The present article is primarily engaged with a summary of the metaphysical and epistemological foundations of Einstein’s work, and only in this restricted domain attempts to relate it to process thought. Appropriate references to the actual physics, at distinct technical levels, are given below.

### 1. Brief *Vita*

Born in Ulm, Germany, Albert Einstein attended the Polytechnic Institute in Zurich from 1896 until his graduation in 1900. While working at a Swiss patent office (1902–1909), he began his rise to become the preeminent physicist of the twentieth century and, by almost universal acknowledgement, one of the greatest scientific minds in history. His papers “On the Electrodynamics of Moving Bodies” (1905) and “The Foundation of the General Theory of Relativity” (1916), which, respectively, lay out his theories of special and general relativity, stand at the beginning not only of a new era in physics, but also, and especially in concert with the contemporaneous development of quantum mechanics, of a new era of philosophical inquiry into the foundations of science and knowledge. Indeed, the classical framework for so much of physics and metaphysics, stretching back millennia and so manifestly given by our traditional ideas of absolute time and space—or duration and extent—was, over a span of fifteen years, entirely reconstructed by Einstein’s theoretical developments and their spectacular verification in the observations of the stars and the planets. This reconstruction is not an incidental feature of the science, but a direct consequence of Einstein’s relentless insistence on grounding his work in concrete epistemological and ontological considerations, no matter what flights of fancy the ultimate mathematical expression might have taken, and no matter how counter-intuitive the eventual results might have been. Hardly ever has a soul been so driven and so capable of seeing through the ultimate victory of logos over chaos.[1]

### 2. Philosophical Leitmotif

Einstein’s overarching and explicit methodological theme was grounded in a composite principle of parsimony, a braiding of three separate strands that we shall attempt to disentangle shortly. We begin however with a few remarks, somewhat closer to the surface, that speak to his view of the nature of physics.

Einstein saw the activity of physics in terms of rational construction, representation and correspondence. Beginning with whatever conceptual foundations the human mind is inspired to create, a physical theory evolves by deduction and computation, and its success is precisely a matter of how well the results represent the actualities of the world. In other words, there must be a correspondence between the actual dynamical features of the world and the logical constructs of a physical theory such that the observed relations among actualities explicitly correspond with the derivable assertions of the theory. Thus, for example, in Newton’s theory of gravitation, physical bodies in space are represented by mass distributions and positions within a coordinate system. Their movements, then, insofar as his theory is successful, correspond with the conclusions one can logically deduce from his laws of motion and the classical inverse-square law for gravitational attraction. The complexities of this sort of representation and correspondence, as widely noted, for instance, by Einstein and Infeld (1938), by Whitehead (in his famous “fallacy of misplaced concreteness”), and more recently—and perhaps most remarkably—by Thomas Kuhn (1962), are enormous, but in broad strokes, this much is sufficient for us to proceed.

The conceptual foundations of a physical theory, Einstein stressed, were themselves *not* a matter of rationality, but of free creativity, independent of any *a priori* justification. Nonetheless, in connection with these foundations he states an unambiguous goal:

These fundamental concepts and postulates, which cannot be reduced further logically, form the essential part of a theory, which reason cannot touch. It is the grand object of all theory to make these elements as simple and few in number as possible, without having to renounce the adequate representation of any empirical content whatsoever (1934, 15).

This, then, is his primary principle in its most concise form, and perhaps there would be little more to say were it not for the complexity of the notion of simplicity or, perhaps more precisely, as regards the methodology and philosophy of science, the notion of parsimony.

#### 2.1. The three faces of Parsimony

##### 2.1.1. Ontological Parsimony

In a metaphysical sense, the most obvious aspect of parsimony is its ontological component, long known as Ockham’s razor. Here Einstein’s most famous example must be the identification of matter with energy via what is undoubtedly the most celebrated equation of the twentieth century: *E* = mc^{2}. Moreover, the very background for this equation already presumes the more subtle amalgamation of space and time into a unified four-dimensional manifold. Beyond these two examples from special relativity, we have his refusal to accept the logically separate notions of gravitational and inertial mass (respectively, the tendency of masses to attract each other and in themselves to resist any change of motion) as coincidentally equal. Indeed, Einstein founded his theory of general relativity on the assumption that these were two manifestations of a single aspect of matter. And we have finally a most curious instance of this commitment to ontological frugality requiring, and certainly worthy of, a bit more space and detail.

