Originally presented as an all-college address at St. John's College in October 1996. Published in the

St. John's Review, XLIV, 2 (1998) 1-59. Copyright © 1998, Peter Suber.Infinite ReflectionsPeter Suber, Philosophy Department, Earlham College

- Galileo's Paradox
- Contradictory or Counter-Intuitive?
- Imagination v. Conception
- Infinity as a Positive Idea
- Do We Experience Anything Infinite?
- The Sublimity of the Infinite
- Conclusion
- Bibliography
- Notes
- Appendix: A Crash Course in the Mathematics of Infinite Sets

Galileo's ParadoxHere's a paradox of infinity noticed by Galileo in 1638. It seems that the even numbers are as numerous as the evens and the odds put together. Why? Because they can be put into one-to-one correspondence. The evens and odds put together are called the natural numbers. The first even number and the first natural number can be paired; the second even and the second natural can be paired, and so on. When two finite sets can be put into one-to-one correspondence in this way, they always have the same number of members.

Supporting this conclusion from another direction is our intuition that "infinity is infinity", or that all infinite sets are the same size. If we can speak of infinite sets as having some number of members, then this intuition tells us that all infinite sets have the same number of members.

Galileo's paradox is paradoxical because this intuitive view that the two sets are the same size violates another intuition which is just as strong. Clearly, the even numbers seem less numerous than the natural numbers, half as numerous to be precise. Why? Because we can obtain the evens by starting with the naturals and deleting every other member. Needless to say, when we delete every other member of a finite set, the result is a set which is half as numerous as the original set.

If the evens and the naturals were finite sets, then these two verdicts would form a strict contradiction. If two finite sets can be put into one-to-one correspondence, then they have the same number of members; but if one can be produced by deleting every other member of the other, then they do not have the same number of members and cannot be put into one-to-one correspondence. So do we have a strict contradiction here?

The evens and the naturals are not finite but infinite sets. By this I only mean that counting them one at a time will never come to the end; there is no greatest even number, and no greatest natural number.

At this point let's introduce the technical term

cardinalityto refer to the number of members in a set. For example, the set of fingers on one hand has cardinality five. The set of faces on Mt. Rushmore has cardinality four. The set of stooges has cardinality three.In the language of cardinality, we may say that any two sets which can be put into one-to-one correspondence are equal in cardinality; they have the same number of members. This is easily verified for finite sets, and we will regard it as the definition of equal magnitude for infinite sets. Using the same language of cardinality, our intuition has given us two additional propositions: (1) that all infinite sets are equal in cardinality,[Note 1] and (2) that if one set can be obtained by deleting members of another, then they have unequal cardinalities. The latter verdict can be paraphrased thus: some infinite sets have a larger cardinality than other infinite sets, or not all infinite sets are equal in cardinality. Therefore, these two verdicts of intuition directly contradict one another and cannot both be true.

Let us introduce one more technical term, our last for a long time. One set is a

subsetof a second set if all its members belong to the second set. It is aproper subsetif all its members belong to that second set and if it omits or excludes some of the members of that second set. The evens are a proper subset of the naturals because they form a subset of the naturals which omits some naturals, namely, the odds. The set of Moe and Larry makes a proper subset of stooges because Moe and Larry are some but not all the stooges; they omit a stooge, namely, Curly. With this terminology, we can offer one more paraphrase of the second verdict of intuition: a set must have a larger cardinality than its proper subsets.If we add Curly to the set of Moe and Larry, then the set grows in cardinality from two to three. What would happen if we added the odd numbers to the set of the even numbers? Would the set grow in cardinality, or would it retain the same cardinality as the set of evens alone? This is the original question in a new form. The first verdict of intuition says no; all infinite sets are equal in cardinality, so adding the evens to the odds would not increase cardinality. The second verdict of intuition says yes, for this verdict is just another way of saying that adding new members to a given set, and especially adding an infinite number of new members, will always increase cardinality.

So which verdict is correct? Before we answer this question, note that we cannot have it both ways. Either all infinite sets are equal in cardinality, or all infinite sets have a larger cardinality than their proper subsets, but not both. Therefore, the truth on this question will violate at least one of our intuitions. For my purposes here, this lesson is at least as important as the mathematical details of the correct answer, for it implies that we should not trust our intuitions in this domain, nor should we expect to confirm mathematical results about infinity with our intuitions. Some true results will violate our intuitions and some false results will be ratified by them.

Now we can point out that both the verdicts of intuition are false. First, it is false that all infinite sets are equal in cardinality. We can prove that some infinities are larger than others (for example, see Theorems 3, 4, 5, and 16 in the Appendix). Second, it is false that all sets have a larger cardinality than their proper subsets. We can prove that some additions to a given set, even infinite additions, do not increase the cardinality of the given set (for example, see Theorems 1, 2, 7, 14, 15, 19, and 22 in the Appendix).

In his original statement of the paradox, Galileo did not use the even numbers; he used the perfect squares, 0, 1, 4, 9, 16....[Note 2] Like the evens, this set is infinite and the set of its natural number omissions is also infinite. But it seems much less likely than the even numbers to equal the naturals in cardinality because, as we move along the series of squares, the interval between members becomes increasingly large. In fact, as we move outward the ratio of perfect squares to natural numbers approaches zero. The evens never peter out, but the squares become infinitely sparse.

Nevertheless, we can put the natural numbers and the perfect squares into one-to-one correspondence. Every distinct natural number has a distinct perfect square; and every distinct square number has a distinct natural number as its square root. Hence every member of one sequence has a unique counterpart on the other, and vice versa. (The same is true of the evens and the naturals.)

This fact is the key to the solution. If the two sets can be put into one-to-one correspondence, then they have the same cardinality, by definition. One intuition ratified this result (namely, that all infinite sets are equal in cardinality) and one opposed it (namely, that all infinite sets have a greater cardinality than their proper subsets). Both intuitions are false in general, but one was accidentally true in this case. The lesson for intuition is: get used to it.

Galileo's paradox is paradoxical only in the weak sense: it violates our intuitions. It is not a contradiction. It is weird and amazing; it is literally counter-intuitive; but it is not contradictory. We were able to choose between the competing intuitions and eliminate the appearance of contradiction once we held fast to the definition of equal cardinality for infinite sets provided by the principle of one-to-one correspondence.

This innovation is due to Georg Cantor, as is set theory itself, the theory of infinite sets, and the modern concept of infinite cardinality. Cantor lived from 1845 to 1918, and worked out his theory of infinite sets from roughly 1870 to 1895. Cantor's verdict is that the set of even numbers, the set of odd numbers, the set of perfect squares, and the set of all the natural numbers have the same cardinality. The key to this solution is simply to define equal cardinality through one-to-one correspondence, and then to show that these sets can be put into one-to-one correspondence with one another. Similarly, we can prove that some infinite sets have a larger cardinality than others by showing that they cannot be put into one-to-one correspondence.

You may know that many mathematicians and philosophers have objected to the very idea of a completed or actual infinity, as opposed to a potential infinity. Cantor's mathematics, however, boldly posits complete infinities. The natural numbers make a potential infinity when we think of counting them out, and never coming to an end; we could always add one more and keep going. They constitute a completed or actual infinity when they are all bundled together and said to form a set of some definite cardinality. Cantor not only flew in the face of the traditional objection to completed infinities, he used completed infinities in the form of infinite sets as an intrinsic part of his solution to the classical paradoxes of the infinite.

