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In the late nineteenth century the mathematician George Cantor dedicated his studies to the mathematical concepts of the infinite.

Woods says:

Thus, after Cantor, there can be no argument about the central place of the infinite in mathematics… Yet despite all the evidence, many modern mathematicians persist in denying the objectivity of infinity. (p358)

How can this perversity of modern mathematics be resolved? Why, if there was to be no argument about the objective existence of infinity, did argument persist “despite all the evidence”? And what evidence? The answer is to be found in discovering what Cantor actually showed. Cantor says that the infinite arises:

First when it is realised in the most complete form, in a fully independent other-worldly being, in Deo [in god] where I call it the Absolute… second when it occurs in the contingent, created world; third… as a mathematical magnitude…

I wish to make a sharp contrast between the Absolute and the transfinite, that is the actual infinities of the last two sorts, which are clearly limited, subject to further increase, and thus related to the finite. (Quoted by Barrow, The Infinite Book, p94)

In Cantor’s mathematics, only infinity in god coincides with Woods’ idea of the objectivity of infinity – an objectively existing infinite god. And indeed there was “no argument” about it.

Next: Einstein and the end of Newtonian absolute space and time

Engels’ materialism

With each epoch-making discovery even in the sphere of natural science [materialism] has to change its form… (Engels, Ludwig Feuerbach and the Outcome of Classical German Philosophy, Selected Works, p597)

Since Einstein, quantum mechanics, and the Big Bang theory, materialism has to change its form. This means dialectical materialism too. We must remove old dogmas and begin again, starting from the recognition that the method of dialectical materialism at base is simply materialism represented dialectically.

Engels’ statement also applies in a general sense to our former concepts of space, time and the infinite universe. But by contrast, Woods criticises specific scientific theories of space and time for departing from the Newtonian concept of time and space as embraced by Engels.

If it appeared to materialists in the 19^th Century, albeit without evidence, that the world must be infinite, the epoch-making discoveries we will shortly discuss have forced a change. These discoveries may not be the final word, but there is no going back to the old dogmas, to the Newtonian outlook.

Dialectical materialism takes from Hegel its “death blow to the finality of all products of human thought and action” as Engels puts it. It is simply wrong, purely from a dialectical standpoint, let alone the current scientific standpoint, to state as if it were an incontrovertible fact that, “Dialectical materialism conceives of the universe as infinite” as Woods does. (p189)

Infinite space and time

It may be argued that Engels’ works set into a kind of Marxist orthodoxy the Newtonian science of his time. In general, Engels was defending and exemplifying, quite correctly, a materialist approach to the world against those such as Eugen Dühring who were seeking to take the socialist movement back to an idealist, eclectic outlook. Engels necessarily proceeded on the basis of those current scientific ideas that appeared to be beyond question.

Woods, on the other hand, defends Engels’ point of view not for its inherent respect for scientific validity, its materialism, but as if Engels was guided by some higher principle (to which Woods elevates dialectical materialism), which in some way lifted the Marxist method above that of scientific investigation. He transforms Engels’ viewpoint into dogma. Woods says Reason in Revolt defends “the fundamental ideas of the movement” in his 2006 obituary of Ted Grant (cited as co-author of Reason in Revolt). In truth, Reason in Revolt defends Engels’ support for Newtonian physics somewhat as others defend the literal truth of The Bible, not as an expression of the Marxist method of historical materialism rooted in a particular epoch, a theoretical method that has to be continually re-evaluated as science advances our understanding of our world, but as a finished philosophical doctrine that literally serves for all time.

Engels’ assumptions of infinite space and time in Dialectics of Nature should be considered in the light of his more detailed, systematic and repeated embracing of the Kantian theory of the origin of all existing celestial bodies (for example in Anti-Dühring and Feuerbach), which Engels, at one point called, “the theory of the origin of the universe” (Dialectics of Nature, p273), and his endorsement of Hegel’s rejection of the spurious infinite. These indicate a method, while Engels’ assumption of infinite time and space merely represents the form, the science of the day. We should also keep in mind that most of Dialectics of Nature consists of unfinished and unrevised drafts and was abandoned unpublished in Engels lifetime, although the introduction appears complete, and the fragment of the chapter entitled Dialectics is invaluable in demystifying Hegel’s dialectics and making it accessible.

