Russell and C. D. Broad on Space  
APA: EASTERN DIVISION (Bertrand Russell Society)
Dec. 2000, New York
by Steve Bayne

Based on a consideration of differences between Russell and Broad, what I hope to achieve here is an improved understanding of the philosophical question, “Is space something real or something we merely add to experience?” It has been a long time since anyone of note in philosophy has denied the reality of space. But hardly anyone has questioned the wisdom in Descartes challenging the obvious fact of their own existence. More to the point is whether regarding space as in some sense a logical construct precludes assigning it any more reality than, as Quine might say, the Greek Gods. Russell and Broad for a time were very close philosophically. To be sure, following Russell’s contact with the emerging mathematical theory of point set topology and quantum theory they would for a time go their separate ways. But later Russell would return to a position much closer to Broad’s, neglecting to reevaluate solutions to problems that depended on their original points of difference.
    It was on January 1st, 1914, that Russell devised his theory of “perspective space,” a rather simple theory, and “elegant” as mathematicians like to say, but reflective of a way of doing philosophy which, as C. D. Broad confessed, would soon become “antiquated without having ... acquired the interest of a collector’s piece” (Schilpp p. 830). More than a sidelong glance in the direction of Leibniz, Russell’s theory drew in sometimes important ways from the writings of Henri Poincare. It was a theory Russell believed to be consistent with Einstein’s Special Theory of Relativity, and later he would extend its philosophical consequences to certain aspects of quantum theory. Curiously, with the important exception of Samuel Alexander, Evander McGilvary, and much later A. J. Ayer it was largely ignored within philosophical circles. Although Broad would review every work where the theory was described and exploited, hardly a mention is made of it, even in Broad’s review of Alexander’s very influential Space, Time and Deity. This studied avoidance, however, conceals differing approaches to common problems.
    Both Russell and Broad wrote extensively on the nature of space, and while Russell is sometimes cited for his work on other subjects, history has not been so kind to C. D. Broad, who has disappeared almost without a trace.
    Broad’s early work on the subject of space indicates his reluctance to abandon the common sense of Euclidean geometry, although his work on the Theory of Relativity was sophisticated enough to be cited by Sommerfeld, Whithrow, and others. One of his best students, Dirac, went on to predict the discovery of anti-matter and of him Broad would later say “I was not fit to lick his boots.” Between 1914 and 1927 few philosophers besides Broad and Russell were competent to discuss the new theories that such physicists as Einstein and Dirac would introduce in light of a philosophical tradition going back to Aristotle. Later philosophers of enormous competence at physics, such as Hans Reichenbach would enter the picture bringing fresh new insights so powerful that the philosophy of science became virtually an autonomous domain, not unlike ethics. But none of these philosophers had been, as had Russell and Broad, immersed in continental metaphysics, none had been - to borrow from Wittgenstein - trapped in the “fly bottle” of metaphysics in the old style.


It is most likely that Russell’s theory of perspective space originated from considerations of the nature of secondary qualities and not geometry or physics. Since Galileo’s first suggested it, the idea had been widely held that properties like color are subjective and do not exist independently of perception as do the geometrical properties of extended objects. During the early part of the twentieth century, however, particularly among the so called “sense data” philosophers it came to be believed that the primary qualities of extended objects are themselves subjective. Suppose we both look at a table top. From where you are, the shape of the table may include four 90 degree angles; but from where I am, it appears to have at least two acute angles. Since we are presented with the same table and no object can have inconsistent properties, it might be surmised that these properties are subjective, belonging to the sense-data of different observers. But for the Russell of 1914 sense-data are physical, and spatial properties are really there and not merely subjective. So how do we avoid inconsistency? T. P. Nunn suggested that the table actually does consist in these conflicting properties but under different conditions of observation. At this time Russell regarded objects as series of “aspects of ” an object (OKEW p.112). So an object is just an assemblage of apspects radiating causally from a center (OKEW p. 100, ML p. 104). Russell took matters a step further, saying that the inconsistency of shape aspects obtains only if the object is thought of as three dimensional, but that if it is viewed as six dimensional there is then enough space to amicably fit an indefinite number of seemingly inconsistent geometric shapes, all in the space of single object. But how do we arrive at six dimensions? Each aspect is located in a perspective having three dimensions; but each perspective is itself a point in perspective space wherein all such points are themselves arranged in three dimensions.
