According to Hermann Weyl, a physicist whose name is almost on par with that of Einstein, philosophy and physics have both been engaged in a long process of sifting properties of objects into subjective and objective categories.1  As philosophy relegated ever more properties (such as "green") to the category of (subjective) sense qualities, physics altogether excluded ever more such properties from its objective world-view. Philosophy, as expressed by Kant’s Transcendental Idealism, eventually declared that even spatial and temporal properties were ultimately subjective.2  Correspondingly – but much later – science, as expressed by Einstein’s Relativity Theory, also assigned spatial and temporal properties subjective status.

…is simply a series of values of mathematical functions in which occur four independent parameters corresponding to the three dimensions of space, and the one of time.3 

 

Weyl’s interpretation of relativity’s world-view is by no means unique. It follows that of Herman Minkowski, the author of the geometric, four-dimensional reading of relativity theory. Weyl’s attempt to embed this reading within a philosophical framework is not unique either. Science may be the offspring of philosophy, but she has repeatedly reimbursed her parent, providing powerful systematic descriptions of reality supposedly in need of foundations. Relativity Theory is perhaps the greatest of all these payments. Indeed, when it appeared in the early twentieth century, numerous philosophers struggled to be the provider of such luminous foundations. To do so would certainly increase the power and respectability of one’s own philosophical system. For instance, the positivists happily read Relativity Theory as vindicating their call to jettison unobservables from physical theory.4 

What is unique is that Weyl embeds his inherited reading of relativity’s world-view within Edmund Husserl’s phenomenological framework.5  This answers the previous question as to the philosophical counterpart to Relativity Theory. Not surprisingly, Husserl was absolutely delighted with this. In a letter to Weyl Husserl wondered just what Einstein, "the positivist," would think after seeing that Weyl grafted relativity onto the tree of phenomenology.6 

Even though Weyl’s embedding of relativity into Husserl’s phenomenology is unique, it is not entirely surprising.7  Weyl was a product of the Göttingen tradition in science. His Göttingen teachers – Felix Klein, David Hilbert and Herman Minkowski – held philosophy in high regard. Husserl frequently corresponded with these mathematicians, especially Hilbert.8  Moreover, within Minkowski’s own reading of relativity’s four-dimensional world-view lie the roots of Weyl’s views.

…the real world, and every one of its constituents with their accompanying characteristics, are, and can only be given as, intentional objects of acts of consciousness.9 

 

Let us call Weyl’s general principle "the intentional object world-view." Admittedly, it is quite strange upon first reading. Thus, some remarks are in order. First, as mentioned, and as we shall see in detail, this has its roots in Minkowski’s reading of relativity’s four-dimensional world-view. Second, it is not a slip, or merely some kind of philosophical window dressing for Weyl. There is no question that Weyl often dressed up parts of his work in philosophical apparel, but this does not apply to the intentional world-view. For the intentional world-view appears in many different parts of the Weyl corpus. 10  In addition, it remains throughout the many editions of Space-Time-Matter, while many other principles were either heavily reworked or abandoned. Hence, strange as it is, it has a history and staying power. I submit that it should be taken seriously.

Two questions come to mind. What does Weyl mean by calling the world and its objects "intentional objects as acts of consciousness?" Moreover, why does he think that relativity theory’s four-dimensional world-view is a particular instantiation of the intentional world-view? I will take up the first one and then turn to the second in the context of Minkowski’s work. Weyl immediately rules out any subjective idealism that reduces the world and its objects to creations of consciousness; he stresses that subjective idealism is no more tenable than naïve realism.11 

Examining Weyl’s ensuing discussion reveals an analysis of perception that mirrors Husserl’s analysis of perception in the Ideas.12  So the first move in clarifying the meaning of Weyl’s general principle would appear to be a comparison between Weyl and Husserl on the nature of perception. This, however, is no easy task since what Husserl means by perception in the Ideas is a thorny issue. There is no consensus as to whether Husserl’s ontological views are even clear in that work.13  My view here is that both Weyl and Husserl clearly held that there is a transcendent world. So in that sense they are both realists. However, the meaning of this transcendence is something that is, for both thinkers, based on consciousness. This sounds, I hold, much more mysterious than it really is. I will return to this later. For now I wish to turn to Minkowski’s work and unearth the roots of the intentional object world-view.

