Causality, Measurement, and Space

Michael Miller

A Journal for Western Man-- Issue XXX-- January 21, 2005


The eclipse of causality in 20th Century thought is one of the leading characteristics of this Dim Age. A revolt against causality began with influential 18th Century philosophers, notably Hume and Kant. The revolt grew throughout the 19th Century and, in the late 19th and early 20th Century it reached physics, where it gave rise to the two central theories of 20th Century physics: relativity and quantum mechanics. 

This may raise some hackles; for while quantum mechanics' disdain for causality is not the least controversial, relativity is usually regarded as a causal theory, a haven of sanity compared to quantum mechanics. Unlike quantum mechanickers, relativists don't crusade against causality; indeed, they occasionally appeal to it. Relativity's sins against causality are more subtle, but no less devastating. 

Causality and Measurement

Relativity's link to the revolt against causality is the movement called positivism, as exemplified in physics by Ernst Mach (of Mach number fame). Mach's inspiration was the French philosopher Auguste Comte, who originated and named positivism. (Comte was also busy in ethics; he coined the term "altruism.") Mach's influence on physics in general, and on Einstein in particular, is well known. 

Positivism rejects causality. According to positivism, causal thinking is merely a pre-scientific relic. Modern science is mathematical; its results are expressed in equations. But these equations merely report what is observed, namely that certain quantities are equal; they say nothing about causes. Since equations are the whole content of modern science, and since they are merely descriptions, and not causal explanations, modern science need not be concerned with causes. Science has advanced beyond causality, and may now discard it. 

This positivist argument is a classic example of the fallacy of the stolen concept. The concepts of "measurement" and "quantity" (not to mention "equation!") rest on causality. To see this, recall that we can only identify quantity by means of measurement, that measurement is a process of counting equal units, and that our only guarantee of equal units is causality. (For the logical necessity of equal units, see my essay, Time, clocks and causality, especially the section "Units, arithmetic and identity: Experimenting on children.") 

The logical requirement of equal units is truly elementary, a matter of everyday experience. For example, it would be ridiculous to measure lengths using a bar of Jello because the length of a bar of Jello would vary with a myriad of causes between measurements; it would not provide equal units. A steel tape provides a better standard because its length varies less than the length of a bar of Jello, and in a more predictable fashion, according to known causes such as tension and temperature. The speed of light in a vacuum provides an even better standard, being unaffected by the run of the mill causes which affect steel tapes and bars of Jello. The less the changes in one's standard, the more accurate one's measurements. 

An absolutely accurate standard would have to be immutable, utterly immune to causal influences: it would have to be acausal. Since nothing real is acausal, measurement requires one to discover the causes which may change one's physical standards, and to correct for them. Immutable standards are not found lying about in nature; one constructs an immutable standard by defining a physical standard according to the best causal knowledge available, and by correcting for its variations. One can validate one's measurements and identify quantities only by reference to causality. 

Causality is prior to quantity. 

This may seem an audacious conclusion, but only in theory. It reflects the universal practice of all who perform measurements, from household cooks to Bureaus of Standards. All of them ensure that their standards are immutable by eliminating as far as possible all the causes which could change them and, in the last resort, by correcting for those changes which are unavoidable. It is only this painstaking care to construct immutable standards which renders measurements meaningful. 

Contrary to the assertions of positivists, causality is essential to mathematical science. Mathematical science may not dispense with causality; all of mathematical science--all the equations, laws, theorems and so on--rests on causality! Only complete and precise understanding of the causal influences on one's measuring standards can give meaning to one's measurements. Those measurements in turn give meaning to the equations based on them. 

It is usual these days to regard a physical theory as no more than the equations it uses, but this is not true. Equations rest on and summarize a vast body of measurements; those measurements are no more valid than the standards by which they were made; and the standards are no more valid than the causal knowledge by which they were defined. That causal knowledge (or error) is the essence of a physical theory; wildly different physical theories can have a great many equations in common. 

If a measuring standard varies who-knows-how, then the quantities measured by means of it vary who-knows-how, and the equations connecting those quantities mean who-knows-what. This is precisely the bog in which relativists have mired themselves; their doctrine of curved space is symptomatic. 

Rigidity and Space

"Curved space" is a staple of 20th Century thought. Space warps are a cliche of science fiction. Generations of science students have tried to make sense of curved space, and succeeded only in warping their minds. Curved space is taken for granted among the learned; if you protest that curved space is absurd, they roll their eyes and shake their heads pityingly. 

