A Critique of Cartesian Metaphysics: The Necessity of Infinite in Extension

Thesis

Descartes’ metaphysics denies the existence of the Infinite in extension, the physical universe, which is a key component of the Cartesian God. In fact, Descartes’ philosophy claims that God is not present in extension, but God only supports Descartes’ extended universe. In this paper, I will demonstrate 2 contradictions that make Classical Cartesian metaphysics untenable and unintelligible; that infinite must exist in the mathematics of extension to be perfect, and the instability caused by Gods absence in the material world.

Introduction

The Infinite must exist in extension if extension is based off geometric and mathematical truths, as the concept of infinite is exalted and must be a dependable concept in mathematics. If the Infinite is not in extension, then extension must run on an independent system of mathematics separate from Infinite, and thus God, which would lead extension to be an imperfect creation, making Descartes’ metaphysics heretical. Finally, if extension is void of Infinite and extension runs off its own mathematical system, then there must be a second “God”, deity, or force that rules over extension, which would contradict God’s omnipresence. This paper intends to show that any attempt to defend Descartes will become incohesive, contradictory, and untenable; furthermore, Classical Cartesian metaphysics should be reassessed and augmented for any level of serious philosophical study.

Part 1: “The First Flaw: Extension Being Void of the Infinite”

Throughout Descartes’ writing, there is a claim that extension, the physical universe, is based on perfect mathematical truths from God. In Descartes’ Principles of Philosophy Part 2, Article 4, he states:

“We shall perceive that the nature of the matter, or body considered in general, consists not in its being something which is hard or heavy or coloured, or which affects the senses in any way, but simply its being something which is extended in length, breadth, and depth.” (Principles of Philosophy pg 190, Selected Philosophical writings)

Further in Descartes’ Meditations, he states in Meditation 6, Section 74:

“But besides that corporeal nature which is the subject-matter of pure mathematics, there is much else that I habitually imagine, such as colours, sounds, tastes, pain and so on- though not so distinctly.”

These passages establish that Descartes’ extension or physical world can only be proven by pure mathematics and geometry. Furthermore, these passages are not limited to Descartes’ assertions that mathematics are the purest description of the physical universe. The Cartesian source of mathematics is stated clearly as the perfect creation of and from God which was placed into extension, and sustained by God. In Descartes’ Letters to Mersenne he states:

“The mathematical truths which you call eternal have been laid down by God and depend on him entirely no less than the rest of his creatures. Indeed to say that these truths are independent of God is to talk of him as if he were Jupiter or Saturn and to subject him to the Styx and the Fates. Please do not hesitate to assert and proclaim everywhere that it is God who has laid down these laws in nature just as a king lays down laws in his kingdom.” (Notre Dame)

This clearly connects that the mathematical truths that are in Descartes’ extension are actually formulated and perfect subjects of God.

              The controversy and contradictions arise that if the infinite can only be attributed to God, which is not present in extension, and mathematics are dependent on the concept of infinite, then extended mathematics would thus be imperfect. Geometry, algebra, calculus etc. all depend on the concept of infinite. If mathematics, which describes and rules Descartes’ extension, is laid down by God, but void of any infinite, then the mathematics that rule extension are thus imperfect. To clarify Descartes’ claims that infinite is only present in God, and God is not present in extension, I will exemplify a passage from Descartes’ Principles of Philosophy:

“God is not corporeal.. the nature of the body includes divisibility along with extension in space, and since being divisible is an imperfection, it is certain that God is not a body.” (Article 23, pg 167-168)

This points at the fact that Descartes believed that God was not extended, and that its being was not corporeal, or rather materially extended. If Descartes’ God is infinite and not in extension, there must be the conclusion that Descartes’ extension cannot hold the concept of infinite, or rather God. Here quite starkly is the extinction of God’s reach. Descartes tries to reconcile this with claiming that God merely sustains extension, rather than populating it. Although this is a strong assertion, it still leaves the concept of infinite outside of mathematics if infinite can only be God; thus, making extended mathematics imperfect, incomplete, and would cause a divide of mathematics of the mind, and physical mathematics.

              The implications for not including Infinite in extended mathematics are devastating for Cartesian metaphysics, as this would imply that extended mathematics operates on their own system. This would open the door to extension being an imperfect creation, which would directly go against God only creating perfect things. Further, if extended mathematics were to operate on a completely separate system than of God’s creation, it would open the doors to an alternate deity, God, or “being” ruling extended bodies, which inherently would make Cartesian metaphysics heretical. If extended mathematics were separate than the mathematics of the mind, then any calculations observed physically would also be different than the calculations of the mind, which would strictly go against mathematical truth, consequently further devastating Cartesian metaphysics and extension as a concept.

