The cure consists in attaining a clear and distinct perception of the essence of God. This cure, however, raises the problem of the "Cartesian circle. The circle would turn on supplying non-question-begging guarantees for both the truth of clear and distinct ideas and the validity of demonstrative reasoning.
Thus, the Deceiving God Argument appears to force upon the meditator two sets of problems. If the meditator is to escape the argument's grip, he apparently must solve the problems posed by an aggravated circle and the problems raised by Dcscartes's doctrine of the creation of the eternal truths. That question is, roughly, whether or not doubting the truths of mathematics implies doubting the laws of logic in the First Meditation.
Broadly conceived, this problem is whether the Deceiving God Argument of the First Meditation casts doubt on all eternal truths.
To twentieth-century eyes, it seems that it would. Surely any reasons which are sufficient for doubting some necessary truths are sufficient for doubting them all. So, if the truths of mathematics, which are necessary truths, are dubitable, then so are all necessary truths, including the laws of logic.
However, this assumes that all necessary truths are of the same epistemic type. But that concession does not end the argument. That is because for Descartes, what makes mathematical reasoning correct is not that it yields the truth , but rather that it yields what cannot be doubted.
This fact is often lost in discussions of the Truth Principle in part because of a crucial passage from Meditations. As Descartes emphasizes, doubts about clear and distinct perceptions are different in type from ordinary doubts about visual or tactile perception. If I hear a rustling noise at night in my garden, but doubt that it is due merely to the wind, I immediately think of alternatives, for example, that it is an animal, or a thief.
Now, Descartes does concede that sometimes it is possible to doubt the product of mathematical reasoning in this sense; that is, it might be that there is an alternative judgment that is more plausible than the initial judgment, but that does not mean that it is possible to doubt just an y mathematical proposition.
Now, let us return to the Truth Principle. It cannot be that I both represent myself as clearly and distinctly perceiving a proposition by the natural light and yet doubt that it is true. That is because to doubt that something is true does not in any way change what is doubted. Similarly, the Truth Principle cannot be doubted, but that, of course, does not prove that it is true—it only proves that I cannot doubt its truth.
Descartes insists that it will do no good to object to the indubitability of the Truth Principle on the grounds that we have sometimes been mistaken in thinking that we clearly and distinctly perceive. Descartes clearly states that when we come to recognize that we have erred in forming a belief, we also come to see that we did not clearly and distinctly perceive the belief by the natural light in the first place.
Indeed, if I come to think that a perception that I once deemed to be clear and distinct may be false , I must also conclude that I did not clearly and distinctly perceive it to be true. What we cannot do is to represent ourselves as having clearly and distinctly perceived a false proposition.
Nor can we deem another to have clearly and distinctly perceived a false proposition. In order to do that we would have to represent that proposition to ourselves as both clearly and distinctly perceived and nonetheless false. The mistake I would make in that case would have been to judge knowledge of the truth to be something weaker than clear and distinct perception. The idea that we can doubt what we clearly and distinctly perceive is a delusion.
Descartes very stringent standard by which truth is judged suggests that we ought to reconsider the question as to whether there really are any propositions that we clearly and distinctly perceive to be true by the natural light.
That brings us to the metaphysical proof of the existence of God in the third meditation. According to Descartes, once we know that God exists, we know that there cannot be systematic doubt because it cannot be both that what we clearly and distinctly perceive is indubitable and yet that we doubt its true. It is only then that we know that what we clearly and distinctly perceive by the natural light is not only indubitable but also that the claim that it is true is also indubitable, which emphatically is not to say that the claim that it is true is itself true.
We know that if God exists, then the Truth Principle is true. Yet as we have discovered, there are important objections to this empiricist line of thought. In the first place, we really cannot tell from any series of finite integers just how to continue them, which is to say that an initial finite series does not determine a unique successor series. Moreover, even if a finite series did determine a unique successor series, we could not carry it out.
