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.

Physical Intuition, Not Mathematics

I ran across an excellent passage in one of Feynman's "extra" lectures about the need to develop physical intuition in learning physics:

Now, all these things you can feel. You don't have to feel them; you can work them out by making diagrams and calculations, but as problems get more and more difficult, and as you try to understand nature in more and more complicated situations, the more you can guess at, feel, and understand without actually calculating, the much better off you are! So that’s what you should practice doing on the various problems: when you have time somewhere, and you’re not worried about getting the answer for a quiz or something, look the problem over and see if you can understand the way it behaves, roughly, when you change some of the numbers.

Now, how to explain how to do that, I don’t know. I remember once trying to teach somebody who was having a great deal of trouble taking the physics course, even though he did well in mathematics. A good example of a problem that he found impossible to solve was this: “There’s a round table on three legs. Where should you lean on it, so the table will be the most unstable?”

The student’s solution was, “Probably on top of one of the legs, but let me see: I’ll calculate how much force will produce what lift, and so on, at different places.”

Then I said, “Never mind calculating. Can you imagine a real table?”

“But that’s not the way you’re supposed to do it!”

“Never mind how you’re supposed to do it; you’ve got a real table here with the various legs, you see? Now, where do you think you’d lean? What would happen if you pushed down directly over a leg?”

“Nothin’!”

I say, “That’s right; and what happens if you push down near the edge, halfway between two of the legs?”

“It flips over!”

I say, “OK! That’s better!”

The point is that the student had not realized that these were not just mathematical problems; they described a real table with legs. Actually, it wasn’t a real table, because it was perfectly circular, the legs were straight up and down, and so on. But it nearly described, roughly speaking, a real table, and from knowing what a real table does, you can get a very good idea of what this table does without having to calculate anything—you know darn well where you have to lean to make the table flip over.

So, how to explain that, I don’t know! But once you get the idea that the problems are not mathematical problems but physical problems, it helps a lot.

This passage makes a point similar to the one in Glen Coughlin's introduction to his translation of Aristotle's Physics: that knowledge and thoughts about the physical world are prior to the abstract knowledge of modern mathematical physics:

To understand Newton's argument for universal gravitation, one must have experience of weight in things and in oneself, of the motion of the stars and planets and moons. Knowing calculus is not enough. This hybrid science [mathematical physics], then, comes after the consideration of nature through non-mathematical means.

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Richard P. Feynman, Michael A. Gottlieb, Ralph Leighton, Feynman's Tips on Physics: A Problem-Solving Supplement to the Feynman Lectures on Physics (Boston: Pearson, 2006), 52-53.

Aristotle, Physics, or Natural Hearing, trans. Glen Coughlin (South Bend, IN: St. Augustine’s Press, 2005), xii.

Fallacious Application of "non-Euclidean" to Physical Space

Physicists sometimes talk about space—by which they of course mean physical space—as Euclidean or non-Euclidean. The problem with this way of speaking is that geometry is timeless. It cannot really apply to physical space.

Notice that if there's one lesson that Einstein's relativity has taught us, it is that space is intrinsically temporal: you can't have one without the other, which is why the combination in the relativistic context is usually called spacetime. Our measurements of length are always in time.

To see this point more clearly (more clearly at least if you are a physicist; I make no guarantees for others), think of the the Minkowski diagram, which plots time on the vertical axis and position on the horizontal (looking at a diagram may be helpful). (It also applies to general relativity with flat spacetimes, that is, regions far from masses.) Light rays are marked at 45-degree angles that divide the plane into four quadrants; the "light cone" consists of the north and south quadrants. There are time-like intervals (points that that lie within the "light cone," that is, that are separated enough in time that they can connect causally) and space-like intervals (points outside the light-cone, that is separated so far in space that they cannot connect causally).

To test whether space is Euclidean, one would have to set out measuring rods in the present, in other words, along the space-like interval parallel to the position (horizontal) axis. (And then test whether parallel lines remain parallel, or else either converge or diverge....)

But relativity has shown us that what one considers the present depends on one's state of motion: the ordering of events is not absolute, there is no unambiguous or absolute "present". On the Minkowski diagram in the frame of a primary, stationary observer, the "present" of a second, moving observer appears as an x' axis tilted obliquely to the x axis.