To also conclude that the mathematics of the extended world is different than that of the mind, God would also have disposed of the truth that the math of extension and the mind are synonymous, and mathematics would be committed to two realities instead of one; that is, one mathematics is of extension, and one of the mind. Mathematics is dependent on the concept of infinite even if infinite is not overtly stated, which would further prove that infinite must exist not only the mind, but extension itself; as it would be incoherent to have two separate mathematical systems in Descartes’ dualism if God made mathematics perfect and synonymous between mind and extension.

              Cartesians may then argue that the concepts of the Infinite were not introduced into mathematics until 1655 by John Wallis in his De Sectionibus Conicus, but surely Mr. Descartes was fully aware of the possibility of indefinite numbers going two directions away from 0. The concept of negative numbers was discovered in the 2^nd century AD, so Descartes must have known of their existence. There could be two indefinite sets of numbers running both directions away from zero, which the concept of infinite would have been plainly obvious. Descartes claimed that extension, which is based off mathematics and geometry, was only indefinite rather infinite numerical values.

An observation on Descartes’ infinite as only being regarded for God can be seen in a couple different places. One is his Principles of Philosophy:

“Our reason for using the term ‘indefinite’ rather than ‘infinite’ in these cases is, in the first place, so as to reserve the term ‘infinite’ for God alone.”(Article 27, pg 169)

This shows that Descartes’ God can only be infinite and have the attributes of infinite, and if God is not extended, this would exclude infinite in mathematical extension, thus basing Descartes’ extended mathematics on an independent system from God and incomplete, as extended mathematics would lack the concept of Infinite. This then opens the door to God actually being a deceiving God, which Descartes claims is false over and over again in his Meditations and Principles of Philosophy. If God created an imperfect extended mathematics that is void of infinite, and also conceptually different than the mathematics of the mind, then God must be a deceiver, as the extended mathematics would be different than that of the mind. If the concept of infinite were to be removed from extension, that would make the conceptual calculations including infinite different inside the mind. If infinite were void in extended mathematics, that would indicate extended mathematics has a limit, which is logically and theoretically impossible; let alone the implications that would have on modern physics, calculus, etc.

Ironically, Descartes in his Principles of Philosophy claims that God must not be a deceiver nearly four articles later after Article 29 with “Yet although God is no deceiver, it often happens that we fall into error” (Principles of Philosophy Article 31, pg 170). If God was not a deceiver, then the mathematics of extension and the mind would be completely similar in all ways, thus inseparable, but by removing the Infinite from extension, extended mathematics becomes deceptive to the mind, making God a deceiver; for it seems that Descartes’ will is overextending his intellect and that extension was relinquished by God to alternative deities.

If infinite does not exist in extension, then the mathematics of physical reality must be different in Cartesian philosophy than that of the mind. What would be physically calculated would never equate to a mental calculation, causing the deception of God’s imperfection via eternal “truths.” This would have implications that would lead to having two separate mathematical systems: one in extension, and one in the mind, which is plainly false and untrue. Calculations of the mind and extension have parallels that cannot be reconciled by removing infinite from extension. Furthermore, if infinite were void from extension, it would create a limited mathematical system, which in turn would be impossible, as numbers and shapes in extension can be divided an indefinite number of times, which hinges on the concept of infinite. Even if Descartes were to claim that he stated indefinite extension was only possible, this argument still hinges on the infinite because to understand the former, infinite must exist and be true. Furthermore, if extension did not have infinite, God’s infinite nature would not be true in all possible worlds, which has dyer philosophical implications on its own, as God must be a tautology in all aspects.

              Descartes must have known that numerical values could indefinitely go either direction of zero. For example, negative numbers, such as -1, -2, -3, etc., would indefinitely go on, while numbers 1, 2, 3, etc. would also indefinitely go on. What seems to be a further contradiction in Cartesian metaphysics is that two indefinites in either direction of zero would inherently create and function as an infinite sequence. Further indefinite number lines mean indefinite sets of numbers, which operate on the very principles of Infinite mathematically.

Had Descartes not overlooked this simple observation, he may have been the discoverer of ‘infinite’ as a mathematical concept rather than Mr. Wallis; but it seems that Descartes’ will to pursue a flawed philosophy blinded his intellect to preserve the righteousness of his ego. To make matters worse, Descartes claims that extension can be “divisible” in several places, which divisibility hinges on the concept of infinite, as well as all other mathematical arithmetic operations, such as multiplication and addition. The fact of the matter seems to be that it doesn’t take a genius to notice that infinite is a necessity for extended mathematics to even operate, as all mathematical operations hinge on this very concept.

