Space and Place: A Scientific History – Part Two

This article is the continuation of the previous one – Space and Place: A Scientific History – Part One -, an inquiry into the concepts of space and place analysed through the scientific perspective of Julian Barbour’s book The Discovery of Dynamics. Now, the scope is to cover the temporal gap from the moment where Barbour left his work (the post-Newtonian clarification of dynamical concepts) to more recent theories. Thus, I shall commence with the presentation of the fifth and the sixth chapter of Max Jammer’s Concepts of Space: The History of Theories of Space in Physics, third enlarged edition, 1993. I hope you are not lost in the passage from Barbour’s to Jammer’s text, given that the approaches and the intents of the two authors were seemingly different: while the intent of Barbour was to unveil the hidden connections between motion and place (Aristotle), motion and space (Descartes and Newton), and motion and matter (Mach), the interest of Jammer is more general: his famous book directly engages with questions of space and/or place from a more global perspective (should I say philosophical, other than scientific?), without being directly focused on the question of motion, which, in this case, is an indirect topic of inquiry. In spite of the differences between Barbour and Jammer (and in spite of the differences between Barbour, Jammer and Weinberg, whose article – The Search for Unity: Notes for a History of Quantum Field Theory –  I also present here to give some information about the field concept, which has an enormous influence on the contemporary spatial/placial debate concerning the ultimate nature of reality), my ultimate plan does not change: to show the convergence between philosophical and scientific perspectives with respect to the modern and contemporary difficult shift from space to place (I refer to the post-Newtonian epoch), that is, the shift of vision from the more abstract idea of space to the more concrete ideas of place and/or matter-as-field as the fundamental concepts to explain the basic nature of reality. From the scientific perspective, I believe the quite recent shift towards the concrete can be epitomized by the concepts of the dynamical centre of gravity defined by the masses of the stars – a conception devised by E. Mach with the intention to supersede the concept of absolute space for dynamical questions – and by the field concept which, at a more basic level, I understand as the ultimate reunion of matter and place.[1]  Since Barbour left his plan suspended exactly in the moment of that shift, we now have to conclude that journey finalizing our historico-scientific inquiry the same way Casey finalized his historico-philosophical inquiry. Therefore, we now pass the proverbial torch from Julian Barbour to Max Jammer – both physicists and both of them highly sensitive to the history of science – and, finally, from Jammer to the American theoretical physicist and Nobel laureate Steven Weinberg, who traced a beautiful and highly readable history of the field-concept,  from Faraday to Quantum Field Theory (as I often say, a physical field is nothing other than a place: to begin with and to stick to the extended conceptualization of place as the concretization of a long chain of processes,  a field is the place where physical processes concretize, properly).

Image 1: Jammer, Max.Concepts of Space: The History of Theories of Space in Physics. New York: Dover Publications, Inc., 1993.

1. The Concept of Space in Modern Science

Chapter Five of Max Jammer’s Concepts of Space is titled ‘The Concept of Space in Modern Science’; it follows the chapter focused on Newton’s concept of absolute space. Given that, in the past article, we have already analyzed that concept through Barbour’s perspective, now, I’m going to summarise Jammer’s exposition concerning the reception of absolute space in the post-Newtonian period.

Jammer speaks of the logical necessity of absolute space when he mentions the work of the Swiss mathematician and physicist Leonhard Euler (Basel, Switzerland, 1707; Saint Petersburg, Russia, 1783) who was well aware of the fact that the laws of motion discovered by Newton presupposed the notion of absolute space. Analogously, with specific reference to the law of inertia, the Scottish mathematician Colin Maclaurin said that the ‘perseverance of a body in a state of rest or uniform motion, can only take place with relation to absolute space, and can only be intelligible by admitting it.’ [2]

Then, Jammer briefly takes into account the positions held by Immanuel Kant (Königsberg, Prussia, now Kaliningrad, Russia, 1724; Königsberg 1804), who shifted his vision on space from a relativist to an absolutist position, before concluding that space was an a priori intuition (an ideal rather than a physical conceptualization, as in the intentions of Newton). So, at first, we read about a relativist Kant saying that ‘I should never say, a body is at rest, without adding with regard to what it is at rest, and never say that it moves without at the same time naming the objects with regard to which it changes its relation.’[3]  Then, five years later, ‘apparently under the influence of Euler, Kant abandons this point of view and declares himself in favor of the Newtonian concepts of absolute space and absolute time. In his essay “On the first grounds of the Distinction of Regions in Space” Kant formulates his program as follows: “space has a reality of its own, independent of the existence of all matter, and indeed as the first ground of the possibility of the compositeness of matter”.’[4]  Finally, the radical change for which Kant became famous: space and time understood as transcendental, a priori, intuitions. We read from Jammer: ‘the idea that intuition lies at the basis of our geometric cognition brings about a radical change in Kant’s attitude toward these questions. The problem of space now appears to Kant in a new light. It ceases to be a problem of physics and becomes an integral part of transcendental philosophy. To Kant from now on space is a condition of the very possibility of experience. In the inaugural dissertation “De mundi sensibilis atque intelligibilis forma et principiis,” the concepts of absolute space and absolute time are considered to be merely conceptual fictions a mental scheme of constructed relations of coexistence and sequence among sense particulars. Not itself arising out of sensations, the concept of space is a pure intuition, neither objective nor real, but subjective and ideal.’[5]  Then, for Kant space is an a priori intuition, a pre-condition for the intelligibility of things, a pre-condition for experience. How far we are from Newton’s physical concept of absolute space! However (without going into details concerning Newton’s greatest contemporary opponents, Leibniz and Huygens, whose contributes Jammer dealt with in Chapter Four), we do not have to wait for Kant to read about other critical remarks on Newton’s concepts.  As Jammer also says, Kant’s theory of space ‘was greatly influenced by the English empiricists Locke, Berkeley, and Hume, and their analytical investigations into the formation of ideas.’ [6]

It was especially Bishop George Berkeley (County Kilkenny, Ireland, 1685; Oxford, United Kingdom, 1753) who criticized the concept of space: with respect to the Treatise Concerning the Principles of Human Knowledge, Jammer says that

Berkeley describes how according to his empiristic view the concept of space is formed by the perception of extension, the notion of space being but an abstract idea of extension. For like other general ideas it is formed in the human mind by abstraction from sense perceptions relating to bodies. Newton’s notion of absolute space, which contains all bodies and is retained if all bodies are thought away, is in Berkeley’s view a false hypostatization of an abstraction. […] The notion of empty space, however, is a mere verbal expression of a state of empirical facts.[7]

I would have said those words since they represent my thinking about the so-called ‘physical space‘: a mere verbal expression – a false hypostatization of an abstraction. That’s what space is for Berkeley (and I fundamentally agree with that position).

… a mere verbal expression – a false hypostatization of an abstraction. That’s what space is…

My ‘sympathy’ for the position held by Berkeley on the concept of space continues. Again, this is Berkeley: ‘When I excite a motion in some part of my body, if it be free or without resistance, I say there is space: but if I find a resistance, then I say there is body: and in proportion as the resistance to motion is lesser or greater, I say the space is more or less pure. So that when I speak of pure or empty space, it is not to be supposed, that the word space stands for an idea distinct from, or conceivable without body and motion. Though indeed we are apt to think every noun substantive stands for a distinct idea, that may be separated from all others: which hath occasioned infinite mistakes.[8]

Curiously, one of my first naive argumentations against the reality of physical space presents some similarities with what argumented by Berkeley, above. It happened to me that I was sitting in balance, dandling on the legs of my chair, while I was at my desk; at a certain moment, I was losing my balance risking to fall down; luckily enough I soon took back my position of precarious balance by quickly opening my arms and touching the wall on my right; the resistance of the wall allowed me to recover my initial position. I asked myself: what if instead of a wall made of stones there was ‘a wall made of water’ offering my arm a different type of resistance? What if there was ‘a wall of air’ or ‘of any other gas’ offering almost no resistance at all to my arm? Then, I began to entertain the idea of ‘a wall of space’ (what can I find below the level of the gas or of the atoms?), but I soon suspected that space was just a name for… nothing, a name for the idea of extension of an immaterial body, properly – therefore, a mere verbal expression, fundamentally, and nothing ‘physical’. My right arm moving through a ‘material’ – ‘physical space’ – which offered no resistance at all, was just an abstraction, something impossible to be realized concretely, since I could find no ‘physical space’ at all, actually: in the best of the hypothesis all I could admit (as an act of faith after certain philosophers and physicists) was the resistance of an ether, which, for me, is certainly another entity, with a different history and meaning, with respect to space. Then, apart from a name – that is, behind that name ‘space’ -, I found nothing that could offer my arm any type of physical or mechanical resistance, at whatever scale of investigation. Names do not offer ‘mechanical resistance’ of whatever degree, even infinitesimal, to physical bodies, independently of the scale of analysis. At a fundamental level, bodies or their components happen to be in place or in places, concretely, and not ‘in names’; the fact that we give names to places is another question (this was just the beginning of a long story that I finally decided to narrate through the articles of this website). Thus, pondering of that personal experience that I had before encountering the thinking of Berkeley about space, I began to think that space was just a name that could occasion infinite mistakes, as Berkeley said. I have narrated that episode since it is representative of the method of my research on space and place: the importance of the first-hand experience which I always used as a vehicle to give direction to my second-hand experience (the experience that others have done before me – like Berkeley in this case – or, in general, the knowledge you learn from others, through their experiences, by reading books, etc.), which in turn can give a positive or negative feedback to your first-hand experience, thus establishing what I consider a virtuous circle of learning (it is obvious that the way to knowledge is a two-way street, where I act upon the others and the others act upon me – or acted upon me before I could even realize – but what I want to say is  that there must be a moment in which one has to look inside him/herself and try to reason in the most independent and naïve manner as possible, by appealing to logic and to the analysis of the language we use to express knowledge; this is especially true and helpful for those who are involved in creative disciplines. I believe this is the only way to guarantee the progress of knowledge, independently of the quality of what the single person produces: anyone should give him/herself a chance).

