August 4, 2015

Beyond Words – and Back

William James was a major figure in 19th century American philosophy. Indeed, he is sometimes referred to as the Father of American Psychology. He was a rigorous thinker but also had a sympathy to religious experience, particularly of the mystical type.

According to James, one of the characteristics of mystical experience is that it can’t be put into words. I’m not sure if it was James who started this idea but certainly many scholars of religion make similar pronouncements: mystical experience cannot be talked about at all, what to say in precise language. At one time I believed such pronouncements because scholarly authorities had made them and also because some Buddhist masters concur. However, at this point in my life, I strongly disagree with the notion that it’s impossible to describe mystical experience precisely.

Of course, it’s true that in order to have mystical experience on a consistent basis, a person has to work through the drive to think in words. So, yes, one part of the mystical journey involves the struggle to get beyond words. But another part involves the struggle to describe in words how to get beyond words, and to describe in words what the experience of getting beyond words is like. There are many ways to get beyond words. You can find one possible description of how to get beyond words by following the ten steps presented here (pp. 39-46). Step 10 - Dance At The Source describes in words (and pictures!) what it’s like to go beyond words. You can find a more detailed breakdown here.

As most of you know, mathematics is a bit of a hobby with me. Recently I discovered a little known byway in the history of early 20th century math—an interesting dialectic between European and Russian mathematicians.

Set theory is the most commonly used foundation for mathematics, and mathematics is foundational for science, so set theory might say something deep about the mind, if not nature itself. One initial problem with set theory was that, if one accepts certain seemingly reasonable assumptions, it can lead to weird stuff and paradoxes. Not just things like Russell’s Paradox (which many people are familiar with), but really weird stuff, like the Banach–Tarski Paradox.

According one historian, Loren Graham, some of Russia’s most famous early 20th century mathematicians were followers of a renegade Eastern Orthodox sect called Imiaslavie. The Imiaslavie theologians firmly believed that God could be precisely named. According to Graham, this emboldened the Russian mathematicians to pursue certain implications of set theory that their more rationalistic European counterparts were unwilling to face.

I’m not sure how relevant this bit of esoterica is to my disagreement with James and other authorities. But, if nothing else, it’s an interesting little byway in the history of science that I thought to share with you.

You can read about it towards the end of this short article.

Also check out this interview with Loren Graham on SoundCloud:


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  2. Greetings Shinzen --- I was delighted to see this, because I've had a lifelong interest in the relation between mysticism and pure mathematics, especially in relation to the infinite. My biography seems to be a kind of opposite to yours in this respect: Fascinated by the infinite as a teenager, I began as a math major, and even got a B.A., but finally realized my path was in the arts. It was years later that I discovered meditation, and even more years before I realized (thanks mostly to Franklin Merrell-Wolff, who was my teacher) that my fascination with the transfinite was very relevant to mystical insight, and to his own efforts to envision a Western non-dual Yoga, one of whose keystones would be pure mathematics, especially modern. Whereas your story seems to have begun as a poor C-student in math, and then many years later, through meditation and concentration, you found that you were actually good at it, and fascinated by it. You even taught courses, if I remember correctly.
    Georg Cantor himself had an intuition of this connection when he wrote the following:
    "When the finite is placed in relation to the infinite, as one can plainly see everything happens. If it (the finite) comes first, it goes into the infinite and disappears there. If however, it knows this, and takes its place after the infinite, then it remains preserved and joins itself to a new and modified infinity."
    It would be good to have a little knowledge of transfinite ordinals to know exactly what he means, but for those who do, he is referring to the inequality:
    1 + ω ≠ ω + 1 (where omega is the ordinal number of the lowest infinity, that of all the counting numbers.)
    What is he saying in more ordinary language? How about this Buddhist translation: only through the realization and practice of no-self (disappearing into the infinite) can the true meaning of self be realized.
    If anyone's interested, I'll post a link to some material about Franklin Merrell-Wolff, and also some videos about transfinite mathematics.

    1. That was excellent and informative. Thank you.

  3. I should have added this: 1 + ω = ω ; in other words when the 1 comes first, it is swallowed up into infinity. When it comes second, it is part of a modified infinity.
    To picture this, you might imagine the infinity of counting numbers along a line like this:
    . . . . . . . . . . ...

    which cluster infinitely at the last. The limit-point to which they cluster is ω.
    1 + ω would just add another point to the left of the leftmost point here, and so the result would look essentially the same as this. However, ω + 1 would look like this:
    . . . . . . . . . . ... .

    which is a different picture.

