Naming the Infinite

Jorge Luis Borges, the intellectual's intellectual, begins his essay on the infinite, "Avatars of the Tortoise," with this:

There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite.

Of course the infinite would be worse than evil to an intellectual! It's a concept that evades all discourse: as soon as you name it, you are no longer discussing it.¹ The real infinite cannot be named. Think about what the word means: it means without borders (Latin, fines), boundaries, limitation, or even form. But as soon as you say that, you've imposed a boundary: at very least you've separated "the infinite" from the strictly finite. How do we resolve this paradox?

When we speak of the infinite, it is usually in the context of quantity. But numbers themselves are forms. For example, we say the counting numbers are infinite, that is, they extend to infinity. But they, like the integers and rational numbers, are ordered. For the integers, at least there is a "next number"; the same cannot be said of the real numbers, since between any two real numbers is another number.² So the counting numbers and integers are not as "infinite" as the real numbers. Cantor says the real numbers have a larger cardinality than the integers, rationals, etc. There are innumerably more irrational numbers than rational.

And it's not just the size of number sets that shows there are bounds added or removed in passing between different sets of numbers. There are also degrees of generality. For example, in passing from counting numbers to integers, we gain negative numbers (and zero, but that is another, quite different matter): we incorporate into the numbers themselves the subtraction operation that is the inverse of the addition operation implicit in what we come to recognize as the positivity of the counting numbers. So now instead of taking away, I can talk of "giving" a negative quantity. So we've expanded our numbers beyond the verb (give/take) to handle giving both "positive quantities" and "negative quantities." We've expanded our concept of "number" to incorporate more of reality, to bound and in some sense homogenize within our numerical discourse what was once outside the boundaries and unbounded. Thus discovering positive and negative numbers, we chip away at the boundless infinite.

But what if we generalize the give and take of the positive and negative numbers? What if we posit a continuum of giving and taking? So instead of just the two discrete operations, "give" (i.e., giving a positive) and "take" (i.e., giving a negative), we create a new continuum, a second dimension alongside or orthogonal to the continuum of the real numbers. So we recognize things in-between giving and taking, a kind of swinging back and forth between these positive and negative poles. One might say an oscillation. This is where "imaginary" (and then the complex) numbers come in, and they can help us keep track of the phase of an oscillation between giving and taking. We derive the full negative by multiplying together these parts, which are complex numbers.

With the successive removals of boundaries, it begins to become clear what mathematics has been doing the past couple centuries is to explore ways to better speak of the unspeakable, and to approximate the infinite. The task of modern mathematics is to name the infinite.

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3 Brown 1 Blue recently hosted an insightful video "What was Euclid really doing?." If really gives a sense of the physicality of ancient mathematics, very different from modern mathematics.

The difference between ancient and modern mathematics is the fundamental point of Jacob Klein's Greek Mathematical Thought And The Origin Of Algebra. With the rise of algebra, numbers lost their clear reference to physical reality, and become for the most part a self-enclosed set of symbols. That move muddles the clarity of how numbers relate to reality. As a result, it is less than clear how one conception of number relates to the other. Yes, we gain in power of articulation in "expanding" our notion of number, but the price we pay is homogenization: we lose the distinct qualities of the things we've ground up and extruded into the growing saugage casing of "number." We become unmindful of qualitative differences between the sundry types we've folded in.

Leopold Kronecker's statement that "God made the integers, all else is the work of man" makes the basic point, while also providing an example of how so sedimentized the conceptions of number had become by the nineteenth century that he included negative numbers and zero as truly natural creations. On that latter point, the inclusion of zero even among the "natural numbers" is an example; zero is not a number (numbers count groups of things), but a place holder, a sign of a currently unfulfilled potentiality for things that can be numbered.

Where do the other sets of numbers come from? Rational numbers (fractions) the Greeks thought of as setting up a new unit. They knew of irrational numbers and connected them with the infinite, the unspeakable or unwritable as a finite fraction, which is why they're also called "surds."

People always wonder about imaginary numbers. But even before that, simple negative numbers are strange, and we do well to notice their oddity, unlike Kronecker. I have provided retrospective justifications for these above, and hopefully they make some sense.

