Descartes's Method of Doubt

3. Descartes's Sum and Nature of Infinity

Book cover: A Brief History of the Soul by Stewart Goetz

All of Descartes's later proofs that he offered in his Meditations on First Philosophy for the existence of God, the dualism of mind and body and the existence of corporeal substance presupposed and required the existence of his sum. On Descartes's schema, without a metaphysical substratum for thought, any attempt at these proofs will not even get off the ground. Descartes's God, though, is of paramount importance in his theory of knowledge. After deducing his cogito ergo sum, Descartes isolated the clearness and distinctness of this idea as validating its truth. Descartes realized, however, that he may be deceived about what he clearly and distinctly perceived. This included such evident truths as those of mathematics and geometry [1641a: 64, 77]. It was only the existence of a God that could not deceive that would save the indubitability of his intuitions. This, then, is what he set out to prove.

Descartes was ambivalent about including his sum in the intuitions it was possible for him to be deceived about. Before embarking on his proof for the existence of God, he grouped it with the questionable axioms of mathematics [1641a: 77]. However, when he came to set out his proof, he included his sum as an indubitable premise [1641a: 79]. First, I will examine Descartes's proof for the existence of God and then see whether it can be used as guarantor for the infallibility of his clear and distinct ideas.

To undertake a thorough review of Descartes's proof would require more words than space here allows. Much of Descartes's proof is clothed in scholastic and neoplatonic garb, the premises of which have been largely passed by with the advances in semantics and logic since Descartes's day. Here, I include Descartes [1641a: 80–5] premise that everything that actually exists has an actually existing cause and his premise that a cause must contain as much perfection and reality as is contained representatively and actually in its effect.

Nonetheless, Descartes raises two pertinent questions that deserve serious consideration. These are, firstly, 'Can our idea of infinity be explained without reference to an external infinite being from which it derives its cause?' and, secondly, 'If not, what predicates, if any, can we validly attribute to this being?' Descartes's answer to the first question is, of course, 'No'. In answer to the second question, he offered such attributes as omniscience, omnipotence, immutability, independence and eternality.

Perhaps the pervasive mystery surrounding the concept of infinity and the intellectual difficulty we experience in coming to grips with it made it very fertile ground from which metaphysicians and theologians historically have sought to grow their grand metaphysical schemes. Descartes cannot be blamed for following this seductive path. In this vein, Descartes [1641b: 268] considered the infinite as 'real and positive', while regarding the finite as a 'non entity, or a negation of existence'. From this, Descartes [1641b: 268] concluded that 'that which is not cannot bring us to the knowledge of that which is'. The Greek neoplatonic scheme that he relied upon, in which things in nature are placed on a graded scale of reality, is no longer required to understand the origins of the concept of infinity. Things in nature are either real or not real, depending on whether its description has a referent or not. In his treatment, Descartes was only relating infinity to the notion of substance. But, of course, it can be applied to other concepts as well, such as space and time.

Is there an alternative, though, to Descartes's [1641a: 85] line of reasoning that 'I could not on that account have the idea of an infinite substance, for I myself am finite; unless, indeed, that idea proceeded from some substance that was really infinite'? He may have had some justification for this belief if it were the case that we finite beings could fully comprehend and imagine, in this case, infinite substance. However, needless to say, this mental feat is impossible.

Book cover: Dialogues Concerning Natural Religion by David Hume

One convincing way to understand the term 'infinite' is to see it as the denotation of the unimaginable result of a series of specific mental operations. Simply stated, the first of these operations is to imagine a given quantity, be it length, duration or any other suitable predicate, by means of a comparable standard. For example, imagine a three metre length (using the international standard metre bar). Secondly, by the mental process of addition, the given quantity is increased by a fixed, finite amount. In our example, imagine adding two metres in length to the original three metre length (yielding five metres in length). Thirdly, the same process of addition is applied to the new, resultant quantity. In our example, imagine adding an additional two metres to the five metre length.

The understanding of the operations just described does not require knowing beforehand what 'infinity' means. Such understanding does not require an infinite regress. The process of addition may begin and continue, as it so often does, from any point in time. Crucially, these steps can be performed by finite human beings. The final step is imagining this process of addition not halting. Here again, knowing the concept of infinity is not a prerequisite for understanding negation. The quantity that results from carrying out these three steps of fixing a finite quantity, successive addition and avoiding stopping, we call 'infinite'. We label the converse result, 'finite'.

By this reckoning, the words 'finite' and 'infinite' are appellations given to two distinct classes, one whose elements have a beginning and an end and the other whose elements have a beginning and no end or no beginning and an end or no beginning and no end. Although the two classes are distinct, the elements of the second class, to use Cartesian terminology, are by no means clear. This account of our understanding of the meaning of 'infinite' renders moot the requirement for the existence of an actual infinite being. In a sense, we have no positive understanding of 'infinity', the term serving as a logical construction fashioned from a finite number of mental operations and designed to aid our understanding of mathematics.

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