There are many other classical paradoxes of the infinite. But Galileo's is enough to get us started. The infinite has been a perennial source of mathematical and philosophical wonder, in part because of its enormity —anything that large is grand, and provokes awe and contemplation— and in part because of the paradoxes like Galileo's. Infinity seems impossible to tame intellectually, and to bring within the confines of human understanding. I will argue, however, that Cantor has tamed it. The good news is that Cantor's mathematics makes infinity clear and consistent but does nothing to reduce the awe-inspiring grandeur of it.

I'll offer reflections on just a handful of the specific questions mathematicians and philosophers have asked about the infinite over the centuries. Has modern mathematics allowed us to speak coherently of "complete" or "actual" infinities, as opposed to merely "potential" ones? Is the very idea of an infinite set (which can be put into one-to-one correspondence with some of its proper subsets) self-contradictory? Can infinite collections be "imagined" or only "conceived" or not even that? Do we have an idea of infinity or only the idea of finitude and its negation? I will discuss how we go about "unlearning" some intuitions, cultivated in our experience of the finite, which make some consistent and demonstrable results literally counter-intuitive. Finally, I will examine why the deep explorers of the infinite, even in its strictly mathematical forms, recurringly find it to be (in Kant's term) sublime.

Contradictory or Counter-Intuitive?Cantor forces us to see that the intuitive notion of a set's size is ambiguous. When we say that one set is smaller than another set, we might mean two distinctly different things. First, we might mean that the "smaller" set is a proper subset of the "larger" set. Second, we might mean that one set has a smaller cardinality than the other set in the sense that one-to-one correspondence between them fails.

These two notions of size are distinct and independent. A set may be smaller than another set by one measure and not smaller by the other measure. Galileo's paradox is a perfect illustration. The set of perfect squares is a proper subset of the set of natural numbers; in that sense it is a "smaller" set. However, the two sets can be put into one-to-one correspondence; in that sense, it's not smaller at all but the same "size".

With finite sets, these two notions of size always and necessarily agree; that may be why they are so easy to confuse with one another when we are dealing with infinite sets. I believe that all the classical paradoxes of the infinite rest on just this confusion of the two notions of a set's size, a symptom of the unwarranted expectation that infinite magnitudes should behave like finite magnitudes. The classical paradoxes set up two infinite sets which are unequal by one test, but equal by the other, and present this counter-intuitive but consistent possibility as a contradiction or impossibility. The classical objections to completed infinities[Note 3] rest on the same confusion. Those who argued that completed infinities are self-contradictory appeal to the apparent contradictions contained in the classical paradoxes like Galileo's. When we recognize the two distinct and compatible notions of size which are at work in these paradoxes, then, we show that the apparent contradiction is not a real one, we dissolve the paradox, and we answer the objections based on it against completed infinities.

To repeat, then, for the sake of explicitness: Cantor's solution to Galileo's paradox is that the set of perfect squares and the set of natural numbers have the same cardinality even though one of these sets is a proper subset of the other.

It follows —have courage!— that some infinite sets can be put into one-to-one correspondence with proper subsets of themselves. This can never happen with finite sets. But it happens, for example, with the natural numbers and its proper subset, the even natural numbers, and again with the natural numbers and its proper subset, the perfect squares.

The very idea that a set can be put into one-to-one correspondence with one of its proper subsets is deeply counter-intuitive. If you're feeling a barrier of resistance, this is probably the cause. For example, an infinite set with this property will not grow in cardinality as we add members to it, one at a time (see Theorem 7 in the Appendix), and will not shrink in cardinality as we subtract members from it, one at a time (see Theorems 8 and 9 in the Appendix).

For the sake of future discussion, let us say that a set which can be put into one-to-one correspondence with at least one of its proper subsets is

self-nesting. (Unfortunately, mathematicians have given no name to this property, so I have to invent one.)Self-nesting sets seemed impossible or contradictory as soon as they were conceived. In the sixth century, John Philoponous of Alexandria argued that if the world were infinitely old, then an infinite number of months would have passed. But thirty times as many days would also have passed. But either the infinite number of months and the infinite number of days are equal or unequal. If equal, then in our terms the infinite set of past days is self-nesting and can be put into one-to-one correspondence with its proper subset, the infinite set of past months. If unequal, then there would be infinities of different sizes. Because Philoponous thought both options contradictory, he concluded that the world must be finitely old.[Note 4]

Cantor's theory faced intense opposition in the late 19th century, from mathematicians as well as from philosophers and theologians. It wasn't just denied and disbelieved; it was hated. Yet despite this heat, no opponent of the theory has been able to show that self-nesting is contradictory for infinite sets. The objections that self-nesting is contradictory for finite sets, or counter-intuitive for infinite sets, are clearly beside the point. Today, Cantor's theory is standard mathematics even though there are still a few holdouts. Beyond consistency, it has the virtue of eliminating the apparent contradiction from puzzles like Galileo's paradox.

When a theory with these virtues is opposed by intuition, the remedy is not to deny the theory but to unlearn our old intuitions.[Note 5] In the task of re-educating our intuitions, I've found three strategies to be helpful.

First, study the proofs for the basic mathematical results. When your intuition is opposed only by someone's say-so, like mine or a teacher's, then intuition can easily win —and perhaps in that case, it ought to win. When it is opposed by an articulate chain of reasoning, then it starts to give —and it ought to give.

Second, remember that our intuitions were cultivated by our experience of finite sets: sets of fingers, sets of coins, sets of people. And for finite sets, self-nesting is a flat contradiction. When we deal with infinite sets, we must accept the fact that most of our "common sense" or "rules of thumb" will either be inapplicable or false, evolving as they did for the more tractable domains of finite experience. This is not a license to disregard or negate our intuitions, which are often valuable clues to mathematically coherent theories. It is simply a reason to put them to one side when they conflict with a consistent theory supported by strong proofs which solves otherwise insoluble mathematical problems.

In the same vein, it is helpful to remember past cases in the history of mathematics in which we mistook counter-intuitive ideas for contradictory ones. The preeminent examples are incommensurable quantities and instantaneous velocities; however we could also cite negative numbers, the denial of Euclid's parallel postulate, and, more recently, incomputable numerical functions. With the passage of time, the acceptance and utility of these ideas have only increased, and their consistency has been more firmly and clearly recognized, while the opposing intuitions have faded away with the world-views which cultivated them.

Third, remember that our intuitions would not be satisfied any better by rejecting Cantor's self-nesting solution to Galileo's paradox. If we didn't accept Cantor's view that Galileo's two sets had the same cardinality, then we'd have to accept the view that they had unequal cardinalities. But this result would contradict the intuitive principle that one-to-one correspondence establishes equal cardinality. When we are at an impasse for intuition, then intuition is no longer a helpful guide, since it pulls as much (or as little) for one side as for the other. That is when we should be looking for another guide, not clinging to the guide which has disqualified itself.

Imagination v. ConceptionWe've seen that intuition disqualifies itself in this domain by endorsing contradictory conclusions. Cantor's conclusions are rigorously proved, and so far (despite some strenuously motivated effort), rigorous proof has not endorsed contradictory conclusions about the infinite. This is one good reason to prefer proof to intuition. The distinction between intuition and proof as reasons for accepting a theory, and the inadequacy of intuition for dealing with the infinite, have many consequences for the philosophy and mathematics of the infinite. For example, even after acknowledging the consistency of Cantor's theory, many people will still insist that we know nothing about infinity. What they seem to mean is that knowing requires some intuition, imagination, or visualization.