Coming back to the concepts of space and place, as Jammer noted, Kant’s greatest achievements influenced the assumptions on the idealism of space in disciplines like philosophy and psychology; but, concerning physics, things went somewhat differently. In fact, as Jammers notes, the progress of mechanics was probably due to the scarce interest that the question about the nature of space aroused in physicists like ‘Lagrange, Laplace, and Poisson, none of them was much interested in the problem of absolute space. They all accepted the idea as a working hypothesis without worrying about its theoretical justification.[9] Jammer also takes the words contained in the Encyclopedie, ou dictionnaire raisonné des sciences, des arts, et des metiers, edited by Diderot and D’Alambert, to support the ‘scarce interest thesis’ about the nature of space: ‘we will not take a position on the question of space; we can see how such obscure question is unnecessary for geometry and physics.[10]  So we come to Jammer’s conclusion: ‘it may even be claimed that this absence, so far from being a hindrance to mechanics in the eighteenth and early nineteenth century, in a certain degree facilitated the development of this science.’[11] Concerning the divergence between philosophy and physics on this specific aspect I redirect you to the paragraph The Relation Between Natural Science and Metaphysics, based on a paper by professor Ivor Leclerc, in the article Place, Space, and the Philosophy of Nature.

Following Jammer’s exposition, there is also another aspect to consider, quite different from the one we have just seen; an aspect we have already considered in the concluding chapter of Barbour’s ‘The Discovery of Dynamics‘: the necessity to surpass the epistemological (and dynamical) limits of the concept of absolute space. As we have already seen, this work of revision was especially done by Ludwig Lange (Gießen, Germany,1863; Weinsberg, Germany, 1936) and Ernst Mach (Brno, now the Czech Republic, 1838; Munich, Germany, 1916). Here, the story narrated by Jammer converges with the one narrated by Barbour. As concerns Lange, I will only briefly quote what Jammer said concerning the concept of ‘inertial system‘ and the reasons for its introduction; Jammer says: ‘In 1885 an important attempt to find a way out of the paradoxical situation (that is, the adherence to the concept of absolute space on the one hand and its absence from practical physics on the other) was made by Ludwig Lange. Lange thought that he had found the way to eliminate the concept of absolute space from the conceptual foundation of physics. In his view, the essential (today he would say the operational) content of the law of inertia, and with it of the whole of mechanics, retains its full physical meaning if the somewhat “ghostly” idea of an absolute space is replaced by the concept of an “inertial system” [which is] a coordinate system in respect to which Newton’s law of inertia holds.[12]

However, rather than on Lange, I’m going to dwell a little bit more on Jammer’s exposition regarding Ernst Mach (Brno, now Czech Republic, 1838; Munich, Germany, 1916), since, according to me, Mach’s contribution is of a paramount importance to put in the right epistemological perspective the traditional concepts of space, of matter, and, consequently, of place (at least place in the extended sense that I call for in this website). At first, Jammer informs us that, according to Mach, ‘the assumption of absolute space for the explanation of centrifugal forces in rotational motion was unnecessary.’[13] In fact, as Mach himself says: ‘Newton’s experiment with the rotating vessel of water simply informs us, that the relative rotation of the water with respect to the sides of the vessel produces no noticeable centrifugal forces, but that such forces are produced by its relative rotation with respect to the mass of the earth and the other celestial bodies.’[14]  The traditional concept of space (i.e., absolute space) should be eliminated from mechanics, as Jammer says, by interpreting Mach’s own view: ‘the very idea of an absolute space, that is, of an agent that acts itself but cannot be acted upon, is […] contrary to scientific reasoning. Space as an active cause, both for translational inertia in rectilinear motion and for centrifugal forces in rotational motion, has to be eliminated from the system of mechanics.’[15]

Jammer continues: ‘the elimination of what Mach calls “the conceptual monstrosity of absolute space” […] is achieved, in his view, by relating the unaccelerated motion of a mass-particle not to space as such, but to the center of all masses in the universe. The assumption of an intrinsic functional dependence between inertia and a large-scale distribution of matter closes for him the series of mechanical interactions without resorting to a metaphysical agent.[16]

the elimination of what Mach calls ‘the conceptual monstrosity of absolute space’ is achieved…

Obviously, the metaphysical agent is space – absolute space, precisely. Then, by interpreting Mach, I would argue that together with the abandonment of the concept of space we simultaneously assist to the return of the concepts of place and matter, provided that we think at the system of all masses as a real place in the extended sense of the term (for physical purposes, the centre of all masses in the universe corresponds to the ultimate place of physical processes, properly; so that, according to this perspective, which modifies and extends the traditional interpretations of place, we assist at the reunion of matter and place). Mach’s position on the ideal or fictitious nature of space is quite explicit; he says: ‘No one is competent to predicate things about absolute space and absolute motion; they are pure things of thought, pure mental constructs that cannot be produced in experience.[17] Jammer continues: ‘Mach summarizes his ideas concerning absolute space and absolute motion in a very clear statement: “For me only relative motions exist, and I can see, in this regard, no distinction between rotation and translation. When a body moves relatively to the fixed stars, centrifugal forces are produced; when it moves relatively to some different body, and not relatively to the fixed stars, no centrifugal forces are produced. I have no objection to calling the first rotation ‘absolute’ rotation, if it be remembered that nothing is meant by such a designation except relative rotation with respect to the fixed stars. Can we fix Newton’s bucket of water, rotate the fixed stars, and then prove the absence of centrifugal forces? The experiment is impossible, the idea is meaningless, for the two cases are not, in sense perception distinguishable from each other. I accordingly regard those two cases as the same case and Newton’s distinction as an illusion”.’[18]  Therefore, by relying on those words Jammer concludes the section dedicated to Mach attributing the Austrian scientist and philosopher ‘the earliest proclamation of the principle of general relativity.’[19]

Has mechanics given up the concept of absolute space (or absolutist positions) after Mach and Lange? Not exactly. In fact, we read from Jammer, another (old) entity soon re-entered the scene with the possibility to preserve the essence of absolute space: the ether. ‘It was suggested by Drude and Abraham, to mention only these names, that the ether, the carrier of electromagnetic waves, should be identified with absolute space. If the ether as an absolute reference system could be demonstrated, the notion of absolute space could be saved.’[20]  Many experiments were devised to detect the ether (the most famous is the one performed by Michelson-Morley, in 1887) but all the attempts failed. Finally, we read from Jammer, ‘physics, and not only mechanics, was ready to abandon the concept of absolute space altogether. Poincare’s words “Whoever speaks of absolute space uses a word devoid of meaning” became an accepted truth.’ [21]

The following parts of Chapter V regarding the nature of the modern concept of space are characterized by the birth of non-Euclidean geometries and by two related questions that the event promoted:  (i) the question concerning the structure of space, and (ii) the question regarding the dimensions of space. Is physical space Euclidean or not? How many dimensions physical space has? These are the two questions now under Jammer’s investigation. It seems that what we took for granted, that is the ideal character of space (Kant or Berkeley), or even its negation as a meaningful entity for physics (Lange, Mach or Poincare) is now put into discussion again by the discoveries in the field of geometry. To put it briefly, it seems it is not possible to give a definite answer to the question of space (and we will see that this is exactly Jammer’s belief at the end of his work, which is not that far from the conclusions drawn by Barbour).

Image 2: Geometry and the Structure of Space.

