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Philosophy of Mathematics

The Cartesian Doubt Experiment and Mathematics

Halil Turan

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ABSTRACT: The view that Descartes called mathematical propositions into doubt as he impugned all beliefs concerning common-sense ontology by assuming that all beliefs derive from perception seems to rest on the presupposition that the Cartesian problem of doubt concerning mathematics is an instance of the problem of doubt concerning existence of substances. I argue that the problem is not 'whether I am counting actual objects or empty images,' but 'whether I am counting what I count correctly.' Considering Descartes's early works, it is possible to see that for him, the proposition '2+3=5' and the argument 'I think, therefore I am,' were equally evident. But Descartes does not found his epistemology upon the evidence of mathematical propositions. The doubt experiment does not seem to give positive results for mathematical operations. Consciousness of carrying out a mathematical proposition, however, unlike putting forth a result of an operation, is immune to doubt. Statements of consciousness of mathematical or logical operations are instances of 'I think' and hence the argument 'I count, therefore I am' is equivalent to 'I think, therefore I am.' If impugning the veridicality of mathematical propositions could not pose a difficulty for Descartes's epistemology which he thought to establish on consciousness of thinking alone, then he cannot be seen to avoid the question. Discarding mathematical propositions themselves on the grounds that they are not immune to doubt evoked by a powerful agent does not generate a substantial problem for Descartes provided that he believes that he can justify them by appeal to God's benevolence.

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The question whether Descartes impugned veridicality of mathematical propositions via the arguments of the First Meditation is of epistemologically significance for an inquiry into the nature of Descartes' doubt experiment with a view to a plausible answer to this question may offer us clues to understand the nature of Cartesian theory of justification and the nature of foundationalistic epistemology in general.

The evil genius hypothesis introduced in the last paragraph of the First Meditation does not seem to call veridicality of mathematical propositions into question: Descartes does not mention mathematical truths when he finalizes the setting of the doubt experiment. The text is ambiguous at this point and the reader is left ignorant whether simple truths of arithmetics or geometry are held exempt from doubt evoked by the evil genius hypothesis. Does this final tool of the doubt experiment put emphasis on the dubitability of judgments of common-sense ontology based on sense perception alone? Or, might Descartes have thought to have called mathematical propositions into doubt as he impugned all beliefs concerning common-sense ontology by assuming with the ordinary man that all beliefs derive from perceptions? (1)

That we may draw a relation between perception and mathematics is incontestable, but confining our thoughts to a context where the ontological presuppositions of unphilosophical reflection on perception are at stake, we must note the significance of perception with respect to the nature of existence that Descartes considers primarily for epistemological purposes. What is the relation between perception and a proposition like 'square has four sides'? Counting the sides of a figure called 'square' may be a sense experiment, but the proposition 'square has four sides' is mere definition. It is evident that Descartes' problem, whether a deceiving God may make me go wrong in counting the sides of a square will be equally meaningful when we consider an experiment of counting the sides of a given figure (a drawing) as it is when we consider an analysis of the definition of square. The case in question is insensitive to the problem whether we operate on sense data or on mere definitions. The same insensitivity may be observed in the case of arithmetical operations: Descartes' question whether I may be deceived by an agent in adding two and three does not concern the ontological status of the objects counted. The problem is not whether I am counting actual objects or images which may not correspond to anything except themselves, but whether or not I am counting correctly.

The ambiguity of Descartes' text may give a plausibility to the thesis that truth of mathematical propositions was never impugned in the Meditations: Descartes seems to abandon the deceiving God argument for the demon assumption and this last hypothesis seems to call into doubt exclusively beliefs related to existence of an external world. Therefore, it is possible to argue that Descartes gave up pursuing the question concerning the veracity of the mathematical judgments in the First Meditation. (2)

The question whether the veridicality of mathematics was put into question by any of the arguments in the First Meditation does not have a straightforward answer. Descartes seems to endow the evil genius solely with the power of deceiving him in matters related to judgments on the existence of external things: "I shall think that the sky, the air, the earth, the colours, shapes, sounds and all external things are merely delusions of dreams which he has devised to ensnare my judgment." (3) Since there is no mention of mathematical propositions in this passage, it may seem natural to conclude that Descartes did not intend to impugn mathematics by the evil genius hypothesis. The evil genius, however, was conceived as omnipotent and therefore capable of deceiving without limit.

