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Problems of Infinity

Charles Seife has written: “Zero…is infinity’s twin. …The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity.” Zero, p.2. I think that the statement is a serious oversimplification, and a misleading one at that.¹²

But, it is correct to suggest that these concepts give rise to problems in scientific applications. I think it is useful to explore why. As we shall see, the situations are quite different with respect to zero and to infinity. And, the differences are important.

The construct of zero

Mathematics certainly had problems with zero,¹³ but at least zero was a concept that could be related to the actual experience of counting (sort of, anyway). Contrary to some popular perceptions, one does not need zero for purposes of counting, i.e., the absence of any As (“no As”) does not have to be characterized as “zero As” in the sense that the presence of A and A is clearly “2 As.”

On the other hand, the use of zero as a "place holder" in larger numbers does make intuitive sense. For example, a number 100 represents one hundreds, zero tens and zero ones. It is clearly a convenient – indeed, a highly useful – convention. In fact,the first use of zero, Seife says, was not as a number but as a place holder to distinguish 1 from 10 or 100. Zero.,pp.26–53. See also, Deutsch, The Beginning of Infinity, p.131. The use of zero as another number is attributed to mathematicians in India. Seife, Zero, pp.66–71; Deutsch, The Beginning of Infinity, p.131.¹⁴

Zero may not have direct obvious application in daily life, but it is not in concept really different from the other cardinal numbers. See Whitehead, An Introduction to Mathematics, p.63. While there may not have been many pressing applications for the addition or subtraction of zero from a positive quantity, the concept does not seem foreign to our physical experiences. Even the notion of zero times a positive number (say, zero groups of five apples) has an intuitive sense.

The mathematical solution for zero has been effective in application to the physical world, with one important exception—division. The concept of division by zero seems of no use in our daily activities. As mathematics developed, however, it was deemed that a positive number divided by zero was infinity. See id., p.71. I say “deemed” to identify this usage as a convention adopted by man. I do not think that it is logically necessary or even logically persuasive to assert that a positive number “contains” an infinite number of zeros (or any zeros, for that matter) or that zero “can go into” a positive number an infinite number of times.

Take the well-publicized issue of “zero divided by zero.” (To get an idea of the extensiveness of the discussions, simply enter the phrase in Google and search.) If we address the question using mathematical conventions, we discover that there are multiple possible answers. For example, zero divided by any number is zero, so the answer is zero. But, any number divided by itself equals one, so the answer is one. So, if you view division as the corollary of multiplication (e.g., 6 divided by 3 equals 2, because 2 times 3 equals 6), then zero divided by zero can equal any number, since zero times any number equals zero. One might simply say that the answer is indeterminate. However, we are told that any number divided by zero is infinity.

Seife asserts that “if you wantonly divide by zero, you can destroy the entire foundation of logic and mathematics. Dividing by zero once—just one time—allows you to prove, mathematically, anything at all in the universe.” Id., p.23. This statement is certainly at least an exaggeration, but it does identify a problem with the mathematical treatment of zero. The structure of mathematics required a convention for the treatment of zero in mathematical operations;; however, the convention adopted for division by zero created problems--the problems of infinity.

The construct of infinity

A mathematical convention was also adopted for the concept of infinity. But, I think that infinity is quite a different type of concept. Infinity is simply not a factor or characteristic that we experience in our physical world. We can conceive of a series of numbers going on forever, with the sum of which series continuing to become ever larger and always capable of being greater than any number one would propound. Indeed, it is very difficult to imagine how such a string could ever end—one can always add one more number. Although, such an endless string can be imagined; it is not something that we actually experience. We can talk about something as being “unlimited” or “unbounded”. See, Deutsch, The Beginning of Infinity, at pp.164–5, 181. Such language is meaningful, even if the underlying concepts have no actual analog in the physical world.¹⁵ But, there is a gap between the conceptualization of such an endless stream of numbers and the concept of the something we have called infinity.

How should infinity be defined in mathematics? For example, is two times infinity greater than infinity? Obviously, it must be, in some sense; but, the concept of infinity would seem to indicate that the answer is no. Similarly, when we add or subtract a positive number to or from infinity, the answer is unchanged—it is still infinity. How can one infinity be larger than another?