Einstein unreservedly revered Maxwell’s great achievement of reducing electromagnetic theory in empty space to a system of four famous equations that encompassed such apparently diverse phenomena as light, electrostatic attraction and magnetism. Indeed this reverence itself speaks worlds about Einstein’s scientific philosophy. Maxwell’s equations exhibited two particular aspects that proved critical to Einstein’s development as a physicist: they replaced the notion of action-at-a-distance with a space-filling *field*—a physical actuality—and they did not change form when transformed from one inertial coordinate system to another; to put is more technically, they were invariant under the Lorentz transformations. We shall return to this point later, but for the moment it suffices to note that Einstein was so impressed with Maxwell’s achievement that he came to believe that ultimately field theory would supersede any field-particle dualism. Writing again with Infeld, he states

[T]here is no doubt that quantum physics must still be based on two concepts: matter and field. It is, in this sense, a dualistic theory *and does not bring our old problem of reducing everything to the field concept even one step nearer realization* (1938, 293; italics added).

The irony here, from the perspective of quantum electrodynamics, as we now understand it, is that we find for once that Einstein’s metaphysical intuition was running ahead of his scientific intuition. In fact, in the small it is a particle ontology that carries the day, with the older field theory of Maxwell being a large-scale approximation, just as the ideal gas law is a large-scale approximation to the behavior of untold numbers of molecules in a sufficiently large volume of space.[2]

##### 2.1.2. Causal Parsimony

A second component of parsimony, distinct from but closely related to the first, might be called *causal parsimony*. Where ontological parsimony is conservative with respect to the kinds of things in the world, causative parsimony is conservative with respect to the kinds of interactions among these very things. Here both Newton and Maxwell provide splendid examples and insights in their work on gravity and electromagnetic theory. Newton’s postulation of an attractive inverse-square force acting along the line connecting the centers of gravity of two given masses is, to a very worthwhile approximation, in itself sufficient to account for the motions of all the celestial bodies that mankind has been observing since antiquity. Maxwell’s equations, as we have noted above, give a compellingly accurate description of the behavior of electric and magnetic fields. Note in connection with these examples that they may both be seen as something of a trade-off vis-à-vis ontological parsimony. Newton and Maxwell each introduce something new into the world—respectively, the gravitational force and the electromagnetic field—to unify not the substance, but the process of the world.

Einstein’s attachment to causal parsimony is shown in two clear examples that descend directly from those of the previous paragraph. In the case of electromagnetic theory, in the framework of special relativity one *derives* the magnetic force as a relativistic transformation of the electrostatic force.[3] In the case of gravitational theory, gravity *as a force* disappears altogether in the framework of general relativity.[4] In the unification of inertial and gravitational mass, Einstein absorbs gravitational effects into the structure of space-time: particles continue to move in geodesic paths (straight lines in the case of Euclidean space), but the notion of a geodesic (path of shortest distance) is dependent on a *metric* (the general mathematical construct that defines distance). For Einstein, gravity was no more than a reparameterization of this metric away from that of classical Euclidean three-dimensional space.

##### 2.1.3. Formal Parsimony

The final and most innovative element of parsimony in Einstein’s approach to science we call *formal*. As we shall see below, it is this element, under the more conventional, but more restrictive name of covariance, that Joachim Stolz (1995) regards as most essential to Einstein’s connection with process metaphysics. The first example comes in connection with special relativity and requires some preliminary discussion.

Newtonian mechanics was long understood to operate in an inertial frame of reference (an unaccelerated coordinate system), and the existence of any such frame immediately implies the possibility of infinitely many similar frames: we need only consider any other coordinate system moving at constant velocity (speed and direction) relative to our given, or so-called standard frame. In classical physics, we observe three features related to the laws of transformation from one inertial frame to another, involving, respectively, temporal measurements (time and duration), spatial measurements (location and length), and the laws of Newtonian mechanics (classical dynamics). Time transforms by application of a constant shift—just as we must add nine hours to transform the time shown on a clock in Los Angeles to that shown on a clock in Munich—and therefore duration remains invariant. Spatial coordinates transform by a shift that varies linearly with time, and therefore length, too, remains invariant. From these results and the further implicit assumption that mass (or, equivalently, force) remains invariant in all inertial frames, it follows that the classical laws of motion similarly retain their usual form in all inertial frames. Insofar as Newton’s laws define mechanical behavior, this last statement is equivalent to a highly restricted principle of relativity: no mechanical experiment performed in an inertial frame can reveal its absolute state of motion.

