Playing infinite chess

Joel David Hamkinsinterviewed by Richard Marshall.

Joel David Hamkins is a maths/logic hipster, melting the logic/maths hive mind with ideas that stalk the same wild territory as Frege, Tarski, Godel, Turing and Cantor. He thinks we all can go there and that we all should. He gives tips about the Moebius strip to six year olds and plays around with his sons homework. He has discovered all sorts of wonders involving supertasks, infinite-time Turing machines, black-hole computations, the mathematics of the uncountable, the lost melody phenomenon of infinitary computability (which really should be the name of a band), set theory and multiverses, infinite utilitarianism, and infinite chess. He's also thinking about whether we really have an absolute notion of the finite and doubts if any of this is brain melting, which is just a testimony to his modesty. He also thinks that although maths is open to all he thinks mathematicians could use more metaphors and silly terminology to get their ideas across better than they do. All in all, this is the grooviest of the hard core maths/logic groovsters. Bodacious!

3:AM:You work in the philosophy of maths and logic. Did you think you’d become a philosopher early on, or was it maths that first took your fancy? Or is your career a surprise?

Joel David Hamkins:Logicians have always been hard to categorise. Even Frege complained that the philosophers took him as a mathematician and the mathematicians as a philosopher, and Tarski sought in Berkeley to forge a unity between mathematical and philosophical logic. For my part, I would be happy in the company of Frege and Tarski, whatever category that may be. Meanwhile, my original training as a logician was in pure mathematics. As a freshman at Caltech, in the midst of my first genuine logic course and pondering registration for the next semester, a friend came running up to me with the news that there were many further courses in logic. There was evidently an entire field of study called “logic,” and I was in rapture; this was to be my future. I was fascinated early on by the incompleteness theorems and the independence phenomenon, by the idea that one could prove things about the nature of proof itself. We can prove that our fundamental mathematical theories are simply unable in principle to validate themselves, to prove their own consistency, like the fantastic pronouncements of a strange gentleman whom we are unsure is a con man or a sage. For every such theory, furthermore, there will be true statements that we are unable to prove in them, and so ultimately none of our fundamental theories can have the whole story. The idea that we can prove such things about the nature of truth and provability was incredible to me, and I sought to get to the bottom of it.

3:AM:Many people find logic and maths very very hard. Philosophy of maths and logic is kind of taking hard to another level! So perhaps you could begin by explaining why you think some people get maths and others don’t. Do you understand people having these feelings? Is it maths that’s the problem, or just the way its been taught or are there just some minds that are better suited to it than others? Have you ever been stumped in a way that is common to most folk?

JDH:I’m not sure that I agree with the premise of the question — that many people simply can’t appreciate mathematical ideas — and I believe instead that a fascinating world of mathematical ideas awaits anyone who is open to discovering it. This morning I went into my daughter’s first-grade classroom, full of inquisitive six-year-old girls, and we all made Möbius bands by cutting out paper strips and taping the ends together, after a twist. The children proved that a Möbius band has only one side by colouring it all the way around, whereas with a simple untwisted band they could colour the outside one colour and the inside another colour. We explored what happens with two twists, or more, and what happens when you cut a Möbius band down the centre, all the way around. Give it a try!Try cutting a Möbius band one-third in from the edge, all the way around, but make your prediction for the outcome before finishing the cut. Mathematics is full of such playful ideas; there is no reason for someone to feel that they just cannot “get” it. These ideas lead, for those who take ideas seriously, to deeper and more abstract mathematical ideas, which can illuminate a greater mathematical truth, while remaining playful for those with the right attitude. The curious properties of the Möbius band lead to further examples of non-orientability, such as the Kleinbottle, and ultimately one lands in the main ideas of algebraic topology. Many sophisticated mathematical ideas similarly have their origin in an intriguing puzzle; we begin with a curiosity or paradox and in resolving it gain genuine mathematical insight. But to answer the last part of your question, of course every mathematician and philosopher finds themselves stumped with the problems that vex them. We persevere, and often enough overcome the initial confusions and find ourselves a way through the tangle. What a great feeling it is when one solves a long-troubling question. It is surely one of life’s great enjoyments. I’d prefer that we get away from the idea that a person may not be a “math person” or may just not “get” math. Such a person is unlikely to assert in the same sweeping manner that they just don’t get literature or humour or art, or that they are not an “idea person”. Meanwhile, I guarantee that there are some fascinating mathematical ideas, at just exactly the right level, waiting for them to ponder. Anyone, even a child, can learn to count into Cantor’s transfinite ordinals, and there are fun elementary problems in graph colouring or game theory. Some beautiful classical arguments, such as the proof that the square root of 2 is irrational or the proof that there are infinitely many prime numbers, stand as pinnacles of human achievement, known for two thousand years, yet are accessible enough for any intelligent person to grasp.