It is interesting though that modern math has proven to be the backbone of modern physics³. It may not be an accident that at the same time modern physics has difficulty connecting to the everyday embodied experience of human beings. Relating these two is the great task that stands before anyone alert to the neo-gnostic chasm standing before the modern world and the doom it represents. I hope that this short discussion in some measure traces the outlines of a bridge between human life and some of the conceptual apparatus that dominates our technological civilization and makes possible its material wealth.

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Notes

1. Similarly, when we talk of "nothing," it's at least a concept and so is something. "Matter" too has many similarities with the infinite, which may be why mathematics is so useful for describing its activities.

2. This is inexact, since between any two rational numbers is another rational number, even though the rationals and the integers have the same cardinality. More precisely, the integers and the rationals can be organized into a sequence, whereas the reals cannot. So the integers and rationals are countably infinite, but the reals are uncountably infinite.

3. "Modern" physics in contrast with Aristotelian physics/natural philosophy, not in contrast with Newtonian "classical" physics that is part of the former.

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Jorge Luis Borges, Labyrinths 1962, at 202.

Jacob Klein, Greek Mathematical Thought And The Origin Of Algebra 1968.

St. Albert was Great, but what are we?

Today is the feast of Albertus Magnus, St. Albert the Great, patron saint of scientists. This week I listened to the recent Jimmy Akin's Mysterious World episode on "St. Albert the Great & Magic," which is a representation of a Justin Sledge episode.

The episode contained the statement, which I've heard many times before, that Albert was probably the last person who knew all of human knowledge at the time. It's a credible statement. But I think it's not only a statement of how remarkable St. Albert was and of how much our knowledge has grown since his time, but also an implicit commentary on how poor the academy has become at integrating its knowledge. Everything today is simply so hyperspecialized, everyone off in their own silo drilling down deeply and narrowly into their itty-bitty subject.

Medieval universities would regularly put on disputations that would call together all the faculties of the institution to debate a given topic. Can you imagine such a thing happening today? Absolutely not! That would take away professors' time from pursuing the purposes of their grants that bring money into the university. Why would they do something so liberal (as opposed to servile, not as opposed to conservative)? But it is supposedly a single universe that all fields of a university study, so light from many different angles should help to illuminate the integral whole. That is presumably why the university is called a university. These days, we don't have universities so much as diversities.

In the past it would have been philosophy that ultimately integrated all knowledge. But philosophy has been dethroned in prestige by physics, which is too busy analyzing the world into parts to do any real integration. (Even its efforts at a Grand Unified Theory suffer too much from the reductionism inherent in its methods.)

So we're left with disintegrated, illiberal institutions of higher learning setting the tone for human society. Hence the fascinating modern world we live in.

St. Albert the Great, pray for us!

Galileo's Children

Today is the feast of St. Robert Bellarmine. He had a famous meeting with Galileo in 1616 that was later popularized as "Galileo's first trial," but wasn't really a trial. There in reference to Galileo's heliocentricism, Bellarmine warned Galileo not to teach as definite what was still conjecture. It's notable that Galileo's "proofs" of the rotation of the Earth were questionable at best. For example, he claimed the single daily tide in the Mediterranean Sea (anomalous among Earth's oceans) supported his thesis. In reality, it would only be centuries later that real proofs like Foucault's pendulum and stellar parallax became available. So in this encounter Bellarmine correctly discerned the certainty of the propositions and was the better natural philosopher, that is, scientist.¹

One of the often-overlooked undercurrents of the case is that Galileo wasn't even a natural philosopher, formally. His title at Padua was Professor of Mathematics, not of the more exalted and substantial (Natural) Philosophy. Mathematics was regarded as inferior to philosophy because it failed to discuss real causes. And Galileo's status at the University was correspondingly inferior.

The fixation on mathematics and the inability to discern certainty is a besetting fault of modern science, and one source of the error of scientism, which places science above all other sources of knowledge. Our culture suffers tremendously because our "science" is not properly contextualized as being of an inferior level of certainty compared to many important truths of philosophy. It's not much of an exaggeration to observe that alongside his many great triumphs, Galileo also bequeathed his errors to us, his intellectual heirs.

May we have the courage to uphold the truth and to give science its proper respect. St. Robert Bellarmine, pray for us!

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Notes

1. The irony is that Galileo might at that time be called in some respects a better theologian than his true adversaries (Bellarmine was not really an adversary). In his 1615 "Letter to the Grand Duchess Christina," he adopted the wise statement, "The Bible teaches us how to go to heaven, not how the heavens go."

