proof to be something thatprovides evidence to understanding proof in the sense of pure formalism[Godel 193?, p. 164]. In his latter writings Godel thinks it is alternative

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(ii) that we should embrace because undecidable number-theoretic sen-tences of the type produced by the incompleteness theorems for a givenformal system are always decidable by evident inferences not expressiblein the given formal system. The new inferences will be just as evident asthose of the given formal system. The result, Godel says, would be that ‘itis not possible to formalize mathematical evidence even in the domain ofnumber theory, but the conviction of which Hilbert speaks [that allmathematical problems are decidable] remains entirely untouched.’ Godelsays in [Godel, 193?] that it is not possible to mechanize mathematicalreasoning. It will never be possible to replace the human mathematicianby a machine, even if we confine ourselves to number-theoretic problems.

The modern mechanist view of mathematics that Godel is addressingin these remarks and in his 1951 disjunction is part of the zeitgeist that isdiscussed in the 1961 paper. It is just a purified formalism. In §6 belowI will have more to say about the zeitgeist and the kind of bias or‘prejudice’ it arguably embodies.

3. From Mechanism (Pure Formalism) to Abstract Objects:The Incompleteness Theorems

How do the incompleteness theorems supposedly lead to these kinds ofconclusions? It is worthwhile to say a little more about this. Hilbert’s earlyidea of proof theory was to show that formalized parts of mathematicswere consistent using only a special theory, let us call it C (for ‘concretemathematics’). C could not be just any theory. What would make Cspecial is the fact that it would possess the kinds of properties needed toinsure reliability or security. It should be finitary and not infinitary, for theworry is that we do not understand the infinite very well and in ourthinking about the infinite we may be led into inconsistencies andparadoxes. It should be possible to understand C as a theory involvingonly concrete and not abstract entities, for abstract entities are supposed tobe mysterious and we should avoid postulating them whenever possible.The concrete entities in this case are finite sign configurations. C shouldbe concerned with what is real and not with what is ideal. Its sentences andproofs should be surveyable in immediate intuition. Immediate intuition isa non-mysterious form of sensory perception.C should not be a creature ofpure thought or pure reason, for the worry is that we may be led by purereason into antinomies and hopeless confusion. C represents the part ofour mathematical thinking that is contentual and meaningful. The contrastis with parts of mathematical thinking that we may regard as purely formaland ‘meaningless’ in the sense that we need not consider their purportedreferences to abstract concepts or infinitary objects.

For formal theories T that contain enough mathematics to make Godelnumbering possible, Godel’s first incompleteness theorem says that if T is


consistent then there is a sentence GT, the Godel sentence for T, such that0T GT and 0T : GT. The second theorem says that if T is consistent then0T CON(T), where ‘CON(T)’ is the formalized statement that assertsthe consistency of T. The first theorem tells us that GT cannot be decidedby T but that GT is true if T is consistent. It does not tell us that GT isabsolutely undecidable. It is only undecidable relative to T. Godelfrequently emphasizes how sentences that are undecidable in sometheories are in fact decided in certain natural extensions of those theories(e.g., by ascending to higher types). T can be, for example, primitiverecursive arithmetic (PRA), Peano arithmetic (PA), Zermelo-Fraenkel settheory (ZF), and so on. The theorems tell us that such formal theories Tcannot be finitely axiomatizable, consistent, and complete. The secondtheorem suggests, generally speaking, that if there is a consistency prooffor a theory T then it will be necessary to look for ‘T0 CON(T), where T is aproper subsystem of T0.

A very likely candidate for C is PRA. PA is arguably less suitable,given the distinctions described in the first paragraph of this section. Inwhatever manner we construe C, however, it follows from the firsttheorem that if C is consistent then the Godel sentence for C cannot bedecided by C even though it is true. It follows from the second theoremthat if C is consistent then C cannot prove CON(C). Given the way wecharacterized C above, it follows that deciding the Godel sentence for Cor proving CON(C) must require objects or concepts that cannot becompletely represented in space-time as finitary, concrete, real andimmediately intuitable. In other words, deciding the Godel sentence for Cor proving that CON(C) must require appeal to the meanings of signconfigurations, to objects or concepts that are in some sense infinitary,ideal, or abstract and not immediately intuitable (see also [Godel, 1972]).Were there no insight into such objects, were we able to proceed only fromwithin C, then we would evidently have to stop with respect to decidingthe Godel sentence for C or obtaining a proof of CON(C). That is, wecould not decide some clearly posed mathematical problem. (I mean wecould not decide it unless the decision were to be made arbitrarily or onnon-mathematical grounds.) However, for some T (e.g., PRA, PA) we infact do have proofs of CON(T) and decisions of related problems. Thehuman mind, in its use, is not static but is constantly developing.

It is worth noting that we can obtain a reasonably good understandingof the sense in which the objects or concepts used in the decisions orconsistency proofs must be abstract, infinitary, and not completelycaptured in immediate intuition. In the case of PA, for example, theconsistency proof requires the use of transfinite induction onordinals < e0, or it requires primitive recursive functionals of finitetype. Thus, some kind of epistemic capacity that takes us beyond C seemsto be required. It seems that cognitive acts of abstraction, generalization,