3:AM:There are some really cool philosophical problems that you’ve been engaged with that I think anyone will appreciate. So can you say something about supertasks and the infinite time Turing machines? In a paper you ask what would we and what could we do with an infinitely fast computer. This sounds like science fiction but its what people are working on. So firstly, what’s a supertask and what could such a computer do? And are we going to be building one anytime soon?

JDH:A supertask is a process involving infinitely many steps, and it is interesting to imagine performing the steps faster and faster in such a way that all are completed in finite total time. That may seem perplexing, but Zeno long ago pointed out that in walking from here to there, we accomplish the supertask of first traversing half the distance, and then half the remaining distance, and then again half the remaining distance and so on ad infinitum. After only finitely many such traversals, we are not quite there, are we, and the full supertask is completed at the instant of our arrival. Similarly, if one performs the first of infinitely many steps in a process in half a minute, and the next in a quarter minute and then an eighth and so on, then the whole infinite process will be completed in one minute, in an instance of the geometric series (1/2)+(1/4)+(1/8)+ … = 1. In the Zeno example, this corresponds simply to walking from here to there at constant speed. Suppose now that we successively add two marbles to a giant bag and then take one out, repeating this action over and over, faster and faster, infinitely many times. What is in the bag when we have completed all these actions? On the one hand, since we seem to be steadily increasing the number of marbles in the bag, it would seem natural to expect that the bag has infinitely many marbles at the end. But suppose that the marbles are numbered 1, 2, 3 and so on, and that at each step we always remove the smallest-numbered marble currently in the bag, never using that marble again. With this procedure, the bag will actually be empty at the end. If there were any marbles in it at the end, then among those marbles, there would have to be one with the smallest number. This marble, if you think about it, would have to have been the smallest-numbered marble at an earlier stage in the process, when all the even-smaller-numbered marbles had been dealt with, and therefore it would have been removed at that earlier stage. But this would have prevented it from being in the bag at the end, and so indeed the bag must be empty then. I find this kind of supertask to show that certain intuitions that we have about the nature of finitary processes are no longer valid for infinitary processes, and so we must be very careful when reasoning about supertask procedures.

Jeff Kidderand Andy Lewisand I developed the theory of infinite-time Turing machinesto provide a theoretical tool for analysing the nature of infinitary computation. What does it mean exactly to carry out a computational procedure with infinitely many steps? The infinite time Turing machine model provides an answer, a theoretical answer, which helps us to understand the resulting class of infinite-time computable functions and infinite-time decidable sets, in the same way that the standard Turing machine model underlies our conception of what it means to be computable in finite time. We don’t actually build these machines, of course, in either the finitary or infinitary realm. Rather, the machines are used in what amounts to thought experiments, guiding our understanding of the resulting concepts of computability and decidability. It turns out that the infinite time Turing machines describe a rich level of complexity that was not yet captured by the other notions of complexity in descriptive set theory, and it is in this context that one should think of the subject. Nevertheless, there are several groups of researchers working on the fantastical idea that it might be consistent with the laws of physics to have a physical realisation of a supertask computational device. This line of thinking can be seen as an attack on the widely-held Church-Turing thesis, which asserts that the Turing machine model of computability correctly captures what it means for a function to be computable in principle or for a set to be decidable in principle by effective means. Turing had argued and indeed even derived his Turing machine model by thinking carefully about what it was that a human did when carrying out a rote computational procedure with paper and pencil, thereby arriving at his original formulation of the thesis. The new objections, however, are aimed at refuting a stronger thesis, for which the notion of computability allows the human computer not only to compute with paper and pencil, but also to take advantage of whatever strange and wonderful computational powers might be possible in a physical universe, using whatever bizarre quantum-mechanical or relativistic effect might be useful for computation. For example, some of the infinitary computational procedures work by exploiting relativistic time contraction (as in the twin paradox), so as to carry out an infinite procedure in finite time for one observer. Similarly, in black-hole computation, a computational device achieves super-Turing capability by falling into a black hole, finding inside another physical universe, while observing whether a certain signal follows or not sent from a component of the device that remains outside the black-hole.