2. On the Structure and Dimensions of Space

If before Newton there was no reason to refute the idea of space as a naturally Euclidean entity, after the birth of non-Euclidean geometries the question whether the space of physics was Euclidean or not became acute. Everything started with a long-standing question regarding the V postulate of Euclid, which states that ‘in a given plane through a given point not more than one parallel to a given line exists. That this postulate is not needed for the demonstration of the first 28 theorems of the Elements was noted early. In antiquity, it was thought possible to prove the postulate on the basis of the other postulates. From Ptolemy and Proclus to Nasiraddin-at-Tusi, the Persian editor df the Elements, and John Wallis, down to Lambert and Legendre, all attempts at such a proof failed. Of all the age-long attempts to solve the problem, the most remarkable is that of Gerolamo Saccheri.’[22]  It was precisely the attempt made by Girolamo Saccheri (Sanremo, Italy, 1667; Milano, Italy, 1733) that influenced the subsequent attempts made by other mathematicians. His study contained elements that arouse the interest of mathematicians like Klein, Gauss, Poincarè, Beltrami, Bolyai, Lobachevsky, Riemann and gave input to the foundation of non-Euclidean geometries, that is, hyperbolic and elliptic geometries.[23]  So, after non-Euclidean geometries were discovered the question arose whether the space of physics was Euclidean or not.

The most important attempt to answer this question was made by Carl Friedrich Gauss who ‘tried to measure directly by an ordinary triangulation with surveying equipment whether the sum of the angles of a large triangle amounts to two right angles or not. Accordingly, he surveyed a triangle formed by three mountains, the Brocken, the Hoher Hagen, and the Inselberg with sides measuring 69, 85, and 107 km. Needless to say, he did not detect any deviation from 180° within the margin of error and thus concluded that the structure of actual space is Euclidean as far as experience can show.[24]  Therefore – Jammer concludes – even if Carl Friedrich Gauss (Brunswick, Germany 1777; Göttingen, Germany 1855) believed non-Euclidean geometry was logically impeccable, experiments seemed to be against its application to physical space.

Another attempt to prove the non-Euclidean structure of ‘physical space’ by experiments was undertaken by Nikolai Lobachevsky (Makaryev, Russia, 1792; Kazan, Russia, 1856) who thought that only astronomical observations and measurements could verify the real nature of space; as in the previous cases, ‘Lobachevsky’s attempt to prove the non-Euclidean structure of space empirically came to nothing. So he concluded that Euclidean geometry alone was of importance for all practical purposes.’[25]  In spite of that, he was convinced that his studies could lead to the foundations of new geometries.

Now, we come to the important related question regarding the dependence of the metrical structure of space from physical data. We arrive at this important consideration through the preliminary engagement of Gauss in geodetic problems, which resulted in a series of studies on the structure of space that also were carried on by Bernhard Riemann (Breselenz, now Germany, 1826; Verbania, Italy, 1866), whose ‘generalization of Gauss’s theory of surfaces, culminating in the concept of “curved space” made it clear that the space of Euclidean geometry and the space of the geometry of Lobachevsky and Bolyai were only special cases of the generalized space, that is, spaces of constant zero curvature [that is flat space, the space of Euclidean geometry] or constant negative curvature [hyperbolic space]. By introducing an appropriate metric Riemann was able to show also that a space of constant positive curvature, a so-called “spherical” space, is conceivable.[26]  Riemann’s argumentations were important since they also contained an anticipation of some central ideas for the theories of Einstein, namely that the metrical structure of space was related to the physical data contained in it, that is, it was related to the distribution of matter; in fact, as Jammer says: ‘for Riemann matter was the causa efficiens of spatial structure.[27] Riemann’s speculations about the relation between space and matter were taken up by the English philosopher and mathematician, William Clifford (he was the English translator of Riemann’s work) who ‘conceived matter and its motion as a manifestation of the varying curvature. He assumed that the Riemannian curvature as a function of time may give rise to changes in the metric of the field after the manner of a wave, thus causing ripples that may be interpreted phenomenally as motion of matter.’ [28]

As suggested by Jammer, it is easy to read those words as a suggestion to the idea that gravitation might be a manifestation of an underlying geometry, anticipating Einstein’s program. I would say that through the words pronounced by Clifford we could read the finalization of a complete overturn of the problem concerning place, matter and space after two millennia and a half: while for Aristotle it was place that determined the behaviour of matter, for Clifford (as well as for many others before and after him) it was space. Is it just a question of language (of misplacing space for place, or, to put it another way, to name the same entity with different names) or is there something more profound?

The new possibilities offered by non-Euclidean geometries had the consequence of stimulating the research about the possibility to prove whether physical space, that is, the space of physics, was ‘flat’ or ‘curved’; it was the French mathematician and physicist Jules Henri Poincarè (Nancy, France,1854; Paris, France, 1912) who demonstrated the futility of such controversy. We read from Jammer: ‘it was only toward the turn of the century that Poincare demonstrated once for all the futility of this controversy and the fallacy of any attempt to discover by experiment which of the mutually exclusive geometries applies to real space. Measurement, he insists, is never of space itself, but always of empirically given physical objects in space, whether rigid rods or light rays. Regarding the structure of space as such, experiment can tell us nothing; it can tell us only of the relations that hold among material objects. What geometry one chooses is, for Poincare, merely a matter of convenience, a convention. We select that system of geometry which enables us to formulate the laws of nature in the simplest way.’[29] I believe these words are a plea for matter/place against space considered as an autonomous physical entity. To say that a physical body roams or moves through (physical) space has a high price to pay: a misplaced concreteness that creates much confusion (‘infinite mistakes‘ according to Berkeley – we have seen above) in the analysis of the actual phenomena of reality.

What was just a suggestion worked out by Clifford, with Albert Einstein became a principle; but before introducing Einstein’s theories and the influence they had on the concepts of space, I would like to point out the eye-opening recommendation made by Jammer at this precise moment in the book:

the structure of the space of physics is not, in the last analysis, anything given in nature or independent of human thought. It is a function of our conceptual scheme.[30]

Max Jammer

I believe we cannot help but subscribe to Jammer’s words. Of course, there are conceptual schemes that are better than others (they explain better the phenomena of the world, or they explain previously unexplained phenomena) and as we have seen in the preceding article – which, in the overall, was about the passage from the Ptolemaic to the Copernican scheme – we could read history as a constant attempt to produce better and better schemes and models. The epoch we are living in is an epoch of transition towards the redefinition of new conceptual schemes after Aristotle (and the Ptolemaic system) and after Descartes/Newton (and the Copernican system); as Casey and Barbour said with different terms, we are living a third peripeteia (a concept I took from Edward Casey) that will lead us to a new model – or a new understanding of the Cosmos – expressed by means of new conceptualizations: ultimately, this is the reason why we necessarily have to rethink concepts of space and place, just like it happened in the past with Aristotle and Newton, the two ‘watersheds’ Barbour has spoken of in his text. Jammer is fundamentally on the same page concerning this basic question.

The exceptional work of Albert Einstein (Ulm, Germany, 1879; Princeton, United States, 1955) represented a turning point in the history of physics (and of mankind); it had far-reaching consequences for the conceptualization of space, time, matter and motion. At first, following Einstein’s intuitions, we passed from the Newtonian idea of absolute space to the idea of relative spacetime: this passage happened thanks to the findings of the Special Theory of Relativity (it is ‘special‘ since it only included the special case of uniform rectilinear motion, that is: the laws of physics remain the same in all the frames of reference that moves with a constant velocity). Within the program of special relativity, the concept of space was still fundamental but, as an inertial system, it lost any individuality. How did it happen that Einstein changed the fate of space from absolute to relative? The laws of mechanics which obeyed the principle of relativity discovered by Galileo and stated by Newton – ‘the motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forward in a right line without any circular motion’-[31] had been used for a long time until Maxwell came up with his theory of electromagnetism; yet, Maxwell’s equations didn’t seem to obey the principle of relativity;[32] to make a long story short, it was Einstein who finally explained this inconsistency between mechanics and electromagnetism (in his third paper of the annus mirabilis 1905) by showing that space and time were not absolute entities, like affirmed by Newton, but relative, deeply interrelated between them and could undergo deformations: only by taking into account space and time deformations (at very high speeds indeed), it was possible to extend the validity of the principle of relativity to include electromagnetic phenomena (years later, in 1908, the three dimensions of space, were combined with the dimension of time into a single entity, the four-dimensional manifold known as ‘Minkowski spacetime’). But this was just Einstein’s first step: in fact, in the passage from the Special Theory of Relativity to the General Theory of Relativity the fate of space was changed again, passing from a relative entity (that is, relative spacetime) to a new physical entity deeply interrelated with matter: a physical field (spacetime became a relative field). Let’s see these passages directly through the words of Jammer and Einstein. Jammer observes: ‘Space as conceived by Newton proved to be an illusion, although for practical purposes a very fruitful illusion – indeed, so fruitful that the concepts of absolute space and absolute time will ever remain the background of our daily experience. […] Thus, Newton’s conception of absolute space and its equivalent, Lorentz’s ether, were shorn of their ability to define a reference system for the measurement of velocities. This was accomplished by the special theory of relativity. Within the framework of this theory, however, space as such was still a basic concept. To be sure, as part of the four-dimensional Minkowski space-time continuum it certainly had lost any individual distinction; an infinite number of coordinate systems were physically equivalent. It had, however, its own representation as an inertial system. Owing to the relativization of simultaneity, furthermore, the notion of action at a distance had to be discarded and the adoption of the field concept as the basic element of the theory had been suggested. This program was carried through by the general theory of relativity whereby the inertial system was replaced by the displacement field, a component part of the total field, “this total field being the only means of description of the real world – it is Einstein who is speaking now, and he continues –

