Space actually

On the disparity between intrinsically discrete thought and intrinsically continuous sensory experience and the resulting misrepresentation of physical space

St. Augustine¹ wrote: “What, then, is time? If no one asks me, I know; if I wish to explain to him who asks, I know not.” The defining character of time, which we express by saying that time passes, cannot be conveyed to someone who lacks the immediate awareness of the passage of time. This, and the fact that we sometimes feel as if time passes fast or slow, suggests that the passage of time is primarily subjective. Moreover, what we mean when we say that time passes is that specific states or conditions change or come into and pass out of existence. While the time indicated by a clock changes, time itself is rather the general and unchanging condition which makes it possible for states to change, be they subjective states of mind or objective states of the world.

It is a bit harder to recognize that space, too, is primarily subjective. What I mean by space is this expanse of which we are immediately aware, which appears to extend without discernable limit in every possible direction. What is actually extended, however, is things and the distances between things. Space itself is rather the general condition that makes it possible for things and their spatial relations to possess the quality of spatial extension.

That spatial extension is a quality that can only be defined by ostentation — by drawing attention to something of which we are directly aware — has already been pointed out in a previous post, in which I introduced you to my imaginary friend Andy. Andy, who lives in a spaceless world, understands you perfectly if you explain to him that space is like the set of all triplets of real numbers, which are used (by physicists and engineers alike) to label the so-called points of space. But if you believe that this gives him a sense of the expanse we call space, you are deluding yourself. We can imagine triplets of real numbers as points embedded in space; he cannot. We can interpret the difference between two numbers as the distance between two “points of space”; he cannot. At any rate, he cannot associate with the word “distance” the quality of remoteness it conveys to us.

All this is indicative of a fundamental disparity between thought and sensory experience. Thinking is by its very nature discrete. It is always about things, individual things or sets of things. A thing is anything that can be the subject of a sentence composed of a subject and a predicate, and a set is what Georg Cantor, the inventor of set theory, defined as “a Many that allows itself to be thought of as a One.” Time and space, on the other hand, are by their very nature continuous, and this not in the faux mathematical sense which allows sets (which are by nature discrete) to be at the same time continuous. The disparity between intrinsically discrete thought and intrinsically continuous sensory experience made the continua of time and space hot topics among philosophers since earliest times.

School of Athens, fresco by Raphael. The two men at the center are Plato (on the left) and his pupil Aristotle (on the right).

The fundamental distinction in Plato’s philosophy is that between the intelligible and unchanging model and the visible and changing copy. While the copy owes its attributes — including the attribute of being — to the model, it owes its physical existence — its being visible and changing — to “a form difficult and obscure.” In the Timaeus,² Plato makes an attempt to describe this form: “What nature must we, then, conceive it to possess and what part does it play? This, more than anything else: that it is the Receptacle — as it were, the nurse — of all Becoming.” The Receptacle is not that out of which things are made; it is that in which qualities appear. Plato has no concept of matter. He goes on to explain:

that which is duly to receive over its whole extent and many times over all the likenesses of the intelligible and eternal things ought in its own nature to be free of all the characters. For this reason ... we shall not be deceived if we call [the mother and Receptacle of what has come to be visible and otherwise sensible] a nature invisible and characterless, all-receiving, partaking in some very puzzling way of the intelligible and very hard to apprehend.

In the Parmenides dialogue,³ Plato makes another attempt to describe this “nature other than the form.” Because every visible or otherwise sensible object partakes of the two forms unity and being — “nothing that is lacks unity, and nothing that is one lacks being” — the Receptacle itself lacks both. Therefore,

those things which come to acquire unity, must, just in themselves, be without limit of multitude. We may see that in this way. Evidently, at the time when they come to acquire unity they are not one and do not possess unity. So they are multitudes which do not contain unity. Now if we choose to take in thought from such multitudes the least portion we can conceive, that portion also, if it does not possess unity, must be not one but a multitude. And if we go on in that way considering, just by itself, the nature other than the form, any portion of it that comes into view will be without limit of multitude.