3:AM:You presented some of your thoughts about this at a conference about effective mathematics of the uncountable. In your paper you liken the uncountability to being able to recognise a tune but not being able to hum it. Is that a picture of what maths of the uncountable is about? Can you say something about this kind of maths that sounds really cool?

JDH:The lost-melodyphenomenon for infinitary computability expresses one of the unusual features of it that Andy Lewis and I discovered, hinging on the distinction between the ability to produce a particular complex object and the ability to recognise the object when it is presented to you. What we realised is that there can be a mathematical object m, a certain infinite string of binary digits, such that the infinitary computers cannot produce mon their own, but they are capable of recognising in a computable manner whether a given object is mor not. The idea is that mis extremely complicated, too complicated to produce by any computable procedure, but at the same time it exhibits certain pertinent internal consistencies, which are sufficient to characterise muniquely, thereby allowing an infinitary computer to verify whether it is dealing with the real mor not. So the object mis computably recognisable, but not computable. It is a lost melody — perhaps from some classical symphony — a tune that you are not able to sing on your own, but nevertheless you can recognise correctly yes-or-no when someone else sings it. This lost-melody phenomenon has now been established for many of the various models of infinitary computability, and one of the necessary features when it occurs is that the computational devices, unlike classical Turing machines, should not be able to undertake a computable exhaustive search of their entire input space.

3:AM:Another big idea that you’re thinking about at the moment is the dispute in set theory between a universe and multiverse. Can you say what this dispute is about? I guess one of the things that interests me is the idea that the multiverse approach challenges the notion that every set-theoretic assertion has a final definitive truth value. Does this then touch on issues outside of maths, such as vagueness? And does the idea intersect with ideas in physics about the nature of reality?

JDH:Set theory can be and often is taken as an ontological foundation for the rest of mathematics in the sense that abstract mathematical objects can be construed fundamentally as sets, and being precise in mathematics often amounts to specifying one’s context in set-theoretic terms. We identify a function, for example, with its graph, a set of ordered pairs, and essentially all mathematical objects can be construed as sets in a similar way. The existence of a common foundation like this for the whole of mathematics has been extremely important for the unity of the subject, for mathematicians often borrow a result from one area for use in another, applying a result of complex analysis, for example, in algebra; this would become incoherent if each sub-area had its own separate foundation. A dominant perspective within set theory itself, what I have called the universe view, holds that there is an absolute background concept of set, giving rise to the unique cumulative universe of all sets, in which every set-theoretic assertion — and hence also every mathematical assertion — has a definite truth value. On this view, statements such as the continuum hypothesis and others have definite final answers, and the goal of set theory is to find these fundamental truths. The widespread independence phenomenon, where such statements turn out to be neither provable nor refutable in our best and strongest theories, is seen on this view as a distraction, telling us about the weaknesses of our theories rather than about what it really true.

Meanwhile, there is tension between the universe view and what is surely the principal set-theoretic discovery of the twentieth century, namely, the unexpected but pervasive phenomenon of diverse set-theoretic possibility. It turns out that almost every nontrivial statement of infinite combinatorics, for example, including the continuum hypothesis and hundreds of other similar assertions, is independent of the axioms of set theory, even when they are strengthened by any of the various large cardinal axioms. What set theory is about is exploring the full range of this set-theoretic possibility. The most powerful tools in set theory, such as forcing, ultrapowers and inner models, are most naturally understood as methods for building alternative set-theoretic universes, realising different mathematical truths. We have discovered an entire cosmos of alternative mathematical universes, related to each other as forcing extensions or through large cardinal embeddings in complex commutative diagrams, like the lines of a constellation in a dark night sky. The competing multiverse position takes these alternative universes to be fully as real as the cumulative universe on the universe view, and the debate on pluralism sets these two perspectives against each other. We may fruitfully consider the analogy with geometry, which for centuries was taken to be about the absolute concepts of point, line and space. The discovery of non-Euclidean geometry in the late nineteenth century was shocking, and the formerly crystalline concepts of geometry splintered into a multitude of non-Euclidean geometries, including spherical geometries, hyperbolic geometries and so on, with different geometrical truths and properties. Nevertheless, geometers today regard these alternative geometrical worlds to be fully as real and geometrically legitimate as classical Euclidean space, and one may find mathematicalvideos showingwhat it is like to wander around in certain hyperbolic spaces. Similarly, the multiverse position in set theory takes those alternative set-theoretical universes as fully real and set-theoretically legitimate.