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idealization, reflection, or imagination are needed if we are not to stopwith respect to solving certain kinds of mathematical and metamathe-matical problems. We need an account of human reason and of evidencethat takes us beyond strict finitism. Intuitionism and some other forms ofconstructivism already recognize cognitive capacities and forms ofevidence that transcend finitism. We also need to countenance rationalactivities, however, in which we rise above intuitionism and theseother forms of constructivism. From the perspective of intuitionism or aconstructivism that is more liberal than finitism, for example, we wouldevidently still have to stop with respect to deciding a clearly posedmathematical problem such as the question whether CH is consistent withthe axioms of ZF. In order to obtain the proof of the consistency of CHwith the existing axioms of ZF it was necessary to take the classicalordinals as given. It was necessary to go beyond the constructive ordinals,and evidently one thereby moves beyond constructivism to a realistposition. In recognizing rational activities that transcend those allowed byconstructivism we obtain a result that would not have been forthcomingotherwise. The classical ordinals cannot be identified with or completelyrepresented by concrete finite sign configurations, nor can they be mind-dependent in the sense that human beings can complete the constructionof these objects individually through step-by-step mental operations.Other kinds of cognitive activities, such as abstraction and meaningclarification, must be involved. The mind-independence of the objectsor concepts involved is further highlighted by the fact that it isimpredicative set theory that is required here and the impredicativityinvolved evidently forces us outside of constructivist bounds. The onlyposition that does not impose restrictions on generalization, idealization,reflection, and the imagination of possibilities of the kind found inconstructivism, predicativism, and related views, would be a rationalistepistemology of some type that is compatible with mathematical realism.Short of preserving consistency, we presumably do not want to imposerestrictions on these kinds of cognitive activities if, for example, we are tohave any hope of confirming or rejecting sentences such as 2@0 ¼ @1.

3

One can point to some clearly posed mathematical problems thatcannot be decided using the resources of C but that can be decided in aprincipled way if we allow forms of abstraction, idealization, reflection,and imagination that take us beyond C. I think that clearly posedmathematical problems in fact are not and ought not to be decided arbi-trarily or on non-mathematical grounds. The existing consistency proofsfor PA, for example, are not arbitrary or non-mathematical. In his 1961paper Godel says that the worries expressed in the first paragraph of this

3 [Hauser, forthcoming] contains further discussion of Godel’s program for set theoryin connection with ideas in phenomenology.


section—the worries that led to the focus on concrete mathematics—wereexaggerated by empiricist skeptics. In particular, the paradoxes of settheory were used as a pretext for a ‘leftward upheaval’. On Godel’sanalysis, the paradoxes were not the result of failing to adhere to leftwardempiricist or materialist strictures. Instead, they resulted from unfoundedgeneralizations or abstractions in set theory, such as dividing the totalityof all existing things into two categories instead of viewing sets asobtained from iterated applications of the operation ‘set of’ to integers orother well-defined objects [Godel, 1964, pp. 258–259].

4. Human Cognitive Functions and Turing Computable Functions

It may not be possible to give a mathematically precise definition of theexpression ‘human cognitive function’. On the basis of our remarks on theincompleteness theorems in §3, however, we can extract the claim that

(*) in human cognitive functioning in mathematics there isa finite, partial, and mediate awareness of abstract finitaryand infinitary objects or concepts, including the meaningsof mathematical expressions.

(I use the term ‘mediate’ here to distinguish the form of awarenessinvolved from Hilbert’s ‘immediate intuition’.) Having made theseobservations about human cognitive functioning, we note that

(**) Turing computability must involve only completely andexplicitly given finite concrete sign configurations inspace-time and purely mechanical (finitary) operationson these objects.

Thinking again of elementary formal systems in terms of Turingcomputability, it is possible to argue that if the claims about humancognitive functioning at (*) are correct and the claims about Turing com-putability at (**) are correct, then not all human cognitive functions areTuring-computable functions. This would mean that there are, asGodel suggests, finite but non-mechanical procedures involved inmathematical decisions that make use of the abstract meaning ofmathematical terms.

If we place any theory T that could count as mechanistic in the schemeof the argument in §3 then, on the basis of the incompleteness theorems,there will be clearly posed but undecidable propositions that are in factdecidable by us on the basis of our understanding of the abstract meaningof the terms involved. This understanding can subsequently be expressedin finite concrete sign configurations that can be added to the existingformalism or machine. It is not known what the limits on this kind of

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insight are. It is however possible to make these kinds of decisions (e.g., todecide GPA) without being omniscient.

5. From Mechanism (Pure Formalism) to Awareness ofAbstract Objects: Remarks on Reason in Mathematics

The question I would like to focus on now is this: how is (*) possible?One might ask how there could be finite procedures involved inmathematical decisions that constantly make use of the abstract meaningof mathematical terms, where this is not reducible to (**). The answer tothis question, I think, requires a central idea in phenomenology that hasappeared in some of the anti-mechanist arguments that have not proceededfrom Godel’s theorems: the idea that human consciousness exhibitsintentionality.

Intentionality is a broader notion than the common notion of intention.It refers to the fact that consciousness, including mathematical con-sciousness, is typically of or about something. We are, especially inscientific thinking, typically directed toward something or other in ourawareness. Once we take this notion of the directedness of consciousnessseriously we see that there are many forms of conscious directedness.Sensory perception is only one form of directedness and it is certainly notthe only form involved in scientific thinking. Let us start with the positionthat in sensory perception we are directed toward sensory objects andstates of affairs, in mathematical thinking we are directed towardmathematical objects and states of affairs, in imagination we are directedtoward imaginary objects and states of affairs, and so on, and let us notautomatically run these types of directedness into one another. It shouldbe a matter for further investigation how they might be related to oneanother.

On the theory of intentionality that I will present here it is not requiredthat there be an object or state of affairs of a conscious act in order for theact to be directed even though, for certain kinds of acts, there frequentlyare such objects or states of affairs. What makes the directedness ofthe conscious act possible is not the existence (or ‘intentional inexistence’or ‘non-existent being’) of an object but rather the ‘content’ or ‘meaning’associated with the act. Let us call the contents or meanings expressed bypredicates concepts.4 Suppose I believe that Pa. Here we have a conscioussubject (‘I’), a type of consciousness (belief), and the content or meaning

4 In [*1961/?, p. 383] Godel says, without much elaboration, that Husserl’s phenom-enology would have us direct our attention onto our own acts in our use of concepts,onto our powers of carrying out our acts, and so on. It is this kind of approach thatGodel thinks is promising. What I do in this paper is to provide some elaboration on thiskind of view.


of the belief, expressed by ‘Pa’. In order for the belief to be justified wewould need to have evidence for it, that is, we would need to haveevidence for the object or state of affairs toward which we are directed.This evidence can come in different degrees and types. In what followsI will call the type of consciousness (e.g., believing, perceiving,remembering, judging, imagining) the ‘thetic character’ of the consciousact. Thus, we might have acts with the same thetic character but differentcontents, or acts with the same content but different thetic characters.