The assumption of what we usually mean by "length measurement" is that one measures both ends at once (as de Koninck points out, in contradistinction from Maritain, there is no absolute notion of length apart from an observer situated in space and time). Length measurements that are simultaneous in one frame are not simultaneous in another. Because of the relativity of simultaneity, "at once," and thus length measurement, becomes tied to the relative states of motion of the measurer and the object measured.

As we have seen, there is no unambiguous "now"; so the application or denial of the qualifier "Euclidean" to physical space confuses physics for pure mathematics (the error of Descartes). It presumes some sort of a timeless frame for making length measurements and it is precisely the existence of such an absolute frame that relativity denies.

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This argument occurred to me when reading Vincent Smith, and was corroborated by de Koninck writing about Eddington.

Vincent Edward Smith, Philosophical Physics (New York: Harper & Brothers, 1950), 355.

Charles de Koninck, The writings of Charles de Koninck, vol 1, ed. & trans. Ralph McInerny (Notre Dame, Ind. : University of Notre Dame Press, 2008), 147-158.

More on the Infinite

As I mentioned before, I'm reading A.W. Moore's The Infinite. (I found out about it through the webpage of a member of the Syndey School of Mathematics.) The book is much clearer than the Zellini book I commented on previously. Whereas Zellini makes a single distinction between the actual infinite and the potential infinite (which are synonymous with "true" and "false" infinites), Moore adds an additional distinction between true and false versions of the actual and potential infinites. The true versions of these he calls metaphysical and mathematical infinites.

Part of Moore's clarity comes from starting off (in the introduction) discussing the paradoxes of the infinite and defining terms.

[O]ne of the central issues concerning the infinite is whether it can be defined. Many have felt that it cannot; for if we try to define the infinite as that which is thus ans so, we fall foul of the fact that being thus and so is already a way of being limited or conditioned. (It is as if the infinite cannot, by definition, be defined....)¹

Two clusters of concepts nevertheless dominate, and much of the dialectic in the history of the topic has taken the form of oscillation between them. Within the first cluster we find: boundlessness; endlessness; unlimitedness; immeasurability; eternity; that which is such that, given any determinate part of it, there is always more to come; that which is greater than any assignable quantity. Within the second cluster we find: completeness; wholeness; unity; universality; absoluteness; perfection; self-sufficiency; autonomy. The concepts in the first cluster are more negative and convey a sense of potentiality. They are the concepts that might be expected to inform a more mathematical or logical discussion of the infinite. The concepts in the second cluster are more positive and convey a sense of actuality. They are concepts that might be expected to inform a more metaphysical or theological discussion of the infinite. Let us label the concepts 'mathematical' and 'metaphysical' respectively. (1-2)

The book is divided into two parts. The first part is an overview of the historical of thought on the infinite, and the second part is an assessment of the various strains of thought. Aristotle has a foundational role in both. Moore points out that Aristotle wasn't saying that the (mathematical) infinite was false, but that it only exists potentially, not actually:

I said at the beginning of §2 that Aristotle appeared to abhor the mathematical infinite. We can now see how profoundly false such an appearance was. What he abhorred was the metaphysically infinite, and (relatedly) the actual infinite—a kind of incoherent compromise between the metaphysical and the mathematical, whereby endlessness was supposed to be wholly and completely present all at once. It was the mathematically infinite that he was urging us to take seriously. Properly understood, the mathematically infinite and the potentially infinite were, for Aristotle, one and the same. Far from abhorring the mathematically infinite, he was the first philosopher who seriously championed it. In so doing he recoiled from earlier thinking in such a way that he set the scene for nearly all subsequent discussion of this topic. (44)

According to Moore, the pre-Socratics had given voice to what was essentially the metaphysical infinite, but Plotinus first articulated it clearly and distinctly:

He called it self-sufficient, perfect, and omnipotent, a complete and pure unity, utterly beyond our finite experience. He also said that it was 'supremely adequate, autonomous, all-transcending, most utterly without need.' Sometimes he spoke of it in a Parmenidean way, implying that it had internal limits. 'Its manner of being is settled for it,' he said, 'by itself alone.' But elsewhere he emphasized its lack of limits, either exeternal or internal. Indeed, in line with this, he insisted that all our attempts to talk about it or derfine it were strictly speaking, and inevitably, inadeqaute. This, in truth, it even transcended such descriptions of it as 'The Good' or 'God'. Its ineffability meant that we had to be content with mystical insight into it. He nevertheless tried to convey as much as possible with words. And in so doing he supplied one of the first explicit identifications of the infinite with God. (46)

Moore very clearly explains the categorematic/syncategorematic distinction originated by Peter of Spain and taken up by Jean Buridan and Gregory of Rimini:

Roughly: to use 'infinite' categorematically is to say that there is something [an actual whole] which has a property that surpasses any finite measure; to use it syncategorematically is to say that, give any finite measure, there is something [another individual] which has a property that surpasses it. (51)

This distinction carlifies an important point about uses of infinite, and furthermore subsumes Aristotle's actual/potential distinction.

