How Long is a Piece of String?

In the cult classic, House of a Thousand Corpses, the infamous Captain Spalding replied facetiously to an overly curious visitor, who had ventured into his roadside country store, "How long is a piece of strang?." Those of you who saw the film will vividly recall these anxious moments. The young man's wry smile quickly gave way to nervous laughter, and ultimately to horror, as he and his friends were treated to an experience of a lifetime! The aptly named Murder Ride would not disappoint. The captain's sarcastic and superficial retort belies a deep philosophical conundrum, which is quite relevant to a serious contemporary inquiry.

To date, no mathematician, no philosopher, no cosmologist, has adequately demonstrated (to my mind at least) that an actual infinite can exist in reality. Of course, infinites are quite malleable and useful in transfinite arithmetic; but, infinites in actuality are much different. In fact, it seems obvious that an infinitude of existing things—apples, quarks, moments or universes—is impossible to instantiate. Note that it is just as impossible that infinite number of bosons exist (in space) as it is an infinite number of elephants do. Each is equally implausible, even though one is much smaller than the other.

To make this point clear, a distinction should be drawn between an actual infinite— this denoted by the aleph symbol followed by an integer such as 0,1,2, etc. in transfinite arithmetic—and a potential infinite (indefinite) which is symbolized by the sideways (lazy) 8, sometimes called a lemniscate. Georg Cantor, the famous mathematician, also introduced us to the idea of "absolute infinity, to make matters worse. Both cosmologists and laypeople often conflate these concepts when speaking of eternal futures, beginningless series', or infinite space-times. Of course, the problem with many of their construals is that any one or more concept of infinity may be in mind. What has been deemed the potential infinite is quite dissimilar in practice to the actual (completed) infinite. With respect to expanding universes like our own, only the potential infinite seems applicable.

We can demonstrate the concept of the potential infinite; or, at least understand the idea of infinity as a limit—some call this indefinite to avoid confusion--when speaking of actually existing things. I can conceptualize, or think about, possessing an infinite number of oranges, for example. Counting or acquiring an infinite number of them is quite a different thing, however. Cantorian set theory—it purports to show that actual infinites do exist—allows us to envision two infinite sets (of oranges), or even an infinity of infinite sets. But, if the one group (of oranges) is infinite, how can another set be added to it, and the sum still be considered infinite? Yet this is exactly the type of thing permissible on transfinite arithmetic. Therein, you could (in theory) add an infinite number of infinite sets (of oranges), even though you began with. No matter how many oranges are added to the existing number, an infinite quantity of Valencias remains. It is quite obvious that this type of infinite should never be utilized to argue for infinitely many oranges.

The potential infinite, or indefinite, is also exemplified within the following scenario: Imagine I wanted to begin counting whole numbers from one to infinity at this very moment. Perhaps I could enlist the services of the world’s fastest computer, which would continue to count for several generations, at millions of calculations per second, or faster. How long do you suppose it would take to reach infinitieth? I dare say the computer would expire long before reaching our goal. Despite all its efforts, after many, many, years, it would have made no relative progress in this endeavor. Even if the computer had an infinite number of days, years, or centuries, at its disposal, no advancement could ever be realized. At any point in the future, the counting computer, would still have infinity to go before reaching its hypothetical limit. Likewise, counting backwards from an infinite future, a first number could never be reached. If a last number cannot be reached, counting down from infinity could never even begin. Ironically, infinitude would be rendered finite if either were acheived, or even conceivably acheivable.

Whether or not a collection of individual universes is viewed as linear or en mass (occurring simultaneously) the infinity problem persists. None of the finite and tangible entities is eternal, nor is the entire collection, which is just a set of definite and discrete parts. As we discussed above, the number of components in the universe constitute a potential infinite, which could be added to or subtracted from. (Noteworthy is the fact that subtraction from infinity always results in a finite number, which is why the action is effectively disallowed in mathematics.) So, a group of finite quantities (universes) can never be actually infinite, as one more universe could always be added to the current number, no matter how many exist at any given moment. Some cosmologists, despite this seemingly intractable problem, persist with the idea that the universe (or multiverse) as a whole is infinite, even if no single constituent is. But, this is logically and physically impossible. You simply can't get to infinity by adding together finites.