The kind of language employed here is all mind language. It is verydifferent from brain language. Brain states, viewed as purely physicalphenomena, do not exhibit intentionality. Physical states are not directedtoward anything. In the philosophy of mind, intentionality is oftenconsidered a distinctive mark of mental phenomena. The logic of mentalphenomena is clearly different from the logic of purely physicalphenomena. We can see how this difference is marked, for example, inthe failure of certain principles of extensional logic (e.g., substitutivitysalva veritate and existential generalization) when we try to apply theseprinciples to mental phenomena.

It is interesting to note that Godel once suggested that the argumentthat mental procedures are mechanical procedures is valid if one assumesthat (i) there is no mind separate from matter and (ii) the brain functionsbasically like a digital computer (see [Wang, 1974, p. 326]). Godelevidently thought at this time that (ii) was very likely but that (i) was aprejudice of our time that might actually be disproved. One of my worriesabout Godel’s suspicions concerning (i), especially given the paucity ofhis remarks on the subject and some of his comments about religion, isthat he perhaps accepts a kind of substance dualism about mental andphysical phenomena. I would like to avoid substance dualism. In anycase, it is not clear that one needs to be committed to it in order to makethe point about minds and machines. In relation to (i), I will notventure beyond the following separation: human consciousness exhibitsintentionality and brain states do not exhibit intentionality.

Now let us focus on the directedness involved in ordinary senseperception. This is what Hilbert hoped to rely upon in his appeals to‘immediate intuition’, and the mechanist view of mind is closely related toHilbert’s approach in the following way. We can clearly be directedtoward concrete finite sign configurations (as tokens), and we can clearlymanipulate these objects in a purely mechanical way on the basis of finitesets of rules. We can do this without needing to understand the meaning(if any), reference (if any), or origin of the sign configurations. Not onlycan we do this but we have obviously developed very powerful technicalextensions of this capacity in our existing computers which, incidentally,no one now regards as intelligent. We can offload this capability andhave our machines execute the operations involved. What we have been

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able to do is to take a capacity we have and amplify and extend it in ourcomputer technologies.

We can be directed toward concrete finite sign configurations and wecan perform purely mechanical manipulations on such objects. This is oneform of conscious directedness for human beings. If we go more deeplyinto the phenomenology of ordinary human sense perception we can makea number of additional observations. One important observation we canmake is that our sensory perception is of concrete objects but it appearsthat a kind of prereflective or automatic abstraction is involved even at thelowest levels of the sensory perception of objects. In our experience of theworld there is a constantly changing flow of sensory input but in ordinarysensory perception we are not directed toward this sensory material. Weare directed instead toward a particular object that is experienced asidentical through this constantly changing flow of sensory input. There isone object that is ‘formed’ or ‘synthesized’ in a non-arbitrary way out ofthe multiplicity of data reaching our senses. Here we can already speak ofa kind of abstraction that takes place passively or automatically in ourexperience. The notion of abstraction here can still be kept fairly simple:‘abstraction’ simply means ‘not attending to’ something. Thus, not toattend to the complex flow of sensory material itself in sensory experienceis to abstract from it. We are not directed toward it even though it is part ofthe concrete whole of the experience. To perceive a sensory object, wemight say, is already to abstract from a difference (or some differences).This might be what Godel had in mind when he wrote about how even inthe case of our experience of physical objects we form our ideas of objectson the basis of something that is immediately given. One of his remarksabout forming our ideas of objects [Godel, 1964, p. 268] is that this isthe function of a kind of synthesis, of generating unities out of manifolds(e.g., one object out of its various aspects).

The role of abstraction in sensory experience is very important but thehuman capacity for abstraction certainly does not stop at this point. Onevery quickly comes upon more active reflective forms of abstraction of thesort found in scientific thinking. It is clear that sensory perception is justone type of conscious directedness. Scientific thinking is arguably built upfrom sense perception on the basis of goal-directed cognitive activitiessuch as abstraction, reflection, generalization, idealization, imaginationof possibilities, formalization, the use of analogies, and other ‘higher-order’ activities of reason. These cognitive activities are indispensableto scientific thinking. We can say that scientific thinkingis founded on sense perception but is not itself sense perception. Thethetic character and the content of conscious acts involved in scientificthinking will be different from the character and content of acts of meresense perception. This is part of what is involved in distinguishing reasonfrom mere sensation.


One of the most important principles in the theory of intentionality iswhat I will call the intentional difference principle:

(IDP) every content (meaning) yields, prima facie, a different kind ofdirectedness.

We are to take the directedness of consciousness just as it presents itselfand not as something else. If I express the content of an act ofconsciousness as ‘x is a primitive recursive functional’ and the content ofanother act as ‘x is a transfinite set’ then these two contents clearly yielddifferent kinds of directedness, although there may be some relationsbetween the contents. The claim that two contents do not yield a differentkind of directedness is something that must be shown. In fields likemathematics and logic it should in fact be proved if possible. In somecases it might be obvious to everyone that two contents do not yield adifferent kind of directedness but, on the whole, this will be the exceptionrather than the rule.

A shift in thetic character also makes for a difference in directedness.Remembering is different from perceiving, and both are different fromimagining. Perceiving is different from judging, and so on. Some theticcharacters are incompatible with some contents. For example, the content7 þ 5 ¼ 12 is incompatible with the thetic character for sensoryperception. It would amount to a category mistake to combine the two.