By way of illustration, consider the following application of the new distinction in a temporal context, noted by Gregory [of Rimini]. If I say, 'An infinity of men will be dead,' and use 'infinity' categorematically, then I mean that there will come a time when infinitely many men are dead; there will then be an actual infinity of dead men. If I say the same thing, and use 'infinity' sycategorematically, then I mean that there is no end to the number of men who will, each in his own time, be dead; there is a potential infinity of dead men. This explains, I think, why so many philosophers have thought that there was something deeper and more abstract underlying straight-forward temporal accounts of the actual/potential distinction. It seems they were right. There is something—something grammatical [cf. later invocation of Wittgenstein]. (Working with this new distinction also has the advantage that one can avoid the false implication in Aristotle's terminology, noted by Aristotle himself, that what is potentially infinite must be capable of being actually infinite.) (52)

Moore includes some provocative reactions of contemporaries to Cantor's transfinite mathematics:

[French mathematician Henri Poincaré] challenged Cantor's claim to have proved that R [the set of real numbers] was bigger than N [the set of natural numbers]. Cantor's proof could just as well be taken to establish merely that we could not devise a way of pairing off the natural numbers with the real numbers, or indeed that R was not a genuine set at all—presumably because the real numbers were somehow too unwieldy to be grouped together into one determinate totality.

This, incidentally, was something urged by the American philosopher and mathematician C.S. Peirce (1938-1914). He had independently discovered that there was no way of pairing off the natural numbers with the real numbers, but he concluded that R did not exist as a completed whole. At most it existed as something potentially infinite. However many reals had been actualized, there were always more waiting to be. A continuum, he felt, was precisely not just a set of points. It was something absolute, consisting of unactualized possibilities, cenmented together in a way that defied description but of which we were aware in experience.

I'm not sure how much credence to give these ideas, but they do seem to lead us back to Aristotle's notion of the continuum not being composed of points.

I'm still re-reading and digesting the book. Moore's advocacy of Kant and Wittgenstein's positions on the subject sound reasonable to me (from what he says), but I'm not entirely certain how kosher they are (especially Kant) and I need to examine them more closely because they have a big part to play in the second half of the book. Being a good Englishman, Moore ends (as I read him) with the empiricist (or more-or-less Aristotelian) position of denying the reality of the metaphysical infinite and affirming that while we can see things about the mathematical infinite, we cannot really say anything about it as actual.

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Notes

1. Not sure this is the right way to note the ellipsis.

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A.W. Moore, The Infinite (New York: Routledge, 1991). All emphases in original.

Paolo Zellini, A Brief History of Infinity, trans. David Marsh (New York: Penguin Books, 2004).

Beach-blanket Natural Philosophy

At a yard sale recently, I bought an old copy of Hubbard's Battlefield Earth for twenty cents. "Beach" reading, perfect for summer. The characters are superficial, the plot movement often contrived, and some of the language confusing, but Hubbard does a good job building that momentum that gets you to turn to the next chapter (or part).

One of the main "science" premises of this science fiction novel is the mysterious workings of the antagonists' teleportation system:

Prior to this [discovery], it was thought that teleportation consisted of converting energy and matter to space and then reconverting it in another place so it would assume its natural form. But this had never been proven. En [the discoverer] had apparently found that space could exist entirely independently of time, energy, or mass and that all these things were actually separate items. Only when combined did they make up a universe.

Space was dependent only upon three coordinates. When one dictated [!] a set of space coordinates one shifted space itself. Any energy or mass contained in that space thereupon shifted with that space shift.

In the matter of a motor such as this freighter had, it was just an enclosed housing in which space coordinates could be changed. As the coordinates changed, the housing was forced to go along, and this gave the motor power.... A series of coordinates were progressively fed to the main motor and it self went forward or backward as the housed space occupied each set of coordinates in turn.