I think I understand the origin of this objection, but I also believe it is easily answered. Just as our intuitions about sets, subsets, and cardinality are cultivated by finite sets, where self-nesting is impossible, our ordinary knowledge of objects is limited to finite numbers of objects of finite size. (In a moment I will look at the question whether we ever experience anything that is truly infinite.) We can visualize objects of finite size and we can visualize finite numbers of them. This means that virtually all of our ordinary knowledge of objects is accompanied by this possibility of visualization. It's natural that we would come to expect that anything we can know, we can also visualize.

Even if this expectation is legitimate for the finite, it is entirely illegitimate for the infinite. Just as intuitions cultivated for the finite are likely to be inapplicable or false of the infinite, so is the expectation that we be able to visualize.

Descartes asks us to imagine, that is, visualize, a chiliogon or 1,000-sided regular polygon.[Note 6] Can you do it? Try it right now. Chances are, you are either visualizing something like a 20 or 30-sided polygon and pretending it has 1,000 sides, or you are visualizing a circle and pretending the sides are too small to see with your mind's eye. We know exactly what a chiliogon is; we can even compute the interior angle of its sides and, for a given edge, its area and perimeter. But we cannot visualize one.

One reason I like Descartes' example is that it is finite. Philosophers who think the infinite utterly beyond human understanding often fail to notice that their arguments, once made specific, also apply to very large finite magnitudes as well. We cannot visualize infinitely many cherries in a tree, but neither can we visualize a billion. Does that disqualify us from using billions intelligibly and accurately?

To Descartes, the chiliogon thought-experiment proved that we have at least two avenues to knowledge: imagination (which I've been calling visualization) and conception. We can conceive the chiliogon, although we cannot imagine it. Once it is pointed out with a concrete example like the chiliogon, this is undeniable and we start to see other examples everywhere. To Descartes, the distinction is more important in theology than in mathematics. The greatest obstacle to true faith, he thinks, is the attempt to imagine God when we can only conceive God.[Note 7]

Let me return to set theory. The

power setof a given set is the set of all its subsets. For example, if I have a set of three stooges, then its power set is the set of all the subsets of stooges I can make from that set of three. There is the set {Moe, Curly}, the set {Moe, Larry}, and the set {Curly, Larry}. There is also the set of {Moe} alone, {Curly} alone, and of {Larry} alone. For technical reasons, we say that every set is a subset of itself, and the null set is a subset of every set. Hence we throw in {Moe, Curly, Larry} and {} to boot. This makes eight. Any set of three objects —any set with a cardinality of three— will have a power set of cardinality eight.We can imagine —visualize— many methods for systematically drawing out all the subsets of a given finite set. These methods will be extremely cumbersome for sets of cardinality 1,000, say, but each method contains an algorithm which we can visualize working out.

Contemplate the set of natural numbers. Here is a set of infinite cardinality. What is the cardinality of its power set?

We saw that the evens had the same cardinality as the naturals, despite appearances to the contrary. We might cautiously generalize that all infinite sets have the same cardinality, but here we find a counter-example. Cantor found an elegant proof that the power set of any set, finite or infinite, possesses a greater cardinality than the original set; this important result is simply called

Cantor's Theorem.It has a short proof of marvelous beauty. The proof is negative, which means that Cantor assumed the negation of his conclusion and derived a contradiction from it. Since the theorem works for any arbitrary set, let's apply it to the set of natural numbers. So, to set up the negative proof, let us assume that the set of natural numbers and its power set have the same cardinality. If so, then they can be put into one-to-one correspondence. Let us suppose we have done so (even though we have no idea how to do so). Now by hypothesis each natural number is paired with exactly one set of natural numbers, and

vice versa. Some numbers will be paired with sets which happen to contain them. For example, 2 might be paired with the set of even numbers. Let us call such numbershappy, and all other numberssad. Now the set of all sad numbers is abona fideset of natural numbers, and so has been paired with some natural number in our infinite list of correspondences. Let's say it has been paired withx. Isxa happy number or a sad one? At this point, I know you'll start to get a little dizzy. That's good; it means you're following along. Ifxis sad, then because it has been paired with the set of sad numbers, it has been paired with set which includes it; but that means it would be happy. But ifxis happy, then it would be a member of the set to which it has been paired; but because it has been paired with the set of sad numbers, that means it would be sad. Hence, ifxis happy, thenxwould be sad, and ifxis sad, thenxwould be happy. Our assumption implied this contradiction, and so must be false. But to deny our assumption is to conclude that the set of natural numbers and its power set have different cardinalities. (See Theorem 4 in the Appendix.)In my view there are two great counter-intuitive results in the mathematics of the infinite. The first is that some infinite sets are self-nesting. (It turns out that

allare; see Theorem 10 in the Appendix.) The second is that some infinities are larger —have a greater cardinality— than others. (See Theorems 3, 4, 5, and 16 in the Appendix.) Now we have seen proofs for both results. The first was proved by one-to-one correspondence, the second by a technique that has been called diagonalization.Cantor's Theorem is not very remarkable if we think only of finite sets.

Of coursefor every finite set the power set is bigger than the original. But for infinite sets Cantor's Theorem is the astounding proposition that for every infinite cardinality, there is a larger one —namely, the power set of the first one. So if the cardinality of the set of natural numbers is one infinite number, and the cardinality of its power set is a distinct, larger infinite number, and the cardinality ofitspower set is a distinct, larger infinite number, then it's clear that what Cantor has really proved is that there existsan infinite sequence of infinite cardinal numbers.Of this remarkable theorem and its remarkable proof, David Hilbert said, "This appears to me to be the most admirable flower of the mathematical intellect and one of the highest achievements of purely rational human activity."

Infinity as a Positive IdeaLet us grant, then, that imagination and intuition are too feeble to grasp infinity. It does not yet follow that conception is strong enough, or indeed that any human faculty is strong enough. We might understand infinity the way medieval Christian philosophers thought we understood God:

via negativa, that is, by understanding what God, or infinity, is not. For example, I know what it is like for a row of trees to come to an end. This exemplifies my concept of finitude. If I say that an infinite row of trees is just like the finite row "except that it never comes to an end", then I am merely negating my concept of finitude.Descartes, again, thought we did have a positive idea of infinity —three centuries before Cantor. This was important to him because he thought that finite human resources could not suffice to give us the idea of infinity, and therefore that the idea could only have been given to us by an infinite being; in short, it was part of another of his arguments for the existence of God.[Note 8] He has two theses here: first, that we do possess a positive idea of infinity, and second that we could not have obtained this idea from our own finite experience or creativity. If both are true, this would be important for just the reason he thought. But are they both true?

In a moment I will take up the question whether we ever experience anything infinite. On the question whether we know infinity positively, or just

via negativa, Descartes is very short. He argues that he would not know that he is finite or imperfect unless he had prior, positive ideas of infinity and perfection.[Note 9] There are many follow-up questions a skeptic would like to ask at this point, but Descartes does not pause for them.Descartes does pause to ask himself the question: Is it possible that I am an infinite being, don't know it, and could therefore be the source of my idea of infinity?[Note 10] Although this is a terribly interesting and important question, he is also very short with it. After a brief look, he answers like Steve Martin in a Saturday Night Live routine, "

Naaaa!"Etymologically, the word "infinite" is "non-finite". This supports that view that perhaps finitude is the primary notion here and our concept of infinity is the negation of our concept of finitude. But we can't get any mileage from etymologies in this inquiry. Etymologically, the word "independent" is "non-dependent" as if

unfreedom were the primary concept and freedom derivative. But the word "unfreedom" is "non-freedom" as if freedom were the primary concept after all. Similarly, the continuum is one of the premier examples of infinity in mathematics, but it differs from other infinities like the rational number series in being "unbroken" or "without gaps". This suggests that we only know the continuumvia negativa, by negating the idea of gaps; but etymologically the terms "continuous" and "discontinuous" suggest the opposite, that continuity is the primary concept here.More telling than etymology is this exercise: define finitude. I often teach a course at Earlham with a unit on the mathematics of infinity, and every now and then I'll throw "finitude" or "finite set" onto a quiz, as a term to define. Invariably, students lose more points trying to define it precisely than they do when defining various infinite cardinalities.