When Woods takes Engels as his authority for what he regards as dialectical materialism, it is doubly misleading, not just because it misrepresents Engels’ methodology, but also because Engels does not make the crude mistakes that Woods does in relation to the concept of an infinite universe. When Engels discusses infinity in space and time in Anti-Dühring, he talks about a “process” which is “unrolling” in the context of an infinity which is “composed of nothing but finites”.

Singularities, “Bad” infinity, and the beginning of the universe

Engels writes:

For that matter, Herr Dühring will never succeed in conceiving real infinity without contradiction. Infinity is a contradiction, and is full of contradictions. From the outset it is a contradiction that an infinity is composed of nothing but finites, and yet this is the case.

Engels reflects Hegel’s dialectic of the infinite – that the infinite is comprised of nothing but finites, for which concept Hegel himself credits Aristotle.

Woods misrepresents Engels, turning his position upside down, when discussing Hegel’s “bad” infinity. For those for whom this may be a matter of some contention, we will pause for a moments’ discussion. In the previous chapter, Hegel on the dialectics of infinity, we noted that the terms “bad infinity” (used by Engels), and “spurious infinity”, are references to the same Hegelian concept. Engels speaks of Hegel’s “bad” infinity in the Moscow translation of  Anti-Dühring whereas we find that the Miller edition of Hegel’s Science of Logic uses the term the “spurious” infinite to represent the same thing. We discussed what Hegel meant by the term.

In the course of his criticism of Herr Dühring, Engels says that:

Infinity – which Hegel calls bad infinity – is attributed to being, also in accordance with Hegel (Encyclopaedia, § 93), and then this infinity is investigated. (Anti-Dühring, p61)

What is perhaps self-evident to Engels and his contemporaries in the above statement, but less than evident to his modern reader, is that when Engels says that “bad” infinity is “attributed to being” in the above quote, he here refers in particular to Dühring’s concept of the primordial beginning of the universe, discussed in his section on ‘World Schematism’. This is a concept which Dühring had plagiarised from Hegel, and which both termed “Being”.

When Hegel appears to describe the world coming into being, at the very beginning of his Science of Logic, he begins with: “Being, pure being, without any further determination”. Hegel later defines this undifferentiated pure being from which the rest of the world appears to emerge as “infinite being” and appears to be a type of pantheistic godhead. (Science of Logic, p100, § 164)

The reader might wonder how Hegel’s beginning of the world in this “undifferentiated pure being” compares to the well known theory, based on the research done by Stephen Hawking and Roger Penrose in 1970, which suggests that the universe emerged from a “singularity”. But the concept of a singularity, although widely accepted for a time, is not, or at least no longer, part of the standard model of the Big Bang. One reason for this is that there is insufficient observational evidence, but another is that the theory of singularities is plagued with infinities, and scientists are usually averse to infinities which arise in theories, treating them as errors in the equations or an indication of a false theory. The standard Big Bang theory is based on well established evidence for a hot dense origin to the universe, by extrapolating the observed universe back in time, but it cannot extrapolate back as far as a singularity. Hawking himself went on to study the possibility that quantum mechanical effects would operate at very small sizes, so that no singularity would arise.

Strangely, Woods considers singularities “extremely unlikely”, despite the fact that if a singularity existed which had achieved infinite density it might be considered proof of the existence of something which was infinite.

Engels takes pleasure in showing that the anti-Marxist Dühring has plagiarised Hegel, including his concept of the world beginning with an undifferentiated essence. In the idealist Hegel’s system, this undifferentiated being is subject to a somewhat vague dialectical process, and develops into a more “determinate” being – that is, develops into the world as we know it. As an idealist, Hegel feels no compelling need for materialist causes to achieve this (and is, at any rate, working on many levels, least of all the physical.)