Teleportation over vast distances worked the same way. Matter and energy were pinned to the space, and when it was exchanged with another space, they simply changed too. Thus matter and energy would seem to disappear in one place and appear in another. They didn't actually change. Only the space did.

There are so many things wrong with this description that it is hard to know where to begin. Now of course, no one takes the "science" mumbo-jumbo in science fiction novels seriously (I mean most of these things technically don't even qualify as novels¹), but for entertainment purposes, let's look at it more closely.

In the first place, this idea of absolute space (apart from an absolute time) was conjured up by Newton to facilitate his mechanics, which was based on Descartes's analytic geometry. It's took us centuries to get over this naive starting point, but as we all know, the consequences of Einstein's notions of space and time continua forming a unified space-time continuum were experimentally confirmed by the time Hubbard wrote his novel in 1982. In his special theory of relativity, Einstein showed how the dilation of time intervals and the correlative contraction of spatial dimensions at speeds approaching the speed of light explain many phenomena, most notably the constancy of the speed of light for all observers, however they are moving. So not only are space and time interdependent, but also there can be no absolute coordinate system---what would define the origin?

Further, in his general theory of relativity, Einstein showed that the geometrical structure of this spacetime depends on nearby masses. So space exists relative to mass-energy as well as time.

But the dubiousness of the idea goes deeper than physics. Coincidentally, I've also been reading A.W. Moore's The Infinite. Of course one of the puzzles of infinity is whether a magnitude or continuous quantity can be composed of discrete, infinitesimal points. Cantor's transfinite mathematics notwithstanding², it is pretty clear that elements with length zero can never add up to a finite length. Philosophically speaking, using a teleporter to move through a continuum requires more sophistication (infinitely more!) than teleporting across the universe. That means that the teleportation motor that Hubbard describes cannot move continuously, but has to move itself through discrete points. Perhaps the resulting vibration is the reason the engines make such noise when "dictating" themselves through space. Of course, if space exists independent of the rest of the universe, then one has to wonder what sort of forces would allow one to shift it.

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Hubbard's novel is interesting also for how it speaks about his personal philosophy, a philosophy that undoubtedly colors the doctrine of the Church of Scientology, which he founded. It shows a naive trust in human nature so characteristic of the 20th century. But in addition, Hubbard also displays a distrust of government and a simple faith in the abilities. More interestingly, Hubbard has at least a rudimentary respect for nature: his villains, a race of aliens called the Psychlos, have been altered at birth to suit arbitrary societal demands (working hard, preserving technological secrets) that also make them sadists.

Wikipedia says that Punch (April 4, 1984) sarcastically commended Hubbard's "excellent understanding of evil impulses, particularly deviousness, which helps with the plot, and [he] is well-enough aware of his weaknesses not to dwell upon frailties like love, generosity, compassion." I might add that Hubbard's detachment from humanity is reflected in his naive, purely quantitative conception of greatness. Beyond the 1000-page length of the book, the Psychlos are much bigger than humans (10-ft tall and weighing 1000 lbs each) and the greatness of their empire is reflected the vastness of the numbers that define it. As Hubbard writes,

Psychlo!

The homeplanet of two hundred thousand worlds.

The center of an empire that had ruled and ruined sixteen universes over the period of three hundred and two thousand years.³

Wanna spice up your novel with more awe-inspiring bad-guys? Just add zeros!

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Notes

1. A novel in the usual understanding centers around the development of a character. The cover of my copy of the present work is a painting in bold primary colors of a blond, bearded bare-chested man (physique of a body builder) firing two futuristic guns, in the background angular spacecraft zooming around (see the book homepage: www.battlefieldearth.com). In this case it seems you can judge a book by its cover. Tragic little of science fiction explores characters with any depth, though later writers have sought to overcome this shortcoming by introducing sex scenes, as if meaningless couplings add human depth! At least Hubbard refrains from the latter.

2. Cantor showed that there are (infinitely) more irrational numbers between any two rationals than there are rationals on the real number line. But his failure to show that the continuum is the next infinity bigger than the rationals leaves an indefiniteness to the place of the continuum in Cantor's hierarchy of transfinite numbers. More on this later.

3. P. 899. Somehow their tremendous evil does nothing to destablize their society.

4. Looks like Battlefield Earth is Presidential candidate Mitt Romney's favorite novel! Just the thing to read while you're having your hair blow-dried.