Descartes always considered mathematical demonstrations among the most evident truths that human mind can attain, and referred to them as examples of objects which can be intuited clearly and distinctly. In the Rules for the Direction of the Mind, he wrote that among all other occupations of man, "arithmetic and geometry alone are free from any taint of falsity or uncertainty." These sciences, Descartes argued "are concerned with an object so pure and simple that they make no assumptions that experience might render uncertain; they consist in deducing conclusions by means of rational arguments." (4) Descartes' discussion in the Rules is in parallel with that which he offers in the First Meditation (5) where the emphasis is put on external existence of the objects in question.

Descartes considered both deduction and intuition as legitimate methods of acquiring knowledge. Intuition, the "indubitable conception of a clear and attentive mind which proceeds solely from the light of reason" is simple and on that account more certain than deduction, but deduction is not epistemologically inferior to intuition for the attentive human mind. No human being can err in deduction, for reason is common to all men. Although mathematics makes extensive use of deduction, Descartes does not say that deduction is the sole legitimate method of this domain and holds that intuition is as indispensable as deduction for the mathematical body of knowledge. He reckons simple mathematical propositions among the intuited truths: "Thus everyone can mentally intuit that he exists, that he is thinking, that a triangle is bound by just three lines, and a sphere has a single surface, and the like." (6)

It is clear that, at least for the early Descartes, mathematical propositions had the same degree of certainty as the indubitable ontological argument cogito. However, although Descartes seems to have held that the geometrical statement that the sum of the interior angles of a triangle is equal to two right angles as evident as 'I think, therefore I am', he none the less always related the evidence of mathematical propositions to exactitude of mathematics which he thought to be deriving from the simplicity of their objects and hence to its ontological unproblematical nature.

Mathematics may be invariable with respect to ontological presuppositions, but once carried into the context of the doubt experiment it is seen that it bears crucial ontological implications: here it appears that mathematical objects and operations presuppose existence. The mathematical proposition '2+3=5' and the argument 'I think, therefore I am' may be epistemologically equivalent, but the notion of counting presupposes the existence of things (i.e. thoughts) to count and therefore the existence of a counting mind. Hence, mathematical objects are functionally dependent on the existence of a thinking being. Considering the question whether the ideas of clear and distinct elements of the ideas of corporeal things could be originating from his own mind, says the following:

"I perceive that I now exist, and remember that I have existed for some time; moreover, I have various thoughts which I can count; it is in these ways that I acquire the ideas of duration and number which I am then able to transfer to other things. As for all the other elements which make up the ideas of corporeal things, namely extension, shape, position and movement, these are not formally contained in me, since I am nothing but a thinking being; but since they are merely modes of a substance and I am a substance, it seems possible that they are contained in me eminently." (7)

It appears therefore, that the fact that 'I count' appearances or thoughts would lead me to the recognition of the cogito argument, for number and other universals are modes of thinking. (8) Counting as an act is a form of thought, and number, a concept derived from this act, is unintelligible unless the substance of which it is a mode is identified as existing. If counting is a mode of thinking, and the expression 'I count' implies 'I think' then Descartes could argue to existence from the fact that he is counting. Number, in a similar vein, being a concept or an idea derived from counting, would be unintelligible without recognizing the existence of a counting being. It goes without saying that this same line of thought can be reproduced for shape and other simple natures once we substitute them for 'number'. Number and other universals are then, functionally and hence ontologically dependent entities.

The priority of the cogito argument in Descartes' epistemology in the Meditations is not a contingency. Functional and ontological dependence of "number and other universals" renders cogito, an instance of thought where both evidence and ontological certainty could be attained in a single step, epistemologically prior to mathematical propositions which may, considered apart from the context of the doubt experiment, seen to embody evidence.

'I count, therefore I am' is epistemologicaly equivalent to 'I think, therefore I am'. Both arguments are immune to doubt. The evil genius can indeed make me go wrong as I count my thoughts or the appearances, but cannot deceive me in the inference I draw therefrom: the fact that I am counting is sufficient to prove that I exist regardless of whether or not I count or add or perform any mathematical operation erroneously. Rendering mathematical propositions uncertain by impugning them via the evil genius hypothesis, could not, therefore, generate a substantial problem in Descartes' epistemology, provided that the philosopher could demonstrate the impossibility of attributing a will-to-deceive to an omnipotent being.