The fact that a cognitive act is directed in a particular way means that itis not directed in other ways. If my thinking is directed by ‘x is a primitiverecursive functional’ then there are a host of ways in which it cannot bedirected. Some contents will be compatible with a given content andothers will not. Contents are obviously intensional entities and, as such,their identity is not to be determined extensionally.5

Contents always present us with a perspective on an object or situation.Consciousness is perspectival and we are finite beings and cannot takeall possible perspectives on an object, state of affairs, or domain. We donot experience everything all at once. Our knowledge is thus typicallyincomplete in certain ways, although we might see how we could continueto perfect it. The incompleteness theorems can be viewed as establishingthis last statement, under certain specific conditions, on a scientific basis.

Not to take the IDP seriously is to lose track of basic facts about humanconsciousness. If we do take the IDP seriously then we must say that withacts of abstraction, reflection, imagination, generalization, and so on,

5 I take intensionality to be linked to intentionality in the following way: expressionsfor mental acts that exhibit intentionality (e.g., knowledge, belief, hope) create inten-sional contexts; that is, contexts in which principles of standard extentional logic, suchas substitutivity salva veritate and existential generalization, fail. Roughly speaking, asindicated above, one can think of contents of mental acts as meanings or intensions.

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we are directed in different ways. Abstraction, generalization, formaliza-tion, and so on, are all human cognitive functions that exhibitintentionality. Formalization, for example, is itself a kind of abstractionthat can be distinguished from other types of abstraction. To illustrate thisbriefly, let us consider the following pair of propositions: ‘all red thingsare colored’ and ‘all rectangles are four-sided’. If we say that these twopropositions have the same logical form, say (8x)(Fx ! Cx), then onceagain we are abstracting from a difference. In this case, however, we aredealing with a kind of abstraction that is different from the abstractioninvolved in moving from, say, species to genus. With formal abstractionwe are shifting from ‘material’ or contentual propositions to the logicalform of those propositions. The relation of form to ‘matter’ is not the sameas the relation of genus to species or of species to instance. If an object isred then it is colored, and if something is colored then it is in some senseextended. There is a kind of material abstraction taking place in ourthinking here as we move from the more to the less specific, but it isdifferent from a formal abstraction. Now if we consider each of the threecases—‘all red things are colored’, ‘all rectangles are four-sided’, and(8x)(Fx!Cx)—we see that the mind is directed differently in each case.In the case where I think that (8x)(Fx ! Cx) the directedness, as purelyformal, is quite indeterminate although it is not completely lacking indeterminateness. It is, for example, different from the directednessinvolved in thinking that (9x)(Fx ^ Cx).

Similarly, if we say that the propositions ‘all perfect numbers are even’and ‘all abundant numbers are even’ have the same logical form then weare abstracting from a difference. Where we say that ‘the pen, the pencil,and the eraser are three’ and that ‘the cup, the cigarette, and the match arethree’ we are abstracting from some differences. In pure number theorythe natural numbers themselves are evidently intended or meant asabstract objects and so are the concepts expressed by predicates such as‘x is perfect’ and ‘x is even’.

Although I cannot discuss the details here I would argue, for reasons ofthis sort, that the phenomenology of human consciousness indicates thatthe human mind can, in effect, abstract universals from concreteparticulars, can subsume concrete particulars under universals, can relateuniversals to universals, or concrete particulars to concrete particulars.According to the IDP, to mean something universal is different frommeaning some sensory particular. We might become aware of the identityof a universal on the basis of different individual intuitions. Thus, wewould have an abstraction from some differences. The universal would begiven as the ideal unity through this multiplicity. Even if an abstractionrests on concrete individuals it does not for that reason mean what isindividual. The awareness of a universal as a universal is not awarenessof a sensory individual.


In mapping out the territory of conscious directedness we are thus ledto distinguish between different types of abstraction or generalization. It isnot possible to analyze acts of idealization, imagination, reflection, etc., inany detail here but it is something that can and should be done. These areall important components of human reason.

Some of the most important concepts that are relevant to Godel’s workare as follows: ‘x is a sign token’, ‘x is a sign type’, ‘x is a natural number’,‘x is an arithmetic function’, ‘x is a primitive recursive functional of finitetype’, ‘x is an ordinal < «0’, ‘x is a predicatively defined set’, ‘x is animpredicatively specified set’, ‘x is a constructible set’, and many otherconcepts of higher set theory. With each of the contents (meanings)mentioned we have a different form of directedness. In the case of ‘x is asign token’ the directedness is toward a sensible physical object. In thecase of ‘x is a sign type’ this is no longer true. There is already a shift incontent and thetic character. There is still a close relation to sensiblephysical objects in this case, however, so that sign types might be referredto as ‘quasi-concrete’ (see [Parsons, 1980]). In the other cases we haveforms of directedness that have been built up in non-arbitrary ways on thebasis of further acts of abstraction, reflection, generalization, idealization,and imagination.

In the case of ‘x is a natural number’ and the other contents thedirectedness is toward entities that cannot be construed as physical ormental. As was suggested above, we might appeal to the incompletenesstheorems themselves to argue for this. In the analysis of mathematicalcognition we also appeal to the IDP. To be directed by ‘x is a naturalnumber’ is not to be directed toward a sensible physical object, althoughthere is a sense in which we can represent the objects toward which we aredirected in this case by sensible physical objects like sign tokens. Theproperties of sensible physical objects, however, are different from theproperties of natural numbers. Sensible physical objects have, forexample, spatial and temporal properties. In pure number theory,however, we are not directed toward objects in physical space. As wesuggested above, there are categories of contents, along with contents andthetic characters, that are either compatible or incompatible with oneanother. In being directed by ‘x is a natural number’ we are also notdirected toward objects with temporal properties. In the science of numbertheory, as it is actually practiced, natural numbers are not meant as entitiesthat have temporal duration. They are therefore unlike mental processessince mental processes do have temporal duration. Of course there will bemental processes and subjective ideas associated with all of these contentsif there is to be directedness at all but, by the IDP, we should not substitutethe mental process or the subjective idea for the content of the processand, hence, for the natural number that is intended by the mental process.The mental process or idea is subjective. It belongs to a particular subject

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at a particular time and place. The natural number is objective. It is, inother words, not immanent to consciousness but is meant as transcendent.To say it is transcendent is not only to say that it is not itself a mentalentity or process but also that we do not know everything we couldpossibly know about it. Our knowledge of it is not complete. It is also tosay that we cannot shape it through acts of will in any way we like. Itresists us in certain ways. All of these features point to the abstractnessand mind-independence of the intended objects in these cases.