The space aspect of real things is then completely represented by a field, which depends on four coordinate-parameters; it is a quality of this field. If we think of the field as being removed, there is no space which remains, since space does not have an independent existence.[33]

Albert Einstein

This is how Albert Einstein contributed to change the fate of space (and possibly of place, if new interpretations of this concept are taken into consideration, as I’m arguing for at RSaP-Rethinking Space and Place, when I say that the field-concept is a physical state of place) in a relatively short amount of time spanning from the annus mirabilis of 1905, passing through the year 1915 (General Relativity), down to the end of his exceptional career when he embraced the working hypothesis of a Unified Field Theory: then, space passed from being considered an absolute, standalone entity (Newton’s absolute space), to a relative entity associated with time (relative spacetime); and from such a relative and correlate entity associated with time to a different entity deeply related to matter and/or energy: the field-concept (that is the relative field, or, relative ether). All of these passages are so important and not completely illustrated (especially, for that which regards the final part of Einstein’s career), to the point that they would deserve a specific article, which is something I’m going to deal with in the coming future proposing you the reading of Ludwik Kostro’s book ‘Einstein and the Ether‘ – first published in 2000 – which has an introduction by Max Jammer himself.

2.1 On the Dimensions of Space

The other field of research that was stimulated by the birth of non-Euclidean geometries regarded the question of the dimensions of space. This is certainly a long-time question that goes far back in time to Aristotle’s concept of place: in fact, it was his understanding of place as a two-dimensional entity (I remind you that Aristotle’s definition of place – topos – understood as ‘the  first unchangeable  limit of  that  which  surrounds’ inevitably leads to consider the Aristotelian place as a surrounding surface, therefore as an extension in two dimensions)[34] that set off a storm of criticism, which, favoured the development of the concept of space. In spite of that, as Jammer noted ‘the three-dimensionality of space as a problem was scarcely discussed in antiquity or in the Middle Ages.’[35]  Jammer introduces the argument of the three-dimensionality of space (or the four-dimensionality of the spacetime continuum)  as ‘an accidental feature, justified only by experience.’[36] He takes the example of Euclid, who explicitly considered solids as three-dimensional entities. This meant that people had the idea (or knowledge) of extension in three dimensions, but – I say – we cannot certainly conclude that to have the idea of a body as a three-dimensional entity implies the idea of three-dimensional space as an autonomous entity that contains bodies, which is what Jammer seems to suggest.  There is a conceptual gap between the concept of substance (matter or body) and the concept of space as a separate entity from substance; and, most of all, we cannot infer the properties of one entity from the other.  Let’s see what Jammer says concerning Euclid: ‘Euclid’s definition I in Book XI of the Elements: “A solid is that which has length, breadth, and depth,” with the implicit identification of solids and bodies, was accepted without further questions. This seemed to be only natural since the notions of surface, line, and point came later to be defined by the process of abstraction from the concept of the solid. Moreover, the problem was dismissed by identifying three-dimensionality with body, as had been done already by Isaac Judaeus.’[37] Then, Jammer directly moves on to Leibniz according to whom the three dimensions of space were derived from geometrical considerations: ‘Even Leibniz, who, as we have seen, submitted the concept of space to a most critical analysis, took little notice of the problem of the dimensionality of space. Recognizing that space has three dimensions, he bases this statement on purely geometric considerations’.[38] With the birth of non-Euclidean geometries the possibility that concepts of space could possess an arbitrary number of dimensions was realized; in fact, the possibility of a geometry with more than three dimensions was already taken into considerations by Gauss in 1844, when, in a letter to a colleague, he refers to a generalization of his considerations on symmetry and congruence for a geometry of more than three dimensions, ‘for which we human beings have no intuition, but which considered in abstracto is not inconsistent.’[39] While as concerns the problem of the dimensionality of physical space, ‘Poincare attempted to demonstrate the three-dimensionality of the space of experience by simple topological consideration.’[40] To solve the problem of dimensionality of space was also the ambition of modern physics. Among various attempts, Jammer cites as the most noteworthy those made by Sir Arthur Eddington and Hermann Weyl. In conclusion of a brief, anticipatory excursion on the problem of dimensionality Jammer says that no satisfying solution has been given to the problem, and, as a final remark, before resuming the question in a subsequent part of the text, Jammer provisionally concludes that ‘our knowledge of large-scale as well as of small-scale properties of physical space is intimately related to the progress in cosmology and microphysics, respectively.’ [41]

2.2 On the Structure of Space According to Quantum Mechanics

As a matter of fact, for that which concerns the domain of microphysics, quantum mechanics had a great influence on how the structure of space could be perceived, and it had a great influence on the development of the idea of a discrete space, setting aside the long-standing idea of the spatial continuum. Let’s see what Jammer says at this regard: ‘Heisenberg, in his attempt to achieve a simplified general representation of quantum mechanics, tried recently to abandon the principle of continuity in Riemannian or Euclidean geometry and introduced the suggestion of a “smallest length” to meet certain difficulties in quantum electrodynamics’[42]  – as Jammer explains, the idea of a discrete space was introduced to eliminate the problem of the infinities, we will see it more in detail when we will refer to forthcoming paragraph on the article of Steven Weinberg. Jammer continues: ‘The concept of a smallest length, or rather a fundamental length, as characterizing the ultimate limit of resolution in physical measurement of spatial extension, has recently gained some popularity amongst theoretical physicists. […] In view of the great mathematical difficulties involved in the construction of a geometry of discontinuous space, however, physics has still to resort to the traditional geometry of a continuous space by a statistical treatment of the concept of length. Thus, continuous space resumes its service, even for nuclear physics, but as a convenient fiction for the statistical mathematization of physical reality.’[43] The success of quantum mechanics and the consequent focus on phenomena at the micro-scale, entailed a certain difficulty of application of the traditional concepts of space  and time: that’s where the necessity to rethinking the traditional concepts of space and time came from (and – I say – this is also the reason why the concept of place, which is historically, ontologically and epistemologically related to those concepts, necessitates the same process of revision, which is the argument of this website). Let’s see what Jammer says at this regard: ‘A profound epistemological analysis of certain quantum-mechanical principles seems to suggest that the traditional conceptions of space and time are perhaps not the most suitable frame for the description of microphysical processes.’[44] After Heisenberg’s uncertainty principle, ‘the impossibility of an exact localization in combination with the determination of the momentum, and the related dualistic wave-particle character of physical reality, can be interpreted as a challenge for a critical revision of the accepted space and time conceptions.’[45] And, needless to say, given that we are speaking of localization, the revision of the concept of place is an immediate consequence. As shown by Jammer, according to Niels Bohr, processes within the atom transcend the traditional frame of space and time, and Louis de Broglie comes to an analogous conclusion. Jammer writes: ‘the problem of an intelligent applicability of traditional space and time conceptions to atomic physics was the subject of a paper by Louis de Broglie [who] admits frankly the difficulties involved in the use of our notions of space and time on a microphysical scale’.[46] Not only concepts of space and time were the subjects of revision after the success of quantum mechanics: the same process of revision regarded the concepts emptiness or void space, as Jammer noted. There are insurmountable epistemological problems that go against the idea of emptiness as showed by P. W. Bridgman, other than physical problems given the nature of ‘the concept of an electrostatic field with fluctuating zero-point’ advanced recently by quantum mechanics.[47]

Among the remaining argumentations of Chapter V, there are some important epistemological and ontological considerations regarding the influence that Mach’s principle had on Einstein’s general theory of relativity. These considerations also regard the main theme of the present research in physics: the conciliation between general relativity and quantum mechanics. Is the field-concept the operative concept that will permit such reunification? This question regards the way we understand the ultimate structure of reality (which is a place – a field of physical processes, to begin with – and not a space). Let’s see what Jammer says on this critical subject.

