Zeno’s Paradox and Our Experience of the Infinite
Note: I owe much in the framing of the issue here to José Benardete, professor emeritus of Philosophy at Syracuse University and brother of University of Hartford Mathematics professor Diego Benardete. In particular, I refer the reader to José’s book Infinity – An Essay in Metaphysics. Clarendon Press, Oxford, 1964. In this book, he gives qualified support for the existence of the actual infinite. Here I do the same, although my argumentation differs.
Zeno’s paradox actually refers to a number of similarly constructed puzzles credited to Zeno, a follower of Parmenides, intended to support his master’s thesis that being or reality is unchanging, that there is no real change or motion, and that all appearance of such is illusory. The idea behind each puzzle is that the assumption that change or motion is real leads to a vicious regress.
Those dealing with the puzzles of Zeno often refer to them collectively as if they were one. One collective label for them is the bisection paradox. In brief, the bisection paradox is that in order for there be motion or change from A to B there must first be passage through the midpoint between A and B; but in order for this to occur, passage must be made through the one-quarter point; and for that to occur requires first passage through the one-eighth point, and so on, ad infinitum. The puzzle is that going from A to B requires as a precondition transit through an infinite quantity of partial “distances”. This is assumed to be logically impossible.
Modern mathematics provides a definitive solution to Zeno’s paradox according to one reading; but many think that it still remains unsolved according to another reading. The sense in which modern mathematics has solved Zeno’s paradox is in the denial of the premise that necessarily, the sum of an infinite set of finite quantities is infinite. It seems clear enough now that certain infinite sets of finite quantities yield a finite sum, in that the latter approaches a certain finite quantity as an asymptotic limit. So, for example, an infinite set that can be arranged into the following sequence has a sum that converges on 1 as a limit: 1/2, 1/4, 1/8, 1/16, etc. Even if one insist that the sum of such a set never quite reaches 1, the fact that it converges on a finite limit as its upper bound proves the sum of the entire set to be finite. This in turn refutes Zeno’s paradox according to its most straightforward reading, by admitting that an infinite regress is implied but denying that it is vicious, since it does not imply the contradiction of having to travel an infinite distance in order to go a finite distance.
In another sense, however, perhaps modern mathematics has not solved Zeno’s paradox. For although in the solution given above recourse was made to an analysis of the puzzle by means of a function converging on an asymptotic limit which by definition is never reached, Zeno’s paradox is arguably teasing us about the notion of what might be referred to as a transit through an infinite number of finite distances to a symptotic point, i.e. a point that is reached: the terminus of the motion or change. This leaves us with a different problem: how can an action actually be executed which requires us first to complete an infinite number of steps and then arrive at a certain point? This seems to imply a contradiction, since an infinite sequence is by definition one which has no end, whereas in this reading of Zeno’s paradox we are expected to get to the end of an infinite sequence in order then to accomplish something else.
As Leibniz himself conceded, there is no such thing as the infinitieth element of a sequence; the ordinality of every single step of an infinite sequence is finite. So if we have to get through an infinite sequence in order to execute any motion or change, it seems that no motion or change is possible.
The sense in which modern mathematics has solved Zeno’s paradox is typically accepted as being based on the well-developed notion of potential infinity. In this sense, we can see the infinite both extensively and intensively; extensively, to generate infinite series such as the natural numbers, etc., and even uncountable infinities; intensively, to ground clear and distinct conceptions of infinitesimal quantities, i.e. quantities which, though technically greater than zero, are less than any specifiable quantity greater than zero. A prototypical example of such is the difference of 1 - .999….
The sense in which Zeno’s paradox has arguably yet to be solved regards the less well developed notion of actual infinity, or infinity fully executed in act; in actual things; in our actual experience. Since mathematics is not about the actual but the possible, it necessarily remains neutral on this point. Solving Zeno’s Paradox in this sense requires affirming the existence of the actual infinite. So the question is left open for us to ponder philosophically: is there in fact such thing as actual infinity, or can we rule it out as impossible?
Right away we see trouble in this query, however. For we already accept potential infinity, that is to say the possibility of infinity is granted. So then if we then turn around and rule out the possibility of actual infinity, how can that not mean that infinity is not possible? If something is potential, that makes it not impossible; conversely, if something is impossible, that should rule it out as potential. How can it make sense to affirm something potential but not potentially actual?
This is why actual infinity was so easy for Leibniz to accept. As an essentialist philosopher, he considered essence or possibility to have primacy over existence, which is to say he accepted that essence causes existence. In the first place, according to Leibniz, God’s essence is the cause of God’s existence. Unlike the medieval existentialist Thomas Aquinas, who insisted God was the uncaused first cause, Leibniz followed the essentialist tradition of John Duns Scotus in seeing God as self-caused first cause. Applying this notion to creation, that possible world will exist which has the most compossible essence. Therefore, there will be as much actual infinity in the actual world as can be fit into it. Actuality is nothing more to Leibniz than optimal possibility.
Of course, the affirmation of potential infinity X can be intended in at least two senses: one affirming that any of X is instantiable; the other that all of X is instantiable. Accordingly, without forcing a decision on whether essentialism is true or false, we recognize that we must be able to distinguish between Claim A: that anything is possible, and Claim B: that everything is possible. Perhaps, for example, our economy is one in which anyone can be rich, but this certainly does not imply that everyone can be rich. This applies to our topic in that potential infinity perhaps is nothing more than the potency implied by claim A: e.g. a line segment may be divided at any point along its length, but not at every point at once. Claim A possibility alone does not provide the grounds for actual infinity; only Claim B possibility does. Aristotle’s combined acceptance of potential infinity and rejection of actual infinity seems to be based on his combined acceptance of Claim A possibility and rejection of Claim B possibility.