Our discussion of mathematical reason, intentionality, and the IDP inthis section has, in short, led us to a kind of realism about mathematicalobjects. Out of an analysis of human subjectivity we derive the objectivityof mathematics. It must be a dynamic of this kind that is involved inGodel’s deep interest in transcendental idealism.6 Perhaps we can have aphenomenological version of transcendental idealism that is compatiblewith a kind of mathematical realism. There is much more to be said aboutthis than I can go into here, but the very short version is as follows:suppose we hold that all knowledge of objects and of truth is constitutedintersubjectively over time by human subjects, where we think of thisview as a form of transcendental idealism. In this process, however, someobjects and truths are constituted as mind-independent, abstract, acausal,a priori, and so on. This seems like a form of mathematical realism. It isthis latter type of constitution that is at work in the sciences ofmathematics and logic. Here the objects are constituted as objects thatwould exist even if there were no constituting or meaning-bestowingbeings in the universe. They are constituted as objects that need not everbe brought to consciousness. ‘Being expressed’ or ‘being cognized’ arenot properties that are essential to mathematical objects. What we add isthe claim that concepts and mathematical objects are constituted in thismanner. This is how they are meant. Thus, the objects and truths ofmathematics are intended as mind-independent but the mind-independence is not so strong that they could not in principle be broughtinto some relation to the mind.

In this section we have presented an outline of how the human mindcan ascend from directedness toward concrete physical objects andmechanical operations on these objects to other forms of directedness. Thesuggestion is that the limitations of human reason are not the limitations offinite machines. What grounds are there for thinking that the human mindsurpasses any finite machine? In sum, the human mind can be directedby ways of its meanings (contents) toward abstract objects, includingmeanings themselves, thanks to founded acts of abstraction, idealization,

6 Godel’s interest in transcendental idealism certainly preceded his study of Husserl.He of course studied Kant but see also his correspondence with G. Gunther (pp. 476–535), along with Parsons’s introduction to it, in Vol. IV of the Collected Works.


reflection, and the imagination of possibilities and determination ofnecessities relative to these meanings. Turing machines do not havemeanings to be directed by or toward. They do not have intentionality.They are not goal-directed. They can, at a suitable level of description,manipulate concrete sign tokens, but they do not have intentional acts ofabstraction, idealization, formalization, imagination, and so on. Theycannot relate universals to particulars, or universals to universals, but theycan only relate certain kinds of concrete particulars to concrete particulars.A kind of informal rigor is possible in human cognition that is not possiblein TM computation.

Human cognitive acts and processes in mathematics are finite at anygiven stage but what they are about is abstract objects and concepts. It isbecause we are directed toward objects by meanings or intensions that thepossibility arises of reasoning with intensions of mathematical expres-sions, even when the extensions of these expressions cannot be fully givenin intuition. The possibility arises of reasoning with meanings ofexpressions that are about infinitary objects. The objects or conceptstoward which we are directed generally transcend the mind or ourintuition, in the sense that there is more to them than we can grasp at anygiven stage of our experience. We nonetheless make decisions against avast mathematical background that is not explicitly or completely known.We are continuously clarifying and extending our partial and incompleteintuition into these objects and concepts.

On the view we have been developing, Godel’s comment about theconstant development of the mind (cited in §2) might be understood asfollows. The mind can constantly develop its grasp of abstract conceptswithout diagonalizing outside of the abstract concept(s) it is intuiting.We do not diagonalize out of the abstract concept(s) we are using themachines to capture (partially). There can be a constant development ofmachines, on the other hand, but only by diagonalizing out of eachparticular machine under consideration. Machines are concrete entitiesthat do not have access to abstract concepts. What amounts to an essentialchange in the machine, a change that requires a new machine, is for usonly an incidental change in (and extension of) the awareness of theconcept(s) we are using the machines to capture. If we start with PA, forexample, then we can think of GPA as an axiom that could be added to PA.We see that GPA is true on the basis of our grasp of the concept of thenatural numbers. More and more ‘axioms’ of this type could be added tothe successive formal systems obtained in this way and one could arguethat these ‘axioms’ are evident to us on the basis of our understandingof the concept of the natural numbers. We are directed through thissequence on the basis of our understanding of this concept. There is noreason to stop with respect to deciding these Godel sentences. We couldnot proceed strictly on the basis of the given formal system or machine,

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however, because the Godel sentence obtained for that system is formallyindependent of the system. Instead, we refer through PA to what is notenclosed in PA. PA does not adequately capture our arithmetic intentionsin the first place.

We have now suggested a way of putting minds and the directednesstoward abstract objects back into the picture. We have a combination ofleftward and rightward features that is different from Hilbert’s. Since thecombination is presumably at odds in some ways with the zeitgeist,I would like to turn attention to Godel’s remarks on the zeitgeist in orderto offer further defense of some of the ideas in this section.

6. The Mechanist Zeitgeist

Godel refers to the bias or prejudice of the modern zeitgeist at severalplaces in his writings.7 He suggests that, philosophically speaking, theincompleteness theorems disrupt the zeitgeist.