  4. Sorry, graphics are difficult here. But below, I've posted a link to an excellent visualization of transfinite ordinals. Discussion is in French, but the picture shows ω
    and ω + 1 towards the left of the picture (and then the mind-boggling infinities that follow...!)

  5. I might add that the controversy aroused by Cantor has never gone away. Probably it is at least as old as Aristotle's dictum that the infinite is always potential, and never actual. Hence, in a strict Aristotelian sense, there is no such thing as the set of all integers, because even that would be a completed infinity. In the 20th century, some mathematicians and philosophers of mathematics attempted to deny Cantor, but they seem a minority. I don't know for sure, but my impression is that the majority of mathematicians who think about these things would agree more or less with Hilbert: "No one can expel us from the paradise which Cantor has opened for us." There are many reasons for this, but one of them is simply esthetic: the mathematics that results from closing and locking the doors of this paradise, and working around its principles leads to a mathematics which is extremely cumbersome, complicated, awkward, and downright ugly.

    1. Hi Joseph,
      Thanks for all the cool remarks. I love this stuff.

      You've probably seen my other math-related blogposts here:

      I also have a math article here:

      And some vids:

      Your quote you give from Hilbert above has always been one of my favorite math zingers.
      All the best,

  6. It's long been one of my favorite math quotes also, Shinzen. At first, I considered the minority opposition to Cantor (and the closely-related opposition to the Axiom of Choice) as philosophically reactionary, and the sign of an anal-retentive imagination --- even if it came from a major mathematician such as Poincaré, who called Cantor's work a "sickness". But subsequently, I had deep encounters (years apart) with two of these doubters --- one in the flesh, and the other through writings --- who changed my mind considerably — not about the value of Cantor's marvelous discoveries, but at least about the depth and subtlety of some of these doubters' positions. It so happens that the one I met in the flesh was the friend you mentioned earlier on this blog, Newcomb Greenleaf (don't know if he remembers me, but my warm regards if the occasion arises), who at that time seemed to have a kind of neo-constructivist position, with a flavor of Buddhist philosophy added. The other challenge, which really hit hard, was Wittgenstein. As far as I know, his critique of infinitist mathematics is the deepest and most challenging ever written. He himself said he felt it was one of his most important writings. If anyone's interested, I'll post a link to a pdf of his long essay on the subject.

  7. My basic position, however, hasn't changed that much. I can't agree with Wittgenstein, though I confess I intend to read that essay more thoroughly someday.
    Briefly, my feeling is this: those who balk at completed infinities often seem to also balk at the notion that pure mathematics (meaning axioms + definitions + theorems) is fundamentally a GAME. Different axiom-systems give different imaginal worlds, whose number is much vaster than theoretical physics. Though some of them include theoretical physics, and are astoundingly useful, they don't have to be "useful" in order to be of great value and beauty. Are these worlds real? Depends on what you mean by "reality." Just because something is a game, doesn't mean it's unreal. And the word "Illusion" is related to ludus, meaning "game, play." The ancient Vedic notion of Lila posits a cosmic game, which is real from one point of view, and illusory from another.
    Now, it seems to me that from a Buddhist, non-dualist view, the most important limitation of this whole math system of games is that the whole thing is based on the principle of the excluded middle. "If you can prove that the non-existence of a mathematical object leads to a contradiction, then you've proved it exists." I understand that people balk at this — I'd balk too, if someone tried to apply it to life in general. But if you're going to accept the excluded middle, then you've got to accept it, too. And so far, all attempts to liberate mathematics from the excluded middle have failed, as far as I know. I could be wrong, or outdated, but the ones I've seen didn't deliver the goods, and were lacking in elegance, to say the least.
    Perhaps category theory will open new possibilities? What is its implication for the infinity debate, do you know?
    In sum, the only concession I'd make to anti-infinitists, is to agree that it's a form of hubris to suppose that human thought, grounded in languages which separate subject from object, can ever fully master the notion of an infinite set, such as the completed set of all counting numbers. When, perhaps as children, we grokked for the first time that incredible endlessness, it left a permanent trace of psychedelic mind-boggle in most of us. In a sense, thought must sooner or later prostrate itself in humility and surrender to this infinity, which it has not created, but discovered (and perhaps invited). Later, with far more sophistication, we attempt to capture and master the dragon for good, fortified by a dose of Cantor's mathematical LSD (that's a quote from no less than Gregory Chaitin!) and soaring off into the transfinite worlds. Then we sooner or later come down, and find ourselves staring at a mind-boggling vista of omega to the omega to the omega power, omega times, to... "inaccessible" numbers, and other strange entities, in an ungraspable, ever-exploding endlessness, beside which the Big Bang seems a mere firecracker, and the whole process reminds us of something ... it begins to feel suspiciously like... the same mind-boggling vista of those simple, endless counting numbers all over again, when we first gazed on them, with a more naive mind... so perhaps this adventure in thought works kind of like a huge koan, forcing thought to the end of its tether, till it snaps ... ".. and arrive where we started, and know the place for the first time." (T.S. Eliot)

    .... just something to think about ... ;-)

  8. Joseph Rowe, could you post the Wittgenstein PDF? I've come across some of his remarks on Cantor in various places, but not sure I've read what you're referring to.