Do try this one at home, however. Define finitude with clarity and precision. There are ways, even brief ways, but they usually don't occur to people with no training in mathematics.

Infinity, by contrast, at least since Cantor, is easy to define with clarity and precision. Remember that Cantor proved that some infinite sets are self-nesting, or can be put into one-to-one correspondence with at least one of their proper subsets. It's not hard to prove that

allinfinite sets, in fact, are self-nesting. (See Theorem 10 in the Appendix.) And we already knew thatonlyinfinite sets are self-nesting, or that no finite sets have this property. Consequently, we can define infinite sets as just those which are self-nesting. Correspondingly, we can define finite sets as just those which are not.Note the neat turning of tables here. Infinite sets have the positive property of self-nesting; finite sets do not.

Finitudeis definedvia negativa.[Note 11]Charles Peirce in 1885, and Richard Dedekind in 1888, proposed to define infinity through self-nesting.[Note 12] According to this proposal, we don't know that infinite sets are self-nesting because of some proof; we know it because infinite sets are

definedas those which are self-nesting. However, we can prove that the Peirce-Dedekind definition is equivalent to a more traditional one by which we know infinitude rather than finitudevia negativa,[Note 13] and for me that fact makes the controversy about logical priority or primacy merely scholastic.What is not merely scholastic is that we have now reduced the question whether we have a positive idea of infinity to the question whether we have a positive idea of self-nesting. I suggest that we do have such an idea, or can, if we study Cantor's transfinite arithmetic. In my own experience, to understand self-nesting at all is to understand it positively. I'm quite sure I don't understand it

via negativaor as the negation of something else like "the failure or impossibility of self-nesting". The failure or impossibility of self-nesting definitely carries for me the status of a derivative idea, one that never comes to mind when I think about self-nesting unless I make a great effort.[Note 14]I realize that my reason derives from my experience putting sets into one-to-one correspondence with proper subsets of themselves, and studying the works of others who have done the same; therefore it begs the question somewhat. I'm saying that if you study the mathematics of infinite sets, this positive idea will come, although perhaps not quickly or in a form you could communicate easily to those who have not undertaken a similar study. But if you haven't studied Cantor this looks like hand-waving. I know that cultists of every stripe say virtually the same thing: Study the book of our inspirational founder and you too will see the light, and until then shut up with your criticisms.

So let me try to do better. I think I can show that we have a positive idea of infinity, in the form of self-nesting, and even that self-nesting can be made somewhat intuitive or visualizable. I owe the following idea to Josiah Royce,[Note 15] one of the first philosophers to make use of Cantor's mathematics. Imagine a perfect map of England, say, somewhere in London. By a "perfect" map I mean one which shows not only the cities and roads, but also the houses, furniture, pennies behind the sofa cushions, bacteria, quarks —in fact, every last particle of matter. Now if the map is perfect in this sense and if it is located in London, then somewhere on the map there will be a perfect image of the map itself. Again, by "perfect image" I mean that every detail of the outer map will appear on the inner map. But if this is true, then like a hall of mirrors the map within the map will also contain a perfect image of the map, and so on

ad infinitum.To use Royce's term, the map will be

self-representing. Of course we can't actually make such a map, and it is useful to think of the reasons why. One obstacle in our way is the fact that the pixels we must use are larger than the smallest particles of matter we wish to represent. It may seem that this fact would not stop us from making a perfect map of England, but only require that the map be larger than England. But if the map were larger than England, then it could not be located inside England, and therefore could not be self-representing or "perfect" in our sense.Another obstacle in our way is that we can only arrange a finite number of pixels to make a picture. Such a 'finitist' map could be self-representing only imperfectly; if it didn't represent London as a mere dot, it would represent the map within London as a mere dot, or the map within the map within London. With only a finite number of pixels to use in composing our picture, we will inevitably run out of pixels before we run out of information. This would not be a problem with an ordinary or non-self-representing map. If we had as many pixels at our disposal as there are quarks in England, then we could (in principle) arrange them to make a perfect map of England down to the quark level —even if the resulting map were larger than England. But once the map itself is put inside England and becomes one of the landmarks to represent on the map, then to be

perfectthe map would have to be perfectly self-representing and therefore infinitely nested; suddenly the number of pixels needed rises from finite to infinite.Now what if we could make a picture using dimensionless points as pixels, and use an infinite number of them? It is strange and wonderful that Leibniz posits just these two conditions in the

Monadologyof 1714. In that work he outlines a new atomic theory in which conventional atoms are replaced by monads, "the true atoms of nature".[Note 16] Monads differ from conventional atoms in many ways, but the most important for our purposes is that they have zero size. They are dimensionless points. And of course there are infinitely many of them. This allows a set of monads to represent England perfectly even if there are infinitely many particles smaller than quarks which would have to appear on the map. It also means that in Leibniz's world it is physically possible for some chunk of matter to achieve perfect self-representation, the way England does in Royce's scenario. It might contain within itself a perfect representation of itself, and hence an infinite series of nested microcosms. But it might do better still: it might be a perfect representation of the universe as a whole, including itself as one of the parts, and therefore contain infinitely many nested perfect representations of itself and the universe. Leibniz thought this was not only possible, but that every chunk of matter of every size is a perfect mirror of the universe, of itself, and of all the other perfect mirrors, in just this way.[Note 17]You don't have to agree with Leibniz that the world is really set up this way —however if truth is beauty, and beauty truth, then there is a lot to be said for the idea. You only have to admit that you grasp his theory, or Royce's.[Note 18] If you do, then you grasp the essence of self-nesting, which is the essence of infinite cardinality. You need no longer approach it

via negativa.[Note 19]

Do We Experience Anything Infinite?So we agree with Descartes that we do possess a positive idea of infinity. If Descartes is correct in his second thesis that we could not have obtained the idea from our finite experience and creative resources, then we feel the pressure he felt to posit an infinite being. So let us face directly the question whether we experience anything infinite.