What is The Expanse?

To the choirmaster. A Psalm of David. The heavens declare the glory of God, and the expanse proclaims his handwork.

I recently ran across the ESVCE translation of Psalm 19, the beginning of which is above. I've taken the liberty of substituting the alternative translation of "the sky above," which the footnote says is equivalent in Hebrew to "the expanse", and invites comparison to Genesis 1:6-8 (sic):

⁶ And God said, “Let there be an expanse in the midst of the waters, and let it separate the waters from the waters.” ⁷ And God made the expanse and separated the waters that were under the expanse from the waters that were above the expanse. And it was so. ⁸ And God called the expanse Heaven. And there was evening and there was morning, the second day.

Well, that's blatant—no substitution by me here. The footnote says that expanse is equivalent to "canopy." Of course the heavenly canopy is reflected in the chuppah at a Jewish wedding.

I've seen a lot of people saying the title of the science fiction novels and TV series The Expanse has to do with mankind expanding out into space. But I haven't encountered anyone who made the connection to the ancient Hebrew Bible.

What does this connection say about the story of The Expanse? Is it some sort of covert Jewish or Christian story? Doubtful. While the story does highlight many real virtues in its heroes, it's not like they're unique to traditional Christianity or Judaism in the modern era. And being as the main hero has eight parents without the obvious psychological issues that would accompany such an artificial upbringing (at least in the TV series; I haven't read the books), the authors don't seem to be much in mind of any sort of Biblical or even natural morality.

Additionally the individual books of the series are titled with literary and historic allusions, e.g., Leviathan Wakes, Caliban’s War, Abbadon’s Gate. So The Expanse fits in with such allusive titles, and obviously the stories take place in "the expanse" of space.

But is there more to the allusion of the series title? For example, is the overall story about a wedding of sorts? What do you think?

Of course, there is Miller's pursuit of Julie, a romance of sorts perhaps. But that's resolved early in the series, no? Or perhaps it's synecdochical, similar to The Thin Man series with the epithet reflecting its first installment? Or is there a larger marriage the authors have in mind?

The Illusion of Love

There's a meme going around observing how Hallmark movies tend to feature a female protagonist who falls for the quirky, useless but entertaining guy over the steady but boring guy who wants to provide for her (and presumably a family). (Not exactly fair: they also tend to make the boring guy a jerk to boot.) But that's a definite tendency in Hollywood romantic-comedies in general too. Clearly a woman with her head on straight would instead be seeking a depenable man.

So we have entertainers advancing useless entertainers as the optimal lovers.

Then I was thinking about how Plato's Socrates concludes in Republic that philosophers would be the best leaders of the city. People make jokes about how self-serving that is ("if garbage men wrote books they'd be advancing themselves as rulers", etc.), but at least (political) philosophers think a lot about politics and should have some sort of expertise. Assuming they're not as scatter-brained as Aristophanes's Socrates.

What Hollywood's actual expertise is, is illusion. So we turn to Hollywood to give us the illusion of love. Problem is that what we see readily becomes our ideal. They've sold us lies. No wonder our society is so matrimonially confused.

Maybe Plato was on to something in recommending to get rid of the poets.

Bohm on self-fragmentation

Physicist David Bohm reflects on how a fragmenting philosophy, like mechanism, even though mistaken, tends to become a sort of self-fulfilling prophecy and to create fragmentation in the objective world:

Of course, there are areas in which the production of fragments is relevant and appropriate (e.g., the crushing of stones for the making of concrete). But what we are discussing here is that irrelevant and inappropriate fragmentation which comes about quite generally when we regard the "parts" as appearing in our thought as primary and independently existent constituents of all reality (including ourselves). A world view, such as mechanism, in which the whole of existence is thus considered as made up of such "elementary" parts will then give strong support to this fragmentary way of thinking, which in turn expresses itself in further thought that sustains and develops such a world view. As a result of this general approach, man ultimately ceases to give the divisions between things their proper significance (e.g., as useful or convenient ways of thinking, indicative of the relative independence or autonomy of these things), and instead, he begins to see and experience himself and his world as actually made up of nothing but separately and independently existing components. Being guided by this view, man then acts in such a way as to try to break himself and the world up, so that all seems to correspond to his way of thinking. He thus obtains an apparent proof of the correctness of his fragmentary self-world view, not noticing that it is he himself, acting according to his mode of thought, who has brought about the fragmentation that now seems to have an autonomous existence, independent of his will and of his desire.