The nineteenth century mathematician Georg Cantor “founded the modern study of infinity.” Id, p.166. The modern approach is based upon “infinite sets”—sets with an infinite number of members. “The defining property of an infinite set [or of an infinity] is that some part of it has as many elements as the whole thing.” Id., p.167. This notion can be illustrated by assuming a rule of one-to-one correspondence tying each member of a subset to a single member of the total set. If there is such a rule, then every member of the subset has a corresponding member in the set, i.e., both the subset and the set have the same number of members. However, at the same time, Cantor demonstrated that some infinities are larger than others, by showing that there is no one-to-one correspondence between certain pairs of infinite sets (specifically, for example, the number of points on a finite continuous line is greater than the number of whole numbers, since for any rule of correspondence between the points and the whole numbers, there are always points between the points that correspond to the whole numbers). Id., pp.170–71. The conclusion is that one infinity may be larger than another.

One consequence of the mathematical definition of infinity is that the concept of probabilities has no meaning in the context of infinite sets. One cannot say that one outcome is more likely than another. Id., p.177. See also, William Byers, The Blind Spot, pp.24–8. Another example is the "ability" of mathematics to model the performance of an infinite number of tasks in a finite amount of time. Here we have another insight into Zeno’s paradox of the tortoise and the hare. Mazur, The Motion Paradox, pp.38–9. “Mathematics tells us that it happens without explaining why [or, in this case, how].” Id., p.25. The infinite steps of the hare, each halving the distance between him and the tortoise, must take place without the passage of time. If time does not pass, then it is not surprising that the hare fails to overtake the tortoise. But, if time passes, then the hare will reach the finish line first.

When the mathematical concept of infinity appears in our physical theories, the mathematics often breaks down. See, e.g., Greene, The Hidden Reality (2011), pp.208–11; see also, Deutsch, The Beginning of Infinity, pp.258–62.¹⁶ And, of course, by convention, infinity is introduced whenever one divides by zero. Indeed, it seems that “[e]very time mathematicians tried to deal with the infinite or with zero, they encountered trouble with illogic.” Seife, Zero, p.113.

We shall return to this subject in the theoretical contexts where it becomes apparent that many of the developments of twentieth century physics, from quantum mechanics to superstring theory, were outgrowths of efforts to cope with the mathematical absurdities that arose because of infinity. See, e.g., Greene, The Hidden Reality; Seife, Zero, pp.157–209. (And, the infinities posed problems even for Newton in the development of the calculus, requiring him to engage in a notational trick that gave the proper results. Seife, Zero, pp.114–7.)¹⁷

There are many examples of other new conventions that were required for the further development of pure mathematics, like imaginary numbers. While the subjects of some of these conventions had no apparent analogue in the physical world, the resulting mathematics did often have applications. For example, imaginary numbers were integral to the development of vector analyses. See Whitehead, An Introduction to Mathematics, pp.87–111. And, the concept of the infinitesimal was at the heart of the development of calculus. Id., pp.217–35.

I suggest that an interesting case can be made for the exclusion of infinity and infinitesimal from our efforts to understand the world or the Universe, as opposed to applications of pure mathematics. We could still think of quantities too small to measure or to notice and of quantities too large to imagine. And, the boundaries of each category can change, and have changed, with increases in our technology and knowledge—we are aware of things formerly too small to measure and of expanses of space and time far beyond our former ability to imagine. But, those dynamics do not create a need for infinitesimal or infinity.

Alternatives

Interestingly, under an approach to the philosophy of mathematics that arose in the late nineteenth century, the scope and content of what we generally consider to be mathematics would be substantially reduced. See Barrow, Theories of Everything, pp.186–88. Such a philosophy would limit mathematics to those statements that can be deduced from the natural numbers.¹⁸ Thus, it would exclude the introduction of human constructs that are not based upon our actual experience of the world, such as the concept of infinity, imaginary numbers and complex numbers. Id. This approach “turns out to have the most dramatic consequences for the whole scope and meaning of mathematics.” Id., p.186.

For example, such an approach to mathematics would eliminate the use of “such familiar devices as the argument from contradiction (the so-called reductio ad absurdum), wherein one assumes some statement to be true and from that assumption proceeds to deduce a logical contraction and hence to a conclusion that the original assumption must have been false.” Id. Under a restricted view, mathematics would not include this commonly accepted method of logical argument. That would be consequential. As Barrow points out, for example, the General Theory of Relativity and current cosmological science depend at various points upon the use of argument from contradiction. Id., pp.186–88.