The best we could do is to say that we would need to carry the series on forever , meaning without end. But the concept of carrying on indefinitely, without end, obviously requires the concept of infinity itself. Yet, at least according to Descartes, infinity is not a negative concept. Indeed, in the case of the transcendental numbers we grasp the concept of infinity directly, for example, as the ratio of the circumference of a circle to its diameter or as the ratio of the diagonal of a right isosceles triangle to its side.
Recall how the Meditations on First Philosophy begins. The key point is that Descartes does not find a reason to doubt mathematical reasoning, which presupposes the concept of infinity 12 He does not doubt that in order to have the concept of infinity from mathematical reasoning it must be that God exists.
Further he finds it to be impossible to doubt that whatever he clearly and distinctly perceives is true provided that God exists. So, he finds it impossible to doubt the Truth Principle, which therefore is justified by his own standard , which is to affirm all and only what he clearly and distinctly perceives by the natural light.
The conclusion of this paper is that Descartes did successfully finish his project for the reconstruction of his knowledge and that he therefore was in a position to sort out just what is commended by reason and what he had been taught by unreliable authorities. His project depends upon his view that mathematical reasoning is indubitable, and that mathematical reasoning presupposes the idea of infinity, which could only be derived from an infinite mind, which is to say the divine mind. This is does not mean that Descartes has proved his thesis and forced agreement by those who disagree with him about what is dubitable.
But then again, Descartes does not claim to satisfy anyone else, he seeks only to satisfy himself and to share his good epistemological fortune with those who care to take notice of it. However, it is only in his final conversation with Burman that he flatly and unequivocally insists that mathematical reasoning is the foundation of all reasoning, including metaphysical reasoning There are at least two ways in which this contribution is limited.
First, it is limited because we ourselves find the concept of infinity difficult to grasp and consequently ever slipping away from our conceptual grip. On the other hand, it was the clear-headed, the tough-minded Bertrand Russell who demonstrated that arithmetic as we know it basically what is central to Newtonian physics , can be axiomatized within ordinary set theory and logic with two additional principles: the Axiom of Infinity and the Axiom of Choice Russell, pp.
In the replies to his critics, he often seems to become impatient with criticism. There are far too many examples of this to detail here, but in this connection, it will perhaps be helpful to remember his voluntarism. Indeed, although Descartes does not doubt that what is clearly and distinctly perceived by the natural light is true, he is completely open to the thought that not everything that is true can be clearly and distinctly perceived by the natural light—at least not by us humans.
After all, did those critics really mean to claim that what they clearly and distinctly perceive might be false after all, and if so, is that because they clearly and distinctly perceive that what they clearly and distinctly perceive might be false?
Did they really mean that they doubted reasoning itself, even the reasoning that led them to think that what they clearly and distinctly perceive may nonetheless be false? Of course, further research includes not only the Cartesians but also empiricists like Locke and Hume, and idealists like Kant. It also includes Newton as well as the rationalists who followed Descartes, especially Leibniz, who prepared the way for Gauss, Lobachevsky, and other great mathematicians of the nineteenth century.
The semantics of formal languages were initially developed by Tarski. For a very helpful introductory discussion of this issue see Beth, pp. The proof depends upon a reductio ad absurdum that purports to show that the opposite supposition that the diagonal is commensurable with its side entails the contradiction that an odd number is even Jenkinson, I, p.
However, the actual proof of incommensurability is attributed by Russell to Euclid. There is no logical reason to think this axiom, which asserts the existence of infinite collections, is true, but neither is there a logical reason to think that it is false Russell, p.
Descartes might well agree that we cannot, for example, prove that there are infinitely many particles in this world.
For Descartes, however, numbers are not in this world. There are collections of things in this world that are numbered, but that does not prove or presuppose that numbers are things in this world. It is true that there is some textual evidence for a different reading. The last part of the quotation seems to imply that pure mathematics includes the study of the nature of corporeal objects. It is certainly true that Descartes thinks that pure mathematics can be applied to the material world because the essence of matter is extension, and pure mathematics describes extension , which is an attribute of matter, but which is not itself corporeal.