    So in order to locate an aspect we need six dimensions. Poincare suggested a similar possibility but within a Kantian framework rather than a Leinbizian one as had Russell. But there is another feature to aspects which is crucial to Russell’s theory.
    Each aspect is uniquely associated with two places in perspective space. Take the case of one of two accute angles given to me when viewing a box top from my own perspective. It can be classified in either of two ways. Either angle can be classified as a member of the series of aspects that make up the object, and therefore as something located at the place of the object to which it belongs; or, it can be considered as an element among all the other aspects present in my perspective, and therefore as located at the place fromwhich that aspect appears. Broad would consider the problem, much as Whitehead did, although Whitehead never addressed this particular problem. Broad would avoid the need of six dimensional space and the at/from distinction, but it required that we revise our ida of what it means for a property to ‘inhere’ in an object. A property would now inhere in an object triadically, where besides the object and property a third term, the position of the percipient, was now introduced. This he called the “multiple inherence theory.” Having an accute angle would, then, inhere in a box top, for example, but only from a position. Having only right angles is a property that could inhere in the very same object but only from a different position.
    Each perspective is a world, an experienced perspective being a private world. Aspects are physical, and on Russell’s theory given in private space. Natural space or public space is for Russell, then, an arrangement of worlds. One quite significant consequence of all this is that my private world is a point in public space. Later under the influence of the Special Theory of Relativity Russell would define points in terms of events, even though after a fashion he retained his theory of perspective space. Nevertheless, while it persisted the idea of my private world being a point is physical space gained some currency in metaphysical writings, particularly it seems those of Samuel Alexander and the early Wittgenstein. Wittgenstein announces

I am my world. (The Microcosm) (T 5.63)
and Alexander, who based much of his brilliant but sadly neglected work on Russell’s theory, says something very similar when in Space, Time and Deity, while discussing what he calls the “microcosm of enjoyed space,” he remarks:
I am my own microcosm (STD p. 110)
It is absolutely fundamental that it be kept in mind that the privacy of the space of my private world is in Russell not the Cartesian privacy of a mental act or sensation, as it was, indeed for Broad. Wittgenstein’s argument against private languages is an argument against private languages in the Cartesian sense of privacy and not the subjective privacy of a private world in Russell’s sense. Roughly forty years before Wittgenstein published the Philosophical Investigations, which contained a systematic statement of his now famous “private language argument,” a very similar if not identical issue was raised by both Russell and Broad. As early as 1916, Broad was arguing that from our ability to communicate we can infer the possibility of the construction of public space. According to Russell, on the other hand, it is the comparative similarity between the space of private worlds that makes communication possible (ML 117) We can re-identify neither places nor things across private worlds but we can construct a public space that makes public language possible. So for Russell the possibility of public language is based on similarity among private places, for Broad our belief in the possibility of public space is based on our ability to communicate (“The Nature of Geometry and Space.” MIND. 1916. v. 25).


The idea of the space of a private world is is the idea of a mind aware of the perspective space that makes up a single point is the space of nature. It is private, therefore in two senses. The question is, What is the relation of mind to the perspective of which it is aware? I am not aware of places in my perspective behind the closed door. Yet the space of my perspective includes that space lying behind it.
    What is the relation of these “ideal” places to my private space? Russell at the time of formulating the theory had not come to grasp the significance of the difference between private space, in his sense, and psychological space. Broad took a different tact. For Broad the space of a private world is not given to us but is as much a logical construct as physical space.