But Dühring takes at face value Hegel’s primordial “being” and then removes the dialectical concept from it. Dühring is scathingly critical of Hegel and castigates his dialectics. Dühring purports to embrace materialism, and has supposedly disproved the existence of God already – albeit, Engels points out ironically, using proofs like  those specious proofs that past philosophers used to prove the existence of God. But without any support from Hegel’s idealist dialectics, Dühring is deprived of any means to enable his second-hand unchanging undifferentiated being to change into the world as we know it, since he has said it is unchanging.

“Herr Dühring’s universe really starts with a being which lacks all inner differentiation, all motion and change” Engels says, and quotes Dühring: “…all periodical processes of nature must have had some beginning, and all differentiation, all the multifariousness of nature which appears in succession must have its roots in one self-equal state. This state may, without involving a contradiction, have existed from eternity… ” (Anti-Dühring, p58 and p62)

So here we have Dühring’s “bad” infinity – an actually existing infinity – an eternity of unchanging, undialectical existence of a “self-equal state” before the world came into being. Scientists today suppose there must be a substratum from which the Big Bang origin of the universe emerged, but current theories picture a very dynamic terrain. Dühring claims that the origin of the universe is an actual, eternally existing “self-equal state” or being. Hegel calls concepts such as the idea of infinity as an actually existing thing, the “spurious infinite”, which Engels translates as “bad” infinity. The one exception Hegel makes to the “spurious infinite” is the same exception that Aristotle makes, as we have explained – the divine. Hegel’s mysterious godhead is “infinite being”.

Engels gleefully points out the contradictions which Dühring has got himself into by plagiarising Hegel, and, as we have seen, gives a reference to Hegel’s Encyclopaedia: “Infinity – which Hegel calls bad infinity – is attributed to being, also in accordance with Hegel (Encyclopaedia, § 93), and then this infinity is investigated.” Dühring has painted himself into a tight spot. Engels refers the reader to paragraph 93 of Hegel’s Encyclopaedia (quoted below) and proceeds to show, with great delight, that Dühring’s investigation of the infinite is also plagiarised – this time from Kant.

Engels proceeds to remonstrate with Dühring’s notion that there can exist, “the counted infinite numerical series”. He challenges Dühring to do the “clever trick of counting it”. This again invokes Hegel’s spurious or bad infinity, because by a “counted” infinite series Dühring intends to indicate a finished, completed, existing infinity, a progress to infinity which is completed with the capturing, so to speak, of an actual, “counted” infinity. This is not the infinite process of counting, which is comprised only of finites. Engels can see how Dühring slips between acceptable notions of an infinite series and the bad or spurious infinity of an actually existing “self-equal state”.

The distinction, however, is subtle. Engels attacks the inconsistencies and vagaries in Dühring’s verbal gyrations around concepts of an infinite series, and these attacks could easily be mistaken for attacks on the concept of an infinite series as such, and this is what Woods does. Nevertheless, two paragraphs later, Engels plainly states that infinity is a contradiction and is “composed of nothing but finites”.

It is in these paragraphs – to add further confusion – that Engels makes a clear assertion of infinite space and time. He does so because that was the overwhelmingly accepted scientific conception of the period, and he keeps his dialectical insights in check where physical evidence is lacking. But Engels does not hesitate to call attention to the passage in Hegel’s Encyclopaedia where he calls infinite space a “barren declamation”, and modern science would tend to agree.

In his Encyclopaedia, paragraph § 93, Hegel briefly refers to the false concept of a progression towards infinity, as opposed to a process which can be infinitely repeated but which gets no nearer to infinity. He then says:

When time and space, for example, are spoken of as infinite, it is in the first place the infinite progression on which our thoughts fasten. We say, Now, This time, and then we keep continually going forwards and backwards beyond this limit. The case is the same with space, the infinity of which has formed the theme of barren declamation to astronomers with a talent for edification.

In the attempt to contemplate such an infinite, our thought, we are commonly informed, must sink exhausted. It is true indeed that we must abandon the unending contemplation, not however because the occupation is too sublime, but because it is too tedious. It is tedious to expatiate in the contemplation of this infinite progression, because the same thing is constantly recurring. We lay down a limit: then we pass it: next we have a limit once more, and so on for ever. All this is but superficial alternation, which never leaves the region of the finite behind.