If quantum mechanics raised speculations on the structure of space at the smallest scale, Einstein’s implementation of the idea suggested by Riemann and Clifford (‘the metric of space structure is a function of the distribution of matter and energy’)[48] promoted investigations on the macroscopic structure of space (at this regard, Einstein published a paper in 1917: Cosmological considerations in general relativity). The mathematical implementation of this critical question was made when Einstein introduced the cosmological constant into his field equations to guarantee the realization of his equations within the domain of the Mach’s principle: ‘the introduction of the cosmological constant […] had to implement Mach’s Principle within the conceptual frame of general relativity or at least to remove certain difficulties that prevented an incorporation of the Principle into the general theory of relativity’, Jammer says.[49] This is a critical passage for our interests on space, place and matter. Regarding Einstein’s proposed ontological subordination of spacetime to matter, we read: ‘Einstein deems the implementation of this Principle as “absolutely necessary” […]. If indeed the structure of space-time is exhaustively conditioned and determined by the distribution of masses, he claims, the absence of matter should imply the nonexistence of a guiding field (spacetime metric)’.[50] By introducing the cosmological constant lambda, Einstein’s hoped to remove the inconsistency with Mach’s Principle. Actually, we learn from Jammer, the question is everything but settled: ‘its compatibility with the theory is still a matter of dispute’, Jammer concluded.[51]  We are approaching the (provisional) end of a long journey, which I would epitomize through these words, pointing at the field-concept as the ultimate datum of reality: ‘Should it become clear that Mach’s Principle is compatible with the general theory of relativity, matter and space-time could be considered as two ultimately distinct physical entities and the Newtonian notion of absolute space would have to be eliminated from the conceptual scheme of theoretical physics. Should it, however, become evident that Mach’s Program cannot be satisfied within the general theory of relativity […], because matter cannot be understood apart from knowledge of space-time, then matter as the source of the field will become part of the field. On the basis of such a unified field theoretic conception as proposed for example by J. Callaway, the field itself would constitute the ultimate, and in this sense absolute, datum of physical reality. As the concluding words in his Foreword to the present book seem to imply, Einstein’s view during the last years of his life was in favor of this conception.’ [52]

I repeat what I think it is a critical passage to understand the ultimate nature of reality (this is also where we have to look at to understand the physical origin of the concept of place in the light of its intimate connection – or unbreakable correlation – with matter/energy, time-as-duration, and space, as abstract extension): ‘because matter cannot be understood apart from knowledge of space-time, then matter as the source of the field will become part of the field.’ I interpret this field as a unique structure – a place – where the temporal and the material are united under the same name or concept, the field concept properly (time understood as duration of the processes that concretize into physical matter; matter understood as that which is concretely extended, this fact meaning that the idea of an ex-tended space existing as separated entity from the ex-tended matter is a misplaced abstraction; a material plenum only exists, which is the meaningful element of a more extended composition or ensemble encompassing space, or place, matter and duration. I believe this is almost the same element – apart from the temporal dimension –  that Descartes epitomized under the ambiguous formula space = internal place = matter).[53] For me, the alternative name for the field of physics is place, given that a physical field is nothing other than a physical state of place (place, in the extended sense I’m arguing for in this website, is a more encompassing concept than the field of physics: it includes it as one of its states, the other states being the chemical, the biological, the social and the symbolic state, which closes the circle of knowledge, promoting the above mentioned unbreakable correlation between the concrete and the abstract). Then, I do not believe in the subordination of spacetime to matter, as Einstein believed at the time of the introduction of the cosmological constant, but I believe in their fusion into a single entity (actually, and originally, they consist of a unity which has been abstractly split into three different entities: space -or place-, time and matter) as Einstein also believed toward the end of his life, if I’m not mistaken (as I have already anticipated, I will deal with Einstein’s different views about space, spacetime and the relative ether in a forthcoming article based on Ludwik Kostro’s book ‘Einstein and the Ether‘).

So we are at the end of Jammer’s Chapter V – The Concept of Space in Modern Science. Curiously enough Jammer closed this chapter with the same proposition with which he began Chapter VI, which he added to the third enlarged edition of the book, in 1993: in fact, Jammer writes in 1954, the problem of space was considered an unfinished business;[54] then, the incipit of the newly added Chapter VI – from the title Recent Developments in the Philosophy of Physical Space – almost 40 years later, is nearly the same: ‘the problem of space is still unfinished business‘, we read.[55]

Image 3: Space understood as a Quantum Foam, whose structure is ruled by probabilistic laws.

3. Recent Developments in the Philosophy of Physical Space: the Structure and the Dimensions of Space

Jammer commences the new chapter with a quotation taken from Einstein’s foreword (written for the first edition of the book, 1954) in which the eminent scientist refers to the ‘victory over the concept of absolute space which became possible only because the concept of the material object was gradually replaced as the fundamental concept of physics by that of the field. […] There is no space without a field’, Einstein concluded.[56] These are also the two main arguments that Jammer developed in the new chapter, added to the third edition of Concepts of Space, 1993: the first statement regards the absolutist/relativist debate that had a burgeoning interest after recent interpretations to Einstein’s General Theory of Relativity (these interpretations reopened a question that seemed to be closed after Einstein’s pronunciation against the concept of absolute space). The other question regarded the development of field theory, or, more generally, of quantum mechanics and the influence it had on the issues regarding the structure and the dimensions of space. Let’s see how the two questions determined the fate of the concept of space (actually, spacetime) in the last decades.

As regards the dispute between absolutism and relativism, Jammer especially refers to the reception of John Earman’s successful work, ‘World Enough and Spacetime: Absolute vs. Relational Theories of Space and Time‘, in which the American philosopher of physics proposes a classification of classical space-time theories (I won’t go into detail – the author speaks about ‘Machian space-time’, ‘Maxwellian space-time’, ‘Leibnizian space-time’, ‘Galilean space-time’, ‘Newtonian space-time’ and ‘Aristotelian space-time’)  ‘in order to cope with the subtleties of the recently revived controversy between absolutists and relationists concerning space, motion and acceleration an issue of central concern in the contemporary philosophy of space. According to Earman “the setting of classical space-time is flexible enough to accommodate coherent versions of both views: that all motion is relative motion and that motion involves some absolute quantities, whether velocity, acceleration or rotation.” […] In his advice not to dismiss absolutism of space, motion or acceleration as empty metaphysical talk, and in his criticism of the logico-positivistic arguments propounded by Reichenbach, Grunbaum and others in favor of relationism, Earman was not alone. Similar ideas have been expressed by quite a few contemporary philosophers’. [57]

As regards the other question – the influence of modern particle physics in the conceptualization of space – Jammer considered different recent working hypothesis, but the issue that gained increasing attention was the one regarding ‘the notions of a possible discreteness and quantization of space and space-time as well as the idea of a fluctuational stochasticity of space-time, partly conceived in analogy with the stochastic interpretations of quantum mechanics.’[58] We read from Jammer that the idea of a quantized space-time, that is the introduction of the smallest length, was proposed in 1947 (by Snyder, Hellund and Tanaka) to remove difficulties in the quantum theory of wave fields;[59] while ‘the idea of a fluctuating metricality and topology, even with the possibility of dimensional variability, attracted attention when in the Sixties Wheeler, in his geometrodynamics, argued that in a unification of general relativity and quantum mechanics, Heisenberg’s uncertainty principle forces space-time geometry to behave like a fluctuating foam-like structure […]. Wheeler showed, e.g., by dimensional analysis, that the fluctuations become physically significant at the scale of the “Planck length”.’ [60] (see image 3, above).

The other issue, which emerged after certain recent developments in the unified theories of the fundamental forces and quantum gravity, ‘is the topological problem of the dimensionality of space or space-time.[61]  Jammer says: ‘physical theory suggests higher dimensionalities than those just mentioned. Such arguments originated in the attempts to unify the field theories of fundamental forces, attempts motivated by the age-old aspirations of explaining all of physics by a single theory.[62]  So, we learn that the attempts made by different theoretical physicists to explain the nature of physical phenomena may necessitate a space of five dimensions (a necessity emerged after the attempt of unifying gravitation and electromagnetism),[63] ten dimensions (after the findings of the superstring theory),[64] up to eleven dimensions (after the attempts of unification between electromagnetic and weak interactions, and after the attempts to include strong interactions and incorporate gravity into a grand unified theory like, for instance, the theory of supergravity),[65] or, even, an undefined and not integer number of dimensions (this is what recent investigations into quantum gravity seem to suggest). [66]

At the end of Jammer’s historico-scientific narration, we are left with the same perplexities raised by Barbour in his book: in the end, it seems scientists cannot say if space is absolute or relative, or cannot agree on the number of dimensions;[67] therefore, it seems opportune to cite Jammer again: ‘the structure of the space of physics is not, in the last analysis, anything given in nature or independent of human thought. It is a function of our conceptual scheme.[68] Then, ‘space is still an unfinished business’ to stick to Jammer’s pronouncement in the introductory part of Chapter VI. Finally, this is Jammer concluding remark: ‘perhaps it is true that the hope that physical research can resolve the philosophical problems of space is just as vain as the hope that philosophical thought can resolve the physical problems of space.[69] I think a possible solution to this fundamental question raised by Jammer can be found in a novel alliance between phylosophy and physics, in the name of a resurgent philosophy of nature (‘nature’ in the classical Greek world is ‘physis‘, whence the term ‘physics‘ derives, ultimately): I have dedicated a few articles at RSaP on these fundamental questions – Place, Space, and the Philosophy of Nature, Place, Space, and a New Conception of Nature, and Concepts of Place, Space, and the Nature of Physical Existence.