3:AM:I noticed you’ve done some work on the theory of infinite utilitarianism. Can you give us some idea about that?

JDH:A simple extension of the theory of utilitarianism from finite worlds to the infinite might direct one to compare two worlds simply by computing the total sum of utility in each world, even if this should be infinite; but such an approach leads to some counterintuitive conclusions. For example, compare a first world with infinitely many people, each having happiness level 1, with a second world, in which we’ve improved each of their lives, so they now have happiness 2 each. The overall sum is the same infinity, but it seems that we might want strictly to prefer the second world over the first. The theory of infinite utilitarianism is about the various principles that we might appeal to in order to do so, and various researchers have proposed and defended a number of very specific principles. In joint work, Barbara Gail Monteroand I have argued that a number of these proposals founder on a similar extension to the infinite of a cardinality principle that holds for finite sets but fails for infinite sets. For example, consider a strong version of the Paretoprinciple, asserting that if we are comparing two worlds with the same people, and everyone in the first world is at least as well off in the second, with at least one person becoming strictly better off, then the second world is strictly better than the first overall. This sounds good at first, but some examples may loosen that intuition. Suppose we have a soccer team with infinitely many players, some very good and some terrible, with their spectrum of abilities given by the integers: ⋯ -2 -1 0 1 2 ⋯ . Now suppose that they go through spring training, and each player improves by one unit. Has the team strictly improved? According to the strong Pareto principle, yes, but if one considers the overall spectrum of ability, it appears to be exactly the same as it was before, ⋯ -2 -1 0 1 2 ⋯, with the only difference being that each player has moved one place to the right, a difference that doesn’t seem to matter in term of the team’s ability. Montero and I argue that the intuition we have for the strong version of the Pareto principle is essentially similar to and ultimately as flawed as the naïve intuition that adding an element to a set always produces a set strictly larger in cardinality. This principle is true for finite sets, but false for infinite sets. A slightly weaker Pareto principle, which we believe is the correct principle, asserts that if every individual in one world is made at least as well off in another (with the same individuals), then the second world is at least as good overall as the first. The topic has numerous papers on these and similar questions, with some of the proposed collections of principles turning out to be subtly inconsistent, as well as dozens of fascinating thought experiments, which test our intuitions about the comparative value of worlds.

3:AM:And a very groovy paper you’ve just produced is about infinite chess. What is infinite chess? Have you proved that there can or can’t be check mate in such a game?

JDH:Last year when I was visiting at NYU, I put up on my office door in the philosophy department a poster of one of my infinite chess positions, a little puzzle, on which there gradually appeared scribbled comments and post-it notes from the graduate students and post-docs who proposed solutions. Infinite chess is chess played on an infinite chess board, a boundless plane tiled with alternating black and white squares on which the familiar chess pieces— kings, queens, rooks, bishops, knights and pawns — move about according to their usual movement rules, striving to place the opposing king in checkmate. Checkmate, when it occurs, does so after only finitely many moves, and infinitely long play counts as a draw. There is no standard starting position, but rather one considers the game beginning from a specified initial position, not necessarily finite. Although unfortunately we cannot sit down in a café for a game of infinite chess, it remains a game of the mind, and there are interesting game-theoretic questions about it remaining open. For example, it is not known whether there can be, even in principle, a computable procedure giving optimal play from any finite position of infinite chess. In ordinary finite 8 x 8 chess, in contrast, like any finite game, there is in principle a computable procedure for optimal play, simply because the entire game tree — the tree of all possible legal moves — although vast, is finite, and one may compute optimal play by recursion backwards from the finitely many positions where the game has been already won or drawn. So as our computers gain power, it is inevitable that they will be able to implement this procedure and when they do, computers will achieve absolutely perfect play in chess. Already computers can beat any human player essentially by searching huge parts of this game tree, which amounts to an approximation of this ideal algorithm.