What would really be meant by a private visual space would be this. Suppose a man were to deal with all his visual experience on the plan of constructing visual bodies and the space in which they move, the space having those characteristics which we have laid down for all spaces; then the space so constructed would be a private visual space. We must assume that the man takes no heed of any information that he gains from anyother sense; the spae is to be constructed so as to deal simply with the data of all his sight experiences on the general plan of distinguishing space from things in space. (Broad IOE p. 47)
Broad may have believed in ESP, he at one time headed the Society for Psychical Research, but one thing he could abide only with reluctance was the idea of “private space.” Taking strong if not rankerous exception to J. E. Turner, who had critiqued his pronouncements on the subject, Broad says that if there were such a thing as private space it would entail there being matter to occupy it; and this is what he found problematic. We must be clear as to exactly why the issue of private space is so important to understanding Russell’s theory and what followed from it both historically and conceptually. It is important because the paramount question is whether space is real, and this is what we must get at. Here is the problem for Russell.
    On Russell’s early, 1914 position, public space or the space of nature is made up of points, each of which is a perspective. Those perspectives which are experienced are private worlds. But what of those perspectives which are not experienced? Are they real? Russell’s causal theory of perception would comple him to accept actual places for unperceived objects. He would be forced by M. H. A. Newman to consider that knowledge of such causes is not merely structural, thus weakening his position with respect to phenomenalism, which for our purpose is merely the belief that not all perspectives are experienced. Russell was steadfast in his endorsement of real space beyond experience, once he came to hld that physical objects are not logical constructs but rather inferred entities.
    Broad in contrast to Russell’s view held that space is something we in some sense add to experience. Kant had implied something similar, but Broad wanted to distinguish his and Kant’s positions. He says that space is added to experience as something constructed from it, and that to say it is a construct is, on his view, to say merely that (1) it is not to be gotten by analysing what is given to experience and (2) that we do not infer it from what is causally given to the senses. (IOE p. 475). What, then, if space is something “we” construct, can we mean by saying that it is real? After all, even Russell, that great believer in the eliminability of definite descriptions, gave license to the idea that some constructs are indispensible (Analysis of Matter p. 315), although what he meant by “indispensible” is yet another story. When we ask whether space is real we are asking one of two things, says Broad (IOE p. 476). We may be asking whether points are of the same logical type as sense data, or we may be asking whether all observable movements can be
...stated as functions of physical bodies with the qualities that have been ascribed to them and of space with the qualities that have been ascribed to it in the particular system of physics and geometry under discussion.
The latter formulation is the one Broad latches onto, and the Russell of 1914 could have little objection although the functions at issue would range over bodies that were merely arrangements of aspects. Although the formulation itself seems clear, there is bound to be some uncertainty over “observable movement” in some cases such as by monsters in our dreams. This leads us to wonder whether Broad’s theory carries with it the suggestion that the space of dreams and waking life are or may be the same? But if not, why not? Indeed, according to him they may be, even though he has said elsewhere that different kinds of space demand correspondingly different kinds of objects. The point is that what makes dreamed objects and spaces different from those of our waking hours is not a matter of necessity. On this matter Russell’s early theory (ML p. 130) is indistinguishable from Broad’s except that the dreamer’s sense-data are physical and thus something “a completed physics would include. Coherentism aside, with his causal theory of perception came the necessity that aspects as objects of direct acquaintance would have to find occupancy at places removed from the objects which caused them.
    If we retain perspective space, what happens to the distinction between apsects as they relate to the places from and at which an object may appear? Objects, which now literally occupy places in the space of nature may appear to be at certain places in natural space by having certain effects (now the aspects) in psychological space.
    Contrary to what Ayer says is assumed ((Russell and Moore: The Analytical Heritage. MacMillan 1971. p. 125) the object does not occupy perceptual space. Nor is an aspect within a spatial neighborhood of a percipient in a private world. Rather a percipient is in some neighborhood of the object which causes those aspects to appear to occupy public space outside the brain. The space of objects and that of percepts are now entirely external to each other just as they had been for Broad of 1915.