I would conclude by saying that, for me, this spatial question could be also put in the following terms: space is an abstract concept, an idea concocted ‘a posteriori’ to explain physical phenomena or situations; as such, it can be absolute, relative or whatever we will, depending upon the conceptual scheme we use, as also Jammer observed (that is, space is an ingenious conceptual invention which, contrarily to what many still believe, has not a true correspondence in the actual world of facts and events: it is a shadow cast on the actual world by our imagination – a fantasy, a dream, or a nightmare as Mach would have probably said; it is not cast by any sort of actual bodies, to keep on using the allegory of the cave); then, space can have three or an infinite number of dimensions, according to the conceptual scheme that one uses to explain actual phenomena of the real world. I believe we should shift our view from space – the shadow without a body – to place-as-matter, the shadow projected by real bodies (place is where physical, chemical, biological, social and symbolic processes concretize into matter, this fact meaning that we cannot split place from matter and time, given that any process requires a time-as-duration to manifest itself), since it is only through that concept that we can recover the correlation between the actual (which is the mode through which objects encounter other objects and subjects) and the ideal (which is the mode through which the subject encounters the objects). Through the concept of place-as-matter (or matter as the place of processes) we can understand the continuity and correlation between ontological and epistemological aspects around which knowledge unfolds (i.e., the correlation of the physical to the ideal or the mathematical). The traditional concept of space is fallacious since it breaks down the correlation between the physical and the ideal, it breaks down the correlation between the shadow and the system of bodies after which the shadow is cast. Could the concept of place (a reformed conception in the sense that I call for in this virtual site) re-enter the contemporary scientific (and philosophical) debate under the guise of the field concept? That is my conviction; in fact, my research and my practical endeavours as an architect are directed to the affirmation of a new mode of thinking about place and space, where scientific and philosophical modes of thinking converge, transcending into a new unity, beyond disciplinary divisions.

So, there remains to investigate the nature of the field concept and the physical theories that emerged after that concept. To do that I will consider the history narrated by the Nobel prize-winner Steven Weinberg in the 1977 popular article ‘The Search for Unity: Notes for a History of Quantum Field Theory’, included in Dædalus, the Journal of the American Academy of Arts and Sciences, whose extended reading I recommend.[70] What follows is a summary of that article.

As the title suggests Quantum field theory (QFT) is the main subject of the article: in the very opening, the author explains that QFT ‘is the theory of matter and its interactions, which grew out of the fusion of quantum mechanics and special relativity in the late 1920s.’[71]  Indeed, ‘before quantum field theory came field theory and quantum theory’, which the author illustrates in their essential points.[72]  Weinberg commences with his exposition by saying that the first classical field theory can be considered the theory of gravitation by Newton, who didn’t exactly speak of the field since gravitation according to him was conceived as an ‘action at distance between every pair of material particles in the Universe.’[73] During the 18th century mathematical physicists found it convenient to replace the notion of action at distance with that of gravitational field, ‘a numerical quantity (strictly speaking, a vector) which is defined at every point in space, which determines the gravitational force acting on any particle at that point, and which receives contributions from all the material particles at every other point.’[74] The reason for introducing the notion of the gravitational field was merely mathematical: as Weinberg says, there were no physical consequences or differences by saying that ‘the earth attracts the moon, or that the earth contributes to the general gravitational field.’[75] Fields began to have an existence as physical entities on their own with the development of the theory of electromagnetism, which was mainly developed by Faraday and Maxwell. The term field was introduced into physics by the British physicist and chemist Michael Faraday (Newington Butts, England 1791; Hampton Court, England, 1867), in 1849. He was among the first scientists to note that different electric and magnetic phenomena had a lot in common. At the time, it was still possible to look at interactions between two electric charges as forces acting at distance, ‘but it became very much more natural to introduce an electric field and a magnetic field as conditions of space, produced by all the charges and currents in the universe, and acting in turn on every charge and current.’[76] Then, a field is a physical entity, made of lines of force, which describes the properties of the regions around an electrically charged, or magnetic, object (image 4, below).

Image 4:  The lines of force passing out a magnet into the regions around constitute a physical field, an electromagnetic field in this case.

As Weinberg says, this field-interpretation of electric and magnetic phenomena became almost unavoidable after the Scottish physicist and mathematician James Clerk Maxwell  (Edinburgh, Scotland, 1831; Cambridge, England, 1879) demonstrated that electromagnetic waves travel at a finite speed, the speed of light; indeed, that physical limit, contributed the expression ‘action at distance’ to lose sense. Maxwell understood that electricity, magnetism and light were the manifestations of the same phenomenon and he unified electric phenomena, magnetic phenomena and the behaviour of light under a single set of equations. However, Maxwell didn’t believe fields existed as independent entities; like many others, he believed fields needed a medium to propagate. In fact, according to a diffused mode of thinking about the existence of a medium – the ether – was necessary for electromagnetic waves to propagate from place to place. Maxwell himself thought his equations were ‘valid in only one special frame, at rest with respect to the aether.’[77] Among physicists, that widespread belief persisted until the beginning of the 20th century. Finally, it was Einstein who ‘removed any hope of detecting the effects of motion through the ether […] leaving electromagnetic fields as things in themselves’ in consequence of the Special Theory Relativity. [78]

It was from the observation of electromagnetic phenomena that Special Relativity and early Quantum Theories developed in the first decade of the 20th century. Eventually, quantum theories led to the birth of quantum mechanics and later to quantum field theory. The conceptual leap made by quantum theories was to approach classical problems in finite or discrete quantities instead of thinking about phenomena at microscopic levels in term of continuity. As we have already seen in the last part of Jammer’s text, this fact implied the abandonment of the idea of phenomena happening in a tetra-dimensional continuum (spacetime), favouring the birth of ideas concerning a discrete space.

At the beginning of the 20th century, one of the classical unsolved problems of electromagnetism was the problem concerning the radiation of black bodies: ‘the classical theories of electromagnetism and statistical mechanics were incapable of describing the energy of electromagnetic radiation at various wavelengths emitted by a heated opaque body. The trouble was that classical ideas predicted too much energy at very high frequencies, so much energy in fact that the total energy per second emitted at all wavelengths would turn out to be infinite![79] In a paper which dates back to 1900, the German physicist Max Plank (Kiel, Germany 1858; Gottingen, Germany, 1947) proposed a solution to that problem, by supposing that the energy released in the form of electromagnetic radiation emitted by oscillating electrons in an opaque body could only assume minimum discrete quantities; the idea that energy can have only certain discrete values is called quantization. This is Planck’s quantum hypothesis where the quantum is a certain fixed discrete quantity; however, Planck believed his idea was just a mathematical trick to avoid the problem of infinities. Again, we have to wait for Einstein before someone understood that the idea was not a pure abstraction (it was not a shadow projected by our imagination…) but a real effect that could explain actual phenomena (it was a shadow projected by actual bodies); as Weinberg says: ‘it was Einstein who, in 1905, proposed that radiation comes in bundles of energy, later called photons, each with an energy proportional to the frequency.[80] Einstein explained that effect by tackling another unsolved problem of classical physics: the photoelectric effect, which is about the modality through which certain metals emit electrons when they get hit by the light. The ideas of Planck and Einstein were put into a coherent frame when, in 1913, the Danish physicist Niels Bohr (Copenhagen, Denmark, 1885; Copenhagen, Denmark, 1962) proposed his theory of atomic spectra. We read from Weinberg: ‘atoms in Bohr’s theory are supposed to exist in distinct states with certain definite energies, but not generally equally spaced. When an excited atom drops to a state of lower energy, it emits a photon with a definite energy, equal to the difference of the energies of the initial and final atomic states. Each definite photon energy corresponds to a definite frequency, and it is these frequencies that we see vividly displayed when we look at the bright lines crossing the spectrum of a fluorescent lamp or a star.’ [81]