Let's assume, nevertheless, that the whole universe, multi-verse, meta-verse, is indeed infinite. Another somewhat lesser known problem arises. If the wider multiverse is infinite, how does it interact with our current finite one? If we have any understanding of cause and effect it is this: A finite cause produces a finite effect; an infinite cause an infinite one. Imagine that the causal conditions, which produce universes is perpetual—never mind that this violates the second law of thermodynamics, and bringing in new mass/energy violates the first law as we are just fantasizing here—how then does it interact with our finite "bubble" universe, on that view? Why is our universe only 13.8 billion years old and expanding? Why hasn't our universe been here from infinity past, if the causal conditions have been? And why hasn't our universe already gained maximum entropy? Why didn't an infinitude of evens transpire an eternity ago? Why is the universe now undergoing change, and events yet taking place? How does it continue to expand, when it should be "infinitely large" already? That is, if there were such a thing as an infinite past.

Infinitely large, with respect to universes, appears problematic on a number of fronts, to say the least. Let's now examine the other extreme: Infinitely small (infinitesimal). A shoestring, for example, may be continuously divided into smaller and smaller strings—whether by actually cutting them, or simply by imagining them being divided in a hypothetical scenario. As we cut the strings in half, with each stroke of the knife or pen, we double the number of (smaller) strings—which by the way are also infinitely long—without ever reaching our goal. In fact, each string could be further subdivided (logically if not physically) an infinite number of times, without ever introducing a new piece of string. When you think about this scenario, it turns out that every piece of string—no matter how short or how long—is potentially infinite in length. i.e., Everything is divisible to an infinite number of parts. So then, how long is a piece of string? Infinitely long, in fact. The absurdity becomes even more apparent when we think about the fact that a centimeter long string can be potentially divided the same number of times as a string encircling the earth: Infinitely many times, of course.

On the above examples, it is quite easy to see that infinity is really just an idea in your mind that is never attainable or quantifiable. Cosmologists have hoped, nonetheless, to portray our universe as eternal, or beginningless, primarily, I think, because of theistic implications (of an absolute inception). With respect to meta-verses, multi-verses, multi-dimensions etc., it is not possible that an infinite number of these exist, as one simply compounds the problem by invoking them. Numerous problems come to the fore when infinite universes are considered. Besides the problems already mentioned, another serious issue regards the idea of spatial extension. An infinite universe would presumably occupy an infinite amount of space. Consequently, only one infinite—even if it could exist—would be possible, because it would necessarily pervade the entirety of infinite space. For me, the idea of multiple infinities is oxymoronic.

 To help illustrate my point, envision an infinitely large balloon—this is obviously a mind experiment that assumes there are infinites, and that those can be added to other infinites, which is not my position—now, add another infinitely large balloon then another, then another, and another, ad infinitum. At first we notice that either (a) the original balloon was not actually infinite—it requires more air perhaps to occupy an infinite amount of space—or, (b) the balloon will have already coalesced with the other infinitely expansive balloons an infinite time ago. Even if we assume space-time is flat, it is clear that multiple spatially extended infinites cannot co-exist, even if space is expanding to accommodate the ever growing space-times—the universes would have surely coalesced infinity ago (If we deny there is interaction among the various dimensions, why bother with the extraneous concepts at all?). Even if we allow that infinite space could accommodate everything by continually expanding, we are right back to a single coalescence occurring infinitely long ago, which is nothing like what we now observe.