To be sure, we should all agree that some things are possible in sense A but not sense B; for example, a line could never be actually divided at all points, since that would imply that it is divided at any two adjacent points along the line. But there is no such thing as two adjacent points along a line; there will always be points in between.
But why should we reject Claim B possibility altogether, that in some cases the entire infinite potential of a function might be activated all at once? To the extent that we may take sensitivity to Zeno’s Paradox as a clue, might such a rejection be prejudiced by an inability to separate our imaginations from the constraints of time, so that we rule out the notion of a fully executed infinity due to time constraints? If so, we should recognize that a fallacy is involved here, which is the failure to take into account the infinitesimal interval. That is, in considering the execution of, say, an infinite convergence, which we recognize as a function “endlessly” converging on a limit which it “never” reaches, we might be failing to take into account that the infinite bulk of the temporal intervals involved in the process are infinitesimal, which means that they collectively would occur in practically no time at all.
In addition, to argue for an a priori impossibility of actual infinity would inappropriately force commitments on matters of fact, such as whether matter/energy exists beginninglessly or not. Modern science does not rule this out, and most who answer it in the negative do so on other than logical or metaphysical grounds. Perhaps the latest Big Bang was in fact preceded by a Big Crunch, and that by a previous Big Bang, with another Big Crunch before it, and so on. The truth about the matter may be for us an empirically undecidable matter from where we stand, but nonetheless it is a meaningful question whose answer, if it could be had, could only come from some kind of observation. It is simply not a question to be answered a priori, as even the great theist Aquinas conceded.
To decide on whether actual infinity exists, it will do us well to specify what we would be looking for when we are looking for actual infinity as opposed to potential infinity. The answer is: discovered empirical or at least discovered noticeable content. The recognition of potential infinity is vacuous of content. It is merely an a priori logical, topological or temporal analysis of the given. Nor could our awareness of infinity be merely by construction. We do have constructed notions of the infinite, but they are secondary in that our invention of them depends teleologically on prior intuitive awareness, so we would know what we were building toward. In this vein, there are two principle strategies: construction by negation and construction by mathematical induction. In the first case, we arrive at a constructed notion of the infinite by negation of our notion of the finite. This is most clear grammatically, as infinite is derived from the negation of finite. In the second case, we intuitively stipulate the existence of at least one element of a class, then provide for the production of the rest of the class by an iterative rule or rules. As such, for example, we define the natural numbers by first stipulating that 0 is a natural number, then adding the iterative rule that any natural number plus 1 is a natural number. It is psychological absurdity to think that our awareness of the infinite comes from our doing such things. Logic does not of itself produce content. We could only accomplish such constructions based on the direction given by an awareness of the infinite already present in us.
In short, to decide whether there is such thing as actual infinity, we have to notice whether here are content-rich cases of infinity in our experience. Here I will give three examples: one regarding human language, one regarding the (“horizontal”) continuity of actual detail, and the third regarding the (“vertical”) bottomlessness of actual detail.
Anthropologists have from time to time made attempts to see whether other animals can speak or meaningfully use human language at least in some rudimentary form. They have made considerable headway in some way, but in another way have been consistently stumped. Whereas they have been able to get various other animals to memorize and more or less consistently use arbitrary labels to refer things and even actions in their experience, this has clearly been limited to their use as proper labels, not as common labels. The difference is that proper labels refer opaquely to a fixed, finite set of “noticeables” in the user’s experience, whereas common labels refer through the mediation of meanings to infinite classes of noticeables. Human language, in short, is made possible only by our pre-linguistic awareness of infinites. I consider this to be an awareness of actual infinity, since it is not a vacuous but a content-rich awareness. To be sure, the infinite classes themselves that we are recognizing pre-linguistically are not populated by infinite numbers of actual existents; but we spontaneously recognize that the meanings themselves are infinite in reference. No one can really know what ‘chair’ means unless she understands it to refer to all chairs, the complete, infinite class of them.
We don’t accomplish this awareness little by little, but all at once, which is the only way to grasp actual infinity. This awareness is infinitely rich, since we can never completely finish saying exactly what chairs are, and exactly how we can so competently distinguish chairs even from chair-like non-chairs.
If someone objects that ideas are not actual, I respond that here we are not focusing on ideas in the platonic sense, or even in any other general or intersubjective sense, but in the psychological sense, as established in time in individual minds. (This point is made without prejudice against the viability of the other senses of the term.) In this sense, ideas are actual because they are acquisitions actually made in the world.
Whatever we notice or observe about the actual world, we recognize spontaneously that it is continuously detailed. As we delve into it, we discover always more detail in between other details previously noticed, in such a manner that we recognize it as infinitely rich. This is due to no mere inventive projection of the mind; for if it were, the result could not be continuous. What humans invent is finite and discreet. If I write a novel, my character development consists in a finite number of words and ideas; I depend on the reader to imaginatively fill in the gaps I inevitably leave. Even the reader will only succeed in filling them in a bit more than I have. But the result will be that there is no real chance for a sane mind with a sufficient finite amount of time to devote to the task to confuse what we invent with what we discover. What we invent has discreet, limitedly rich detail; what we discover has continuous and therefore infinitely rich detail.