As is well known, Albert Einstein first put forth the Special Theory of Relativity in his 1905 paper "On the Electrodynamics of Moving Bodies," which expresses the theory in terms of particle motion.14  Three years later Herman Minkowski (one of Einstein’s mathematics professors) presented his paper, "Space and Time," which expresses Special Relativity in terms of geometry. Minkowski begins his paper as follows:

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.15 

 

After admitting the radical nature of the space-time view, Minkowski backtracks, arguing that this union is within everyday experience; perceptual objects always include "places and times in combination."16  (Minkowski insists that we never notice a place without a time nor a time without a place.) Minkowski then idealizes this experiential connection of space and time calling a "space-at-a-time" a "world point." A world point is a system of values, three for space and one for time, x,y,z,t, "The multiplicity of all thinkable x,y,z,t systems of values we will christen the world."17  Now, "…(n)ot to leave a yawning void anywhere, we will imagine that everywhere and "everywhen" there is something perceptible." 18  Minkowski calls this "perceptible something" at a particular world-point a "substantial point."–In sum, Minkowski fleshes out his world in terms of "thinkable" systems of values and "perceptible" substantial points. Here are the roots of the intentional object world-view. There is, according to Minkowski, a deep connection between consciousness and the world. This connection is much like the rationalist line: that the order of the world and the order of the mind ultimately coincide.

Note that t is also the same for S and T. As Minkowski notes, it does not matter which direction we point the time axis for these systems, so long as it is pointing towards the "top of the world."19 

c2t2 – x2 – y2 – z2 = 1

c2t2 – x2 = 1

Let us now take the x,t axes as orthogonal. Call this system, Q. The transformations of Q will result in other sets of axes, whose axes are no longer mutually orthogonal, but oblique. This set of permissible transformations of Q Minkowski calls Gc ; this set depends upon the parameter, c.

Now, suppose, Minkowski says, that we permit c to grow arbitrarily large – to approach infinity. There result is that the hyperbola flattens; its asymptotes bend towards the x axis. At the limit, the t axis vanishes and x’ conforms to x. Basically, if you change the parameter, c, you modify the group of transformations. Minkowski is simply asking what happens when you allow c to grow infinitely large. The group that results from c becoming infinitely large Minkowski labels G8. In other words, G8 is the set of transformations associated with Newtonian physics.

Minkowski then makes the following point regarding G8 and Gc. He stresses that Gc is more mathematically intelligible than G8 and so it is possible that a mathematician could have anticipated that natural phenomena would have the invariance group Gc and not G8. "Such a premonition would have been an extraordinary triumph for pure mathematics."20  What I want to stress here is that, according to Minkowksi, the real invariance group of phenomena could have been presented as a possibility within the context of pure mathematics and, on the basis of simplicity within this context, decided upon.

Another way to look at this is that even though the recognition of Gc over G8 actually blossomed from the soil of experimental physics, this was by no means necessary; it was simply a historical contingency. Although mathematics missed this great opportunity and so can now only display "staircase-wit," it has had its senses sharpened and is able "..to grasp forthwith the far-reaching consequences of such a metamorphosis of our concept of nature."21 

Minkowski then stresses that to call the recognition of Gc over G8 simply a "relativity postulate" is far too weak of a term. For this recognition ultimately means that phenomena are given as a four-dimensional space-time world. (Of course we can, with limited freedom, project this given world in terms of a separate space and time.) To fully express the connotations of this recognition, Minkowski suggests that we call it the postulate of the absolute world or the world-postulate.22  This postulate permits treating all four values, 3 space and 1 time, on the same footing.

The complete details of the four-dimensional view, of course, transcend immediate experience. Even if we grant that space and time are inseparable in our experience that does not mean that they should be thought of as identical. To do so entails that space-time is some kind of "solid block," existing all at once. Thus, temporal flow is merely subjective.23 

At the end of his paper Minkowski suggests that the world-postulate "lies open in the full light of day." Thus, there will be plenty of opportunities to experimentally confirm it. These future confirmations (of which Minkowski has no doubt) will serve to sooth physicists that are reluctant to abandon the traditional view of space and time. Why will these future confirmations placate old fears? It is because they will serve to reinforce another old idea: "…the idea of a pre-established harmony between pure mathematics and physics."24  I doubt that one could ask for a stronger rationalist sentiment.

These two exist within in Weyl’s writings. First, he stresses that the four dimensional world, the objective world of physics, is in fact a world that physics "…endeavours to crystallize out of direct experience."25  So, in some sense, this structure is implicit within the experiences of ordinary consciousness. In a similar vein, Weyl stresses that there must be a primordial link between the world and consciousness. This link appears within consciousness as a "felt causality," which is our deepest connection to the world; it is prior to that connection we call "perception."26 