To create the special theory of relativity, Einstein extended the narrow principle above by dropping the qualifier on its domain of application: no experiment *whatsoever* performed in an inertial frame can reveal its absolute state of motion, and thus all inertial frames are equivalent. This is the so-called *principle of special relativity.* When the observation that the speed of light remains constant in all inertial frames is now included among the facts of the universe, it follows that the simple coordinate transformations assumed by Newton must fail, and with that failure the fundamental notions length, duration, and mass were no longer invariant across frames of reference. In particular, and perhaps most famously, the whole notion of absolute simultaneity—that is, simultaneity independent of any spatial frame—disappears.

Clearly one pays a high price in intuition for the principle of special relativity, and something of a price in algebra, too: the classical transformations between inertial frames are replaced by more complex *Lorentz transforms*. What does one gain by this mathematical complexity? For one thing, the effects predicted by the new theory are observable: at high velocities yardsticks shrink, clocks run more slowly and mass increases. Thus “the representation of empirical content” (to paraphrase Einstein and Infeld) has been enhanced in the new system. But beyond that, something else of a different stripe altogether has been achieved, and this points us toward the heart of this third principle of parsimony: when Maxwell’s equations are rewritten for a second inertial frame via the Lorentz transforms, they retain their form exactly (see Lindsay and Morgenau 1936, Chapter VII). To use the technical term, they exhibit *covariance* with respect to the Lorentz transforms, and in our current context we see covariance as parsimony applied not to objects and their interactions, but to forms themselves—a point to be further defended below.

In light of the spectacular success of the assertion of the full equivalence of inertial frames and the attendant covariance features in the domain of special relativity, it is natural that Einstein would seek further success by further extension of both notions to a *general principle of relativity* (or a general equivalence principle) and a *general covariance principle*.[5] The driving force here was gravity, and the springboard was the assertion that an inertial system subject to a uniform gravitational effect is indistinguishable from an accelerated system subject to no gravitational effect whatsoever. Mathematically, the consequences of this manifest themselves in the search for a theory of gravity that satisfies the requirements of general covariance, a goal that Einstein achieved, after some false starts, in his paper of 1916.

Whereas general covariance is a rather precise mathematical requirement for a physical theory, beyond it there remains something perhaps less well defined, but more general, in connection with the idea of formal parsimony in Einstein’s work. This is not a requirement about the nature of any given physical theory; it rather concerns the observation that physical theories, which we may here regard as an integrated deployment of ontology, causal categories, and formalisms, tend to be applicable *across the various domains of physics*. In commenting on the development of quantum mechanics, Einstein and Infeld write:

It has often happened in physics that an essential advance was achieved by carrying out a consistent analogy between apparently unrelated phenomena. In these pages we have often seen how ideas created and developed in one branch of science were afterwards successfully applied to another (1938, 270).

This is of particular importance for us here because it suggests a contemporary mathematical interpretation of formal parsimony that is important to Einstein’s philosophical connection with Whitehead. We discuss this in Section 3 below.