The mechanist view of the human mind or of human reason is certainlypart of the spirit of our times. We see many forms of this view all aroundus. There are of course many technical issues about the relation of mindsto machines that one can go into, but what I want to do in this section is totake Godel’s references to the bias of the zeitgeist seriously and thendevelop them in a certain direction. If we do take these referencesseriously then we might say that the extent to which one is under theinfluence of the zeitgeist is the extent to which it is difficult to grasp someof the ideas about the human mind that have been presented in §5. Ideasabout how the human mind surpasses finite machines, in other words,tend to be concealed by the zeitgeist. The predominant perspective of ourtimes seeks, as Godel suggests, to eliminate the mind and abstract entities.

Modern mechanism is part of the empiricist, scientific, perspective thathas led to incredible success in many domains of human endeavor. Thisperspective has revealed many things to us that we would otherwise nothave seen. One can argue that it has also concealed important things. Inorder to have a firmer grasp of what it conceals we need to consider how itoriginated. Thus, in this section I will briefly discuss the philosophicalorigins of the modern mechanist view in order to make clearer thestructure of revelation and concealment that it embodies. One might thinkthat Godel would have to look far into the future of human development tosee a way out of the hypothesis that all human cognitive functions aremechanizable (see also [Webb, 1990]). Perhaps we can learn just as muchabout the view that human minds are machines by looking to the past.

7 See, e.g., [Godel, *1961/?] and the letter to Rappaport quoted above. Hao Wangalso discusses this in [Wang, 1974, 1987, and 1996].


As Godel suggests, the modern mechanist view is a perspective on theworld that has been in a process of development since the Renaissance.One could trace its development in detail. For example, a very importantstage in its development occurred during the Scientific Revolution (see[Husserl, 1936]). The mechanist perspective is, in effect, an interpretivescheme that has been built up as part of the general development of thescientific worldview. It is a scheme we use to interpret the world andobjects in the world. Central to the development of modern natural scienceis the mathematization of our experience. One takes number, shape,magnitude, position, and motion to be primary qualities and colors, tastes,smells, and warmth/cold to be secondary qualities. The former propertiesare seen as objective features of experience while the latter are viewedas subjective. Indeed, the primary qualities are just those that aremathematizable and are absolute and immutable while the secondaryqualities are sensory, allegedly relative, and fluctuating. Knowledge isconcerned with primary qualities, but opinion and illusion are associatedwith secondary qualities. One might hold that the primary qualities inherein the objects themselves while secondary qualities do not.

One builds up the interpretive schemes of the modern sciences bydistinguishing the quantitative from the qualitative aspects of ourexperience and then employing calculation or mechanical techniques withthe quantitative aspects. There is a shift to structural or formal features, indistinction from contentual or ‘material’ features of experience. As part ofthe package, various idealizations of nature emerge. With mathematiza-tion a kind of exactness or precision arises that would otherwise not bepresent in our thinking about nature and the things in nature. Pureformalism and modern mechanism are developed against this background.With pure formalism and mechanism there are further abstractions: oneabstracts from the meaning, reference, and origins of the symboliclanguages that are essential to the mathematization of nature. We discoverwith mathematization that we can make various aspects of our thinkingamenable to purely mechanical, computational, techniques. A vasttechnization and mechanization emerges in the sciences. One of theoutstanding results of this entire development is the emergence of a sharpconcept of mechanical procedure, as characterized by Turing machines,along with the fact that a host of alternative characterizations ofthe concept of mechanical procedure can be shown to be equivalent.Many other interesting and important results emerge in the theory ofcomputation.

The distinctions that lie behind the empiricist, scientific, worldviewand modern mechanism allow us to separate the subjective from theobjective. They are in fact used for just this purpose. With quantification,calculation, formalization, idealization, and exactness we obtain inter-subjective agreement on methods and results, including repeatability of

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calculations, experiments, and procedures.We obtain a kind of objectivity,and objectivity is what we seek everywhere in the modern sciences.We have in the zeitgeist an alignment of objectivism, empiricism, andmechanism.

As we have developed and used the mechanist perspective we have, asalready mentioned, met with many successes in science and technology.When we turn this interpretive scheme back around on ourselves,however, what do we find? True to our intentions to eliminate humansubjectivity, we find that we have eliminated human subjectivity, alongwith all of its complexity and detail. In its stead we have a purelyobjectified subject, merely the outer shell as it were. Consciousness, thevery essence of subjectivity, disappears. The zeitgeist thus tends towardthe elimination of the human mind and its most important features. Thefeatures that are ignored or eliminated include intentionality, the meaning-bestowing nature of consciousness, the perspectival character ofconsciousness, the Gestalt characteristics of consciousness, internal timestructure, the existence of expectations that may be either fulfilled orfrustrated, the underdetermination of perception and thinking bysensation, the capacity for acts of idealization and imagination, and soon. At earlier stages in the development of the modern zeitgesit the humanbody was interpreted as a machine, with the effect that the ‘lived body’and bodily intentionality were ignored. As the scheme was extended andaugmented the human mind also came to be interpreted as a machine.

Thus, we develop in the sciences an interpretive scheme the goal ofwhich is to minimize subjectivity absolutely and to maximize objectivity,and when we apply this interpretive scheme to the human mind we see thatwe achieve just this effect. The problem is that we are forgetting what thisinterpretive scheme abstracts from or leaves behind in the first place. It isnot a foundation but is rather already a founded reflective scheme thatdepends on making the abstractions we have noted (e.g., quantitative fromqualitative features, primary from secondary qualities, form from content)and then forgetting about the whole from which they were abstracted.Hence, it can become a limited or one-sided view that conceals much thatis important about human cognition. The key point is this: the claim thatthe human mind is a finite machine depends on the fact that human beingswhose cognitive acts exhibit intentionality have developed a particularinterpretive scheme in the first place, a scheme which they have thenapplied to themselves. We have, in effect, taken an important and fruitfulinterpretive scheme and applied it beyond its legitimate boundaries. Inso doing, we substitute a part of what we can do—namely, act ascomputers—for the whole. At the founding level of all of this, however,we have human subjects with intentionality who build up ways ofunderstanding the world through their manifold capacities for interpreta-tion. This is my answer in this paper as to why we cannot reduce (*) to


(**). The claim that the human mind is a finite machine (a TM) restson a development that presupposes human intentionality, directedness, thecapacity for acts of abstraction, and so on. Our awareness of our ownconsciousness, however, does not depend on building up layers ofscientific theory, abstraction, idealization, and so on. At the pre-reflective,pre-scientific, level humans are already conscious interpreters of theworld who are directed toward various goals.8

The mechanist zeitgeist is thus an interpretive scheme that both revealsand conceals. This is part of how I would explain the one-sidedness orprejudice (in a hermeneutical sense) of the zeitgeist that Godel mentionsin some of his remarks.