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L. Ron Hubbard, Battlefield Earth: A Saga [!] of the Year 3000 (Los Angeles: Bridge Publications, 1984), 187-8.

A.W. Moore, The Infinite (Routledge, 1989).

Comprehending the Infinite

As part of the commentary on Aristotle’s Physics that I am composing, I’ve been researching infinity, which is one of the topics in book III.

David Foster Wallace’s Everything and More: A Compact History of Infinity was the first I read. As I’ve noted here before, Wallace’s writing style is extremely mannered. His treatment of the mathematical concepts and ideological camps is very understandable, but unfortunately, his thinking and writing are not so clear on the implications and meanings of the mathematical developments. The book could use circumscription by a table of contents and an index, though perhaps the lack of these is intended as an artsy way of embodying the “infinite.”

Brian Clegg’s Infinity: The Quest to Think the Unthinkable was an accessible and straight-forward recounting of the history of conceptions of the infinite. It wasn’t particularly deep or probing and covered roughly the same ground as Wallace’s book, but in less detail, and certainly less manically.

Paolo Zellini’s penetrating A Brief History of Infinity was reminiscent of Jorge Luis Borges’s writings, and not merely because he starts the book with a quotation from Borges. The book supplies a sweeping treatment of the implications of conceptions of infinity, and if I could tag it with any flaw, it would be that its depth combined with its brevity (200-pp.) flirt with impenetrability of language—but then infinity is an obscure topic (and it’s difficult to translate from Italian). The Wallace book’s treatment of the paramount proofs was good background for this book.

I thought it would be helpful to summarize the most significant points of Zellini’s work. Then I'll conclude with a brief consideration on how Aristotle's conception of infinity compares with modern developments.

Preliminary Clarification of Terminology

Part of the difficulty of Zellini’s book was keeping track of various terms for the two kinds of infinity.

These terms are roughly synonymous with each other: potential infinity, syncategorematic infinity, improper or false infinity, infinity ex parte materiae, material infinity, negative infinity

These terms are antonymous in a sense with the previous list, but synonymous with each other: actual infinity, categorematic infinity, real or true infinity, infinity ex parte formae, formal infinity, positive infinity.

This last set of terms is sometimes synonymous with the absolute or simple infinity actualized in God.

It might be helpful to recall through an example how potentiality and actuality are linked to temporal succession. A car is able to move because it is potentially in another place; once it has moved to that place, it is there actually and not potentially. Thus change in general is a succession of potentiality and actuality.

The Classical-Medieval Understanding

The Greek word for infinite, apeiron, means indefinite or unlimited. A connotation of perfection is completely foreign to it. Aristotle understood the term as denoting incompleteness. Limits make any object exist concretely, individually and with proper form; apeiron denotes a privation of limits—formlessness.

Aristotle writes, “The infinite [apeiron] is not what has nothing outside of it, but what always has something outside of it.” The infinite is always potential, because more can always be added to it. The infinite by division also exists only potentially: a concrete thing can potentially be divided an infinite number of times.

Numbers to the ancient Greeks were intimately tied to concrete things. Even if these things were conceived as abstract entities, they were nevertheless embodied in a sort of intelligible substance. Thus the Greeks could not conceive as a number as an entity in itself, but only as the result of counting a concrete set of things, and of course, one can never complete (or actualize) the counting of an infinite number of things (that is, in time).

St. Thomas Aquinas agreed with Aristotle’s basic point, but with the additional information of the Christian revelation of God’s infinitude, added a new layer of meaning to the infinite. As Zellini writes,

Aristotle had excluded any possibility of confusing the false infinite of apeiron with the infinite divine perfection, simply by denying the latter any infinite attributes. Divine perfection was designated by terms referring to its ‘totality,’ its ‘plenitude’, and to its ‘eternity’, but not to its unlimitedness. The last term belonged exclusively to the realm of quantity, and was therefore completely extraneous to God.

Aquinas dares not follow Aristotle’s formulation of the problem [because of the condemnation of Arabic Aristotelianism in 1277], and instead embraces the thesis that ‘God is infinite and eternal and boundless’. But he immediately adds that the infinite can have two opposite natures: one derived from the idea of form, and the other from the idea of matter.