But what the heck does "curved space" mean, and how does it measure up against the principle of immutable units? 

Instruments of geometry

Geometry seeks to measure objects. It deals with such questions as, "How big is this thing here? How far is this thing here from that thing yonder? Which is bigger: this thing here, or that thing yonder?" The answer to them all is straightforward in principle: grab a measuring stick and go find out. 

A gas, a liquid, or a bar of Jello will not make a suitable measuring stick. If your measurements are to be valid, you will need a measuring stick which doesn't shrink, expand, shiver, bend or etc. as you move it from place to place. You need a measuring stick which is immutable in these respects, one which is rigid.

Measurement requires an immutable standard, and geometrical immutability is rigidity. Rigidity is prior to geometric quantity. 

This is no new thing, as is shown by the traditional instruments of geometry: the compasses and straight-edge. The distance marked off by compasses is assumed not to change as the compasses are translated and rotated; the continued straightness of the straight edge is similarly assumed. All of classic geometry rests on the assumed rigidity of compasses and straight edges. These rigid instruments are the traditional standards of length and straightness. 

If any particular compasses and straight-edge are discovered not to be rigid, that just means they are not adequate to the task we demand of them, and we must find better ones or apply corrections as needed. This much is apparent to any beginning student struggling with slipping compasses and sagging straight-edges. The geometer assumes rigidity in proving his theorems, but the experimenter must ensure the rigidity of his standard by reference to causality. 

Given rigid standards, the theorems of geometry enable us to compare near things with far things without actually moving the things, by defining units, counting them and performing calculations. We can meaningfully calculate only because our units are rigid. 

Unfortunately, the classic system of geometry formulated in Alexandria by Euclid (fl. ca. 300 BC) is silent on the issue of rigidity. Thereby hangs a tale. 

Axiom of parallels

Euclid succeeded in deriving the geometrical knowledge of his day from five geometrical axioms and five "common notions." (A common notion is a principle common to all sciences, e.g., "if equals be added to equals, the results are equal," and "the whole is greater than a part.") The manifest success of Euclid's system in fields ranging from carpentry to astronomy testifies that there are no contradictions among Euclid's axioms, but it has never been clear whether all the axioms were necessary to the system. Could one of them be discarded, derived from the other axioms, or replaced with a simpler axiom? 

Euclid's first four axioms are simple and obvious, so attention focused on his fifth axiom: the axiom of parallels. There are various formulations of Euclid's fifth, but one of the simplest states that precisely one straight line can be drawn parallel to a given straight line, through any point not on the given line. 

There are a number of problems with this axiom. It is messy--its self-evidence is not obvious. Worse, it is negative; it refers to what will not be found where you can never look--i.e., that two lines will not meet at infinity. 

Thinkers have long been queasy about Euclid's fifth axiom. There is clear evidence that issues involved in the axiom of parallels were discussed in Aristotle's school. (This evidence was collected in a Scientific American article, published in the 1970s or 80s, on non-Euclidean geometry in Aristotle. I would welcome the exact reference.) Euclid was probably connected with Aristotle's school, and he himself had qualms about the axiom of parallels, as shown by the fact that he used it sparingly; he proved his first 28 propositions without invoking it. 

Euclidean or non-Euclidean?

In the 18th Century, some geometers set out to test the axiom of parallels by denying it and checking if a contradiction would result. The axiom of parallels can be denied in two different ways, either by saying that no parallel to the given line can be drawn, or by saying that more than one parallel can be drawn. In the 19th century it was found that both denials led to internally consistent non-Euclidean systems when combined with the other Euclidean axioms. 

It was soon recognized that two-dimensional non-Euclidean systems applied to curved surfaces rather than planes--provided one was willing to accept the violence this did to the idea of a straight line. But what of the three-dimensional case? Could it be that real, observed, physical space was non-Euclidean, and therefore in some sense curved? Since the three (kinds of) systems contradicted one another, they could not all be true. A decision between them was required, but on what basis? 

Mathematicians (under the influence of bad philosophy) had come to regard coherence, or logical consistency, as the standard of geometrical truth. But for this problem, coherence was apparently a non-starter: in terms of the accepted axioms, all the systems were equally coherent. This was taken to mean that logic could not decide the issue. 

Furthermore, measurement could not decide between systems. Euclid's system was obviously consistent with all measurements that had been made, but to exclude all non-Euclidean geometries would require infinitely precise measurements, and infinitely precise measurements are not to be had. With logic and measurement ruled out, a consensus grew (egged on by philosophical irrationalism) that any decision between systems would be arbitrary. 