As Cottingham et al. However, there are other arguments in mathematics that may be sound but are not so easily validated by the natural light. Descartes acknowledges that many arguments require detailed critical analysis and are open to doubt.
Only after extensive criticism and revision are finally perceived clearly and distinctly by the natural light. It is not part of Descartes theory that every mathematical question can be easily or definitively settled.
Perhaps the following is an example of what would appear to be a simple problem, but which actually is puzzling. It is the problem of accounting for the exponent 0 and, in particular, the problem of finding the value of positive integers that are raised to the exponent 0. But we can be sure that God exists only because we clearly and distinctly perceive this. I mention it here only because the troubles caused by equating the affirmation of an idea and the affirmation of its truth plagues all seventeenth philosophy.
Higher order infinities are generated from the infinity of the reals by the powers set operation Russell, pp. One may claim that these axioms are not clearly and distinctly perceived by the natural light, but I do not think Descartes would agree.
That is because these axioms are presupposed by the truth of familiar mathematical principles for example by the algebra of the conic sections without which we could not even state the principles that describe the motion of the planets.
Indeed, if we think of something as infinitely good or powerful or wise, we do not seem to be making quantitative judgments. I believe, however, that Descartes would insist that at bottom, all references to infinity must be reduced to quantitative judgments. That is essentially what it means to say that metaphysical reasoning is subordinate to mathematical reasoning. Perhaps it will be readily granted that this view is plausible when it comes to space and time.
Space and time are measured quantitatively; so, if we say that Euclidean space is infinite, we must mean that its measuring stick must contain infinitely many marks. But what are the marks by which we measure infinite goodness or wisdom or power? Even so, good deeds can be counted, both with respect to frequency and comparative value, as can the number and comparative significance of truths that are known and finally as well as by the comparative potency and frequency of acts of will.
Without some form of measurement , goodness, knowledge, and power are essentially incomparable. It is right to attribute this type of view to scientifically-minded early moderns like Descartes. The authors declare no conflicts of interest regarding the publication of this paper. Fourth Set of Objections. Cottingham et al. II, , pp. Cambridge: Cambridge University Press. Antoine, P. Buroker Trans. I Think, Therefore I Am. Doney Ed. Reprinted From: Ayer, A. The Foundations of Mathematics 2nd ed.
Amsterdam: North-Holland Publishing Co. Conversation with Burman. Cottingham Trans. Oxford: Clarendon Press. The Philosophical Writings of Descartes Vol. I, pp. Rorty Ed. Discourse on Method. Meditations on First Philosophy. Second Set of Replies. II, pp. Principles of Philosophy. The Circle, from Descartes Point of View. Open Journal of Philosophy, 7, Fifth Set of Objections.
Cogito Ergo Sum: Inference or Performance. The Philosophical Review, Vol. LXXI, No. The Third Set of Objections. Aristotle Stagiritis Son of Nicomachus.
McKeon Ed. We admit that those dreaming sensations do not correspond to reality, so why are we any more certain of our waking sensations? How do we know that any particular sensation is not just a dream, a sensation stemming from causes unbeknownst to us?
This second argument is popularly referred to as the "Dreamer Argument. Descartes next casts doubt onto our mathematical demonstrations and other self- evident truths. In order to do this, he first points out that people are sometimes known to make mistakes when it comes to these subjects. In addition, he claims, for all we know, God or some lesser being is manipulating our thoughts, causing things to seem certain when really they are not.
This argument is commonly referred to as the "Evil Demon Argument". After attempting to undermine all of our beliefs, Descartes identifies one belief that resists all such attempts: the belief that I myself exist.
This stage in Descartes' argument is called the cogito, derived from the Latin translation of "I think. The cogito is arguably the most famous argument in philosophy, but what is it really supposed to prove? What is Descartes' purpose in beginning his magnum opus with such a trivial piece of knowledge?
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