From Planck’s initial idea of quantization in 1900 to Bohr’s idea of atomic spectra in 1913 -, passing through Einstein’s photoelectric effect explained in his first paper of 1905, early quantum theory was settled. As Weinberg says, in the following years – and especially in the period between 1925 and 1926 – ‘quantum theory became the coherent scientific discipline known as quantum mechanics’[82] through the works of Louis de Broglie, Werner Heisenberg, Max Born, Pascual Jordan, Wolfgang Pauli, Paul Dirac, and Erwin Schrödinger. At the time, the theoretical foundations of quantum mechanics were laid but despite its origins in the theory of thermal radiation, until that moment quantum mechanics mainly dealt with material particles and not with radiation itself. We have to wait until 1926 before quantum mechanics principles were applied to electromagnetic fields, so that it was really possible to speak of the birth of Quantum Field Theory (QFT): that happened when Born, Heisenberg and Jordan presented a paper about the behavior of electromagnetic field in empty space in the absence of electric charges or currents. With that work, they confirmed that Planck’s idea of quantization could also be adapted to electromagnetic fields and important consequences were derived: ‘they were able to show that the energy of each mode of oscillation of an electromagnetic field is quantized […].The physical interpretation of this result was immediate. The state of lowest energy is radiation-free empty space, and can be assigned an energy equal to zero. The next lowest state must then have an energy equal to the frequency times Planck’s constant, and can be interpreted as the state of a single photon with that energy. The next state would have an energy twice as great, and therefore would be interpreted as containing two photons of the same energy. And so on. Thus, the application of quantum mechanics to the electromagnetic field had at last put Einstein’s idea of the photon on a firm mathematical foundation.’[83] That paper dealt only with the electromagnetic field in empty space; it did not lead to quantitative predictions. As we learn from Weinberg, the first practical use of quantum field theory was made in a 1927 paper of the British physicists Paul Dirac (Bristol, England, 1902; Tallahassee, Florida, 1984): the problem was ‘how to calculate the rate at which atoms in excited states would emit electromagnetic radiation and drop into states of lower energy.’[84] A tentative formula had been already given by Born and Jordan, but the real problem was to understand that formula as the mathematical outcome of quantum mechanical processes; ‘this problem was of crucial importance, because the process of spontaneous emission of radiation is one in which particles are actually created. Before the event, the system consists of an excited atom, whereas after the event, it consists of an atom in a state of lower energy, plus one photon. If quantum mechanics could not deal with processes of creation and destruction, it could not be an all-embracing physical theory. […] Dirac’s successful treatment of the spontaneous emission of radiation confirmed the universal character of quantum mechanics.’ [85]

Even if quantum field theory was giving theoretical and practical results, physical reality was still conceived of being composed of distinct fields and particles. The decisive steps to leave behind this material dualism toward a truly unified vision of nature was taken between 1928 and 1930 thanks to the works of Jordan and Eugene Wigner, Heisenberg, Pauli and Enrico Fermi: ‘they showed that material particles could be understood as the quanta of various fields, in just the same way that the photon is the quantum of the electromagnetic field. There was supposed to be one field for each type of elementary particle. Thus, the inhabitants of the universe were conceived to be a set of fields-an electron field, a proton field, an electromagnetic field-and particles were reduced in status to mere epiphenomena. In its essentials, this point of view has survived to the present day, and forms the central dogma of quantum field theory: the essential reality is a set of fields, subject to the rules of special relativity and quantum mechanics; all else is derived as a consequence of the quantum dynamics of these fields.[86]

the central dogma of quantum field theory: the essential reality is a set of fields, subject to the rules of special relativity and quantum mechanics; all else is derived as a consequence of the quantum dynamics of these fields.

Steven Weinberg

This turn, which has relevant ontological and epistemological implications, had immediate physical consequences as well: given enough energy, material particles could have been created the same way a photon is created when energy is released by atoms. In 1932 Enrico Fermi used this aspect of QFT to formulate a theory of the process of nuclear beta-decay (the nucleus emits an electron, changing its chemical properties); the question to answer was to find where did the electron come from when a nucleus suffered beta-decay: ‘Fermi’s answer was that the electron comes from much the same place as the photon in the radioactive decay of an excited atom-it is created in the act of decay, through an interaction of the field of the electron with the fields of the proton, the neutron, and a hypothesized particle, the neutrino.’ [87]

Weinberg continues the article introducing a final argument – a problem to be solved for quantum field theory to take its modern form: the question of antimatter. Everything began in 1928 with Dirac’s theory of individual electrons for which he discovered equations with ‘solutions corresponding to electron states of negative energy, that is, with energy less than the zero energy of empty space. In order to explain why ordinary electrons do not fall down into these negative-energy states he was led in 1930 to propose that almost all these states are already filled. The unfilled states, or “holes” in the sea of negative energy electrons would behave like particles of positive energy, just like ordinary electrons but with opposite electrical charge: plus instead of minus. Dirac thought at first that these “antiparticles” were the protons, but their true nature as a new kind of particle was revealed with the discovery of the positron in cosmic rays in 1932. Dirac’s theory of antimatter allowed for a kind of creation and annihilation of particles’.[88] In 1934, the theory of antimatter, the creation and annihilation of particles and antiparticles, was confirmed by the works of Oppenheimer and Furry, Pauli and Weisskopf. The question was settled and ‘antiparticles are now seen as coequal quanta of the various quantum fields.’[89]

It is important to point out that quantum field theory contributed to offering a different interpretation of particles and of the forces acting between them: ‘we can think of two charged particles interacting at a distance not by creating classical electromagnetic fields which act on one another, but by exchanging photons, which continually pass from one particle to the other. Similarly, other kinds of force can be produced by exchanging other kinds of particle. These exchanged particles are called virtual particles, and are not directly observable while they are being exchanged, because their creation as real particles (e.g., a free electron turning into a photon and an electron) would violate the law of conservation of energy.[90] Then, in the current understanding of particles and forces that act among them, ‘the real problem is to determine what are the fundamental quantum fields and what are the interactions among them’, Weinberg concluded. [91]

Up to now, the narration of Weinberg about the history of quantum field theory was a narration of success; as a matter of fact, there were very difficult moments as well for QFT: it had to face several internal inconsistencies before acquiring the current status of an independent discipline. One of such inconsistencies regarded the so-called problem of infinities – a mathematical problem to the solution of equations predicting the shift in atomic energy levels after interactions among particles -, which puzzled physicists for a long time before a solution to the problem was given ‘eliminating infinities by absorbing them into a redefinition of physical parameters‘, a method called renormalization.[92] The problem of infinities haunted the mind of physicists for many years and it contributed to casting shadows on quantum field theory. Interestingly enough, apart from technical reasons, according to Weinberg there was also another fundamental reason for which QFT took so long before being taken seriously (an explanation that regards epistemological and psychological levels of inquiry): the huge distance between mathematical formula and physical reality. In fact, Weinberg argued, ‘it takes a certain courage to bridge this gap, and to realize that the products of thought and mathematics may actually have something to do with the real world.’[93] Soon after the solution to the problem of infinities was found, there came another phase of depression: in fact, other internal inconsistencies of the discipline were related to weak interactions (they cause the radioactive beta decay of nuclei), which could not be described by a renormalizable field theory, and strong interactions (such interactions hold atomic nuclei together against the electrostatic repulsion of the protons they contain) for which there was no way to use QFT ‘to derive reliable quantitative predictions and to test if it were true.’ [94] As Weinberg said, these operational limits led to a ‘widespread disenchantment‘ with quantum field theory in the early 1950s. In spite of that, thanks to the work of many physicists QFT regained popularity in the subsequent decades: some of the problems were surmounted by gauge theories, which are quantum field theories of weak and strong interactions, and are not subject to the old problems of nonrenormalizability and incalculability. ‘The essence of the new theories – Weinberg says – is that the weak and the strong interactions are described in a way that is almost identical to the successful older quantum field theory of electromagnetic interactions. Just as electromagnetic interactions among charged particles are produced by the exchange of photons, so the weak interactions are produced by the exchange of particles called intermediate vector bosons and the strong interactions by the exchange of other particles called gluons. […] This picture represents a nearly complete triumph of the field over the particle view of matter: the fundamental entities are the quark and gluon fields which do not correspond to any particles that can be observed even in principle, whereas the observed strongly interacting particles are not elementary at all, but are mere consequences of an underlying quantum field theory.’ [95]

Weinberg closed his 1977 article with a glimpse into the future – ‘there are hopes of a unified gauge theory of weak, electromagnetic, and strong interactions’ –[96] which ended up like sort of premonition: it was not long after the article was published in the Journal Daedalus that physicists found a way to make those hopes come true. Glashow, Weinberg himself and Salam delivered a model for the electroweak unification. This model was then combined with results and discoveries concerning strong interactions to form the current Standard Model of Particle Physics and Interactions. This unified model, implemented by quantum field theory, deals with electromagnetic forces, weak and strong nuclear interactions, which mediate the dynamics of the known subatomic particles; the only exception regards gravitational forces: they are not included (the explanation of gravity falls within the domain of General Relativity). Then the present programme of physics is to find what is sometimes called a Theory of Everything, an all-encompassing model which can explain all of the physical phenomena of the Universe, from micro to macro scale. Thereby, such unifying theory will be able to include and supersede General Relativity and the Standard Model implemented by QFT. Among the present researches that are trying to reach for such a Unified Theory, we find String Theory and Loop Quantum Theory. Again the placial/spatial understanding of reality is susceptible to further analysis and modifications.


[1] The concept of field in Physics has an old tradition that goes far back in time to Faraday and Maxwell, but I specifically refer to the most up-to-date scientific versions of that concept, as I’m going to show by referring to the article by the American theoretical physicist Steven Weinberg.

[2] Max Jammer, Concepts of Space, The History of Theories of Space in Physics (New York: Dover Publications, Inc., 1993), 130.