The infinite ‘on the part of matter’ (ex parte materiae) thus had to correspond to an analogue of the false infinity of Aristotle’s apeiron. By contrast, the infinite ‘on the part of form’ (ex parte formae), by referring to a sort of formal perfection, could indicate in what sense one could speak correctly of God’s infinity. (59–60)

Infinity on the part of form means that God is infinitely articulated (delimited in a positive sense)…and yet Aquinas’s Five Ways show God to be infinitely simple in another sense….

Formal and material infinities parallel actual and potential infinities. As to whether the actual infinite could exist outside of God, Aquinas contrasts the divine infinite by essence or simplicity (per essentiam or simpliciter) with the relative infinite (infinitus secundum quid), corresponding to a specific nature. Aquinas seemed to think the latter could only exist potentially and thus seemed to identify it with the potential infinite. (61)

In the interest of clarity, Peter of Spain introduced novel terminology. The syncategorematic infinite is the potential infinite, but stripped of the residual connotation in “potential” of an actual orientation to an end or possibility of actual realization. The categorematic infinite, in contrast, is an actually realized whole that is simply larger than any finite quantity. (67)

Enlightenment Turn

Descartes distinguished the infinite from the indefinite. The latter roughly parallels the false or potential infinite, but with a significant attitudinal shift. For Descartes the imperfection of the indefinite is not its boundlessness, but the boundedness that remains. “It was the same continual opening which Boethius had rejected as a ‘monster of malice’, and which Aristotle had associated with non-being and privation. Instead…Descartes perceives in it the unequivocal sign of a divine imprint.” (103) While Descartes did not believe in the existence of the actual infinite, he paved the way for its use by speculating about the mind’s openness to the infinite as a standard of perfection.

Renaissance painting discovered perspective, which manifests the convergence of lines on the horizon, that is, at infinity. This innovation grounded the change in the concept of the infinite. Descartes’ rationalization of space is premised on “the affirmation of existence culminating in the visibility of a point in which the entire infinity of visual space appears enclosed and unified.” (111)

The shift in attitude is dramatically apparent by contrasting Aristotle with Leibniz, the originator of infinitesimals. Whereas Aristotle writes that “Nature avoids what is infinite, because the infinite lacks completion and finality, whereas this is what Nature always seeks,” Leibniz writes that nature “involves the infinite in all it does” (112–113)

Modern Developments

While the Leibniz’s conception of the infinite incorporated some flaws, its practical application convinced Bernard Bolzano of its reality. Bolzano saw that the infinite, as with other mathematical entities such as zero and imaginary numbers, could be an objective idea without corresponding to a concretely existing thing; a thing could be determinate without existing in reality: the infinite can be defined unambiguously even if it is impossible to enumerate its elements. (143) As Zellini writes,

When we speak of the set of inhabitants of Peking, we individuate a well-defined set without being required to enumerate separately all its components one by one. Analogously, the terms of an infinite sequence can all be specified by the law governing the formation of the sequence, which renders superfluous enumerating its terms: it is the law that specifies the sequence, and not ‘all’ its terms counted one by one. (148)

“Dedekind did not scruple to declare that arithmetic evolves perfectly independently of a priori intuitions of space and time, and that the concept of number is an immediate result of the laws of thought.” (160)

Like Bolzano, Dedekind said that an intellectual entity could be determined by all that can be said or thought about it. Similarly Cantor believed that mathematical entities can be considered actual insofar as they, as he wrote, “assume a perfectly determinate place in our knowledge, are clearly distinct from all other constituents of our thought, stand in definite relation to them, and therefore modify the substance of our mind in a definite way.” But (as any Platonist) he also believed that we don’t arbitrarily construct these entities, but receive them from “the voice of nature.” Still, he put the existence of these entities outside the competence of mathematics—such metaphysical questions have no bearing on mathematics. Furthermore, he saw clearly that transfinite numbers cannot even approach an understanding of the Absolute, which can only be acknowledged but never known. (160–161)

Yet, in the 12th chapter on “The Antinomies, or Paradoxes of Set Theory,” Zellini argues that Cantor’s achievements are not unqualified. Hilbert tried to preserve them by formalizing mathematics in a rigid system of “symbols without significance,” but Gödel showed that symbolic systems were not able to express a complete and closed world of mathematics. (177)

In 1932, Weyl condensed this finding in a neat summary: the infinite is intuitively accessible as an indefinitely open field of possibility, and in this respect would seem analogous to a series of numbers that can be extended unlimitedly. Yet completeness, the so-called actual infinite, lies beyond our reach. Nevertheless, the exigencies of totality impel the mind to imagine the infinite, by means of symbolic constructions, as a closed entity. (179)

It may be my lack of mathematical expertise that keeps me from understanding how later limits to mathematics concerned Cantor’s achievements (or perhaps Zellini is simply vague). Perhaps it is fair to summarize the conclusion of the book as saying that modern mathematics shows that transfinite numbers are never completely formless—that they can be handled in a determinate way, even though unable to completely exclude paradox.

