A Mathematician’s Shortcomings. Descartes’ Faulty Use of Math and Science.

Descartes lays the groundwork for mathematics as an example of truth a priori in Meditation One (Concerning Those Things That Can Be Brought into Doubt), but not without an asterisk. The outlier is that if “God is a deceiver,” we cannot be certain of anything mathematical because a deceitful God may just be making them appear as absolute truths. To counter this, Descartes presents his Ontological Argument for the existence of a non-Deceiving God in Meditation Five (Concerning the Essence of Material Things, and Again Concerning God, That He Exists). He reaches affirming conclusions using the following methodology: A) God is the greatest possible being; B) Being the greatest possible being requires the possession of perfection in every aspect; C) Existence itself is a perfection; and thus D) the greatest conceivable being must exist. Therefore, the conclusion that God is all encompassing and non-deceitful yields the respective conclusion that mathematical statements are true a priori and can consequently be used to perceive and understand the corporeal world. We can thus determine that Descartes views math as the idealization of knowledge—specifically, what we can truly know about the external world has mathematical or scientific backing given that mathematics is the language of science. As such, his ideas must account for advancements in both fields to be considered indubitable. Sadly for Descartes, his theories are not consistent with some of the most widely regarded, influential, and themselves indubitable mathematical and scientific concepts. These include but are not limited to Albert Einstein’s concept of General Relativity, scientific regression of causes, plus Kurt Gödel’s Incompleteness Theorems. I shall follow with discussions of how Descartes’ impressions of the physical universe and of God do not follow each of the aforementioned theories’ guidelines. 

            In Meditation Three (Concerning God, That He Exists), Descartes ascribes God the quality of having fulfilled all potential; in other words, God is highest as there is no greater than or equal to. He reasons that human’s potential knowledge can be somewhat satisfied, but not infinitely as having this gradual increase is in and of itself indicative of imperfection. As for God, he states the following:

            While it is true that my knowledge is gradually being increased and that there are many things in me potentially that are not yet actual, nevertheless, none   of these pertains to the idea of God, in which there is nothing whatever that is potential (M3, 47).

 

Descartes is reasoning that because God is utmost in all respects, God cannot have anything potential included in him. By the nature of potentiality—namely, that it exists and is imperfect—one who has no potential has the property of having fulfilled all potential perfectly. If this be the case, God’s fulfillment of all potentiality precedes potentiality itself by Descartes’ own logic that one cannot infinitely gain knowledge due to the inherent imperfections. Looking at this through the indubitable scientific lens, the only way for God to exist is if he is capable of transcending time—thus being able to satisfy all potentiality before potentiality exists. The Theory of General Relativity does not forbid time-travel… given that that time travel is in the forward direction. However, Descartes’ logic assumes a reversal of the standard direction of time as forbidden by Einstein. The flaw here is obvious once exposed. An absolute truth (this being Descartes’ opinion of math and science) is one that is unbreakable with no outliers. Here though, in order to prove not only that God exists but also that he is perfect, he forces his concept of God to break an absolute truth—the scientific knowledge that time cannot travel backwards. If God were to break this supposed truth, it would thus make him a deceitful deity and counter Descartes’ argument from the outset.

            Regarding the infinities, Descartes states his idea of cause and effect in Meditation Three.

            I will once again inquire in similar fashion about this other cause: whether it got its existence from itself or from another cause, until finally I arrive at the    ultimate cause, which will be God. For it is apparent enough that there can be no infinite regress here […] (M3, 49).

 

Concisely, that the observable cause-and-effect chain cannot infinitely reduce; thus necessitating an initial cause—this being God. This is flawed in that even if infinite regression were impossible, the initial cause is not necessarily God. The logic lies in recent developments in scientific research that have proven that nothing (in this case, nothing can represent an empty, hypothetical space with no initial cause) is not actually nothing anymore. It is really a stew of “virtual particles” that pop in and out of existence at immeasurable time intervals. Despite the inability to time these particles, this “nothing” has observable and measureable mass; so much mass in fact that over ninety percent of all mass in the universe comes from this “dark matter.” This research is so accurate (to 10 decimal places) and high-regarded that the scientists were awarded a Nobel Prize for their efforts. The contradiction this causes in Descartes’ argument is that because given this, the universe does not need a self-sustaining first cause and can begin from nothing (if this “quantum foam” can begin from nothing, why can’t all life?). Thus, if there were a first cause, it is just as likely that it is scientific as it is theological. Science has taken us back to the beginning of time without a God as a first cause.

            I will not attempt to discern and entirely explain Gödel’s Incomplete Theorems as I have neither the necessary given knowledge nor the paper-space. For argument’s sake, it will suffice to say that these complex theorems establish inherent limitations on truly knowing anything but the most trivial arithmetical systems. Gödel argues that even the seemingly simple concept of the number one has an extremely elaborate logical underpinning. Still, I shall use the number one as the building point of my argument and progress towards applying infinity to Gödel’s theorems. Gödel argues that addition is the most basic arithmetic concept (and that all other subsequent concepts—multiplication, division, et cetera are derivatives of addition) as it is discernible into observable zeros and ones. Specifically, that 1 + 1 = 2 is an absolute truth; however, 3 + 1 = 4 is only an absolute truth because it can be written subsequently as 1 + 1 + 1 + 1 = 4. When this number increases towards infinity, these larger numbers are impractical as they are not observable. This is not only a mathematical concept that is not observable in corporeal reality, but also argues that mathematics is not necessarily always absolute. This digs at the very root of Descartes’ philosophies and makes his entire methodology inconsistent and reliant on observation.

            Within Descartes’ proofs, he consistently relies on math and science to guide his arguments. However, the three aforementioned ideas clearly contradict his conclusions. In essence, mathematics has failed Descartes. He perceived it to be the ultimate truth—the indubitable—and thus entrusted the field with the entire groundwork of his study; but he negated taking the dynamic and chaotic nature of mathematics into account. Today, proofs like Gödel’s prove that almost nothing is absolutely provable with math—a catch-22 of sorts. This faultiness counteracts all Descartes’ statements I have mentioned and lead to the following antithetical conclusions:

1)    Descartes has not proven God is non-deceiving. If he is indeed non-deceiving and did fulfill potentiality before potentiality existed, then science is an imperfect field and Descartes’ reasoning is wrong. If he is deceitful, he is imperfect and has therefore not fulfilled all potential.

2)    If an infinite regress of causes is indeed impossible, this first cause does not necessarily have to be God. His assuming it is God negates a scientific concept and therefore creates discontinuities in his methodology.

3)    That math itself is an open-ended and imperfect field. This makes Descartes use of math in philosophy at all look somewhat ill conceived as imperfect foundations cannot yield perfect conclusions.