There was progress when Gauss (1777-1855) discovered the intrinsic geometry of surfaces. He found that the shape and curvature of a surface (in Euclidean space) could be discovered by measurements of length made entirely in that surface. For example, surveyors on the Earth can deduce from their surface measurements that the Earth is approximately spherical. This showed that questions of straightness or curvature can be settled in some contexts by an appeal to a standard of length. 

Riemann (1826-1866) generalized Gauss' result to show that a "curvature" of higher dimensional spaces could be defined by measurements made solely within them. He found that these measurements characterized the many-parallel class of non-Euclidean geometries as an infinity of spaces of constant negative curvature, the no-parallel class as an infinity of spaces of constant positive curvature, and Euclidean geometry as a unique space of constant zero curvature. This neatly displayed all the contending systems on a continuum of curvature, but it raised the daunting prospect of yet more infinities of non-Euclidean spaces of non-constant curvature! 

Apparently Riemann had only compounded the muddle by introducing new infinities of contending geometries. In fact, he had all but settled the issue; all he lacked was the philosophical principle of immutable units. 

Euclid wins!

How does Riemann's work enable us to select the correct system of geometry? 

I cannot resist remarking that when--amid infinities of candidate geometries with curvatures which range from minus infinity to plus infinity--and amid further infinities of candidates with curvatures which vary in more ways than you can imagine--there is precisely one, single, unique system with a constant curvature of precisely zero--then it might as well be emblazoned with flashing neon signs and loudspeakers blaring out, "Pick me! Pick me!" But this is not my argument. 

Riemann's measurements are based on "metrics." A metric is a certain function of position that Riemann was able to define for each of the candidate systems, and which is that system's standard of measurement. In the Euclidean case, the metric is independent of position and yields ordinary length. In non-Euclidean cases, the metric varies with position. It is precisely this variation which expresses the "curvature" of the space; a space is "curved" only if its metric varies with position. 

If we now appeal to the principle of rigid units, this settles the issue of flat vs. curved space! For we have just seen that curved spaces have standards of length which change with mere change of position, and the principle of rigid units forbids this! I.e., curved space diddles the standard of length, and so must be rejected. The rejection of curved space is neither arbitrary nor dependent on physical measurement: it is logical. 

Only in Euclidean geometry is there a standard of length which does not change with position, i.e., which is rigid. Flat, Euclidean space is the system of geometrical measurements made with rigid units. 

Curved space is simply a system of measurements made with squidgy units. (E.g., Jello measuring sticks.) That is the trivial secret behind the gaudy curtain of curved space theories. Despite the mighty theatrics, there is nothing behind the curtain but a dishonest little man who refuses to admit that he fudged the units. 

Absolute length

Riemann's work serves as a reductio ad absurdum of curved space, but it is much more useful than that. Once you grasp that curved space implies variations in your measuring sticks, you can recognize Euclidean geometry as a logical standard for judging real measuring sticks. 

If your measurements show space to be "curved," you can validly deduce that your measuring sticks are not rigid--even if you have no other evidence of their variations. Then you can ask, "What causes them to vary?" By adopting the logical standard of Euclidean space you can discover causes which a curved space theory would whisk behind the curtain of "space curvature." 

Furthermore, there is a mathematical transformation which reduces a general Riemannian system to a Euclidean one. The transformation to Euclidean geometry reveals the variations in the measuring sticks, and the lengths in the Euclidean system are lengths. In other words, we can extract knowledge of absolute lengths from our measurements even if our physical measuring sticks are not rigid! 

This transformation is a precise mathematical form of the common sense thought that we don't need absolutely rigid sticks, provided we can figure out how they vary. It tells us that we can figure out how they vary, so that we can make absolute length measurements. 

Notice the logical order involved. First, we must demand rigid units. Then we can select Euclidean geometry as the only system in which there are rigid units. Then we can identify deviations from Euclidean geometry as signs of systematic errors in our measurement standard. Then we can discover and correct for such errors. The principle of rigid units is the keen sword that slashes the Gordian knot of curved space. 

Just as we can cite contradiction as proof of an error in logic, and as we can cite dilatory time as proof of an error in time measurement, so we can cite curved space as proof of an error in length measurement. (For the case of absolute time, see Time, Clocks and Causality.

We can surmise that variations in other standards will reveal themselves by similar signs. As we identify such signs for one standard after another, we will demonstrate the absolute nature of measurement in field after field--in terms specific to each field. Our warrant in each case will be the laws of identity and causality, and the principle of immutable units. 