The new combination of leftward and rightward ideas I have beensketching in this paper does not imply any kind of irrationalism. On thecontrary, it is possible to embrace it while adopting a healthy skepticismtoward superstition, mysticism, prophecy, and the like. We adopt arationalist skepticism to accompany our broader view of reason, science,logic, mathematics, and evidence, instead of an exaggerated empiricistskepticism that eliminates the mind and its directedness toward abstractobjects in the sciences of mathematics and logic. We include criticalreason but not pure uncritical reason. We see the worries that led tofinding a foundation for mathematics exclusively in concrete mathematicsas exaggerated. Of course we do not want idealizations, abstractions, andforms of reflection that lead to inconsistency. Instead, we want to open upa space in which we allow the broadest set of mathematical possibilitiesand necessities compatible with scientific thinking.

A skepticism according to which everything about human subjectivityis illusory, relative, or fluctuating is skepticism gone awry. There can besome objectivity in the phenomenology of the human mind, for certainlythere are common structures of cognition that make sciences such asmathematics possible. We can look for the invariants in human cognitionand at the same time do justice to the mind and its forms of directedness,its capabilities for abstraction, and so on. In short, we need to keep bothsubjectivity and objectivity in the picture.

7. Hilbertian (Godelian) Optimism and Decidability as an Ideal

How does the non-mechanist view of the human mind we have beendiscussing point us in the direction of option (4) instead of option (3)

8 It is sometimes claimed that even if the human mind is not a Turing machine it isnonetheless some other kind of machine. I think the argument presented in this sectioncan be extended to cover any other conceptions of machines that involve the abstractionsdiscussed.

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among the possibilities allowed by Godel’s disjunction? I would like tomake a few brief comments about this before concluding.

Now that we have sketched an alternative combination of leftward andrightward views the contrast with blind formalism or blind mechanism isstriking. Why will it not be possible to replace human mathematicianswith machines? In blind mechanism there is no directedness by way ofmeaning, including no directedness by way of acts of abstraction,idealization, reflection, and so on. This makes a difference to questionsabout absolute undecidability. Open problems in mathematics can, inaccordance with what we have said, be viewed as expressions ofintentions that can be either fulfilled or frustrated. We can see a sentencesuch as 2@0 ¼ @1 as an expression of an intention that we expect to beeither fulfilled or frustrated instead of viewing it as no more than anuninterpreted string of signs such that either 2@0 ¼ @1 or : 2@0 ¼ @1 canbe derived from other uninterpreted sign configurations in a mechanicalway on the basis of uninterpreted sets of ‘rules’. It is the nature of humanreason in its scientific practice to proceed from what is given andconstantly hypothesize or intend states of affairs that are not yet realizedbut that may be realized. This drive toward the resolution of openproblems may be viewed as part of our cognitive makeup. In the contextof our remarks above, it amounts to the effort to come to know moreabout a transcendent or mind-independent mathematical reality on thebasis of continuous analysis and clarification of the abstract concepts(or meanings) by virtue of which we are directed toward that reality.Optimism about solving open problems is, on this view, accompaniedby a kind of realism.

I would like to suggest that Hilbert’s optimism about mathematicalproblem solving, an optimism that Godel does not want to abandon, can beviewed as a regulative ideal in a Kantian or Husserlian sense.9 The ideal ofdecidability regulates our scientific thinking. To say that it is a regulativeideal does not imply that we will actually complete the task of solving allopen mathematical problems at some point in human history. It isprecisely because the real completion of our knowledge is not or may notbe possible that it is called an ‘ideal’. Nonetheless, it stands as a postulateof reason. It is a projection of completeness, ‘lying at infinity’. It is the

9 Thus, I am suggesting that the quotation at the beginning of this paper, as well asthe famous list of open problems that Hilbert presented in 1900 [Hilbert, 1900] be readin this manner.I have discussed regulative ideals in mathematics in other places, e.g., [1989, pp. 39,

180–182, 193]. In Godel’s case, the optimism about problem solving may also be con-nected with ideas in Leibniz’s philosophy. See, e.g., the remarks at the end of [Godel,1944] or Godel’s remarks on the principle of sufficient reason. On the basis of what Ihave said, however, decidability as an ideal of reason cannot be construed as decidabil-ity according to a purely mechanical conception of reason.


idea of perfection of fulfillment as regards the aims of mathematicalreason. Such an ideal is what lies behind not stopping with respect todeciding clearly formulated mathematical problems. We might say that itis only because we are aware of such an ideal in the first place that werealize that we have fallen short, that our knowledge is incomplete. It isthis fact about human cognition, this awareness of an ideal, that is acondition for the possibility of the awareness of open problems.

Measuring our knowledge against an ideal of decidability is, in effect,a condition for the possibility of mathematics as a science, in the sensethat the scientific endeavor would collapse were such an ideal notregulating our behavior. Why would we pursue a problem if we thoughtthat there could be no answer to it? We presumably do not want to give upon Hilbertian optimism as such a regulative ideal. If we were not driven bythis ideal would we perhaps have given up on solving problems such aswhether there is a consistency proof for arithmetic or real analysis,whether CH is consistent with ZFC, or whether :CH is consistent withZFC? These are problems I mentioned earlier that appear to requiredevelopments in our rational capacities, or what Godel calls an ascent to‘higher’ states of consciousness.