In infinite chess, however, this game tree argument breaks down, because the tree is infinite and cannot be searched in finite time. And not only is the game tree infinite, but it is also can be infinitely branching, because queens, bishops and rooks may have the choice of infinitely many legal moves from a given position. Thus, we cannot hope even to search the game tree to finite depth. This observation had strongly suggested that the mate-in-n problem for finite positions in infinite chess the problem of determining whether a designated player can force checkmate from a given position in at most n moves — might not be computably decidable. Meanwhile, Brumleve, Schlicht and I proved that the mate-in-n problem of infinite chess actually is decidable. Our algorithm does not search the game tree, which as we’ve said is impossible, but rather it interprets the mate-in-n problem as an assertion in what we call the structure of chess, whose theory we proved is a regular language in the sense of finite automata theory and therefore decidable. The fact that checkmates, when they occur, do so after only finitely many moves makes infinite chess technically what is known as an open game and therefore subject to the theory of transfinite ordinal game values, a powerful theory that applies to any open game. In this theory, the transfinite ordinal value of a position is a kind of abstract measure of the distance of a position from a win for a designated player. A position has finite value n, for example, when the player can force checkmate in at most nmoves. The position has value ω, for example, when any move by black leads to a mate-in-n position winning for white, but black may choose nas large as desired. Such transfinite values and higher occur in infinite chess, but it is totally open how large they may become. The omega one of chess is the supremum of the game values that arise. In our recent paper, Evans and Woodin and I prove that the following infinite position has value omega squared times four, with black to move, and we have other positions with value omega cubed and higher. We conjecture that the game values will go considerably higher. We had an idea for establishing this, but unfortunately, it didn’t quite fit into the two dimensions of ordinary infinite chess. But the idea does work, if you can believe it, in infinite 3D chess! The history of 3D chess, you may be surprised to learn, spans several centuries, going back at least to Kieseritzky’s Kubikschack in 1851, including Maack’s raumschach chess clubs in Hamburg from 1919, and running well into the twenty-third century, in light of the games played by Spock and Kirk on several Star Trek episodes. We may be the first, however, to have established nontrivial facts about the infinite version of 3D chess. What we proved is that every countable ordinal arises as a game value of an infinite position in infinite 3D chess, and thus the omega one of infinite 3D chess is true omega one, as large as it could possibly be.

3:AM:So can you introduce the lay folk to some of the really interesting ideas that you’re brooding about in the philosophy of maths? What are the cutting edge thoughts around and why are they significant? And are there some way out, weird calculations that would melt our brains?

JDH:One idea I am brooding on — I’m not sure it will melt anyone’s brain — is the question of whether we really do have an absolute notion of the finite, as many seem to say we do, for example, when describing the natural numbers as the set containing, 0, 1, 2, and so on. My question is whether that “and so on” is really as meaningful as some people take it to be. Mathematicians, of course, give some substance to the idea by proving that the natural numbers are categorical, in that there is one and only one structure satisfying the second-order Peano axioms, and this is the structure of the natural numbers. My objection to this explanation is that this second-order proof in effect establishes the uniqueness of the concept of finiteness by relying on the presumed absoluteness of our concept of set of natural numbers, which is surely a murkier realm in comparison. So how can people think this approach succeeds? Ultimately, I find myself increasingly led to the idea that there may be an undiscovered plurality of finiteness concepts, going along with the plurality of set concepts, that different mathematical universes may have incomparable concepts of the finite, each of them improved by a stronger concept of the finite. Indeed, I speculate that this kind of plurality is ultimately what underlies the pervasive logical independence we find in our theories as well as the pervasive phenomenon of computable undecidability. I await technological developments in set theory that will provide an arithmetic analogue to forcing, allowing us to modify the arithmetic of a model of set theory in a controlled way that does not seem to give preference to the legitimacy of one model over the other, just as forcing does in the higher-order realm.

3:AM:One thing that strikes me is that maths is dead important but the higher end of maths is largely inaccessible to ordinary folk like myself. However, it is this higher stuff where a lot of maths becomes both significant and really weird and exciting. I noticed that your intro to your paper ‘Set-Theoretic Geology’ had a very neat picture that helped guide a reader someway into what you were analysing, so do you think perhaps pictures are underused in making higher maths comprehensible to the folk? So is there any way that I could become maths literate without being able to do the calculations? Could there be a use for maths teaching for the non-specialist so the sorts of issues you are examining become more widely known about, in the way that science literacy might be promulgated?