How Aristotle Falls Short

As we have seen, infinite numbers aren’t unequivocally infinite, that is, without bound or indeterminate. Merely the fact that they are numbers constrains their properties. For example, that they are ordered bounds them from complete chaos. They needn’t be actualized in matter to be determinate.

As is well known, Aristotle failed to separate the actuality of form from the actuality of existence. The great advance of Aquinas was seeing that a determinate essence still required an act of being (esse) to be real. For example, you can conceive of a unicorn in a perfectly consistent and rather complete way, which means the unicorn has a certain actuality; but this actuality doesn’t mean such a creature actually exists. It seems to me that mathematical entities similarly can be fully determinate (formally actual) without being actualized in matter (possessing an act of being).¹ Potentiality and actuality in the fully real sense of existing in the world have no part in such beings, so the inability to actualize infinity in the world is irrelevant to the actuality (or determinateness) of infinity.

Thus, Aristotle’s mistake was part of his general failure to distinguish formal actuality from existential actuality.

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Notes

1. But then it would seem that only actually existing things can be determinate (or consistent) in the fullest sense.

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Paolo Zellini, A Brief History of Infinity, trans. David Marsh (New York: Penguin Books, 2004).

David Foster Wallace, Everything and More: A Compact History of Infinity (New York: W.W. Norton and Company, 2003).

Brian Clegg, Infinity: The Quest to Think the Unthinkable ( (New York: Carroll & Graf Publishers, 2004).

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17 Feb 2006: Minor edits.

Zero vs. Nothing

Lately I've been reading about conceptions infinity, which is an important topic to natural philosophy and which Aristotle discusses in Book 3 of the Physics.

In any event, this is the reason I picked up David Foster Wallace's popular treatment of the mathematics of infinity (from Zeno up through Cantor). Wallace's writing is definitely mannered. He maintains a modern bias against Aristotle and in favor of the actuality of infinity, both of which points the book inadequately supports. (Sometime I'll have to do a full review.) Because of these flaws, the excellence of his explanation of the difference between zero and nothing is quite surprising:

It's a tricky difference [between the number 0 and the abstract word 'nothing'], but an important one. The Greeks' inability to see it was probably what kept them from being able to use 0 in their math, which cost them dearly. But 0 v. nothing is one of those abstract distinctions that's almost impossible to talk about directly; you more have to do it with examples. Imagine there's a certain math class, and in this class there's a fiendishly difficult 100-point midterm, and imagine that neither you nor I get even one point out of 100 on this exam. Except there's a difference: you are not in the class and didn't even take the exam, whereas I am and did. The fact that you received 0 points on the exam was thus irrelevant—your 0 means N/A, nothing—whereas my 0 is an actual zero. Or if you don't like that one, imagine that you and I are respectively female and male, both healthy 20-40 years of age, and we're both at the doctor's, and neither of us has had a menstrual period in the past ten weeks, in which case my total number of periods is nothing, whereas yours here is 0—and significant. End examples.

I suppose the difference can be summarized by noting that with zero, there is at least to start out with a possibility of having a something. Then of course the notion of possibility (vs. actuality) is critical to the whole notion of infinity....

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David Foster Wallace, Everything and More: A Compact History of ∞ (New York: W.W. Norton and Company, 2003), 142.

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Also of interest: Nothing Comes from Nothing

Nothing Comes from Nothing

While I was hunting down that Hawking quotation for my previous post, I ran across a book review of Kitty Ferguson's The Fire in the Equations. The author of the review is an excellent writer Stephen M. Barr, University of Delaware physicist.

As I might have expected... Barr very clearly (more clearly that I have) untangles the muddle of something and nothing that befuddles scientific atheists.

Another class of ideas involve explaining the Big Bang as a quantum event. In quantum mechanics one can have particles being "created out of the vacuum." That is, there can be transitions from a state with no particles to a state with one or more particles. By analogy it has been suggested that spontaneous transitions can occur from a state with "no universes" to a state with one (or more) universes.