If we eschew blind mechanism, if the human mind is not a finitemachine, then it is not clear that there are pressing reasons to hold thatthere are absolutely undecidable diophantine problems. It has beensuggested that perhaps set theory will split around CH, so that there will beone set theory in which 2@0 ¼ @1 and another in which : 2@0 ¼ @1. It ispossible to say enough about the meaning of CH to cast doubt on thisview. In the case of diophantine problems of the type Godel has in mind, itis highly unlikely that there could be an analogous kind of split. One mightthink the case for rejecting absolute undecidability is more compelling inconnection with diophantine problems but this kind of optimism mightalso be extended to any mathematical problem that can be clearlyformulated, for we must allow any method for producing conclusiveevidence that human reason can conceive and it is difficult to determine inadvance what this might include.

I do not want to deny outright that it is possible to construct openproblems in mathematics that are a fool’s task. When is it pointless topursue an open problem? When is the hope that a particular scientificproblem is solvable misguided or even ridiculous? Perhaps it is possible tomake some progress in addressing these questions. Careful, detailedstudies of such questions would be helpful. Most of my attention inthis paper has been focused on raising questions about the thesis thatthe human mind is a finite machine. I have not eliminated option (3) fromthe possibilities allowed by Godel’s disjunction. I do think enough hasbeen said to indicate how, in the spirit of Godel’s interest in some of theideas of Husserl and Kant, a case can be made for favoring option (4).

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REFERENCES

GODEL, K. [1934]: ‘On undecidable propositions of formal mathematicalsystems’, in [Godel, 1986], pp. 346–371.

—— [*193?]: ‘Undecidable diophantine propositions’, in [Godel, 1995],pp. 164–175.

—— [1944]: ‘Russell’s mathematical logic’, in [Godel, 1990], pp. 119–143.—— [1946]: ‘Remarks before the Princeton Bicentennial Conference on

problems in mathematics’, in [Godel, 1990], pp. 150–153.—— [*1951]: ‘Some basic theorems on the foundations of mathematics and their

implications’, in [Godel, 1995], pp. 304–323.—— [*1953/9, III and V]: ‘Is mathematics syntax of language?’, in [Godel,

1995], pp. 334–363.—— [*1961/?]: ‘The modern development of the foundations of mathematics in

the light of philosophy’, in [Godel, 1995], pp. 374–387.—— [1964]: ‘What is Cantor’s continuum problem?’, in [Godel, 1990],

pp. 254–270.—— [1972]: ‘On an extension of finitary mathematics which has not yet been

used’, in [Godel, 1990], pp. 271–280.—— [1972a]: ‘Some Remarks on the Undecidability Results’, in [Godel, 1990],

pp. 305–306.—— [1986]: Collected Works, Vol. I. Feferman, S. et al., eds. Oxford: Oxford

University Press.—— [1990]: Collected Works, Vol. II. Feferman, S. et al., eds. Oxford: Oxford

University Press.—— [1995]: Collected Works, Vol. III. Feferman, S. et al., eds. Oxford: Oxford

University Press.—— [2003a]: Collected Works, Vol. IV. Feferman, S. et al., eds. Oxford: Oxford

University Press.—— [2003b]: Collected Works, Vol. V. Feferman, S. et al., eds. Oxford: Oxford

University Press.HAUSER, K. [forthcoming]: ‘Godel’s program revisited’, Husserl Studies.HILBERT, D. [1900]: ‘Mathematische Probleme. Vortrag, gehalten auf dem

internationalen Mathematiker-Kongress zu Paris 1900’, Nachrichten von derKoniglichen Gesellschaft der Wissenschaften zu Gottingen, 253–297.English translation by M. Newson in Bulletin of the American MathematicalSociety 8 (1902), 437–479.

—— [1926]: ‘Uber das Unendliche’, Mathematische Annalen 95, 161–190.English translation by S. Bauer-Mengelberg in [van Heijenoort, 1967],pp. 367–392.

HUSSERL, E. [1936]: ‘Die Krisis der europaischen Wissenschaften und dietranszendentale Phanomenologie’, Philosophia 1, 77–176. See The Crisis ofthe European Sciences and Transcendental Phenomenology. D. Carr, trans.Evanston, Illinois: Northwestern University Press, 1970.

PARSONS, C. [1980]: ‘Mathematical intuition’, Proceedings of the AristotelianSociety 80, 145–168.

—— [2003]: ‘Introductory note to correspondence with Gotthard Gunther’,in [Godel, 2003a], pp. 457–476.


TIESZEN, R. [1989]: Mathematical Intuition. Dordrecht: Kluwer.—— [2005]: Phenomenology, Logic, and the Philosophy of Mathematics.

Cambridge: Cambridge University Press.TURING, A. [1937]: ‘On computable numbers, with an application to the

Entscheidungsproblem’, Proceedings of the London Mathematical Society42, 230–265.

VAN ATTEN, M., and J. KENNEDY [2003]: ‘On the philosophical development ofKurt Godel’, The Bulletin of Symbolic Logic 9, 425–476.

VAN HEIJENOORT, J., ed. [1967]: From Frege to Godel. Cambridge, Mass.:Harvard University Press.

WANG, H. [1974]: From Mathematics to Philosophy. New York: HumanitiesPress.

—— [1987]: Reflections on Kurt Godel. Cambridge, Mass.: MIT Press.—— [1996]: A Logical Journey: From Godel to Philosophy. Cambridge, Mass.:

MIT Press.WEBB, J. [1990]: ‘Introductory note to Godel 1972a’, in [Godel, 1990],

pp. 292–304.

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