Fragmentation is thus an attitude which disposes the mind to regard the divisions between things as absolute and final, rather than as ways of thinking that have only some relative and limited range of validity and usefulness. It leads to a general tendency to break things up in an irrelevant and inappropriate way, and so, it is evidently inherently destructive. For example, though all parts of mankind are now actually fundamentally interdependent and inter-related, the primary and overriding kind of significance generally given to the widespread and pervasive distinctions between people (family, profession, nation, race, ideology, etc.) is preventing human beings from working together for the common good, and indeed, even for survival. When man thinks of himself in this fragmentary way, he will inevitably tend to put his own separate Ego first, or else his own group. He cannot seriously think of himself as internally related to the whole of mankind and therefore to all other people. Even if he does try to put mankind first, he will tend to think of nature as something separate, to be exploited to satisfy whatever desires people may happen to have at the moment. Similarly, he will think body and mind are independent actualities, and he will go on in his thinking to divide these further, into various parts and functions, each to be treated separately. Physically, this is not conducive to over-all health (whose root meaning is "wholeness"). And mentally, it is not conducive to sanity (which has basically a similar meaning), as is indeed shown by an ever-growing tendency to the break-up of the psyche, as neurosis, psychosis, etc.

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David Bohm, "The Implicate Order: A New Approach to the Nature of Reality" in Beyond Mechanism: The Universe in Recent Physics and Catholic Thought, ed. David L. Schindler (Lanham, New York: University Press of America, 1986), 13-36 at 36-7.

Dijkstra on Language

A video I saw on "Dijkstra on foolishness of Natural Language Programming" (April 11, 2025) helpfully exposed me to some of Dijkstra's ideas on language. It's a short piece, but I'd like to focus on the central paragraphs:

In order to make machines significantly easier to use, it has been proposed (to try) to design machines that we could instruct in our native tongues. This would, admittedly, make the machines much more complicated, but, it was argued, by letting the machine carry a larger share of the burden, life would become easier for us. It sounds sensible provided you blame the obligation to use a formal symbolism as the source of your difficulties. But is the argument valid? I doubt.

We know in the meantime that the choice of an interface is not just a division of (a fixed amount of) labour, because the work involved in co-operating and communicating across the interface has to be added. We know in the meantime —from sobering experience, I may add—that a change of interface can easily increase at both sides of the fence the amount of work to be done (even drastically so). Hence the increased preference for what are now called "narrow interfaces". Therefore, although changing to communication between machine and man conducted in the latter's native tongue would greatly increase the machine's burden, we have to challenge the assumption that this would simplify man's life.

A short look at the history of mathematics shows how justified this challenge is. Greek mathematics got stuck because it remained a verbal, pictorial activity, Moslem "algebra", after a timid attempt at symbolism, died when it returned to the rhetoric style, and the modern civilized world could only emerge—for better or for worse—when Western Europe could free itself from the fetters of medieval scholasticism—a vain attempt at verbal precision!—thanks to the carefully, or at least consciously designed formal symbolisms that we owe to people like Vieta, Descartes, Leibniz, and (later) Boole.

The virtue of formal texts is that their manipulations, in order to be legitimate, need to satisfy only a few simple rules; they are, when you come to think of it, an amazingly effective tool for ruling out all sorts of nonsense that, when we use our native tongues, are almost impossible to avoid.

Recall that Dijkstra was primarily known as a computer scientist, and most famous for the shortest-path algorithm named after him.

What really caught my attention here was the knock on the Scholastics (who admittedly sometimes got to rigid about terminology) and the exaltation of Viete, Descartes, and company. Then the "ruling out all sorts of nonsense" phrase. I don't know much about the man's personal beliefs, but the latter statement inclines me to think that he would include "unscientific" beliefs like creation and God among the "nonsense", but, if not him, then there are plenty of people out there who would happily apply his statement in such a way.

And yet Dijkstra is perfectly right about this. He's perfectly right in what he explicitly says about this language needing to be unambiguous.