JDH:Mathematics is a very tall subject, in that new knowledge often builds on earlier knowledge, and it can be difficult to understand or appreciate the new knowledge without mastering the old. Euclid famously told Ptolemy that “there is no royal road in geometry,” no shortcut for kings. Nevertheless, I also believe that mathematicians can often do a much better job of explaining their ideas, in particular, by making more room for metaphor and soft explanations, which help to explain the idea behind an argument. Such explanations are useful not only for beginners, but also help experts to organise their thoughts about a topic in a more enlightening and higher-level manner, allowing them to apply those ideas elsewhere. How frustrating it is to read an incomprehensible mathematical argument, which contains numerous details while omitting the most important part, a high-level description of the argument. I try in my own writing to explain my ideas on many levels, using pictures, metaphors or whatever I think might be helpful. I’ve realised in retrospect, however, that this has often caused me to introduce odd or even silly terminology, such as the lost-melody theorem, the hypnagogic digraph, and buttons and switches in the modal logic of forcing, to name a few. I am trying in each case to convey a particular idea as best I can, but does the silly terminology make me absurd? Probably, but if it helps people understand the idea, then so be it.

3:AM:I’ve noticed that you are the top-rated user on MathOverflow, the new online forum for mathematics questions and answers. What can you tell us about that?

JDH:MathOverflowis a new online forum for asking and answering research-level questions in mathematics, with dozens of new posts every day on diverse topics, including many on logic and foundations, including the philosophy of mathematics. The forum provides a vastly more focused manner for mathematicians to interact than was previously possible, connecting researchers who are interested in or knowledgeable about a particular topic, drawing from a large pool of users ranging from graduate students and post-docs up to senior faculty, including a surprising number of extremely prominent mathematicians. Faced with a puzzling mathematical issue, you can tap into this collective expertise simply by posting a question. In very short order, you will find that someone, somewhere in the world, who has thought deeply about that very issue, will post an expert reply. Indeed, you may receive several answers giving perspectives on your topic that hadn’t occurred to you. Young researchers can experiment with the ideas they are learning, by posting questions right at the boundary of what they understand. I have used MathOverflow not only in this ordinary way to learn a topic I was curious about, but also to confirm, for questions arising in my own work that I suspected were difficult, that they did not admit of an elementary solution I had missed. People now say that a problem is “MathOverflow-hard” — a pun on the terminology such as NP-hard from complexity theory — to mean that a problem was asked on MathOverflow, but not answered there, a rare occurrence indicating that a question is difficult. Part of the success of MathOverflow, in addition to efforts to maintain a high-level of discourse, is attributed simply to its manner of social design, using principles or even tricks of social engineering to motivate people to participate, and the software model, built by Stackexchange, seems to have hit upon some successful ideas. The result, though, is a slightly game-like nature for the forum, since contributors earn “reputation” points when other users vote up their contributions. This may sound silly, and some may find it off-putting, even if it does stimulate the competitive instincts of others, but the voting and particularly the sorting of answers by votes does seem effectively to communicate the community’s perspective on a post, because quality answers generally rise to the top, while junk answers inevitably sink to the bottom. So people are motivated to provide explanations that other users find valuable. Personally, I am engaged on MathOverflow because I find the mathematics there to be compelling. I was surprised to find such a strong interest in mathematical foundations among non-logicians there, with numerous questions about set theory and the philosophy of set theory coming from other parts of mathematics, and of course I was pleased to find that my explanations were appreciated. So, yes, I am the top-rated user there, and I can joke with my colleagues that at least in this sense I can claim to be the most “reputable” mathematician in the world. In any case, I’ve learned an enormous amount of mathematics there.

Meanwhile, there is also the companion site, Math.Stackexchange, which is open to lower level material. You can check out a question I asked there about the large numbers in what I call the googol-plex-bang-stack hierarchyand another concerning a homework assignmentmy fourth-grade son brought home from school. There are also an astounding number of fantastic mathematics blogs, at every level of expertise, many of them written by enthusiastic youngsters who take pains to explain an idea very well; search MathBloggingto find one to your taste. And please climb into Cantor’s Attic, a site I founded with Victoria Gitman, a Wikipedia-style compendium of information on all notions of infinity, large and small.

ABOUT THE INTERVIEWER
Richard Marshallis still biding his time.