To suppose that by stepping out and away into that infinity we release ourselves from the finite, is in truth but to seek the release which comes by flight. But the man who flees is not yet free: in fleeing he is still conditioned by that from which he flees.

If it be also said that the infinite is unattainable, the statement is true, but only because to the idea of infinity has been attached the circumstance of being simply and solely negative. With such empty and other-world stuff philosophy has nothing to do. What philosophy has to do with is always something concrete and in the highest sense present. (Encyclopaedia, para 94)

Hegel returns us to the concrete from false notions of an infinite progression through space and time. Hegel applies the term “spurious” or “bad” infinity not only to a notion that an infinity of space actually exists, but also to the supposedly “sublime” descriptions of a “progress” to infinity through a tedious repetition, for instance of galaxies beyond galaxies stretching to infinity. His point is that there is no such thing as an independently existing infinity and one does not make any progress towards it.

It is remarkable that Woods missed this since Engels specifically refers to it at the beginning of the chapter from which Woods quotes. In fact, Woods twice uses the same large quote from Engels to back up his assertions about his notion of infinity. (p218 and p358) But he twice stops short of the paragraph which immediately follows and casts an entirely new light on what preceded it:

For that matter, Herr Dühring will never succeed in conceiving real infinity without contradiction. Infinity is a contradiction, and is full of contradictions. From the outset it is a contradiction that an infinity is composed of nothing but finites, and yet this is the case.

The limitedness of the material world leads no less to contradictions than its unlimitedness, and every attempt to get over these contradictions leads, as we have seen, to new and worse contradictions. It is just because infinity is a contradiction that it is an infinite process, unrolling endlessly in time and in space. The removal of the contradiction would be the end of infinity. Hegel saw this quite correctly, and for that reason treated with well-merited contempt the gentlemen who subtilised over this contradiction. (Anti-Dühring, Part V, p66)

Here Engels refines his support for infinite space and time with Hegel’s concept of the infinite (as being comprised of nothing but finites) clearly in mind. This passage bears some, slightly forced, comparison with modern science, which currently calculates that the universe will expand indefinitely – an infinite process, unrolling endlessly.

Cosmology

For Engels, the “Kantian theory of the origin of all existing celestial bodies”, which we discussed in the chapter, Kant’s cosmology and Engels’ commentary, means not just that the galaxies of our universe have a history in time, coming into being at some definite time in the past, but also that they will pass away.

“Nature,” Engels emphasises, “does not just exist, but comes into being and passes away.” And he discusses exactly this passing away, basing himself on some scientific speculation of the period.

Nevertheless, ‘all that comes into being deserves to perish’… instead of the bright, warm solar system with its harmonious arrangement of members, only a cold, dead sphere will still pursue its lonely path through universal space. And what will happen to our solar system will happen sooner or later to all the other systems of our island universe; it will happen to all the other innumerable island universes, even to those the light of which will never reach the earth while there is a living human eye to receive it.

And when such a solar system has completed its life history and succumbs to the fate of all that is finite, death, what then?

Later, Engels adds:

The view is being arrived at that heavenly bodies are ultimately destined to fall into one another… so that all motion in general will have ceased. (Dialectics of Nature, p38, p50-2)

Surely, all this demonstrates that Engels would have expressed little surprise if told that our universe was discovered to have a history in time – a birth perhaps ten to twenty billion years ago? Surely he would say that Kant predicted it centuries ago?

In fact, in note form, he did: “Kant – the theory of the origin of the universe before Laplace”. (Dialectics of Nature, p273) Engels emphasises “before” because the work of Laplace provided a proof of Kant’s insight, at least in relation to the development of solar systems, now typically termed the Kant-Laplace nebular hypothesis. This note appears in the fragment headed “Büchner”, which is thought to be the first fragment or set of notes which Engels made for the composition of Dialectics of Nature, written in 1873.