#### 2.2. Parsimony and Symmetry

All of the foregoing species of parsimony can be summarized in terms of a reciprocal relationship between objects and perspectives, or ontology and epistemology, that chiefly implicates the idea of symmetry when properly generalized and understood. We meet this term in geometry in making naïve and intuitive statements of the following sort: an equilateral triangle is more symmetric than a general isosceles triangle; a square is more symmetric than an equilateral triangle; a regular octagon is more symmetric than a square; a circle is more symmetric than any regular polygon. Upon deeper reflection, symmetry is intimately a matter of equivalent frames of reference. To illustrate this, let us imagine a standard two-dimensional coordinate system with a square centered at its origin, sides parallel to the coordinate axes. The coordinates occupied by the square do not change at all if we rotate the coordinate axes by any multiple of a right angle; we have thus essentially four distinct rotations all of which preserve the square in this sense. However, should we rotate by only 45 degrees, the square occupies a very different position in the plane—whereas a regular octagon similarly centered at the origin would not. Hence we say that this octagon has more rotational symmetries than the square (eight, in fact). The circle clearly has even more such symmetries (indeed, an infinite number) and it is precisely in this way that one makes precise sense of the statement that a circle is more symmetric than a regular octagon, and the octagon more symmetric than the square. The point is that symmetry is a matter of invariance under transformation of coordinates, or, to put it more casually, a kind of perspective invariance.

The connection between ontology and invariance may now be made via this specific characterization of symmetry; it operates in two directions that exhibit the promised reciprocity. In one direction, given a collection *T* of coordinate transformations, we may hold that the only legitimate objects of discourse are those that are invariant with respect to every transformation in *T*. Thus, if *T* is limited to the four rotations by right angles as above, we may talk of squares, octagons and circles (centered at the origin). But if we expand *T* to include also rotations by angles that are multiples of 45 degrees, our corresponding ontology must surrender the squares. If we expand further to *all* rotations, only the circles survive. The point is that as we demand more symmetry, we limit our ontology.

Taking now the reciprocal approach from the other direction, we might declare two objects identical if one can be obtained from the other by a transformation in our given collection *T*. (Here it may be more natural to think of the transformation applying to the object, rather than the coordinate system.) Thus in our example of polygons and circles, with respect to the collection *T* of *all* rotations, any two squares of the same size and centered at the origin are taken as identical. Effectively this inflates the symmetries of the square to include all of *T*. Again, the size of the ontology and the number of symmetries vary inversely.

In reviewing the varieties of parsimony we have discussed above in connection with Einstein’s work, we see the priority he placed on symmetry, working as above, in both directions. In asserting the principle of special relativity, or in elevating covariance as a key feature of Maxwell’s equations, Einstein replaced the ordinary scheme of coordinate transforms familiar to Newton with the Lorentz transformations. In other words, in consolidating the *form* of fundamental physical laws as they applied in an inertial frame, he introduced a new set of symmetries in space-time. However, with these in place, certain classical quantities which were of use precisely because they were conserved in nonrelativistic physics must be redefined accordingly. Thus for instance, Newton’s law of conservation of momentum is made invariant under Lorentz transformations by a redefinition of mass, which is no longer a constant attached to an object, but now dependent as well on its state of motion within a particular frame (see Resnick 1968, 111-17). Mass is thus no longer conserved, but mass-energy is. Hence in the case of special relativity we see that a contraction of forms (via covariance) leads to an expansion of symmetries, which then in turn exerts itself on the objects of the theory themselves. In general relativity, the equivalence of gravitational and inertial mass, or the requirement for general covariance, again expands the symmetries of space under consideration, this time vastly beyond those encoded in the Lorentz transformations. As a result gravity becomes an aspect of the structure of space itself as we pass from the “flat” world of classical or special relativistic physics into a universe with an intrinsic curvature.

### 3. Semantic Transfer

In the domain of science itself, the very existence of Whitehead’s book *The Principle of Relativity with Applications to Physical Science* (*R*) is proof of Einstein’s influence on Whitehead.[6] The book develops Whitehead’s own approach to the physics of relativity founded on a metaphysical awareness quite equal to Einstein’s (and perhaps better informed). Whitehead even takes Einstein to task on a subtle circularity implicit in postulating a space-time metric that is ultimately dependent upon the distribution of mass in the universe (see *R*, Chapter IV; *cf*. Stolz 1995 Chapter XIII).[7] Yet insofar as Whitehead’s theory has been eliminated as a viable alternative to general relativity (Stolz 1995, 194), our interest in semantic transfer must move to the philosophical domain, where it is at best speculative. In this regard, Joachim Stolz has put forth an arresting thesis in his monograph *Whitehead und Einstein* (1995). In order to facilitate a few comments on it, we return briefly to the idea of covariance in its most abstract form, as it can now be given by contemporary mathematics.