Second, Weyl also has a version of how the "mathematical consciousness" could have anticipated the results of physics. As we have seen, Minkowski formulated the geometric structure of special relativity and then claimed that it could have been anticipated by mathematics, that it turns out to be a mathematically simpler structure than the Newtonian system. Weyl claims that a similar event has occurred with respect to the mathematical foundations of general relativity. In its first, Einsteinian formulation, it was presupposed that parallel transfer of vector direction on Riemannian manifold could only be formulated infinitesimally. But this, Weyl insists, only goes "half way:" the notion of parallel transfer also has to be formulated infinitesimally for vector length as well. Upon doing this, a pure infinitesimal geometry will finally have been achieved and the old Euclidean demon of "comparison at a distance" finally exorcised. It was not anticipated because of the pull of prejudice on the human mind. In principle, according to Weyl, mathematicians could have directly seen his own extension of Riemannian space long before he himself in fact saw it. But mathematicians did not see it since the "mathematical consciousness" is ultimately tied to human existence, a domain of historical, non-essential contingencies.27 

Since there is a deep connection between the concepts ultimately surfacing in physics and human everyday experience and mathematical experience, it comes as no surprise that Weyl would turn to a philosophy like that of Husserl’s, which was always concerned with the experiential origins of scientific concepts.28  Indeed, Weyl, at least during the time of writing Space-Time-Matter, held that his mathematical investigations into the nature of space was very similar to Husserl’s phenomenological investigations of essences.29 

Husserl’s discussions of perception are notoriously complex, although his basic results are in fact quite clear. Suppose that we wish to consider our experiences of a table. The table is never present "all at once." The table is only present to consciousness in terms of its profiles. Two things must be stressed. First, Husserl admits that the real table exists; it is not a creation of human consciousness or of any consciousness whatever. Hence he rejects both subjective idealism and Berkeleyan idealism.30  Second, Husserl rejects any realism that understands there to be an object existing in itself "…with which consciousness or the Ego pertaining to consciousness has nothing to do."31  A real, spatio-temporal object is always something that must be experiencable by consciousness, not merely in the logically possible sense but could be presented to consciousness in intuition. However, a spatio-temporal object is such that it cannot be entirely present to consciousness – not even to a divine consciousness. It is a fundamental error to assume that there could be such a consciousness to which a spatio-temporal object could be completely present.32  In Husserl’s words:

It is neither an accident of the own peculiar sense of the physical thing nor a contingency of "our human constitution," that "our" perception can arrive at physical things themselves through mere adumbrations of them. Rather it is evident and drawn from the essences of spatial physical things (even in the widest sense, which includes "sight things") that, necessarily a being of that kind can be given in perception only through an adumbration.33 

Weyl writes that in every perception lies the "thesis of reality."34  But to understand what is meant by this thesis, we cannot simply presuppose that there is a reality "out there." Rather, the meaning of this thesis must be understood based on the "data of consciousness." To abandon simple realism and turn towards examining this data of consciousness, we are in essence following Husserl’s phenomenological reduction.35 

Now, what emerges after we introduce the phenomenological reduction? Weyl clearly states that we can bring the essence of the supposed external object to intuition.36  When we do so for supposed external objects, Weyl says, we can never be sure that the current experiences of that object provide us with certainty. That is, we can never be sure that experience is providing us with a "…conclusive right to ascribe to the perceived object an existence and a constitution as perceived."37  There is always the possibility that future experiences will force us to alter our conclusions. Nonetheless, Weyl, maintains that there is an essence regarding external objects that is ultimately grounded on our experiences:

It is the nature of a real thing to be inexhaustible in content; we can get an every deeper insight into this content by the continual addition of new experiences, partly in apparent contradiction, by bringing them into harmony with one another. In this interpretation, things of the real world are approximate ideas. From this arises the empirical character of all our knowledge of reality.38 

 

I end on a brief note. First, the details of this paper are indeed sketchy, but I hope to have made the point that reading Weyl as trying to embed his understanding of relativity theory into Husserl’s phenomenology clarifies many of Weyl’s obscure comments. This reading reveals that Weyl, despite his scattered philosophical comments, is actually a fairly systematic metaphysical thinker. A study of his mathematical work in The Continuum will reveal a similar approach to the foundations of mathematics.39 

Moreover, this reading of Weyl, I suggest, helps us to further our understanding of Husserl. By his approval of Weyl’s grafting relativity onto phenomenology, we gain an insight into just how closely science and philosophy can operate, a close cooperation that Husserl himself stresses.40  Husserl insists that his phenomenological analyses of the foundations of human experience in no way prevent such cooperation. It comes as no surprise how pleased he was with Weyl’s work.

  1. Weyl, Space-Time-Matter, tr. H.L. Brose, New York: Dover Publications, 1952.

  4. For discussion of this see Michael Friedman, Foundations of Space-Time Theories: Relativistic Physics and the Philosophy of Science, Princeton: Princeton University Press, 1983, Ch.1.