    Aspects would survive but only as entities much more akin to the ordinary “sense-data” of the period and questions raised by the spectre of six dimensional space disappear like shadows at high noon. Although there would no longer be “private worlds” as distinct from perspectives, mental events would not become as they had for Broad and Alexander emergent phenomena. Russell’s 1927 definition of a ‘point’ in the space of nature in terms of copunctual quintets would be realized over a domain of events that included some which were mental. This notwithstanding Russell’s peculiarly inconsistent views on neutral monism.
    Concommitant with his denial of realism and the acceptance of space as something real and not merely constructed Russell pursued an interest in causation independently of the primarily Humean conception encouraged by Broad, although Broad was almost visceral in his attacks on Hume’s theory of space. This may have been owing to the criticism of his Analysis of Matter lodged by M. H. A. Newman (Mind 1928). Newman was a first class topologist who along with Oswald, who was himself pre-eminent in the field, had provided the mathematical impetus for Russell’s interest in the subject. If Russell’s own writing is any indication, it was, however, the 1914 publication of Felix Huasdorff’s Mengenlehre wherein he sets forth his theory of “neighborhood space” that appears to have affected Russell most as a post Principia mathematician. Hausdorff’s work was no doubt more comforting to Russell the geometrician than related work in quantum theory, even though there may have been interesting connections to the work of Felix Klein. Hausdorff’s “separation axioms” had been taken up by Urysohn who had shown that ‘coordinate properties’ - to use a Carnapian turn of phrase - presupposed Hausdorff’s axioms. Although we know from letters (at the Bertrand Russell Archives at McMaster University) that Newman urged Russell’s to abandon the view that events are fundamental, which Russell did not, he would nonetheless devalue the concept of distance in seeming pursuit of a metaphysics based on topological properties.
    Distance would become a way of talking about ‘causal ancestory’ (Analysis of Matter. p. 380) since it obtained between events not things. Russell’s deft application of topology combined with his excitement over quantum theory and relativity led him to a definition of space-time order as a consequence of the denial of action at a distance (Human Knowledge 223) Here Russell relied on a principle supplied by A. A. Robb who is now largely forgotten but who appears to have influenced Eddington. Indeed, Russell’s approach is something of a novelty so I will conclude by discussingone implication that seems to follow from it.
    Suppose we follow Russell in his denial of action at a distance and give it the following informal characterization: Every neighborhood of effect b must overlap some but not all neighborhoods of its cause a. Now if a immediately precedes b, every neighborhood of b will overlap every every neighborhood of a, and vice versa. So a cause, a, cannot immediately precede its effect; hence there is no such thing as instantaneous causation. But now consider this: from the same principle it follows that there can be no action at a distance, for this this would be as much as to say that there is some neigborhood of the effect, b, that does not overlap its cause, and this contradicts the principle. I will not make the judgment, presently, that instantaneous causation and action at a distance are equivalent and that this may have something to do with Bell’s theorem. I only mention it as evidence that Russell did not abandon mathematical interests later in his career. Broad would not approach these topics, and later interest in topology in philosophy would wane as model theory took its toll on problems of what Russell had in Principles of Mathematics called “general philosophy.”
    In summary, we see that beginning in 1914 Russell and Broad held radically different views on the reality of space. Russell regarded one space, private space, as real and public space as constructed; whereas Broad held both to be in some sense constructed from visual fields - in fact the “frames” of Poincare. Russell would later espouse a causal theory of perception, permitting him to escape the idea of objects and their space as mere logical constructs; but this would have the result of moving him closer to Broad’s much earlier view, although Broad would never actively engage as did Russell advances in topology and quantum theory. Philosophy would take a linguistic turn brought on by developments in proof theory and model theory, leaving it to future generations to rediscover a heritage unwilling or unfit to sustain a philosophy presently devoid of anything resembling its untempered ambition.