Axiom of rigidity, or a new common notion?

It is now clear that the axiom of parallels can be replaced. By insisting on a rigid standard of length, we single out precisely that set of theorems which was derived by means of the axiom of parallels. In other words, the non-obvious, negative axiom of parallels can be demoted to a theorem, and replaced by an axiom of rigidity. 

But an axiom of rigidity would simply be an application to geometry of the principle of immutable standards which is common to all measurement. Therefore it would be better to make this explicit by invoking the principle of immutability as a new common notion. 

Code breaking

Where does all this leave curved space theories? It leaves them with neither an epistemological nor a geometrical leg to stand on! 

Relativity has encoded (encrypted!) its physical content in terms of curvaceous space and dilatory time. This procedure is not merely odd, but flat out wrong--as wrong as constructing a theory out of contradictions. Freeing modern physics from the 20th century hash of syncopated clocks, Jello compasses and squirming straight-edges will be a massive job, but we can leave it to physicists. 

The rest of us can heave a sigh of relief as we abandon the hopeless task of trying to imagine curved space. And the next time someone tries to sell us an option on curved space, we can roll our eyes and shake our heads pityingly! We can also use our new understanding of space to draw some long overdue distinctions. 

Space, Void and Vacuum

Space, void and vacuum are usually regarded as synonyms for emptiness, but this will not be an empty discussion. 

Quite the opposite! By drawing distinctions between space, void and vacuum, we are able to re-affirm that reality is full, that it is a plenum. This re-affirmation is no mere philosophical nicety, of concern only to those who take an eccentric interest in obscure issues of metaphysics. It points to a neglected physical reality of fathomless importance for human knowledge and action. 

Space: a concept of method

Space is often supposed to be a sort of box in which existence is placed, or a sort of insubstantial stage on which the drama of reality unfolds. These notions have the fatal flaw of making space into something prior to existence or apart from existence. But nothing exists apart from existence, so these notions boil down to the idea that space is simply emptiness, non-existence. 

Men tend to think of space as a box because they think of it as metaphysical--as something which is intrinsic, which exists independently of consciousness. We draw a firm distinction between space and the entities in space, from yon cat to stars and galaxies. We distinguish space from all the "stuffs" which may "fill" space. But if space is regarded as metaphysical, these distinctions leave it with nothing whatever that it can be but a shadowy, unperceived box or stage. 

The puzzle is instantly resolved by recognizing that space is not metaphysical, but epistemological. Space is a product of human method just as numbers and concepts are products of human method. Like them, space does not exist independently of consciousness; like them it is neither arbitrary nor intrinsic: it is objective.

Space is a grid of reference lines which we imagine to be constructed according to geometrical method. We imagine these lines to run through reality in the same way that we imagine lines of latitude and longitude to overlay the globe of a planet. We construct them to help us to visualize geometrical measurements. 

One indication that space is a concept of geometrical method is that it grew up with geometry. In Aristotle's pre-Euclidean time, geometry was relatively new; and Aristotle is silent on the subject of space. Instead, he speaks of "place," which he defines in terms of bodies: "the innermost boundary of the containing body." (Phys., IV, 4) Only after geometry had won men's confidence did they boldly extend their reference lines through the entire universe. The concept of universal space was the result, as was our modern ability to define "place" in terms of our reference lines, and thus explicitly to relate all parts of the universe to each other. 

Another indication that space is a concept of method is the fact that the controversy between flat and curved space is a controversy over method; should geometry employ rigid units or squidgy units? If you employ rigid units, then you construct a flat space. If you use squidgy units that vary with position, you construct a curved space. The allegedly unknowable changes in your units are revealed by the flatness or curvature of the resulting space. (See above.) 

Space is an epistemological construction, a product of human method. 

This lets us solve the puzzle that although every fragment, scrap and particle of the universe is in space as in a box, yet the universe itself is not in space. The reason is that we draw our boxes in the universe. The universe is not in space, space is in the universe. 

Our reference lines do not affect real things; they are, after all, imaginary. Lines of latitude and longitude cannot trip you; you cannot stumble over the equator. Even if you resort to squidgy units and take pains to draw your lines around rocks, you still cannot stumble over your lines, but only over the rocks. Space is not a cause because space is not an entity. 

It is idle to debate the infinity of the universe by appealing to concepts of space. Some curved spaces are indeed finite; but that depends on your choice of units, and your choice of units cannot dictate to reality. Indeed, those who employ a finite space must face the jibe that their space is too small for reality, leaving some parts unmeasured! 

