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As we discovered last week, the concept of infinity is one that has far-reaching implications not just for mathematics, but for philosophy and theology as well. Far from being a mere mind-bending theoretical concept, infinity has actual bearing on the real world, on the finite bound “here and now” man. As mathematician John Byl explains it:
The concept of infinity is the key to the philosophy of mathematics. We can distinguish between potential infinity and actual infinity. Potential infinity is the notion of endlessness that arises from human counting. We soon realize that, given any large number, we can always obtain a yet larger one by adding 1 to it. There seems to be no largest number. Potentially, we can go on forever.
Actual infinity, on the other hand, is the notion that numbers exist as a totality, as a competed set. Plato believed in actual infinity; his student Aristotle held to only a potential infinity. Theological considerations led medieval philosophers to postulate an actual infinity. Today, however, humanism views actual infinity with suspicion.
The distinction between potential and actual infinity is quite revealing to one’s ultimate view of reality. Plato’s belief in actual infinity, led him to further ascertain the concept of Forms, or the ultimate perfection. Plato was the first philosopher to look beyond the material world and postulate a perfect metaphysical “Good” (not God) of which the physical world is only an imperfect “copy.” “Whereas pre-Socratic philosophers thought of reality as material stuff of some sort, Plato now designated the nonmaterial Ideas or Forms as the true reality. Similarly, whereas the Sophists thought that all knowledge is relative and changing, Plato argued that knowledge is absolute because the true object of thought is not the material order but the changeless and eternal order of the Ideas or Forms.” Plato’s metaphysical basis for ultimate reality made logic possible because it was fixed on “something” outside of the observable material world of change and flux.
Aristotle on the other hand, broke with the beliefs of his teacher and postulated a potential infinity. For Aristotle infinity was something that was strictly theoretical, ever-expanding without completion. To his mind it was essentially useless for any practical application. He was “less interested in mathematics than Plato and more interested in empirical data.” In this sense, Aristotle is something of the patron saint for modern science and especially the “New Atheists.” He was attempting to maneuver the razor-thin fence between the singularity of Plato on one side and the plurality of the Pythagoreans on the other. The Pythagoreans “rejected the basic assumption…that reality is one. Instead, they believed in a plurality of things, that there exist a quantity of separate and distinct things and that, therefore, motion and change are real.” Aristotle understood and tried to avoid the logical trap of the Pythagoreans due in large part to Zeno’s “paradoxes of motion.”
Almost a century before Aristotle, Zeno had shown the absurdity of the Pythagoreans assumptions by using the presuppositional method. Since the Pythagoreans believed that the material world was composed of an infinite amount of “stuff,” they concluded that all things could be divisible infinitely. Zeno cleverly revealed that according to this belief, motion is absurd, because movement would require traversing an infinite amount of points in a finite amount of time. His imaginary footrace between Achilles and the tortoise is an example that Douglas Hofstadter uses repeatedly throughout his book Gödel, Escher, Bach; which the author himself describes as “a very personal attempt to say how it is that animate beings can come out of inanimate matter.” Hofstadter’s attempt is precisely that and he is no more successful than the Pythagoreans, despite his more than 700 pages of technical arguments and co-opting of Zeno. No matter how hard the materialists try, they cannot marry the infinite with the finite. They cannot “have their cake and eat it too.” Their quest for a “theory of everything” is actually over before it begins, as Zeno masterfully points out, because of their faulty assumptions and illogical starting point. The finite will never contain the infinite, no matter how hard the materialists push, pull, and obfuscate. But, as we will see next week, the finite fits quite easily into the infinite, which is the starting point and first premise of the logic of the Bible.