[3] Ibid., 132.

[4] Ibid., 132.

[5] Ibid., 134, 135.

[6] Ibid., 135.

[7] Ibid., 135, 136.

[8] Ibid., 136.

[9] Ibid., 139.

[10] This is the extended quotation reported by Jammer, which I partially translated from the French: ‘Nous ne prendrons point de parti sur la question de l’espace; on peut voir, partout ce qui a ete dit au mot Etemens des Sciences, combien cette question obscure est inutile a la Geometrie & a la Physique‘, 139, 140.

[11] Ibid., 140.

[12] Ibid., 140, 141. An inertial system is a system of reference built on real ‘free-force bodies’ – see Barbour’s ‘The Discovery of Dynamics‘ for an extended technical explanation.

[13] Ibid., 141.

[14] Ibid., 142.

[15] Ibid., 142.

[16] Ibid., 143.

[17] This is the translation from the German phrase reported by Jammer on page 143 of Concepts of Space; the translation is taken from Ernst Mach, The Science of Mechanics – A Critical and Historical Account of Its Development (London: The Open Court Publishing Co., 1919), 229.

[18] The German phrase reported by Jammer on page 143 and translated into English is taken from Mach’s ‘The Science of Mechanics’, 542, 543.

[19] M. Jammer, The Concepts of Space, 143.

[20] Ibid., 143, 144.

[21] Ibid., 144.

[22] Ibid., 145.

[23] Specifically, we read from Jammer that Saccheri’s work had a great influence on subsequent investigations into the nature of the fifth postulate. The problem attracted many mathematicians, among them Gauss – ‘who seems to have recognized the logical possibility of a nonEuclidean geometry even before Lobachevski and Bolyai came out with their sensational discoveries- and Klein, – who succeeded in showing that […] non-Euclidean geometry is certainly as consistent as Euclidean geometry. So Euclidean geometry stood as one system among others with no privileged position, at least from the point of view of logic’, 146.

[24] Ibid., 147.

[25] Ibid., 150.

[26] Ibid., 151.

[27] Ibid., 162.

[28] Ibid., 162.

[29] Ibid., 164, 165.

[30] Ibid., 173.

[31] Corollary V in Newton’s Scholium, see Julian Barbour’s The discovery of Dynamics, pages 32, 577, 630.

[32] See The Feynman Lectures on  Physics, Vol. I, pp. 15-1/15-4

[33] M. Jammer, Concepts of Space, 173, 174.

[34] ‘That is what place is: the first unchangeable limit (peras) of that which surrounds’, see Edward S. Casey, The Fate of  Place, 55.

[35] M. Jammer, Concepts of Space, 175.

[36] Ibid., 174.

[37] Ibid., 176.

[38] Ibid., 176.

[39] Ibid., 177.

[40] Ibid., 182.

[41] Ibid., 214.

[42] Ibid., 186.

[43] Ibid., 187-188.

[44] Ibid., 188.

[45] Ibid., 189.

[46] Ibid., 189.

[47] Ibid., p. 190.

[48] Ibid., 190.

[49] Ibid., 192.

[50] Ibid., 194.

[51] Ibid., 197.

[52] Ibid., 197, 198.

[53] ‘space or internal place and the corporeal substance, which is contained in it are not different’ Descartes says in Principle X. For me, the same entity that ‘contains’ space, place (I do not make a distinction between internal and external place the way Descartes does) and matter, also contains time as duration (of the processes through which a material substance becomes actual). Then my definition of place contains and extends Descartes’ definitions.

[54] M. Jammer, Concepts of Space, 214.

[55] Ibid., 215.

[56] Ibid., 216.

[57] Ibid., 220, 221.

[58] Ibid., 240.

[59] Ibid., 240.

[60] Ibid., 240.

[61] Ibid., 241.

[62] Ibid., 243, 244.

[63] ‘the earliest attempt at unifying two fundamental forces, namely gravitation and electromagnetism, the only forces known prior to the development of modern nuclear physics, was made two years before the general theory of relativity was published. In 1914 Gunnar Nordstrom extended four-dimensional space-time to a five-dimensional manifold in order to unify these two forces […] The first tensorial five-dimensional theory, unifying these forces, was proposed in 1921 by Theodor Kaluza’, p. 244.

[64] ‘the superstring theory […] combines the string theory of the early Seventies with the supersymmetry between bosons and fermions. According to this theory, the fundamental objects of nature, as is well known, are one-dimensional strings whose oscillation modes correspond to quantum particles. Each term in its perturbation expansions is finite without being subjected to any renormalization, and at distances large compared to the Planck length it yields general relativity as an approximation. But in contrast to supergravity, superstring theory works only in ten dimensions. The question of how, precisely, the six additional spatial dimensions become unobservable is still a controversial issue.’, p. 247.

[65] Let’s see what Jammer precisely says at this regards: ‘With the discovery of the weak and the strong interactions in the Thirties the Kaluza-Klein attempt at unification of the gravitational and electromagnetic forces lost much of its attraction, and physicists focused their interest on exploring the nature of the newly discovered forces. […] the unification program […] culminated in the remarkably successful Glashow Salam-Weinberg unified theory of the weak and electromagnetic interactions. This breakthrough, in its turn, encouraged the search for a further unification with quantum chromodynamics, which describes the strong nuclear force, and resulted in the construction of several versions of “grand unified theories”. Finally, the ambitious attempt to incorporate gravitation as well began in the Seventies […]. From the outset it was clear that such a program requires additional dimensions, the exact number of which depends on which version of the grand unified theory is chosen. This arbitrariness is constrained in the theory of supergravity, an extension of general relativity […] which restricts on algebraic grounds the number of dimensions to a maximum of eleven.’, 245, 246.

[66] ‘In quantum gravity, recent investigations […] indicate that the fluctuations of the space-time foam are so violent that they require a new quantum topology according to which at the scale of the Planck length there are regions of different dimensionalities and even regions to which a dimensionality cannot be assigned at all. Certain quantum-gravity arguments about the small-scale topology of space also challenge the traditional, and presumably indisputable, assumption that the dimensionality of space is an integer.’, 247.

[67] Julian Barbour, in The Discovery of Dynamics, asked: ‘is motion absolute or relative? No definite answer to this question can be given’, p. 18; or, again: ‘conclusions… are always liable to radical revision when new objective relationships are discovered in the world’, p. 46

[68] Ibid., 173.

[69] Ibid., 251.

[70] Steven Weinberg, “The Search for Unity: Notes for a History of Quantum Field Theory”, Daedalus, Vol. 106, 1977, 17-35.

[71] Ibid., 17.

[72] Ibid., 18.

[73] Ibid., 18.

[74] Ibid., 18.

[75] Ibid., 18.

[76] Ibid., 19.

[77] Ibid., 19.

[78] Ibid., 19.

[79] Ibid., 20.

[80] Ibid., 20.

[81] Ibid., 20, 21.

[82] Ibid., 21.

[83] Ibid., 21.

[84] Ibid., 22.

[85] Ibid., 22.

[86] Ibid., 23. This means that the current vision offered by quantum mechanics consists of fields (the old concept of matter) existing in a spacetime structure (the old concept of space, or place, which includes the temporal dimension).

[87] Ibid., 23.

[88] Ibid., 23.

[89] Ibid., 24.

[90] Ibid., 24.

[91] Ibid., 24.

[92] Ibid., 27.

[93] Ibid., 30.

[94] Ibid., 31.

[95] Ibid., 32, 33.

[96] Ibid., 33.

Works Cited

Barbour, Julian B. The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories. New York: Oxford University Press Inc., 2001.

Casey, Edward S. The Fate of Place: A Philosophical History. Berkeley: University of California Press, 1997.

Einstein, Albert. “The Problem of Space, Ether, and the Field in Physics”. In Ideas and Opinions. New York: Crown Publishers, Inc., 1954.

Feynman, Richard P., Leighton Robert B., Sands Matthew. The Feynman Lectures on Physics, New Millennium Edition, Volume I. New York: Basic Books, 2010.

Jammer, Max. Concepts of Space – The History of Theories of Space in Physics. New York: Dover Publications, Inc., 1993.

Kostro, Ludwik. Einstein and the Ether. Montreal: Apeiron, 2000.

Mach, Ernst. The Science of Mechanics, A Critical and Historical Account of Its Development. London: The Open Court Publishing Co., 1919.

Thorne, Kip S. Black Holes and Time Warps: Einstein’s Outrageous Legacy, W. W. Norton & Company, Inc., 1994.

Weinberg, Steven. “The Search for Unity: Notes for a History of Quantum Field Theory”, Daedalus, Vol. 106, 1977.

Image Credits

Featured Image from the article: Gravitational Waves Should Permanently Distort Spacetime; by Alfred Paseika on

Image 3 from Kip S. Thorne, Black Holes and Time Warps: Einstein’s Outrageous Legacy, W. W. Norton & Company, Inc., 1994.

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