Flat space on the other hand cannot be too small for reality. When you use rigid units, you can extend your grid as far as the universe extends--and the universe is boundless. Existence exists everywhere, and flat space enables us to extend our lines everywhere. Everywhere is somewhere! 

Employing flat space, you can draw your lines in any directions you wish, passing through any points you wish, because flat space is based on rigid bodies, and they are rigid under translation and rotation. 

Flat space is thoroughly and completely acausal. No space is a cause, but in addition, flat space is not subject to causes. Nothing whatever can curve, bend, deflect or tangle our flat space reference lines, for the simple reason that we exclude causes by employing rigid units. Flat space is utterly acausal; we know it is, because that's the way we made it. 

Our use of acausal, flat space does not hamper our discovery of real causes: it makes it possible. Curved space can hide causes behind the squidginess of its units, but flat space is the incarnation of a null hypothesis. It embodies an assumption of rigidity, of acausality. Any discrepancy between geometrical knowledge of rigid bodies and physical measurements of a real body implies that the real body is not rigid. That's our cue to ask, "What caused that?" 

Void: an epistemological error

"Very well," you may ask, "if space is our visualization of geometry, what are we to call a place from which everything has been removed?" The name for this notion is "void," but like the unicorn, there ain't no such animal! 

If everything were to be removed, what would be left would be nothing. As Parmenides pointed out about 2500 years ago, and as Ayn Rand reminded us more recently, there is no nothing. To say that a void exists is to say that there is a place where non-existence nevertheless exists. Void is absurd--an epistemological error, a figment. There is something everywhere; reality is full. It has no "gaps." This conclusion has puzzled thinkers since ancient times, and their struggles are instructive. 

If reality is full, how can we see gaps all over the place? To perceive a number of entities is to perceive that they are separate; to see this cat and yon dog is to see a gap between them. 

Faced with this, Parmenides himself lapsed into collectivism and rationalism: he declared that there are no separate entities, that our senses deceive us; there is only a mystic unity: The One. 

Ancient atomists sought to preserve individuality and the evidence of perception by ditching Parmenides' axiom: they declared that everything is made up of atoms and the void, and that void--non-existence--exists every bit as much as the atoms! Their desperate expedient was doomed from the start, for a trivial exercise in logic will extract from it the same rationalistic, collectivist conclusion: "Void is nothing, and void separates the atoms; so nothing separates the atoms. So all is One, and individuality is mere sensory illusion." Atomists raised the specter of a real void, and it has haunted the outer reaches of science ever since. 

Parmenides and the atomists share the error that perceptual gaps are voids. They differ only in the way they use the error. Parmenides says, contrary to perception, that gaps do not exist; because voids do not exist, and gaps are voids. The atomists insist, contrary to the axiom of existence, that void exists; because gaps exist, and gaps are voids. 

The solution is to admit--on the warrant of perception--that perceptual gaps exist, and--on the warrant of the axiom of existence--that gaps are not nothing: something exists between perceived entities. 

What is it? Void is not an option, and space is no answer. Space is merely our system of reference lines. Our new question is "What is the stuff through which we draw those lines between entities?" This stuff is prior to our lines, prior to space. What's the stuff? 

Rationalists may as well leave right now, for this question cannot be answered by deduction; we have no premises from which the deduction could proceed. The only positive fact we know about our "stuff" is that it exists, and you cannot deduce what a thing is from the premise that it is something. To learn more, you must observe more. 

Bricks, to air, to vacuum, to ...?

Suppose we observe a cat and a dog on opposite sides of a brick fence. What is between them? Obviously there are bricks between them. That's no problem for anyone: we all know that bricks exist. 

But if we remove the bricks from between our critters, they don't merge into the mystic unity of The One. They are still apart; they are still distinct entities. Now what's between them? 

You might hazard the suggestion that there is air between them. Congratulations! That recognition marks a great and difficult advance of science. The existence of air was not always obvious; as late as Alexandrian times, experiments demonstrating the reality of air were thought to be necessary. After all, air is shapeless, colorless, invisible, non-dog, non-cat, non-etc. If you focused on these negatives, you would be led to think that air is mere void. But you would be wrong; there is air between our cat and dog; air exists. 

Gradually, by further observation and experiment, man learned that air is not a fundamental, elemental constituent of reality. Rather, air is made of entities: air is a mixture of molecules, which are made up of atoms, which are made up of subatomic particles. But those particles aren't merged into mystic unity; there is something between them. What is it? 