Part 2: Infinite and Eternal Mathematical Operations

Even if Descartes were to argue that mathematical operations themselves need a starting point such as a number, and the initial operation imposes itself onto the number; the operations, such as division and multiplication, are timeless and infinite, as mathematical operations were neither created nor destroyed, but merely discovered. Take for example the number 113, and it was divided or multiplied over an uncountable number of times. The operation of multiplication or division doesn’t have a starting point, it simply is an operation of mathematics that is eternal and infinite that is imposed onto a beginning number, such as 113. This means that operations do not have a beginning nor end, but are rather eternal and infinite aspects of mathematics that are embedded into extension. The reason this is an important point to make is because nature herself operates off of the mathematical operations that man imposes onto number. Simply look how nature divides, multiplies etc. this would show that nature operates on the infinite and eternal truths that Descartes was pointing too. These operations hinge on the very factor of infinite via operations, further reinforcing that indefinite must hinge on infinite principles. Thus, the principles of infinite are actually embedded into the operations themselves.

At which point Descartes may argue that operations must have a starting point, I will counter with the contingency that the operation acting on the number also proves an infinite is present in extension, as the number itself must come from an indefinite, and the operation itself can be indefinite. As explored earlier in the essay, two indefinites must create an infinite. So therefore, operations acting on numbers, including those in extension, must be infinite as two indefinites must create an infinite.

Part 3: God as a Deceiver

Denying the concept of infinite does not exist in the extended world would perpetuate doubt into Descartes’ extension, causing a paradoxical contradiction between the mathematics of the mind and extended mathematics. This would lead an individual into a perpetual state of doubt, leaving them with neither clear nor distinct ideas, which would cause pure and utter chaos in a mathematical and mechanical system that Descartes’ God intended to create to cause order. This would perpetuate a doubt not only between extension, but also Descartes’ “mental substance”, causing clear and distinct perception to be continually contradicted; thus leaving one in utter disarray, distrust in God, and lack of certainty.

The implications of having two separate mathematical systems would then make God a deceiving entity. This would leave Descartes’ extended universe not mechanically perfect and flung into an imperfect mathematical system. Moreover, the extended mathematical system that would only exist to deceive and cause doubt to the mental mathematical system, as one would be perfectly true, and the other imperfectly false. If the continuation of doubt perpetuates between the broken system of mathematics that is void of Infinite in extension and the mathematics of the mind, divine truth becomes only deceptive, and well, unreconcilably untrue and false.

If Infinite would be void inside extension, then mathematical consistency would also be void in the extended universe. There would be a specific cap on extension’s “indefinite” reach and would also cause unreconcilable shifting in extension’s mathematical patterning, such as having a limit to potential shapes and extended calculations, thus making extended mathematics imperfect. Mathematician Georg Cantor, who devised infinite sets, states “the essence of Mathematics lies in its freedom.” If extended mathematics is limited, and excludes infinite, it is not completely free.

              An issue that does arise for this argument and Descartes also lays in the perfect geometry of the extended circle. The circle often associated with and representative of infinite, cannot be perfectly represented in the extended and material world; as it is impossible to create a perfect circle in extension. The circle can only be theoretically perfect, ergo, that the circle can only be absolutely perfect in the mind. Not only does this present the same issue for Descartes -- that there are imperfections in extended mathematics-- but also that for my own argument as well, due to the relationship of the circle and infinite.

A reconciliation to this issue is even though Descartes claimed that infinite may not exist in the extended universe which is false by modern mathematics utilization of the concept, but had he claimed the perfect circle is not in extension would have been more appropriate. Then again, Descartes would need to reconcile why extended mathematics is not perfect and different than the mind, still leaving him in a debacle. Even if a circle were to be made an indefinite amount of times in the extended universe, it would be imperfect, proving that extended mathematics are actually in fact deceptive, as the perfect concept of the circle in the mind is imposed onto the extended world. Alas, the imperfect circle may prove the separate mathematics of extension, which ultimately needs aiding by the perfect calculations of the mind. An example of this can be seen with how calculations that include a circle are perceived as perfect during the calculation, but in reality in extension the circle is in fact imperfect.

Part 4: “Extension Void of Infinite Infers God Lacking Providence”

              If the Infinite is void of extension, then the mathematics of extension is imperfect, and God is void of extension, leaving God completely absent from any type of sustaining power in the extended world. If Infinite can only be attributed to God, then it would be impossible for infinite to even support extension as it would be imperfect; let alone have a perfect mathematics influence it, as mathematics necessitates infinite to be complete. If infinite is absent from the material world, God’s providence, or even sustaining power in extension, is void, leaving extension to “the styx and fates.”