The ontological situation established by the Cartesian experiment of doubt brings in serious epistemological constraints. The experimenter discovers that any epistemological means he may want to employ for a further ontological move must necessarily be one available from the proper resources of the ontological situation he has confined himself to for epistemological purposes. In other words, the epistemological standards the experimenter must conform to are determined by the ontological setting of the doubt experiment. Hence the experimenter finds himself alone with things which we may call perceptions or thoughts, at a standpoint from where he attests to happenings of perceptions and thoughts and cannot know well how they are procured. Descartes could therefore depend solely on the thought that he has perceptions or thougts in his epistemological inquiry to establish a certainty which may not be affected by the arguments of the doubt experiment. Once the view that things exist independently and are the causes of his perceptions is rejected, the world closes on the experimenter. It is evident that the horizons of this ontological perspective will render any external epistemological anchorage illegitimate.

It must be noted that the ontological setting which the arguments of the First Meditation bring forth is not that of solipsism proper. Cartesian experiment becomes conceivable by assuming an agent, a being which is represented as the cause of the experimenter's perceptions. The Cartesian inquiry concerns, among other things, the clarification of the nature of substrata with a view to a certain ontology and one of the major steps towards establishing it is taken by questioning the legitimacy of attributing (transferring) all qualities we find in our perceptions to independently existing substances. Seen through this perspective, Cartesian experiment does nothing other than substituting a 'temporary' ontology for the received one.

Given that Descartes had recourse to a certain schema of causality even for setting up the doubt experiment, and that his first demonstration of the existence of God rests on the assumption that causality is operative in a world defined by the constraints of the First Meditation, it is possible to hold that the universe to which Descartes confines himself is a universe for two, a world constituted by the experimenter and a certain being whose attributes he does not yet know well. Descartes does not conceive himself as the cause of his perceptions and thoughts, instead he assumes a powerful being which produces these thing as if by projecting them onto the screen of his imagination and occasions him to think what he thinks by secretly enjoining the course of his thoughts.

That Descartes' doubt experiment rests on a non-solipsistical ontology is not an objection to the thesis that the experimenter is under the constraints of this peculiar state. The evil genius may be a causal agent but since it is only an 'agent' it may never be thought as operating in the same manner as an object conceived as partaking in the causal schema of perception. The evil genius cannot be construed as a substratum of qualities that I can abstract from my perceptions. This powerful agent of the experiment is assumed to be projecting perceptions on my mind and enjoining my thoughts, but although thus represented as a sort of 'origin' of ideas, it is never seen as a substance of the qualities which may be discerned in those perceptions and thoughts. It must be observed that the evil genius was merely a methodological tool, a hypothetical entity which would never admit any positive attribute as the substance of the received metaphysics did. The evil genius or the deceiving God are not alternatives proposed for the old substance. Further, it must be observed that Descartes never saw this agent as a being of absolute power. The evil genius may be the 'origin' of perceptions or even of the simple natures like size, shape and motion, but thoughts of the experimenter regarding these simples, or his operations on these simple natures are not absolutely under the influence of this agent. The experimenter enjoys a certain degree of freedom of thought in reflecting on or drawing relations between his experiences: he can, for example, assure himself that he himself perceives them, or that he himself counts them, or again, that he perceives himself to count them. It is probable that the experimenter may be deceived in his judgments regarding the true nature of these things or the outcome of his operations as the evil genius may interfere in his train of thoughts, nevertheless he still may conceive himself as a free agent reflecting or operating on the given material. It is through this consciousness that the experimenter can envisage an epistemological inquiry. Freedom is an important postulate of Cartesian metaphysics and it can be discerned in a rudimentary form even in the First Meditation: the experimenter conceives himself as endowed with freedom, for how could, if it were otherwise, the experiment be conceivable? Would it not be absurd to ask questions regarding knowledge if I disclaim responsibility for my thoughts? The doubt experiment becomes meaningless without a subject who resists the idea that all his thoughts are under the patronage of an all-powerful agent. This would be negating his own existence and would put an end to all discourse.