The words "infinite" and "infinity" are often used loosely in street English to suggest that we do experience infinites. For example, we may say that a film is infinitely clever, a coral reef has an infinite variety of wildlife, a spouse has infinite patience, or that a vinyl upholstery cleaner has infinitely many applications. (That's why it's called a miracle product.) Before cameras were automated, they had a focal-length setting called "infinity", presumably for photographing the arrow Lucretius shot into the edge of space. In these cases we speak loosely, and "infinity" means very many or very large, perhaps indefinitely many or large.[Note 20] On a clear day the sky may seem infinitely deep, but it's really just a

wild blue yonder—anindefinitelydeep 'out there'.[Note 21]Do we ever experience something which is

literallyinfinite? If time, space, or matter are infinitely divisible, then to experience a finite chunk of any one of them is to experience its infinity of parts. Having said this, I would like to put to one side the question whether time, space, or matter really are infinitely divisible. Not only is it very thorny, it is unnecessary to answer the question on the table. For even if time, space, and matter are infinitely divisible, we experience their infinite parts bundled into chunks most of whose parts are indiscernible to us. When a movie runs at 24 frames per second, it appears continuous, its separate frames indiscernible to us. We certainly experience 24 chunked frames, but not the 24-ness, or even the finitude, of the chunking. Once the eye is fooled into seeing continuity, the number of frames per second could increase to a billion, or to an infinite number, and we would not notice the difference.[Note 22] This is the sense in which we could experience something infinite without experiencing its infinitude. Similarly, if time, space, and matter were continuous and infinitely divisible, then the spectacle of life would be like a movie run at an (uncountably) infinite number of frames per second; but while we would experience expanses, durations, and objects with infinitely many parts, but we would not experience the infinitude of those parts.As the movie shows, the same is true of

finitedivisibility. If my car has (say) 5,000 parts, I experience it as an object with many parts; but I don't experience the 5,000-ness of the parts.Motion seems to introduce new issues. If I open a pair of scissors and close them again, then the blades produce an infinite number of different angles, and in a sense I saw them all. But when we think about it we realize the we are dealing with the same issues all over again. First, an infinite number of distinct angles is produced only if time and space are both continuous; if either one is composed of irreducible quanta, then only a finite number of angles is produced. Second, even if time and space are continuous, and the angles infinite, we don't experience the infinitude of the angles. This is shown by the fact that we cannot tell from the experiment whether time and space are continuous; that is, we cannot tell whether we saw an infinite or merely a huge finite number of distinct angles.

Similarly, if space is continuous, then walking any distance at all is to traverse an infinity of spatial units. Or if time is continuous, then it is to traverse an infinity of temporal units. But even if so, we only experience the chunked, finite meter we traversed, in the chunked finite second, not the infinitude of dimensionless points inside them.[Note 23]

When Descartes said we experience nothing infinite, I think he meant that we see nothing infinite in any given scene, and nothing infinite in a lifetime of scenes. But how do we know this? Because we only live a finite time? Actually, it depends on how you count. If you count in years or months or days or seconds, then yes, the duration of our lives spans only a finite number of those units. But if we divide time into dimensionless points, such as points on a time line, then we live an infinite number of them —and we would still do so even if we lived for only one second.

The same holds spatially within a given scene. Whether a scene is finite depends on how we divide it. No panaroma covers an infinite number of miles or meters or nanometers. But every scene, even a pinhead, covers an infinite number of dimensionless points of space. Hamlet was thinking of something else at the time, but he made this point very well when he said, "O God, I could be bounded in a nutshell and count myself king of infinite space...."[Note 24]

Still, while the spatial points would be infinite, our experience would never notice or recognize their infinitude.

Past time might be infinite. But even if it is, living in the present would be like treading water over an infinite depth. We would not experience the infinitude except in the form of buoyancy —which could, of course, have a finite explanation. The time in which we exist may rest upon, and be continuous with, an infinite prior time, but we will never know whether this is so simply from our experience of present time.

Performing an infinite number of tasks in finite time has always been a mathematician's dream. If I could count one number in half a second, the next number in the next quarter second, the next number in the next eighth of a second, and so on, then I could count an infinite number of numbers in one second flat. So far nobody has managed to pull this off. However, a mathematician at Bell Labs, named Peter Schor, has come close by showing that the kind of parallelism possible on a quantum computer is indefinitely large if not infinite.[Note 25] We could in effect peform an infinite number of simultaneous computations using only finite hardware, allowing us to compute otherwise intractable functions. Schor proved that quantum indeterminacy makes this kind of parallelism mathematically possible; but notably, it has not yet been realized in a physical machine.

An analog signal as opposed to a digital signal contains an infinite amount of information. But when we make an audio recording of a single piano keystroke, the digital nature of the molecules of air carrying the waves, and the digital nature of the molecules of the magnetic coating on the tape, mean that we can preserve and send to the ear only a finite subset of the information which the keystroke would have registered in a continuous medium.[Note 26] And even if we could hear the note played back after being perfectly recorded in a continuous medium, we would at best hear an analog signal with an infinite amount of information in it; we would not experience the infinitude of that information.

This is precisely why Leibniz posits a continuous medium (a

plenumof monads) rather than discrete molecular air to mediate causal influences like the propagation of sound waves.[Note 27] Leibniz thinks we are continuously bombarded by an infinite amount of information from the universe at large, and that we register all of it, although not all of it consciously. This is his famous doctrine of minute perceptions.[Note 28] Without going into its details here, we can at least see that it unabashedly implies that we do experience something infinite; in fact, we do so continuously.Until we got to Leibniz, there was a pattern in these examples. There are several ways in which the objects or theaters of our experience might be infinite. But we can't tell from our experience whether they are or are not infinite, and this means at the very least that we don't experience their infinitude. By positing minute perceptions, Leibniz posits the experience of infinitely faint influences. He admits, even insists, that not all of these experiences are conscious,[Note 29] but likewise insists that without them conscious experiences would not exist, just as finite line segments would not exist without their constitutive dimensionless points.

Elegance is the chief reason to believe Leibniz's theory. After positing an infinite number of infinitesimal monads

a priori, Leibniz surprises us by making the theory remarkably subtle and adept at explaining the world and experience. If Kant is right, however, we should hesitate to affirm or deny infinitiesa priori.In Kant's diagnosis, Leibniz fell victim to a natural, even rational temptation. It's extremely tempting to think that time, space, and matter

reallyare, in themselves, apart from the limitations on human knowledge, either infinitely divisible or finitely divisible. We may not know which one they are, and we might not perceive their internal infinitude if they are infinitely divisible, but they must really be one way or the other. Kant argued that this is a mistake; in fact, this assumption leads to a special kind of contradiction which he called an antinomy.[Note 30] It also leads to contradiction or antinomy to assume that past time is really either infinite or finite, or that space is really either infinite or finite.[Note 31] There are two reasons, briefly, why these assumptions lead to contradiction: first, they treat the world as a thing in itself, rather than as a phenomenon partly constituted by the act of knowing it; second, they area prioriclaims, based on no empirical evidence, and the oppositea prioriclaims are equally compelling to reason. Kant concludes that to avoid these contradictions, we must regard the extent of space, the depth of past time, and the divisibility of time, space, and matter as indeterminate. We know them as far as we have inquired into them, and tomorrow we may know more. We must speak of the world (time, space, matter) as growing in extent, duration, and divisibility as we find it to be larger, older, or finer; to say that the world consists of something in and of itself which fixes its size, age, and ultimate particles is a demand of reason but ultimately a contradictory one. This is one place where reason must be reigned in, disciplined, or subject to critique.What follows from all this for Kant is the strange-sounding doctrine that in its spatial extent, temporal duration, and material divisibility, the world is

neither finite nor infinite.[Note 32]For myself, I find that I am attracted to the view that time and space are continuous; at the same time I suspect that the question whether time and space are continuous cannot be settled empirically. When I am inclined to soar in the sky of unfettered conjecture, I am attracted by the elegance of Leibniz's theory of minute perceptions, which arguably follows from the view that time, space, and matter are continuous; when I am inclined to discipline my conjectures and hold them inside the bounds of verification, I heed Kant's admonition. I'm no closer to a resolution than this.