The term *functoriality* refers to the preservation of abstract formalisms under transformation. To be more precise, imagine first a class of objects, referred to as the *source class*, together with certain given (directed) relations among those objects.[8] Imagine next a second class, the *target class*, of objects with its own given set of relations. Finally suppose that to every object *A* in the source class we can assign an object *A’* in the target class, and, moreover, to every relation *R* between two source class objects we can assign a relationship *R’* to the corresponding target objects; that is, a relation *R* between source objects, say, *A* and *B* is assigned to a relation *R’* between the corresponding target objects *A’* and *B’*. Such a relationship across classes is called a *functor* provided that certain other natural restrictions are met. When a functor preserves the direction of source and target relations (a relation from *A* to *B *becomes a relation from *A’* to *B’*), we call it *covariant*. When it reverses the direction of the source and target relations (in this case a relation from *A* to *B *becomes a relation from *B’* to *A’*), we call it *contravariant*. [9]

General covariance in physics is a cognate of mathematical covariance insofar as it maintains that the form of a physical law should likewise persist across changes in coordinate systems. Stolz argues that even though Whitehead never explicitly mentions covariance in his philosophical writings, it is the defining leitmotif of the etiology of process thought (1995, 156). Indeed, process metaphysics is characterized by a set of dynamical objects in processive flux against an invariant background of unchanging forms. Stolz emphasizes physical covariance more as a matter of symmetry and formal homogeneity and claims correspondingly that Whitehead had implicitly imported these ideas into his process metaphysics to develop a system that is “perspective invariant” even as physical laws should be “coordinate invariant.” Thus Stolz speaks of process thought as a “covariant natural philosophy,” and, indeed, we might even more abstractly characterize the relational elements of Whitehead’s philosophy as functorial.[10]

### Notes

[1] For the layperson, Lincoln Barnett’s *The Universe and Dr. Einstein* (1950) remains an unsurpassed account of the origins and meaning of theory of relativity, as well as the mind of its creator. For an only slightly more technical audience, Gamov’s *Thirty Years That Shook Physics* (1966) is a splendid exposition of the development of quantum theory and Einstein’s part in it.

[2] Along these lines, Richard Feynman remarked, “When Einstein and others tried to unify gravitation with electrodynamics, both theories were classical approximations. In other words, they were wrong” (1985).

[3] See Purcell 1965, for a stellar development of this approach to electromagnetism.

[4] If one chooses to regard forces as entities in themselves, certainly both notions of parsimony so far introduced reduce to flavors of what we have called ontological parsimony. But in this regard, we might observe first that an intuitive distinction still remains and, second—noting the splendid self-referential irony—just how strong the forces of parsimony truly are!

[5] The relationship between an equivalence principle and a covariance principle, easily enough illustrated for Newtonian mechanics, becomes a matter of unsuspected subtlety at this point, with an attendant literature that is both surprising and beautiful (see especially Friedman, 1983, and Norton, 1993). Suffice it to say here that ultimately general covariance is the weaker (more easily satisfied) of the two, and for reasons to be made clear in the following section, the more important in a Whiteheadian context.

[6] But see Stolz 1995, Chapter XII for the claim that Whitehead was independently investigating questions similar to those of Einstein. He claims also that Whitehead’s distress at the incorporation of the concept of the Euclidean point (having neither structure nor extension) into physical theories anticipates the more sophisticated subsequent formulations that incorporate quantum uncertainties.

[7] Whitehead summarizes and concludes his misgivings as follows: “[W]e could not say how far the image of a luminous object lies behind a looking–glass without knowing [in advance] what is actually behind that looking-glass” (*R* 83) To rephrase his point, the very notion of a mass distribution presupposes the possibility of uniform spatial measurements *without reference to the mass properties of the body or bodies in question*. Yet in Einstein’s system, the *a priori* dependence of such measurement on the very distribution in question seems to render this impossible; hence the alleged circularity.

[8] A good example to keep in mind might be the class of two-dimensional manifolds (spheres, donuts, and such) with the relations taken simply as continuous functions between such objects.