The piece missing is that the application of the language he's talking about is not the standard use of language. Instructing a machine requires language with no ambiguity. It is imperative language: do this, then do that. It's not even simply scientific language in the usual sense, which itself strives to rule out ambiguity and embrace univocity, but something much narrower. It might be the narrowest use of language there is.

This imperative language contrasts with the more prosaic use of language to make observations about the world, or even to instruct another person. These uses of language admit of ambiguity. Indeed, ambiguity is in some ways an advantage. Regarding even scientific language, Werner Heisenberg put it this way:

Furthermore, one of the most important features of the development and the analysis of modern physics is the experience that the concepts of natural language, vaguely defined as they are, seem to be more stable in the expansion of knowledge than the precise terms of scientific language, derived as an idealization from only limited groups of phenomena. This is in fact not surprising since the concepts of natural language are formed by the immediate connection with reality; they represent reality. (quoted more completely here)

So Dijkstra's statement is correct about computers, but would be sorely incomplete if it were about language in general. Computer languages separate themselves from normal, natural human language by being exclusively about control. There is no room for freedom of interpretation in instructing a machine, which is both its glory and its limitation. But this separation of computer languages from human language, makes them a whole other thing. And the exaltation of computer languages over human languages slides us further down the path of dehumanization.

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Edsger W. Dijkstra, 'On the foolishness of "natural language programming".'

Werner Heisenberg, Physics and Philosophy: The Revolution in Modern Science (New York: Harper & Row Publishers, 1958).

The Face Palm of Hossenfelder on Qualia

Sabine Hossenfelder, whom I generally like and who's probably one of the most sane of the public physicists, thinks scientists have managed to measure qualia.

I suppose it's inevitable a physicist will incur into philosophical territory. In this case, Dr. Hossenfelder has done so in the way that seems best calculated to display her ignorance of philosophy. Qualia are by their nature unmeasurable. They are experiences, subjective if you will, that correspond to qualitative parts of the external world.

The difference is between the inside and the outside of an experience. Measure all you want, you'll still be poking around the outside of experience, and never penetrate to the inside, what it's like to have the experience. Being "approximately inside," as it were, doesn't make the real thing. Suppositions and creative renaming don't make the real thing.

In reality the scientists have measured the way brains react to similar stimuli. As a friend helpfully pointed out, the experiences of different organisms are incommensurable, even when those organisms are nearly identical; while the technique described might be a useful tool for, say, helping compare pain levels between patients, it falls short of explaining what pain feels like.

Dr. Hossenfelder would do better to study a little good philosophy before spouting off like this again. Lately I've been reading Michael Augros's very good book The Immortal in You, which would be an appropriate primer on the issues that Hossenfelder tries to discuss here.

A Promising New Formulation of Quantum Mechanics

I've been watching the series of interviews that Curt Jaimungal conducted of Jacob Barandes from Harvard. The first, which is probably the most exciting (bang for the buck), here, runs over two hours. I found the discussion scintillating and probably the best thing on approaches to quantum mechanics in quite some time.

So far, I don't know much about Barandes's approach beyond what I've seen in these videos, but my understanding is that he derives the textbook axioms of quantum mechanics, which have always appeared rather strange and arbitrary, from a couple much simpler and entirely natural axioms involving undivided stochastic systems. And that this broader picture does away with the need for mystical interpretations of QM, including and most especially Many Worlds.

For those more interested in the formalism, there is this two-hour online seminar Barandes gave on the derivation of Hilbert Space from his stochastic expressions.

A friend who's an active theoretical physicist thinks this approach can provide some new tools for understanding physics. He says Barandes knows what he's talking about. I like Barandes's philosophical approach to physics, though I do think it's too limiting to restrict philosophical physics to the analysis of physics, theory and experiments. Why not also allow it the analysis of nature itself?

To me the most appealing point of Barandes's Undivided Stochastic Systems approach is the "undivided": the fact that motions are treated as indivisible wholes, and that imposing the requirement that these wholes be treated as divisible is what produces much of the quantum craziness. Recall that in the latter half of Physics, Aristotle discusses the wholeness of motions, and further, that the unity of motion is closely related to the unity of substance. I'm not sure there's a close connection of Barandes's work to Aristotle, but the former does seem to provide an opening for an incipient unity or teleology. More study is needed.