Plato is trying to describe the genuinely continuous (hence non-discrete) nature of space. How does one speak, how does one even think of something that lacks unity? It cannot be a single thing, nor can it be many things. “Those things” to which Plato refers are not things at all. Lacking discreteness, they are “multitudes which do not contain unity.” Even the least conceivable portion of them will be “without limit of multitude.” (Unity, including that of a Cantorian set, is needed to delimit a multitude.) There can hardly be a clearer demonstration of the disparity between thought and sensory experience than Plato’s struggle to express in the intrinsically discrete language of thought what is intrinsically continuous.

What Plato was trying to put in words soon came to be elegantly expressed by his pupil Aristotle, in terms of his freshly minted concept of potentiality. While it is the intrinsic nature of a continuum to be without parts, it does appear to admit of division into ever smaller parts. Aristotle does not make the point that space, rather than being itself a divisible thing, is the general condition that allows us to divide material things. (Aristotle does have a concept of matter.) The point he is making is that the division (whether of space or of things in space) into ever smaller parts, being a process of finite thought, cannot be completed. If the continuum contains infinitely many parts, it contains them potentially only. An actually completed infinity is an impossible thought. And because the division of a continuum into ever smaller parts cannot be completed, a fortiori it cannot terminate in unextended, not-further-divisible parts or points.

Two millenniums later, Immanuel Kant⁴ — easily the most important philosopher of the modern era — insisted that no part of space or time “can be given except as enclosed between boundaries.... Space therefore consists only of spaces, time of times.” From mere points, conceived “as components that could be given prior to space or time, neither space nor time can be composed.” This is important, not least because it is universally ignored by the community of ontologizing physicists: because the concept of a point presupposes the intuition of space, points cannot be compounded to form the continuum of space.

Half a century after Kant, Carl Friedrich Gauss (who ranks among history’s most influential mathematicians) likewise insisted that infinity can only be conceived as potential. Neither Kant nor Gauss reflected on infinity and the continuum as merely potential in an historical context, as a conception we owe to Aristotle. In treating it as “so self-evident that it no longer had to be linked to his name, they paid Aristotle the greatest homage a thinker can gain,” von Weizsäcker⁵ wrote.

In the second half of the nineteenth century, mathematicians nevertheless shifted to thinking of the continuum as an actual infinity of points. “So successful has this shift been,’’ von Weizsäcker remarked, “that it is nearly impossible to disabuse the contemporary student of mathematics of the superstition that this conception is the only possible, indeed the theory of the infinite and of the continuum.”

In 1872 two papers appeared which ushered in the set-theoretic conception of the continuum, one by Georg Cantor,⁶ the other by Richard Dedekind.⁷ In his paper, Dedekind articulated the idea that “[t]he line is infinitely richer in point-individuals than is the domain ... of rational numbers in number-individuals.” In other words, if the rational numbers (which can be written as fractions of whole numbers) are mapped onto a straight line, then between any two rational numbers there are infinitely more points than there are rational numbers. The rational numbers manifest “a gappiness, incompleteness, discontinuity,” in contrast to the straight line’s “absence of gaps, completeness, continuity.”

While Dedekind regarded the gappiness of the rational numbers as essentially indemonstrable, Cantor (sort of) demonstrated it. His so-called diagonal argument is well-known.⁸ One considers the totality of real numbers (which can be written as non-terminating decimals) in the interval between 0 and 1, and one assumes that they can be arranged in sequential order. It is then readily seen that one can construct a number between 0 and 1 which is not part of the sequence, demonstrating thereby that the numbers between 0 and 1 cannot be arranged in a sequence. Because the rational numbers can be so arranged, this proves that there are more points on a line corresponding to real numbers than points corresponding to rational numbers.

Cantor’s proof was revolutionary. It established the existence of a non-denumerable subset of the real numbers without furnishing a procedure for constructing a single element of this subset. Previously it was taken for granted that arguments for the existence of a mathematical object had to be constructive. Cantor’s argument was purely existential (i.e., non-constructive). To appreciate how far-reaching were its implications, consider this: since the set of numbers that can be defined in a finite number of words is countable, and since the set of all real numbers is not countable, there must be a non-denumerable subset of the real numbers none of whose elements can be defined in a finite number of words. So shocked was the great mathematician and theoretical physicist Henri Poincaré by Cantor’s new method of proof that he made it a rule “never to consider objects that cannot be defined in a finite number of words.” Yet when the application of transfinite sets to outstanding mathematical problems began to yield important results, opposition to its non-constructive methods began to subside.