[9] Those with a nodding acquaintance with abstract algebra may take the class of groups as the target class, with homomorphisms between groups as the requisite relations. A good deal of algebraic or geometric topology may then be considered as finding functors from manifolds to groups, thereby converting continuous functions to group homomorphisms. See Valenza (1993) for a very brief introduction to categories and functors. Category theory was invented in 1945 by mathematicians Samuel Eilenberg and Saunders MacLane.

[10] See Henry and Valenza (1994) for an independent development of this position based on abstract graph theory.

### Works Cited and Further Readings

Of the references listed below, the following require little or no background in mathematics or physics: Barnett (1950), Einstein (1934), Einstein and Infeld (1938), Feynman (1985), Gamow (1966), and Stolz (1995). Resnick’s text on special relativity (1968) and Purcell’s on electricity and magnetism (1965) are excellent resources for those with some familiarity with vector calculus; the other scientific references, including the more historical work of Lindsay and Morgenau (1936), are technically more demanding. The book by Friedman (1983) and the long article by Norton (1993) are both exemplary of the proposition that the philosophy of science is at its best when it is most informed by actual scientific work. Kuhn’s famous work (1962), equally well informed, remains, of course, essential to any serious discussion of the history and epistemology of science.

Barnett, Lincoln, 1950. *The Universe and Dr. Einstein*, revised edition (Mattituck NY, American Reprint Co.).

Einstein, Albert. 1989. *The Collected Papers of Albert Einstein*, Volume 2 (*The Swiss Years: Writings, 1900–1909*), translated by Anna Beck (Princeton, Princeton University Press).

_____. 1997. *The Collected Papers of Albert Einstein*, Volume 6 (*The Berlin Years: Writings, 1914–1917*), translated by Alfred Engel (Princeton, Princeton University Press).

_____. 2004 [1934]. *Essays in Science* (New York, Barnes & Noble).

Einstein, Albert and Leopold Infeld. 1938. *The Evolution of Physics* (New York, Simon & Schuster).

Feynman, Richard P. 1985. *QED: The Strange Theory of Light and Matter* (Princeton, Princeton University Press).

Friedman, M. 1983. *Foundations of Space-Time Theories: Relativistic Physics and the Philosophy of Science *(Princeton, Princeton University Press).

Gamow, George. 1966. *Thirty Years That Shook Physics* (Garden City NY, Doubleday). Reprinted by Dover Publications.

Hartle, James B. 2003. *Gravity: An Introduction to Einstein’s General Relativity* (San Francisco, Addison Wesley).

Henry, Granville C. and Robert J. Valenza. 1994. “The principle of affinity in Whiteheadian metaphysics,” *Process* *Studies* 23, 1.

Kuhn, Thomas S. 1970 [1962]. *The Structure of Scientific Revolutions*, 2^{nd} Edition (Chicago, University of Chicago Press).

Lindsay, Robert Bruce and Henry Morgenau. 1936. *Foundations of Physics* (New York, John Wiley & Sons). Reprinted by Dover Publications, 1963.

Norton, J. D., 1993. “General covariance and the foundations of general relativity: eight decades of dispute,” *Rep. Prog. Phys.* 56, 791-858.

Purcell, Edward M. 1965. *Electricity and Magnetism*: *Berkeley Physics Course*, Vol. 2 (New York, McGraw-Hill).

Resnick, Robert. 1968. *Introduction to Special Relativity* (New York, John Wiley & Sons).

Stolz, Joachim. 1995. *Whitehead und Einstein* (Frankfurt-am-Main, Peter Lang).

Valenza, Robert. 1993.* Linear Algebra: An Introduction to Abstract Mathematics* (New York, Springer).

Zwiebach, Barton. 2004. *A First Course in String Theory* (Cambridge, Cambridge University Press).

### Author Information

Robert J. Valenza

Dengler-Dykema Professor of Mathematics and the Humanities

Department of Mathematics

Claremont McKenna College, Claremont, California 91711

Robert_valenza@mckenna.edu

### How to Cite this Article

Valenza, Robert J., “Albert Einstein (1879–1955)”, last modified 2008, *The Whitehead Encyclopedia*, Brian G. Henning and Joseph Petek (eds.), originally edited by Michel Weber and Will Desmond, URL = <http://encyclopedia.whiteheadresearch.org/entries/bios/contemporaries/albert-einstein/>.