    If you have time to read it, I'd be curious to hear your thoughts on this explication of Wittgenstein's objections (or at least some of them) to the diagonal argument:

    In the case of Cantor's proof, not merely actual infinity but denumerability of the reals is I think a key point of contention: in the case of the cardinals, we know what the process of indefinite denumeration looks like, but not so for the reals. I think for Wittgenstein this is an extra problem on top of the "rule" vs. "object" problem (treating, say, pi as an object when we really only have a rule to construct additional digits, that is, 4 - 4/3 + 4/5 - 4/7 . . . = pi, but the devil is in those dots!)

    Incidentally, Wittgenstein answered Hilbert, "If one person can see it as a paradise of mathematicians, why should not another see it as a joke?"

  9. Greetings, Anonymous... thanks very much for your link. Actually, it was more useful than the one I thought I had, and I was glad to see it. It's the best interpretive resumé of the mathematical core of LW (Wittgenstein)'s objection to Cantor I've ever seen. Also, I regret to say that my link to the complete pdf of LW's Remarks is no longer free! There used to be several sites where you could download a free pdf of his "Remarks on the Foundations of Mathematics", but now the only ones I can find have been taken over by vampires like I do have this link though: ... it's a good overview, but your link is better, because it gets to the core of LW's attack on Cantor's diagonal argument.
    I need to think about this more, but I do have these preliminary remarks: LW uses polemics against Cantor such as "logical sleight-of-hand", but I see a repeated sleight-of-hand in his own arguments: over and over again, he ASSUMES the old Aristotelian doctrine that infinity cannot be complete, but only a potentiality. He seems to forget that we (pro-Cantorians) do not accept this doctrine. On the other hand, his deconstruction of the diagonal argument is brilliant, challenging, and worthy of much thought. But my quick & dirty question is: why can't one construct a new Cantorian diagonal on that lower B rectangle (which becomes a square when extended to infinity). The confusing thing is that we're also supposed to be listening to his more fundamental objection, which is that set theory consistently confuses lists with laws for making a list. But I see no reason why I should accept his claim that a set such as {1,4, 9, 25} is fundamentally different from, say, "the set of squares of the first four counting numbers." In any case, we declare them to be the same set because it makes for a simpler and more elegant mathematics.
    By the way, LW not only rejects Cantor, he rejects Gödel's proof as well. It puts him in an eccentric, fringe position, and that's probably why mathematicians tend to ignore his work. But they are wrong to do so, for this guy is a very deep and challenging thinker, and he knows his math, too. As I said, I need to think about it more. What do you say to my question about constructing a new Cantorian diagonal on the lower square, after extending the whole business to infinity by adding zeros at the end of each finite expansion? (Of course it means I have to just steamroll over LW's attempt to forbid "lawless lists", such as this sort of infinite decimal fraction.)

    My response to LW's "joke" remark: And why can't a third person with a still larger view see it as both a paradise and a joke? (However, for me it is a great, wondrous cosmic joke, and not at all the kind of derisive joke he probably means.)

  10. ooops! ERRATUM: I should have said {1,4,9,16}

  11. I think I may have found a refutation of LW's refutation, but I need to think about it more.( Never mind what I said about a new diagonal, it's a red herring)

    Meanwhile, for interested novices in this debate (no advanced math really necessary), here's a good intro:
    (see also link to controversies at end of article)

    and here's a lively, well-written page by mathematical physicist John Baez (cousin of Joan) on Cantor's ordinals, beginning with ω

  12. I have an online version of the Remarks, but it seems pretty *cough* illegal *cough* as far as I can tell, so maybe I shouldn't link it. I'll start reading it today!

    I'm not sure what you meant about another diagonal -- did you mean to change the inclination of the first one (if so, how?) or to make a second one (but what would that prove?).

    On the subject of infinity in general, do you think the concept of actual infinity is logically coherent?

  13. "I’m not sure if it was James who started this idea but certainly many scholars of religion make similar pronouncements: mystical experience cannot be talked about at all, what to say in precise language."