The acceptance of pure-existence proofs, however, made it necessary to rethink the presuppositions of mathematics and the meaning of mathematical truth. Before Cantor, axioms were supposed to be self-evident. Now they are merely formal propositions about essentially meaningless symbols. Now it makes sense (or no longer has to make sense) to call a self-adjoint operator “elephant” and a spectral decomposition “trunk.” While this makes it possible to prove a theorem according to which every elephant has a trunk, nobody should be fooled into thinking that this theorem has anything to do with real elephants. Now it makes sense (or no longer has to make sense) to call a real number “point” and the set of real numbers “continuum.” While this makes it possible to talk about the points of a continuum, nobody should be bamboozled into thinking that this phraseology is adequate for dealing with the actual continua of space and time. Alas, the vast majority of physicists think that it is.

The gappiness question may also be asked with respect to the real numbers: if the real numbers are mapped onto a line, are there more points than real numbers between any two real numbers? The rote answer is No, and the standard argument is that the real number 0.999... (with the digit 9 repeated ad infinitum) equals 1. There are, however, arguments to the contrary. Imagine a finite line divided into ten segments, the endpoints of the last segment labeled 0.9 and 1. Imagine this segment divided into ten segments, the endpoints of the last segment labeled 0.99 and 1, and in your mind’s eye enlarge the last segment by a factor of 10. If you could keep on doing this, you would end up with a segment whose end points are labeled 0.999... (with the digit 9 repeated ad infinitum) and 1, yet in your mind’s eye this segment would appear as large as the one whose end points are labeled 0.9 and 1, which suggests that there is room for more points.

In Cantor’s theory, if you have an infinite sequence a₁ < a₂ <a₃ < ... of increasingly greater numbers, there exists a transfinite number that is greater than all the numbers in this sequence. (“Transfinite” means greater than any finite number.) In particular, there will be a number ω that is greater than every natural number. Hence, logically, there also ought to be an infinitesimal number 1/ω that is smaller than any positive number. Yet when a fellow mathematician attempted to use Cantor’s transfinite numbers to develop a theory of infinitely small quantities, Cantor accused him of trying to “infect mathematics with the Cholera-Bacillus of infinitesimals.” Cantor even constructed a “proof” that no number can be infinitely small. This proof, however, was just as circular as were attempts by his contemporaries to prove that no number can be infinitely large. In both cases, the desired conclusion was smuggled in as part of the definition of “number.”

Be that as it may. The more relevant question here is whether the set-theoretic conception of space can be an adequate representation of physical space. Regarding this question, the distinguished physicist and philosopher Carl Friedrich von Weizsäcker⁹ wrote:

The conception of the continuum as potential, which originated with Aristotle, appears to be more suitable for the quantum theoretical way of thinking than is the set-theoretical conception of an actually existing transfinite manifold of “real numbers,” or of the spatial points they designate. The “real number” is a free creation of the human mind and perhaps not conformable to reality.

The theoretical framework of modern particle physics is relativistic quantum field theory. As the word “field” suggests, it describes physical space as a transfinite manifold of (triplets of) real numbers that supports additional mathematical structure. David Malament, Gerhard Hegerfeldt, and Rob Clifton (with Hans Halvorson)¹⁰ have shown with increasing generality that if, in such a manifold (ℳ), a free particle is located at a time t₁ in a bounded region R₁ then it has a non-zero probability to be found at a later time t₂ in a bounded region R₂, even if in the time between t₁ and t₂ no light signal can travel from R₁ to R₂. Because this is inconsistent with the special theory relativity, the conclusion is drawn that particles cannot be localized (i.e., situated at any time in any bounded region of space). Clifton and Halvorson thus conclude that particle talk is “strictly fictional”:

The argument for localizable particles appears to be very simple: Our experience shows us that objects (particles) occupy finite regions of space. But the reply to this argument is just as simple: These experiences are illusory! Although no object is strictly localized in a bounded region of space, an object can be well-enough localized to give the appearance to us (finite observers) that it is strictly localized.