But Engels has more to say on the matter. One must allow for some naturally out-dated science and scientific terminology, such as the use of the term “motion”, which was the very latest terminology at the time but was shortly to be superseded by the term “energy”. We also remember the fact that Engels predates Einstein and Hubble, both of whom gave a much more definite material meaning to time and space, matter and energy, so that they can only arise together. Making this reasonable allowance, Engels would not encounter many objections from modern scientists in seeking a substratum from which our universe emerged:

This much is certain: there was a time when the matter of our island universe had transformed a quantity of motion [i.e. energy] – of what kind we do not yet know – into heat, such that there could be developed from it the solar systems appertaining to (according to Mädler) at least twenty million stars, the gradual extinction of which is likewise certain.

How did this transformation take place? We know just as little as [the astro-physicist] Father Secchi knows whether the future caput mortuum [worthless remains] of our solar system will once again be converted into the raw material of a new solar system.

But here either we must have recourse to a creator, or we are forced to the conclusion that the incandescent raw material for the solar systems of our universe was produced in a natural way by transformations of motion which are by nature inherent in moving matter, and the conditions of which therefore also must be reproduced by matter, even if only after millions and millions of years and more or less by chance but with the necessity that is also inherent in chance. (Dialectics of Nature, p51)

Leaving aside the exact terminology of ‘matter’ and ‘motion’, most scientists would not deny the main thrust behind such speculation, which is that whatever substratum gave rise to our universe, the processes are natural and are likely to reoccur. The difference is that scientists today do not take for granted that any newly emerging universe needs to have the same physical constants, so that it may not be comprised of energy, matter, space and time in the way that our universe is.

Does not Engels’ approach show that Woods’ scheme has no history in time? That in Reason in Revolt’s cosmological speculations we find just an endless repetition: “only galaxies and more galaxies stretching out to infinity”?

And does not Engels also see that there are processes going on inside these stars? That they and the galaxies they inhabit are consuming and changing their elemental constituent atoms, so that they must have a history in time? This history is written in the current make-up of the elements (for instance hydrogen, helium, oxygen, carbon, nitrogen, silicon, calcium, and iron) found in stars, which can be detected in their light. It tells us whether they are first generation stars, or stars that are consuming in part the remnants of a previous generation of stars. Observation can in practice tell how old galaxies are. How could an infinite universe be a never-ending replication of our current state – galaxies stretching to infinity? Would this not be a rather static viewpoint?

Woods poses a challenge, which might as well be directed at Engels: “The problem is: how to get from nothing to something? If one is religiously minded, there is no problem; God created the universe from nothing.” (p210)

In his notes, Engels quotes Angelo Secchi without comment:

Secchi (p 810) himself asks: When the sun and the whole system are extinct, “are there forces in nature which can reconvert the dead system into its original state of glowing nebula and reawaken it to new life? We do not know.” (Dialectics of Nature, p367, quoting from Secchi’s book Die Sonne)

Engels’ reply in general terms is the same reply that modern science and materialist scientists generally give: “we do not know”, although there are many interesting possibilities. (Undoubtedly, there are some scientists who, despite investigating materialist reality, hold religious beliefs and in some cases support ‘creationist’ theories.) But Woods forgets that coming in to being and passing away is one of the fundamental laws of dialectics. From what substratum, we do not know, but “Nature,” says Engels, “does not just exist, but comes into being and passes away.” (Dialectics of Nature, p38) From a purely dialectical point of view, this presents no problems.

Engels shows that dialectical materialism refuses to “conceive of the universe as infinite”, except perhaps as an infinite process which is composed of the finite and, perhaps, is comprised of universes which come into being and will have an end.

But as for a cyclical universe, such as the theory that Woods endorsed in 2002, the infinite repetition of Big Bangs, Engels refutes it in advance – its authors, Steinhardt and Turok, now recognise their mistake here – when he says that:

… in the last resort, Nature works dialectically and not metaphysically; that she does not move in the eternal oneness of a perpetually recurring circle, but goes through a real historical evolution. (Socialism, Utopian and Scientific, Selected Works, p407)

Next: The infinite in mathematics

Throughout his life, the German philosopher Hegel was an enthusiastic supporter of the French Revolution of 1789. Widely thought of as the most difficult of all philosophers to understand, Hegel followed in the radical philosophical tradition begun by Kant, who established a school of philosophy called German Idealism.