Continuing our evaluation of the potential infinite—this notion of indefinite alluded to earlier—let's consider just our own space-time reality. The universe we reside in, the only observable one we'll ever appreciate, is demonstrably finite and potentially infinite, in at least one other regard. Our space-time continuum began a finite time ago: 13.8 billion years in the past (If the universe is actually infinite, this age is just an arbitrary measure, and we can discard it readily, along with the age of the earth, carbon dating of other objects etc.). Our universe couldn't be actually infinite, because it had a beginning, and is now expanding (indefinitely). Though all known evidence points to the fact that the universe will expand forever, never being infinitely large, it should not be characterized as actually infinite in duration, as the attainment of infinitude is impossible, on grounds we have already discussed. Neither can our universe be claimed to extend infinitely in the earlier than direction, logically or scientifically (See Borde, Guth, Vilenkin 2003.). Though it seems plausible that a physical universe like ours could have a beginning without an end—it could expand from now on. The universe could not have an end, or even the capability to progress, without a beginning. Since we know that infinitude involving actual physical objects is really indefinite, we can logically surmise that because physical entities, including universes, are measurable, and quantifiable, it is improper to characterize them as infinite.

If a multi-verse does exist, it too is finite. Of this, we can be quite certain, again based on what we know about causality. If, for example, water was to exist in a universe, in which the temperature were perpetually below 32 degrees, it would be frozen for the duration of that universe. In any hypothetical meta-verse, a universe contained in, or interacting with, the others would necessarily be of the same sort, whether finite or infinite. The fact that all things contained in our universe, including us, are demonstrably finite is very strong evidence that there is no multi-verse, or the multi-verse itself had a beginning. If the causal conditions are always present, so too the effect. If the effect (our 13.8-billion-year old universe) is finite, so is any proposed multi-verse—at least one that could interact with or cause our space-time reality. One proposed solution to this apparently intractable problem is the introduction of volition, or will, in what philosophers refer to as agent causation. A person sitting from eternity could will to stand up, for example. From a timeless state, a universe might be created in time, via this volitional agent. Though controversial, the idea is at least plausible, and there is no logical preclusion. The proposed agent—perhaps like our minds—may not be spatially extended, but capable of interacting with objects in space-time. No physical state involving perpetual affects is afforded the luxury of maintaining the quality of infinite, while producing its effects in time. In the case of the latter, time could not have ensued, and no state of affairs involving events now occurring could ever have not obtained. In other words, all things that will ever occur have already done so, if the physical past is indeed actually infinite.

Still, some cosmologists proffer the idea that the universe is infinite in the sense that a hypothetical observer could travel up in a straight line, forever, and end up back where he started—multiple times in fact—because of the curvature of space-time. This suggestion is comparable to how one might travel the surface of a balloon, or hoop, and go round and round forever. But, what does such a scheme have to say about the infinity of the universe? Very little, I think. I can travel around the globe indefinitely, but few will argue that the earth is infinite in any respect. Not only do these representations not answer questions about infinity, they raise additional questions, which are problematic in their own right. Questions like, how did this demonstrably finite space/time originate from an infinitude, and why does the universe exist at all?, come to mind. Our potentially infinite universe still requires a cause in order to begin—

inflation began on modern views—a finite time ago. It cannot be causal to itself, or the universe would have to exist before it existed. Nor can our universe be the result of a beginningless series, which, as we have seen, must have a terminator, in order to progress.

In summary, it turns out that every constituent of the universe—and the whole—can be considered infinitely large or infinitely small, depending on one's perspective. Nevertheless, if all things within the universe, or even the universe itself, are infinite, there just is no perspective one might take; no specific time or place from which we might make an observation--no foothold to stand. We are everywhere and nowhere at once. In order for material physical objects to be called such, and to interact with the whole, there must be a quantifiable relationship to the entirety, else the object can't be said to exist, in any meaningful way. If the universe is indeed infinite, we cannot be, as finite entities, related to the extremes. Nor could any other objects, irrespective their collective number. So then, how long is a piece of string? According to some scientific views, infinitely long. I and Captain Spaulding know, however, a string is only as long as you cut it.