So if we have a positive idea of the infinite, how do we obtain this idea? We make this question harder to answer, not easier, if we say that the world is neither finite nor infinite, or that if it is infinite, then we do not experience its infinitude. My disappointing, pedestrian answer is that we may not possess the positive idea of infinitude until we study self-nesting, and during that study, we get the positive idea of infinitude from the exercise of putting an infinite set into one-to-one correspondence with one of its proper subsets. This exercise, I should add, is a finite experience. We take the first few even numbers, 2, 4, 6..., for example, and pair them off against the first few natural numbers, 1, 2, 3.... We know that each sequence is rule-governed, because we know exactly how to generate the next member of each. Hence, we know that the

nth member of one sequence will have a partner in thenth member of the other, no matter how largenis, or no matter how far out we take the sequences. This is the finitistic way to put infinite sets into one-to-one correspondence. But if one set is the proper subset of the other, then we have established self-nesting, which is impossible for finite sets. Until we undertake this exercise, and think about what it means, our notion of the infinite may well be nothing more than the negation of the idea of finitude.While we do not experience the infinitude of time, space, or matter, even if they are infinite in extent or divisibility, neither do we experience large finite magnitudes. I've seen estimates of the number of sub-atomic particles in the universe ranging from 10

^{65}to 10^{85}. But to be conservative, let's say that nothing in the universe, including the universe itself, has more than 10^{100}parts. The name for 10^{100}, or 1 followed by 100 zeroes, is a googol. So even if there are more than googol of ultimate particles, it's fair to say that no collection of physical objects thatwehave ever experienced —grains of sand on a beach, snowflakes in a storm, stars in the sky— has more than a googol of members.[Note 33] If true, then we did not obtain our idea of a googol from experience. But it does not follow that we must posit a very large finite being —Googolzilla— to be the source of our idea. We know exactly what a googol is as a concept, even if we have never experienced it manifest in a sensation or image. We can list the million natural numbers which are its closest neighbors, we can do arithmetic with it, and we know infallibly whether an arbitrary natural number is larger or smaller than it. If we may export the lesson of this to the infinite, then we may suggest that while we have no experience of the infinitude of anything, we have a perfectly good concept of infinity, and that the ultimate explanation of this fact lies not so much in anything special about infinity as in the distinction between concepts and images.

The Sublimity of the InfiniteI am profoundly grateful that understanding infinity does not deprive it of its majesty. If the infinite were only interesting because of the paradoxes it generates, and the absorbing academic issues raised by the need to resolve them, then it would not be studied any more than self-reference, a prolific but more pedestrian engine of paradox. But the infinite is also majestic, one might say infinitely majestic.

An hour under a clear sky at night, looking up, gives some sense of this. The depth of space is a wild blue yonder, not a true, perceived infinity.[Note 34] But it inspires contemplation of the true infinite, and the slightest brush with that idea is breath-taking, invigorating, expanding, lifting, calming, but also agitating, alluring, but also distant and magnificently indifferent. One reason to study mathematics is that you can get these feelings in broad daylight or indoors.

There are many ways to become precise about these feelings, and many ways to praise and honor the infinite. I'd like to use Kant's term: it is

sublime.[Note 35]Just for comparison, Cantor had a different set of numinous feelings about the infinite. He was not only a great mathematician, but a very religious man and by some standards a mystic. Yet his mysticism was supported by his mathematics, which to him was at least as strong an argument for the mathematics as for the mysticism.[Note 36] Apart from claiming divine inspiration for his work, we don't know exactly what spiritual views he linked to his mathematics, but his theorems[Note 37] give support to the following. Measured in meters, we are tiny specks compared to the universe at large. But measured in dimensionless points, we are as large as the universe: a proper subset, but one with the same cardinality as the whole. Similarly, measured in meters, we may be off in a corner of the universe. But measured in points, the distance is equally great in all directions, whether universe is finite or infinite; that puts us in the center, wherever we are. Measured in days, our lives are insignificant hiccups in the expanse of past and future time. But measured in points of time, our lives are as long as universe is old. We are as small as we seem, but simultaneously, by a most reasonable measure, co-extensive with the totality of being in both space and time. This is truly (as Blake put it) "[t]o see the world in a grain of sand and a heaven in a wild flower, hold infinity in the palm of your hand and eternity in an hour."[Note 38]

Kant's theory of the sublime does not rest on these Cantorian theorems. His chief thesis for our purposes is that, "

That is sublime in comparison with which everything else is small."[Note 39] Clearly the infinitely large is a perfect fit for this definition.[Note 40]The sublime is not an easy notion, and the best approach to it may be

via negativa, showing how it differs from something familiar, the beautiful. Sticking only to those differences which bear most on the sublimity of the infinite, Kant says that the beautiful concerns a bounded object while the sublime object can be unbounded; the beautiful is compatible with charms while the sublime is not; the beautiful attracts the mind while the sublime both attracts and repels it; and the beautiful "seems as it were predetermined for our power of judgment" while the sublime is "incommensurate with our power of exhibition, and as it were violent to our imagination, and yet we judge it all the more sublime for that."[Note 41]The infinitely large meets these criteria almost by design. The infinitely large is unbounded, incommensurate with our powers of imagination, and to engage and satisfy us it no more needs charm than spring water needs sugar. It is so large that some of its proper subsets are just as large, a property shared by no finite magnitude.

What triggers the feeling of the sublime most is immensity. Immensity in turn makes us feel a tension between two aspects of ourselves. On the one hand it makes us feel the inadequacy of our senses and imagination. On the other it makes us feel that there is more to us than senses and imagination, whose adequacy cannot be brought into question by immensity, no matter how spectacular or infinite. This second dimension of ourselves is not conception but moral vocation. While physically the immensity dwarfs us into insignificance, this very fact highlights that within us which is not dwarfed. As long as we are physically safe when viewing the sublime immensity, Kant argues, it helps us know our moral dignity and nonphysical invulnerability undiminished, even accentuated, by our forceful acknowledgement of our physical smallness and frailty.[Note 42]

ConclusionProperly understood, the idea of a completed infinity is no longer a problem in mathematics or philosophy. It is perfectly intelligible and coherent. Perhaps it cannot be imagined but it can be conceived; it is not reserved for infinite omniscience, but knowable by finite humanity; it may contradict intuition, but it does not contradict itself. To conceive it adequately we need not enumerate or visualize infinitely many objects, but merely understand self-nesting. We have an actual, positive idea of it, or at least with training we can have one; we are not limited to the idea of finitude and its negation. In fact, it is at least as plausible to think that we understand finitude as the negation of infinitude as the other way around. The world of the infinite is not barred to exploration by the equivalent of sea monsters and tempests; it is barred by the equivalent of motion sickness. The world of the infinite is already open for exploration, but to embark we must unlearn our finitistic intuitions which instill fear and confusion by making some consistent and demonstrable results about the infinite literally counter-intuitive. Exploration itself will create an alternative set of intuitions which make us more susceptible to the feeling which Kant called the sublime. Longer acquaintance will confirm Spinoza's conclusion that the secret of joy is to love something infinite.[Note 43]

Mark Twain came to love mathematics as an adult and always regretted that he didn't have a stronger foundation for it. He once said that if he could live forever, he'd spend 8,000 years studying mathematics. I've never been able to decide whether this remark shows his wit or his weak foundation in mathematics. If he could live

forever, then he could spend infinitely many years studying mathematics, and have infinitely many years left over for other pursuits. That's the way I'd like to do it.BibliographyBlake, William.

The Viking Portable Blake. Ed. Alfred Kazin. Viking Press, 1946,Cantor, Georg.

Contributions to the Founding of the Theory of Transfinite Numbers. Trans. Philip E.B. Jourdain. Dover Publications, 1955. (Translation originally published 1915.)Copeland, Jack.

Artificial Intelligence: A Philosophical Introduction. Basil Blackwell, 1993.Dauben, Joseph Warren.

Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton University Press, 1979.Dedekind, Richard.