In the context of knowledge sketched above, the answer is vacuum, or if you prefer, ether. The description of vacuum involves even more negatives than the description of air; but no list of negatives, however long, can justify the conclusion that vacuum is void: void is a mere figment. To the contrary, we have positive evidence for the existence of vacuum, namely, the separateness of particles. 

There is vacuum between particles: vacuum exists.

If we look ahead to a hypothetical future, it may turn out that vacuum, too, is made up of some kind of entities. Then the axiom of existence will oblige future scientists to ask what exists between those entities. Or, if future scientists find they can remove even vacuum from a vessel, then the axiom of existence will oblige them to ask what exists between the walls of the vessel. Or, perhaps, vacuum will turn out to be elemental, a primary constituent of reality. Only further evidence can decide the issue. 


What can we say about vacuum? Not much, but some. Vacuum transmits electrical and magnetic forces with a time delay which depends on distance. Vacuum transmits gravitational force. If one assumes that gravitational force travels through vacuum at the speed of light and is aberrated like light, one arrives at the correct orbit for the fast-moving planet Mercury. (Paul Gerber published this calculation in 1898. See Petr Beckmann's "Einstein Plus Two," Sec. 3.1) Vacuum transmits light and, near massive bodies, it deflects light. Certain kinds of clocks run more slowly as they move faster through vacuum. Particle masses increase with their speed through vacuum. 

These facts are all certified by uncontroversial experiment. They are conventionally "explained" in terms of relativistic space-time curvature, but such explanations are worthless. Curvaceous space and dilatory time are means to delude yourself by using squidgy measuring sticks and inconstant clocks. What delusion might one seek by means of variable units? Relativists choose their variable units to maintain the delusion that vacuum does not exist. 

I'm not guessing about this; it is implicit in their procedure. They begin by denying a real vacuum--in their jargon, an "ether" or "preferred reference frame"--and they derive units which vary precisely as required by that dogma. Their fudged units obediently conceal much of the evidence for vacuum. (The evidence that they have fudged the units--namely, their curved space and inconstant time--remain manifest to all who look.) Relativists' denial of vacuum revives the irrational metaphysics of the ancient atomists, for it amounts to the assertion that non-existence exists between particles. 

We can now understand why relativists must postulate the speed of light in vacuum to be a universal constant. They equate vacuum with void; and if vacuum were nothing, there would indeed be nothing which could change the speed of light in vacuum! Relativists have no grounds to be smug about this fragment of consistency, for it comes at a terrible price: it banishes reason and causality from physics. 

Just as a void would be unable to cause any change in the speed of light, it could not cause light to have a speed, and certainly could not cause it to have one particular speed rather than another. Instead of regarding the motion of light in vacuum as an experimental fact to be explained by its causes, relativists must regard it as a metaphysical miracle, forever and in principle inexplicable: absolutely causeless. Indeed, light itself and all forces between particles become miraculous; for they would have to propagate through a void, i.e., through nothing at all! To equate vacuum with void is to spawn an endless torrent of contradictions, for void is itself a contradiction. 

Back in reality, variations in the speed of light in material media are commonplace; they cause the everyday effect of refraction. Refraction between air and glass or plastic makes eyeglasses work, and refraction between air and water makes a stick which is partially immersed in water appear sharply bent at the water surface. Refraction between warmer and cooler air causes the "heat waves" that you can see over a paved road on a sunny day. 

There is positive evidence that vacuum (or ether) is pretty much like any other medium, being affected in specific ways by specific causes. For example, starlight passing near the sun is deflected from its usual course. The straightforward conclusion is that vacuum is a refracting medium, i.e., that the speed of light in vacuum is reduced near massive bodies. 

It's high time for physicists to expel the void from their minds, and to admit that vacuum exists. They will then be free to standardize their measuring sticks, to steady their clocks, and to use these tools to study vacuum. 

There's no telling what they'll find! The potential of vacuum studies for human progress and prosperity is boundless. Vacuum occupies most of the volume of the universe, and even most of the volume of every atom of ordinary matter. If men can devise methods to make this ubiquitous stuff (or stuffs!) serve human purposes, what might they achieve! 

(C) 1999, Michael Miller.

Michael Miller is an engineer and Objectivist filosofer with thirty years of experience. He had been a member of Boycott Alberta Medicare in 1969 and of the Association to Defend Property Rights from 1973 on. He writes in-depth philosophical theory at his publication, Quackgrass Press, which can be accessed at

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