    I think that idea goes back at least as far as Plotinus, perhaps further!

    Borges has some great stories about the secret name of God: Death and the Compass and The Secret Miracle come to mind. :)

  14. Borges is one of my favorite authors. He was fascinated with the notion of actual infinity, which is the theme of "The Library of Babel."
    Do I find the concept of actual infinity logically coherent? The short answer is Yes. But first, I need to be clear about how I'm using the word "actual". I'm speaking about mathematical existence and mathematical realities, not about physical realities. If I were using "actual" in a way that included the latter, my answer would have to be "I don't know." But if we are speaking of actuality in the sense of the existence of the complete set of ALL counting numbers, (and the logic of the foundations of mathematics as given by ZFC set theory) I would say there is no logical incoherence whatsoever about that actual infinity, nor about the far larger infinities that result from ZFC. But it does lead to plenty of paradoxes! As far as I know, none of them imply logical incoherence, though they may seem to. There are simple, long-established ways of "fixing" the latter, such as adding notions such as classes, types, etc.
    For those unfamiliar with this debate about the diagonal argument, please remember that it's NOT a debate about whether actual infinite sets exist. Cantor assumed they do, and so do we. The debate is about whether or not Cantor really proved that, given the simplest infinite set N, that of all the counting numbers, that the infinity of N is a "weaker infinity" than that of R, the set of all real numbers. In more technical terms, the cardinality of R is higher than that of N. This opens up a mind-boggling vista of stronger and stronger infinities ... ad infinitum... (A vista which some people find wondrous, and others , such as LW, nonsense.) END PART I

  15. I believe I've found a fundamental flaw in Guido Imaguire's interpretation of LW's argument against Cantor's diagonal proof. (Definitions: for simplicity in the following, all decimals are considered as infinite decimals; i.e., instead of .029 we write .02900000.... to infinity . Whenever I say "us" I mean those who agree with Cantor's diagonal proof, and probably also in the validity of ZFC).
    For clarity, let us not forget that we don't know whether this really was the argument intended by LW! Imaguire himself admits that the original notes are "obscure." And the error I've found in his interpretive argument is so basic that I hereby withdraw my previous praise of this monograph. It doesn't seem to be very well-thought out, after all. (either that, or I've made an error, and if so, I'd be grateful if someone can show it.). I don't think LW himself would have let such a fundamental mistake pass. So I guess what I need to do is what you're doing, A., and find the time to really pore over "Remarks ..."
    THE ERROR: Imaguire seems to think that Cantor's argument depends in some way on a progressive construction of larger and larger finite arrays of decimal expansions and generalizing from them to ω rows of numbers, each of ω decimal places. But this is not so. Cantor's array is not a constructible one. The value of his own progressively larger finite arrays is mainly illustrative. To show the limits of an illustration is not to show the limits of the argument, which is quite different in this case. The infinite array on which Cantor forms his diagonal is not the result of extrapolation from the finite. No: the infinite matrix is posited "all at once," as the necessary implication of assuming that there could exist a one-to-one mapping of the counting numbers onto all the set of all real numbers --- an assumption which would imply the existence of such a list.
    Here is the core of Imaguire's contention:

    I think that Wittgenstein’s point is exactly this: If, for any finite n, it is
    impossible to enumerate a set of numbers with n decimals in just n lines,
    why should we suppose that this method would work for an infinite n? If it
    is not possible to enumerate the set of all numbers with n decimals using n
    numerals, why should we be able to enumerate the set of numbers with ω
    decimals in ω lines?

    But the phrase "this method" in that paragraph is not Cantor's method! Imaguire is apparently so attached to constructive methods that he assumes they're the only valid way to posit mathematical existence. But to us, they are not the only way. Furthermore, he doesn't seem to realize that, even if it somehow turned out that there existed a missing number that couldn't be found anywhere on this list, it would only strengthen Cantor's argument that such a countable list cannot be complete. He wouldn't even need his diagonal argument! But back to reality.
    On the other hand, it IS legitimate to ask: "Why should Cantor suppose that every infinite decimal expansion corresponds to one and only one real number?" In other words, how does he know:
    1) that every one of his horizontal rows of "lawless infinite enumerations" (LW's pejorative term ;-) must converge somewhere to only one real number?
    2) that for every real number, there exists an infinite decimal (i.e., an infinite sum, expressed in the decimal system) that converges to it, and it only?
    Well, he knows both 1) and 2) because of the axioms and theorems of the field of real numbers, as formalized by Peano and Dedekind. (Among many other things, these axioms guarantee that the line of the reals doesn't have any "holes" in it; i.e., no converging series that do not converge to a real number.)
    Voilà. END PART II