Essays on the Theory of Numbers.Trans. Wooster Woodruff Beman. Dover Publications, 1963. (Translation originally published 1901.)Dedekind, Richard.

Was sind und was sollen die Zahlen?6th ed., Braunschweig, 1930 (original 1888).Descartes, René.

Philosophical Essays. Trans. Laurence J. Lafleur. Bobbs-Merrill, 1964. (Discourse on Method, original 1637;Meditations, original 1641.)Fraenkel, Abraham A.

Abstract Set Theory. North-Holland Pub. Co., 1953.Hofstadter, Douglas R.

Gödel, Escher, Bach: an Eternal Golden Braid. Basic Books, Inc., 1979.Galilei, Galileo.

Dialogues Concerning Two New Sciences. Trans. Henry Crew and Alfonso de Salvio. Dover Publications, 1954 (original trans. 1914; original work 1638.)Kant, Immanuel.

Critique of Judgment. Trans. Werner S. Pluhar. Hackett Pub. Co., 1987 (original 1790).Kant, Immanuel.

Critique of Pure Reason. Trans. Norman Kemp Smith. St. Martin's Press, 1968 (original 1781).Kant, Immanuel.

Foundations of the Metaphysics of Morals. Trans. Lewis White Beck. Bobbs-Merrill Co., 1959 (original 1785).Kant, Immanuel.

Observations on the Feeling of the Beautiful and Sublime. Trans. John T. Goldthwait. University of California Press, 1960 (original 1763-64).Keyser, C.J., "Charles Sanders Peirce as a Pioneer,"

Galois Lectures (Scr. Math. Library, No. 5, 1941) 87-112Kleene, Stephen Cole.

Introduction to Metamathematics. North-Holland Pub. Co., 1988 (original 1952).Kleene, Stephen Cole.

Mathematical Logic. John Wiley & Sons, 1967.Leibniz, Gottfried Wilhelm.

New Essays on Human Understanding. Trans. Peter Remnant and Jonathan Bennett. Cambridge University Press, 1981 (original 1704).Leibniz, Gottfried Wilhelm.

Philosophical Papers and Letters. Ed. and Trans. by Leroy E. Loemker. D. Reidel Publishing Co., second ed., 1969. (Discourse on Metaphysics, original 1686; Monadology, original 1714.)Lloyd, Seth, "Quantum Mechanical Computers,"

Scientific American, (October 1995), pp. 140-145.Locke, John.

An Essay Concerning Human Understanding. Ed. Peter H. Nidditch. Oxford University Press, 1975 (original 1690).Maor, Eli.

To Infinity and Beyond: A Cultural History of the Infinite. Princeton University Press, 1991 (original 1987).Moore, A.W.

The Infinite. Routledge, 1990.Peirce, Charles Sanders.

Collected Papers. Ed. Charles Hartshorne and Paul Weiss. Vol. III, 1933.Pickover, Clifford A.

Keys to Infinity. John Wiley & Sons, 1995.Royce, Josiah, "The One, The Many, and the Infinite," Supplementary Essay in his

The World and the Individual, First Series, Dover Publications, 1959 (original 1899), pp. 473-588.Rucker, Rudy.

Infinity and the Mind: The Science and Philosophy of the Infinite. Birkhäuser, 1982.Rucker, Rudy, "One of George Cantor's Speculations on Physical Infinities,"

Speculations in Science and Technology, (October 1978) pp. 419-421.Schor, Peter, "Algorithms for Quantum Computation: Discrete Log and Factoring,"

Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, IEEE Computer Society, 1994, pp. 124ff. This paper is available by FTP from the menu of papers at the Quantum Information Page, http://vesta.physics.ucla.edu/~smolin/.Shakespeare, William.

Hamlet. Ed. Edward Hubler. New American Library, 1963 (original 1600).Spinoza, Benedict.

Ethics, Treatise on the Emendation of the Intellect, and Selected Letters. Trans. Samuel Shirley. Hackett, 1992. (Treatise, original date unknown, posthumously published;Ethics, original 1675.)Vilenkin, N. Ya.

In Search of Infinity. Trans. Abe Shenitzer. Birkhäuser, 1995.Notes

1.Locke argued for this verdict of intuition thus: "[I]f a Man had a positiveIdeaof infinite...he could add two Infinites together; nay, make one Infinite infinitely bigger than another, Absurdities too gross to be confuted." Locke,Essay Concerning Human Understanding(1690), at p. 222. [Resume]

2.Galileo,Dialogues Concerning Two New Sciences(1638) at 31-33. [Resume]

3.Here I mean the classicalmathematicalobjections. In this paper I put to one side theological objections such as that completed infinities contradict the doctrine that God is both infinite and unique. [Resume]

4.See Moore,The Infinite(1990), at p. 48. Many ancient and medieval scholars, however, accepted the view that infinite sets permit self-nesting; Kleene,Mathematical Logic(1967) at p. 176.n.121 cites various authors who point to Plutarch in the first century of the common era, Proclus in the fifth, Adam of Balsham in the twelfth, and Robert Holkot in the fourteenth. [Resume]

5.When a theory oflesservirtue is opposed by intuition, the remedy is not as clear. For example, when Zeno argued through his four paradoxes that motion and change were impossible, and hence illusory, his conclusions were opposed by everyone's intuitions about the reality of motion and change. In this case it's not clear whether we should trust Zeno's logic more than our intutions, orvice versa. [Resume]

6.Descartes,Meditations(1641) at pp. 126-127 (Meditation VI). [Resume]

7.Descartes,Discourse on Method(1637) at p. 28 (Fourth Discourse); see also hisMeditations(1641) at pp. 64, 69, 71, and 73. [Resume]

8.Descartes,Meditations(1641), at pp. 101-102 (Meditation III). Note that this argument would work just as well with very large finite magnitudes. [Resume]

9.Descartes,Meditations(1641), at p. 102 (Meditation III). [Resume]

10.Descartes,Discourse on Method(1637), at p. 26 (Discourse IV), andMeditations(1641), at p. 103. [Resume]

11.Descartes, who thought we had a positive and not merely a negative idea of the infinite, draws the same conclusion: "[M]y notion of the infinite is somehow prior to that of the finite...." Descartes,Meditations(1641), at p. 102 (Meditation III). [Resume]

12.Peirce,Collected Papers(1885), at pp. 210-249, 360; and Dedekind,Essays on the Theory of Numbers(1888), at p. 109 (theorem 160). Bernard Bolzano may have been the first to suggest this idea in hisParadoxien des Unendlichen, Section 20, published posthumously in 1851. [Resume]

13.For a proof that the Peirce-Dedekind ("reflexive") definition of infinity is equivalent to a more traditional ("inductive") one, see Fraenkel,Abstract Set Theory(1953), at pp. 41-42. [Resume]

14.One might argue that "the failure or impossibility of self-nesting" is simply a negative way of describing Euclid's positive principle that the whole is always greater than its (proper) parts, and that therefore the idea of self-nesting is equivalent to the negation of the positive Euclidean idea. While this is true, it remains the case that self-nesting, at least after Cantor, has taken on a positive life of its own and may be thought in its own terms, directly, and no longer as the mere failure of the Euclidean logic of parts and wholes. [Resume]

15.See Royce, "The One, The Many, and the Infinite," (1899), esp. pp. 503-507. [Resume]

16.Leibniz,Monadology(1714), at §3. For Cantor's physical speculations on similar topics, including his views on mass-monads (which were infinite but not continuous) and aether-monads (which were infinite and continuous), see Rudy Rucker's translation of Cantor in Rucker, "One of George Cantor's Speculations on Physical Infinities," (1978). Cantor's views are briefly summarized in Rucker,Infinity and the Mind(1982), at p. 90. [Resume]

