Potential versus Completed Infinity:
its history and controversy

an essay by Eric Schechter (version of 5 Dec 2009)

I am not a leading researcher on infinite sets, but I nevertheless attract a fair amount of email on the subject; I imagine this is mostly because I have posted several web pages on related subjects. Much of the email consists of arguments against the notion of a completed infinity. On this web page I will try to clarify that subject, so that I don't have to spend so much time answering email. (The first version of this web page was written on May 7, 2005; the page probably will evolve over the next few months as I respond to comments.)

History and controversy

Nearly all research-level mathematicians today (I would guess 99.99% of them) take for granted both "potential" and "completed" infinity, and most probably do not even know the distinction indicated by those two terms. Some of these mathematicians may be impatient with the few students who still have difficulty with completed infinities. But their impatience is not justified; they are forgetting what difficulty the mathematical community had in reaching its present perspective. Completed infinity has only been part of mainstream mathematics since the work of Georg Cantor (1845-1918), and his ideas initially were met with resistance, because they were not supported by what we see in the physical world around us. Before Cantor's time, mathematicians had struggled with the notion of infinity for many centuries, mostly without success. Indeed, the fact that the ancient Greeks turned to geometry rather than algebra can be attributed in part to the difficulty they had with infinite processes. For instance, the square root of two can be constructed geometrically in just a few steps, but to define it algebraically takes some understanding of an infinite procedure.

Infinity cannot be experienced in our everyday lives, but infinity might be a good "approximation" to some of the quantities that we read about in the news. There are 7 billion people in the world, and the annual national budget is several trillion dollars, and the national debt is many trillions of dollars; all of these numbers are much bigger than most of us -- even mathematicians -- have any real feeling about. And the number of atoms in the earth is much much bigger than trillions; I don't even know the name for that number. But still these numbers are finite.

Nor can we experience the infinitely small in our lives. In fact, the currently prevailing theories of quantum physics tell us that there is a lower limit, a smallest physical object.

If we don't see infinity in the physical world around us, then where do we see it? Why, in our heads, of course. Actually, we see all of mathematics in our heads. We may see three airplanes or three apples in the physical world, but the abstract notion of "3" does not exist in the physical world -- it only exists in our minds. The notion of "3" is simple enough, and is an abstraction of enough concrete objects, that there is little chance of our disagreeing on the notion. Our conversations seem to suggest that the "3" in my head is very much like the "3" in your head (though we will never be 100% certain of that). But more complicated notions such as infinity, less grounded in physical reality, are harder to explain; it is harder to be sure that we are successfully conveying a concept from the inside of one head to the inside of another.

Cantor's discoveries about infinite sets were just part of a deeper philosophical revolution that affected all branches of mathematics, not just set theory. New conventions became fashionable, governing what kinds of imaginary worlds mathematicians would permit inside their heads. In effect, formalism replaced Platonism. Many mathematicians today still believe themselves to be Platonists, and perhaps they can afford that luxury if they work in a small enough portion of mathematics; but the predominant paradigm of mathematics as a whole has shifted toward formalism. The birth of mathematical formalism is most often associated with David Hilbert (1862-1943), but I think much credit for it is owed to Cantor, and also to a less well known geometer, Eugenio Beltrami (1835-1900).

Formalism and its consequences were controversial at first. One of the more visible battle lines was between the group now known as classicists (who believe that mathematics is a collection of statements) and constructivists (who believe that mathematics is a collection of constructions or procedures). The overwhelming majority of mathematicians today are classicists, but this is merely a matter of personal preference (like one's favorite color), not a matter of someone being right or wrong. Nearly any mathematician today who understands both sides of the issue agrees that both sides make perfectly good sense. (On the other hand, many classicists today are entirely unfamiliar with the constructivist viewpoint.)

A striking example is the Axiom of Choice (described in greater detail on another web page). This axiom, acceptable to classicists but not to constructivists, is a nonconstructive assertion of the "existence" of certain sets or functions. The use of the word "exist" is merely a grammatical convenience here; mathematicians and nonmathematicians do not mean quite the same thing by this word. Unfortunately, we mathematicians don't have a better word; to be more precise we would have to replace this one word with entire paragraphs. If we assume the Axiom of Choice, we are not really stating that we believe in the physical "existence" of those sets or functions. Rather, we are stating that (at least for the moment) we will agree to the convention that we are permitted write proofs in a style as though those sets or functions exist.

Whether those sets or functions "really" exist is actually not important, so long as they do not give rise to contradictions. Mathematicians are perfectly willing to use devices that may be fictional, as intermediate steps in getting from a real problem to a real solution. Perhaps the most striking example of this is the use of so-called "imaginary numbers" such as i, the square root of -1 (described in greater detail on another web page). Such numbers were first developed for the purpose of solving certain polynomial equations. Initially, the attitude mathematicians took was, "there cannot really be a square root of -1, but if such a number did exist, what would its properties be?" Many decades later, it was discovered that those properties correspond, in a natural way, to the process of rotating the Euclidean plane through a quarter turn. The number i is very useful to engineers, for solving differential equations involving sines, cosines, and other functions related to rotation. That's very real, not at all fictitious. Nevertheless, the name "imaginary" stuck.

The formalist revolution took longer to reach some branches of mathematics than others. One of the late arrivals was mathematical logic. One type of logic, now known as "classical logic," was given almost exclusive sovereignty until perhaps as late as 1960, and only gradually began to share its power with nonclassical logics during the last decades of the 20th century. Perhaps this delay was caused by the fact that, around 1930, Kurt Gödel made some highly interesting and important contributions to classical logic, thereby distracting people away from other logics. Classical logic is adequate for the needs of most mathematicians, and it is computationally the simplest of the main logics, but it disregards qualities such as constructiveness, relevance, and causality. The study of those qualities has led to alternative logics, some of which are discussed further on the web page advertising my logic book.

Though the formalist revolution is an undeniable fact of mathematical (and perhaps scientific) history, some questions about it still remain -- e.g., is formalism good or bad? Some scientists and mathematicians have suggested that mathematics, no longer tied to its origins in physics, is developing into a baroque art form, a thing of great embellishments and few uses; that mathematics has been reduced to a mere game of meaningless marks on paper. Others have argued that mathematics turns out to be useful in surprising and unexpected ways, just because mathematicians have concerned themselves with the investigation of the fundamental properties of basic mathematical objects, such as numbers. Perhaps the most famous essay on this subject is The Unreasonable Effectiveness of Mathematics in the Natural Sciences, published in 1960 by Eugene Wigner.