  16. And now, I hope to find the time to look at LW's Remarks. Surely he was able to do better than this in his argument against Cantor. But if I were looking for a way to refute Cantor, I'd concentrate on logic, "language-games", and the hidden assumptions that underlie mathematical infinity. In other words, I'd look for a flaw in Peano/Dedekind and/or ZFC. I'd forget about trying to find a flaw in the diagonal proof, because once you accept Peano and ZFC, it seems like a damn solid proof to me.
    A parting shot from LW:
    "a giddiness attacks us when we think of certain theorems in set theory ... when we are performing a piece of logical sleight-of-hand. This giddiness and pleasant feeling of paradox may be the chief reason it was invented."
    To which I reply, "Hey, Ludwig, Bro --- as Gregory Chaitin said, "Cantor is mathematics on LSD." If it only makes you 'giddy', and doesn't expand your consciousness and get you high, well... what can I say?"

  17. Thanks for your interesting posts, Mr. Rowe!

    Yes, I too find Imaguire's argument less convincing the more I go over it. Most importantly to me, I don't think LW is raising the sort of "logical" objection to the diagonal argument that Imaguire thinks he is since LW himself says after drawing the picture with the lower rectangle:

    "Here what we have is different pictures; and to them correspond different ways of talking. But does anything useful
    emerge if we have a dispute about the justification of them? What is important must reside
    somewhere else; even though these pictures fire our imagination most strongly."

    I think LW is less arguing with the diagonal proof itself and more with the standard language/understanding of it. He's saying that we're tempted to find some mystical significance (as Cantor actually did) in the transfinite cardinals, but that really ~nothing~ has been discovered here, as in mathematics nothing ever ~really~ is -- we only unfold the consequences of our assumptions.

    A critical point here for me (and I think for LW) -- what justification is there for saying that alelph c is GREATER than aleph null? (Am I wrong to think this language is often the way the difference is expressed?) Isn't this often where those dastardly quotation marks enter -- it must be "greater" (somehow...) since no one-to-one correspondence is possible? Whereas Wittgenstein would say that we have indeed hit upon differences between the reals and the natural numbers, but we haven't discovered one to be greater than the other.

    I think this becomes important for the theorems themselves (not just interpretations) when Cantor (or Cantorians) want to prove Cantor's Theorem or the Continuum Hypothesis. Wittgenstein would throw these problems out off-hand since ~there is no coherent object "the set of all reals" at all~ (for him what the diagonal argument shows). So there's no going past aleph null in terms of cardinality. Another paper, if you'll forgive me:

    (I link the papers because the math itself is often unknown to or forgotten by me. :) )

    I think we're on the same page that LW is ultimately thoroughly constructivist -- I think he thinks all math is constructed by people, there's no mathematical object "out there."

  18. Certainly LW is a constructivist, no doubt about that! But I think that most constructivists, even he, would agree that it's perfectly OK to call a set B "greater" (or of greater cardinality) than a set A, if it can be proven that no 1--to-1 mapping exists between them; and yet it can also be proven there does exist a 1-to-1 mapping between A and some proper subset of B. The differences do not lie in this principle so much as in how these sets are produced. Constructivists insist (though to different degrees) on the constructibility of such sets, whereas we Cantorians say that proof by contradiction (reductio ad absurdum) will do the job.
    Are the entities and truths of mathematics invented, or discovered?
    This is of course the ancient debate, going back at least to Aristotle vs Plato. I'm not going to get into it here, except to say that it seems obvious to me that both descriptions have a truth that must be acknowledged. If there's anything I'm sure of, it's that mathematics can't be reduced 100% to either invention or discovery.
    But I feel I've been hogging this conversation enough. I'm going to retire for awhile (unless someone has a very specific question of clarification, or something), hope to find time to really read LW's "Remarks on the Foundations of Mathematics" (not a very high priority in my life right now, I admit) and hope that other people who are hopefully familiar with this subject and interested in it, will post here.

  19. But to do a reductio properly, we need to make sure of the existence of the identities involved, right? I can't meaningfully say "A unicorn is just or unjust" without first assuring myself that there really are unicorns.

    In this case, the argument might be that the set of all real numbers is neither equal to the set of rationals (no 1-to-1 correspondence) nor lesser (since the reals contain the rationals), so it must be greater. But Wittgenstein is saying "I don't know how to interpret this phrase 'set of all real numbers,' and in fact your proof seems to demonstrate that such a concept is unintelligible." I was just reading today that Brouwer (like Euclid?) refused to use reductios to construct mathematical objects since we can't be sure we're not committing absurdities when we do...


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