17.Leibniz,Discourse on Metaphysics(1686), at §§8-9, 14, andMonadology(1714), at §§62-68. [Resume]

18.Leibniz is not alone in arguing for the truth of this vision, as opposed to its mere possibility or consistency. Royce, "The One, The Many, and the Infinite," (1899), at pp. 538-554 argues that the entire "realm of reality" is a self-representing system, just like England conceived as the home and subject of its own perfect map. [Resume]

19.The positive idea of self-nesting not only frees us from the indirectness and incompleteness of knowing infinityvia negativa, but as a bonus it decisively answers one line of objections to the idea of a completed infinite. This line of objections asserts that the very idea of a completed infinite is unattainable by finite human beings, or incoherent and contradictory, or meaningless. The positive idea of infinity, if it exists and we possess it, and its consistency, are standing refutations to this line of thought. [Resume]

20.The examples show that sometimes we want terms of indefinite largeness rather than infinitude. That is why the American Indian expression that a promise will hold as long as the grass grows and the rivers flow is more accurate and credible than a declaration for eternity, even if it is still an overstatement. [Resume]

21.When Kant speaks of the human person as a being of "infinite worth", is this another figurative or exaggerated use of the term "infinite"? A tool may be used as a means to an end, and nothing more, without violating its dignity; the reason is that a tool has only "finite worth". As Kant is wont to say, a tool has a price, while a person has a dignity. Kant,Foundations(1785) at p. 53. If we measure the "worth" of these entities with a unit of finite size, such as the dollar, then the tool has finite worth. But it's not clear whether the person has infinite worth or whether the person is beyond measure the way she is beyond price. To say that a person is worth an infinite number of dollars may be as much a category mistake as to say she is worth a finite number of dollars, and just as far from capturing Kant's meaning. This is why I don't use the human person as an example of something we experience which is literally infinite. [Resume]

22.At 18 frames per second, old silent films look jerky. The jerkiness alerts us to the fact that we are viewing a rapid succession of frames, not a continuously changing image. But our ability to discern 1/18th second intervals of time, and see the jerks, is not the same as the ability to discern that we are seeing 18, rather than 17 or 19, frames per second. It is, however, enough to tell us that we are experiencing a finite number of frames per second. But once the speed increases to the point where the jerkiness disappears, and the appearance of continuity sets in, we cannot know whether the underlying pace of frames is infinite or finite but huge. [Resume]

23.Several of Zeno's paradoxes of motion are best solved by using the commonplace notion of the calculus that we can traverse an infinite number of spatial units in a finite time. Note, however, that those who object to the use of completed infinities cannot answer Zeno in this way, for it is to appeal to a completed infinity of spatial units successfully traversed. [Resume]

24.Shakespeare,Hamlet, Act II, Scene II, line 258. The quotation continues: "...were it not that I have bad dreams." [Resume]

25.Peter Schor, "Algorithms for Quantum Computation: Discrete Log and Factoring," (1995). See also Seth Lloyd, "Quantum Mechanical Computers," (1995). [Resume]

26.One unexpected reason why this matters is that if potential brain inputs through the senses are only finite, then artificial intelligence is definitely possible. That is, we could in principle create a computable function that duplicated the brain's operation flawlessly. Whether AI is possible when potential brain inputs are infinite is still unsettled. See Copeland,Artificial Intelligence(1993), pp. 233-238. [Resume]

27.Leibniz,Monadology(1714), at §§8, 61-62. [Resume]

28.Leibniz,New Essays(1790), at pp. 53-58. [Resume]

29.In my view, Leibniz is the first thinker to posit unconscious experience. It is important, then, that his theoretical motivation is not to explain memory, dream, or neurosis, but the infinitely small sensory influences that constitute all sensation and the infinitely large number of sensory experiences. [Resume]

30.Kant,Critique of Pure Reason(1781), at B.462. [Resume]

31.Kant,Critique of Pure Reason(1781), at B.454. [Resume]

32.Kant,Critique of Pure Reason(1781), at B.533. The world would be either finite or infinite if it were a thing in itself, B.532.Here is one way to paraphrase Kant's view here. There is no empirical way to ascertain whether time and space are infinite, or whether time, space, or matter are infinitely divisible. So on empirical grounds we can say neither that they are infinite nor that they are finite. To try to decide these questions on

a priorigrounds is precisely what leads to contradiction. Hence ona priorigrounds as well we can say neither that they are infinite nor that they are finite. [Resume]

33.Even if this is not true of a googol, it is true of 10^{googol}. The point is that there is some large finite number which is larger than the cardinality of any collection we've ever experienced. [Resume]

34.Kant,Critique of Judgment(1790), at p. 124: "[T]he infinite...for sensibility is an abyss." Cf. pp. 115, 130. [Resume]

35.In this section I will speak only of theinfinitely large. [Resume]

36.See Dauben,Georg Cantor(1979), at pp. 288-291, 294-297. [Resume]

37.See Theorems 12, 13, 18, 20, 21, 23, and 24 in the Appendix. [Resume]

38.William Blake,Auguries of Innocence, lines 1-4, inViking Portable Blake(1946) at p. 150. Also see hisMarriage of Heaven and Hell: "If the doors of perception were cleansed every thing would appear to man as it is, infinite. For man has closed himself up, till he sees all things thro' narrow chinks of his cavern,"ibid. at p. 258. [Resume]

39.Kant,Critique of Judgment(1790), at p. 105. The italics are Kant's. [Resume]

40.Kant,Critique of Judgment(1790), at p. 114: "The infinite, however, is absolutely large (not merely large by comparison). Compared with it everything else...is small" —at least if "everything else" is limited to finitely large objects. Here Kant mistakenly assumes that all infinities are equal, a common mistake before Cantor. If one were larger than another, then the latter, although infinite, would indeed be small in comparison with something. In Kant's defense we may offer Moore's view that Kant was one of the first thinkers to acknowledge that it is no contradiction to suppose that one infinity can be larger than another; Moore,The Infinite(1990) at p. 90. (Moore does not make clear on which passages in Kant he bases his reading.) [Resume]

41.Kant,Critique of Judgment(1790), at pp. 98-99. In an earlier work Kant says the sublime brings "enjoyment" but sometimes with "horror", while the beautiful is a "pleasant sensation but one that is joyous and smiling"; "[n]ight is sublime, day is beautiful"; the sublime "moves", the beautiful "charms"; the face of a person feeling the sublime is "earnest, sometimes rigid and astonished" while the face of a person experiencing the beautiful shows "shining cheerfulness [and]...smiling features";Observations on the Feeling of the Beautiful and Sublime(1763-64), at p. 47. [Resume]

42.These mixed feelings are in tension. Unlike the beautiful, the sublime does not yield pleasure. Because the mind is both attracted and repelled, it responds more with admiration than liking, which Kant calls a "negative pleasure",Critique of Judgment(1790), at p. 98; cf. pp. 129, 131. It includes a note of displeasure, with our inadequate sensory and imaginative resources, pp. 114, 116, leading Kant to call it "a pleasure that is possible only by means of a displeasure", p. 117. [Resume]

43.Spinoza,Treatise on the Emendation of the Intellect, at p. 235. [Resume]

Peter Suber,
Department of Philosophy,
Earlham College, Richmond, Indiana, 47374, U.S.A.

*peters@earlham.edu*. Copyright © 1998, Peter Suber.