Whether this makes sense as physics is not yet clear. But if it does, will it give us creation ex nihilo without God? Only if one equivocates about what "nothing" and "universe" mean. A quantum state without any particles or even without any "universes" is not nothing-it is a quantum state.

Perhaps the distinction can be illustrated with an analogy. There is a difference (if not a spendable one) between a bank account with no dollars in it and no bank account at all. To have a bank account, even one with a momentarily zero (or negative) balance, requires having a bank, an agreement with that bank, a monetary system, a currency, and banking laws. Similarly, to talk about states with various numbers of "universes" requires having a quantum system with different possible "states," and laws determining the character of those states and governing the transitions among them. The term "the universe" should really be applied to this whole system with its laws, and not, as is misleadingly done in such discussions, to "space-times" that are coming into and going out of existence.

Hawking had it right: having equations that describe a "universe" (or anything else) coming into being does not mean that these equations must be describing anything real. Having a story about fairies does not mean there are fairies.

Barr concludes with a classic quotation:

The Latin apologist Minucius Felix, writing around 200 a.d., said, "If upon entering some home you saw that everything there was well-tended, neat, and decorative, you would believe that some master was in charge of it, and that he himself was superior to those good things. So too in the home of this world, when you see providence, order, and law in the heavens and on earth, believe there is a Lord and Author of the universe, more beautiful than the stars themselves and the various parts of the whole world." The greatest contribution of science to the "search of God" has been to bring into fuller view the grandeur of this providence, order, and law.

This is just common sense: you can't get something from nothing. Or, as that soulful street-sage Billy Preston sings,

Nothin' from nothin' leaves nothin'
You gotta have somethin'
If you wanna be with me

This ain't rocket science....

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Stephen M. Barr, "The Gods of the Physicists," First Things 65 (August/September 1996): 54-57.

Kitty Ferguson, The Fire in the Equations: Science, Religion, and the Search for God (Grand Rapids Michigan: Eerdmans, 1994).

Minucius Felix, Octavius, chapter 18. [The ancients were masters at rheortic, weren't they?]

Billy Preston (and Bruce Fisher), "Nothing from Nothing," Kids and Me (1974).

Stephen M. Barr, Modern Physics and Ancient Faith (Notre Dame, IN: University of Notre Dame Press, 2003). [In case you're interested in reading more by Barr, this book is great!]

The Null Solution

Perhaps you have been following the Terri Schiavo case lately, in which her husband, claiming she is a vegetable, seeks to kill her by having her feeding tube removed. The U.S. Senate recently debated the issue:

Mr. WYDEN. Mr. President, the Senate is now addressing probably the most gut-wrenching decision that an American family can ever face. Without even a single hearing, without any debate whatever, the Senate is tackling an extraordinarily sensitive concern that involves morals and ethics and religious principles, and this troubles me greatly. (Congressional Record, March 17, 2005)

The life of a woman unable to speak for herself has become a "decision" instead of an inalienable right.

The idea of death as a solution has gained wide acceptance in our self-indulgent culture that finds no meaning to life outside momentary enjoyment. Michael Medved wrote a great column in USA Today last week, in which he asks "Has suicide become the pop culture flavor of the month?" and enumerates recent instances of the zeitgeist's promotion of suicide as "brave."

The idea of "solving" life's problems by refusing to live reminded me of the most underrated field of mathematics, linear algebra. (The name is boring, you say. I know: "linear" sounds soporifically straightforward and "algebra" reminds you of your nightmare high-school class.) Trust me: next to geometry, this is the most insightful field of mathematics requiring only limited mathematical abstraction.

There are a number of useful insights in linear algebra that provide a wonderful basis for better understanding how the world works. (I'll leave a neat example to the comments and get back to the point of this post.)

There is a common solution for every* set of linear equations, called The Null Solution, which consists of setting every parameter to zero. The Null Solution is also known as the trivial solution because it is uninteresting: any moron can slap it down.

Death is the Null Solution to all of life's problems. What is the one sure way to rid the world of hunger, poverty, illiteracy, anxiety, terrorism and bad manners and bad breath? Global suicide.

A professor will very rightly flunk a student who enters the null solution for every problem of an exam. As in linear algebra, life's null solution is "trivial" and no solution at all.

Choosing death is not brave. It's cynical and selfish.

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*Homogeneous equations, the most common form.