What Clifton and Halvorson have actually shown is that particles cannot be localized relative to ℳ. But this conclusion is irrelevant, for such a manifold is not (nor can it represent) the expanse in which position measurements are made. Because positions are contextual, they owe their existence to detectors, by whose sensitive regions they are defined, and what holds for “objects (particles)” also holds for objects that are detectors. If particles cannot be localized in finite regions of ℳ then neither can detectors. What is strictly fictional, therefore, is ℳ. Such a manifold is useful, even indispensable as a calculational tool, but as a feature of an ontological quantum reality, it is a recipe for disaster.

What about the expanse in which position measurements are made? It is a consequence of the contextuality of positions that space cannot be intrinsically partitioned. To define a partition of space, we need an array of detectors, and no physically realizable array of detectors can partition space ad infinitum. The reason for this is the uncertainty principle, which implies that material objects (including position-defining detectors) cannot be sharply localized relative to each other. So if (in our minds) we keep cutting up physical space into ever smaller regions, we will reach a point where the distinctions we make between regions cease to correspond to anything in the physical world, a point where such distinctions can no longer be objectivized.

This, incidentally, is another argument why physical space cannot be conceived as an expanse that exists independently of experience, mind, or consciousness. This was first stressed by Kant,¹¹ who insisted that “space and time are only forms of sensible intuition, and therefore only conditions of the existence of the things as appearances.” (“Intuition” is the standard translation of the German word Anschauung, which covers both visual perception and visual imagination. Perception is sensible, i.e., filled with actual sensations, imagination is not.)

But if, by conceptually dividing physical space into ever smaller regions, we reach a point where the distinctions we make between regions cease to correspond to anything in the physical world, then physical space — the space that can be objectivized — by itself is an undivided expanse. The spatial relations and distinctions that exists — those that can be objectivized — are relations between material objects or distinctions between material parts. That physical space — the expanse which contains the physical universe — is intrinsically undifferentiated, that it has no parts, is an affront to reason, yet it accurately expresses an ontological truth. As Sri Aurobindo¹² writes, spatial extension and temporal duration

are only symbols which represent to the mind something not translatable into intellectual terms, an eternity which seems to us the same all-containing ever-new moment, an infinity which seems to us the same all-containing all-pervading point without magnitude. And this conflict of terms, so violent, yet accurately expressive of something we do perceive, shows that mind and speech have passed beyond their natural limits and are striving to express a Reality in which their own conventions and necessary oppositions disappear into an ineffable identity.

Isn’t it remarkable that quantum mechanics, via the uncertainty principle and the contextuality of the properties of quantum objects, leads to the same conclusion as the most penetrating spiritual inquiry?

1. St. Augustine, Confessions, Book XI, Chapter XIV.

2. F.M. Cornford, Plato’s Cosmology, pp. 177, 186 (Hackett Publishing Company, 1935).

3. F.M. Cornford, Plato and Parmenides, pp. 142, 210 (Kegan Paul, Trench, Trubner & Co. Ltd., 1939).

4. I. Kant, Critique of Pure Reason, translated and edited by P. Guyer and A.W. Wood, p. 292 (Cambridge University Press, 1998).

5. C.F. von Weizsäcker, Major Texts in Philosophy, edited by M. Drieschner, p. 128 (Springer, 2014).

6. G. Cantor, Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, Mathematische Annalen 5, 123–132 (1872).

7. Continuity and Irrational Numbers, in: R. Dedekind, Essays on the Theory of Numbers, pp. 1–27 (Dover Publications, 1963).

8. See, for instance, M. Kac and S.M. Ulam, Mathematics and Logic, pp. 12–14 (Dover Publications, 1992).

9. C.F. von Weizsäcker, The Unity of Nature, p. 130 (Farrar, Straus, and Giroux, 1980).

10. D.B. Malament, In defense of dogma: Why there cannot be a relativistic quantum mechanics of (localizable) particles, in: R. Clifton (ed.), Perspectives on Quantum Reality, pp. 1–10 (Kluwer, 1996); G.C. Hegerfeldt, Particle localization and the notion of Einstein causality, in: A. Horzela and E. Kapuscik (eds.), Extensions of Quantum Theory 3, pp. 9–16 (Apeiron, 2001); R. Clifton and H. Halvorson, No place for particles in relativistic quantum theories, Philosophy of Science 69, 1–28 (2002).

11. Critique of Pure Reason, p. 115.

12. Sri Aurobindo, The Life Divine, p. 81–82 (Sri Aurobindo Ashram Publication Department, 2005).