The experimenter assures himself of the fact that he is not a plaything of an evil genius, or a mere screen for what happens, and he can only count upon his thoughts to prove to himself that he has some power. He conceives himself as enjoying a certain degree of freedom of will, and it is only through this assumption that the catastrophic hypothesis of a deceiving agent can be discarded. Descartes cannot, until his argument that God exists and is not a deceiver, think that a simple mathematical proposition like '2+3=5' may constitute a point d'appui for any gain of knowledge. The operation of addition (or any mathematical or logical operation) however, belongs to a different class, it is an experience of an act of thought and does not assert anyhing other than the presence of a thought. Cogito, the exemplar of certainty, is a piece of thought which represents all acts of thought, including all operations on thoughts or perceptions. All consciousness of operations are represented by the statement of the fact that 'I think' which cannot be impugned on the basis of the imperfection of my memory as could be any assertion regarding the outcome of an operation.

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(1) Harry. G. Frankfurt held that veridicality of mathematics was put into question by the demon hypothesis and supported his case by arguing that Descartes assumed a naïve empiricism throughout the First Meditation. According to Frankfurt, Descartes cannot be seen to bring into view the "clear and distinct perception" of these propositions within a context where he purportedly confines himself to an empirical view and therefore he cannot keep the mathematical truths immune from doubt (Demons, Dreamers and Madmen, New York, 1970; cf. p. 73). Although Frankfurt's analysis offers a profound solution, it seems to impose a sort of empiricism to Descartes' metaphysics which the philosopher himself would hardly have endorsed. The fact that arithmetics and geometry can be applied to nature does not imply that nature is an external existence - this is true for Descartes who assumes a non-realistic point of view throughout the First Meditation. Hence Frankfurt's statement, "To say that [mathematics] scarcely cares [about existence] suggests that it does care to some extent, and this would not be so if it did not depend in any way on the existence of anything (op.cit. pp. 75-6)", could entail a misinterpretation of Descartes' view that arithmetic and geometry deal with simple natures "regardless of whether they really exist in nature or not" unless we construe the depence on existence in question only as dependence on the existence of a thinking substance.

(2) John Cottingham holds that Descartes left the task of investigating the nature of God to later stages of the Meditations and that the question concerning the truth of the propositions of logic and mathematics is left "hanging in the air" since the evil demon argument is not intended to show that they are dubious. Cottingham supports his thesis by arguing that the "therefore" of the opening sentence of the last paragraph of the First Meditation, refers to the practical difficulties of maintaining the state of doubt. ("The Role of the Malignant Demon", in René Descartes, Critical Assessments, ed. Georges J.D. Moyal, London and New York, 1991; cf. pp. 130-32 [first published in Studia Leibnitiana, 8(2) 2, 1976, pp. 257-64].) Recently Robert Wachbrit argued for the same case in "Cartesian Skepticism from Bare Possibility", Journal of the History of Ideas, 57(1), J 1996.

(3) First Meditation, AT VII, pp. 22-3. CSM II, p. 15 (For our quotations from and references to Descartes' works we will give the page numbers of the Adam & Tannery edition of Descartes' works (Oeuvres de Descartes, ed. Charles Adam & Paul Tannery , Paris, Vrin, 1996) by the abbreviation AT, Volume number in Roman and page number in Arabic numerals; we will quote Descartes in English and will refer to The Philosophical Writings of Descartes, Vol. I, translated by John Cottingham, Robert Stoothoff, Dugald Murdoch [abbreviated as CSM I], New York, 1990; The Philosophical Writings of Descartes, Vol. II, translated by John Cottingham, Robert Stoothoff, Dugald Murdoch, New York, 1990 [abbreviated as CSM II].)

(4) Rules for the Direction of Mind, AT X, p. 365. CSM I, p. 12.

(5) First Meditation, AT VII, p. 20; CSM II, p. 14.

(6) Rules for the Direction of the Mind, AT X, p. 368; CSM I, p. 14.

(7) AT VII, pp. 44-45; CSM II, pp. 30-31.

(8) Cf. Principles of Philosophy, AT VIII A, p. 27; CSM I, p.212.

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