Thursday, February 15, 2007

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.


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

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).

17 Feb 2006: Minor edits.


CrimsonCatholic said...

Thanks for such a detailed summary and perspicuous analysis. Tracing Aristotle's difficulty here to a confusion between form and the act of being seems entirely correct. It may be that he wasn't confused, but rather, that this was a self-imposed constraint in studying physics rather than metaphysics. Perhaps his conclusions might have been different, but it remains the case that many people failed to distinguish the physical and metaphysical in following Aristotle.

One quibble I might have with Zellini is that he doesn't appear aware of several Christian thinkers noted by Leo Sweeney who applied apeiron and (more often) numerous cognate terms as a positive power (in Plotinus and others, such as Albertus Magnus), and later, as positive being (Richard Fishacre, Thomas Aquinas). I think the purely pejorative concept of infinity had probably been rejected long before Descartes, although it probably took St. Thomas's more detailed metaphysics to articulate the idea. The framework for productive discussion of infinity seems to have been there already, and Nicholas of Cusa appears to have started down the road even before Descartes. That's a relatively minor quibble, though, with an impressive overall picture.

Lawrence Gage said...

A friend of a friend notes that, contrary to the idea given in the final section, Aristotle believed numbers could be determinate without being actualized in (sensible, extra-mental) matter, that is, that numbers could be understood apart from sensible matter and motion.

On this point, I entirely agree. Perhaps I should have been clearer in that concluding section. But it has to be remembered that Aristotle believed that numbers would be at least actualized in a sort of intelligible matter that was (evidently, given Aristotle's objections to infinity) subject to potency and actuality in a temporal sense.


Zellini does make note of Nicholas of Cusa, though his name was left out of my brief review. Can you say more about Sweeney's work?


CrimsonCatholic said...

Prof. Gage:
I've been meaning to say more about that for some time, so I decided to put it into a blog entry. Please see:

Lawrence Gage said...

Great! I'll take a look.