              As it stands in Cartesian metaphysics, extended mathematics must be void of Infinite, which would lead to a deceptive mathematical system that could only be created by God. God then becomes a pure deceiver, leaving man in a perpetual state of doubt un-sustained or supported by divinity; but, also allows extension to be governed by “demonic force” or another deity. Descartes’ claims in Meditation 3 “By the word God I understand a substance that is infinite, eternal, immutable, independent, supremely intelligent, supremely powerful, which created myself and everything else.” Truly, he is stating that God is independent of extension, and therefore infinite, which would create a collapse on extended mathematics.

Surely, God is “independent,” but would not be supremely intelligent if He created a broken mathematical system. If infinite were to be void of extension, this would also cause God to lack providential sustenance, even relinquishing Descartes’ claims that God is only a sustainer; but to mut, God would not be able to sustain an imperfect mathematical system if God was truly perfect. This would also have implications of math not being a divine truth, or rather a way to prove a connection between the mind and extension, and thus God could only be a deceiver that chooses to cause deception; this would lead sensory perceptions to be more truthful than God had intended.

              If God were to sustain extended mathematics without infinite, it would lead to God supporting imperfection, which would make God imperfect. If Descartes’ and Cartesian philosophers wish to base a metaphysical system on mathematics, it must be coherent; it is of mathematical necessity to include the Infinite, and if the Infinite then is not an exalted quality of God, Descartes’ conception and properties of a deity outside of extension thus collapses. This would go to show that if a deity exists that is truly infinite that created mathematical truths as a bridge between extension and the mind, infinite must be included in mathematics, or it creates a fractured truth between both extension and the mind.

Part 5: Reconciliation and Possibilities for Cartesian Philosophy

              To reconcile a metaphysical system that is based on mathematics being a deified truth between both the extended physical world and the mind; but, exclude infinite in extension would be to try to have imperfect mathematics for theological comfort, which is impossible. Either mathematics is unified under God including infinite, or it is disconnected, either choice leaves no choice for Cartesian scholars but to augment classical Cartesian philosophy to include the concept of infinite into mathematics, or simply say Descartes was wrong.  

              One way to reconcile such an overlooking would be to say that Infinite in the framework of mathematics must be included, but in a spiritual sense, God is the only thing that can be truly “infinite”; although this has implications if the soul can be eternal and infinite, which most traditions accept. Another solution would be to accept that mathematics of the mind is more perfect than that of extension; but at this point Cartesians would have to accept that Mathematics is no longer perfect or a divine truth in extension. The best way moving forward would probably be that the broken mathematics of extension simply represent the perfect mathematics of the mind, but this would mean God is supporting an imperfect mechanical extended world. This can be proven and seen with the concept of the circle, as the perfect circle in extension is impossible, while it is perfect of the mind.

              The final and most radical move for Cartesian philosophers would be to admit that the infinite God is both present in extension and of the mind. Accepting this would be the most radical move for modern Cartesians, and would demolish Cartesian dualism as scholars know it, and prove monists the victor. Anyway possible moving forward would demand a change in Descartes’ system of metaphysics.

Conclusion

Even if Descartes may be defended by a historically contextual defense, allowance, and grievance, such as Descartes would not have known about the mathematical necessity of infinite; Descartes’ metaphysics would not hold up to today’s standards and would need refinement or augmentation. If Descartes’ metaphysical system cannot accommodate modern mathematics, it must be deemed inadequate for metaphysical study, as metaphysical systems must be able to stand the tests of time. This is not to say that Cartesian philosophy does not hold any value, but rather this may be the final card in Cartesian metaphysics that makes the house of cards fall down.

This has been an exemplification of how extended mathematics must accommodate infinite, and that refuting Gods presence in extension renders the physical world metaphysically unstable due to God’s absence. The solution maybe to include infinite as a concept mathematically but also hold that God as a deity is also infinite, or quite possibly that the concept of infinite is God itself represented in extension. Wherever Cartesian philosophy leads, time may only tell, but as more eyes are placed on Cartesian metaphysics, the void of the infinite in extension must be addressed.

Works Cited

Cantor, G. (n.d.). Quotations "Georg Cantor". Math History of the United Kingdom, University of Saint Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Cantor/quotations/

Descartes, Rene, et al. Descartes: Selected Philosophical Writings. 2nd ed., Cambridge University Press.

Descartes, R. (1630). Letters To Mersenne. Notre Dame University. https://pls.nd.edu/assets/309772/

Wallis, J. (1655). De Sectionibus Conicus.