Yet Hegel’s idealist philosophy, and in particular his dialectics, when placed on a materialist basis by Karl Marx and Friedrich Engels, became one of the cornerstones of Marxism. In their youth both Marx and Engels were Young Hegelians, radical opponents of the old autocracy of the German nation.

Engels comments that, “the true significance and the revolutionary character of the Hegelian philosophy [was] that it once for all dealt the death blow to the finality of all products of human thought and action.” Engels continues:

Truth lay now in the process of cognition itself, in the long historical development of science, which mounts from lower to ever higher levels of knowledge without ever reaching, by discovering so-called absolute truth, a point at which it can proceed no further, where it would have nothing more to do than to fold its hands and gaze with wonder at the absolute truth to which it had attained. (Engels, Feuerbach and the End of Classical German Philosophy, in Marx and Engels Selected Works, p588)

This alone should give Woods pause before asserting, as a statement of absolute truth, that “Dialectical materialism conceives of the universe as infinite”, folding his hands and gazing with wonder at the discovery he has made.

There are more than a few respects in which, as Engels comments, the materialist outlook penetrated into Hegel’s philosophy. “Hegel laboured to discover and demonstrate the pervading thread of development” in history, religion, the arts and the sciences, Engels writes. In doing so, “he played an epoch making role in every sphere”. The forced constructions of Hegel’s “system” are only the frame and scaffolding of his work, Engels says:

If one does not loiter here needlessly, but presses on farther into the immense building, one finds innumerable treasures which today still possess undiminished value. (Feuerbach and the End of Classical German Philosophy, Selected Works, p590) 

Hegel on Aristotle’s ‘potential’ and ‘actual’ infinity

Hegel explicitly defends Aristotle’s point of view on the infinite – that there is no “actual” infinity, only a potential infinity. Hegel says that: “The solutions propounded by Aristotle of these dialectical forms merit high praise”. (Science of Logic, p198)

Hegel criticises the seventeenth century French philosopher, Pierre Bayle, who, Hegel points out, argued that “if matter is infinitely divisible, then it actually contains an infinite number of parts… [it is] an infinite that really and actually exists.”

“On the contrary”, Hegel responds, this is only a “possibility, not an existing of the parts”. Infinity, an infinity of parts, does not exist. (Here Hegel uses the word “possibility” where Aristotle would use the word “potential”).

Hegel says that Bayle commits the “error of holding mental fictions, such abstractions, as an infinite number of parts, to be something true and actual”. (Science of Logic, p199, para 427)

Hegel and Newton’s calculus

It appears that Woods is unfamiliar with what Hegel had to say on the infinite, although there are some seventy references to Hegel throughout Reason in Revolt. Hegel in general takes a position closer to materialism than Woods on this question; Woods is more idealist.

Hegel illuminates his views on the infinite by considering the following stanza of poetry by the eighteenth century German scientist and poet Albrecht von Haller.

I heap up monstrous numbers,

Pile millions upon millions,

I put aeon upon aeon and world upon world,

And when from that awful height

Reeling, again I seek thee,

All the might of number increased a thousandfold

Is still not a fragment of thee.

I remove them and thou liest wholly before me.

Quoted in Science of Logic, p230

Hegel remarks:

When this heaping and piling up of numbers is regarded as what is valuable in a description of eternity, it is overlooked that the poet himself declares this so-called terrifying journey into the beyond to be futile and empty.

Hegel argues in various passages that it is futile and false to conceive of an infinite which exists somewhere “beyond”. In fact, Hegel appears to entirely reject the notion that the universe extends infinitely. Woods strives to make a complete distinction between the finite and the infinite, describing the infinite as, at first sight, “beyond all human experience”, but Hegel rejects that separation.

Woods says:

The idea of the infinite seems difficult to grasp, because, at first sight, it is beyond all human experience… Mathematics deals with definite magnitudes. Infinity by its very nature cannot be counted or measured. This means there is a real conflict between the two. (p353)

But Hegel says:

Thus the infinite does not stand as something finished and complete above or superior to the finite, as if the finite had an enduring being apart from or subordinate to the infinite. (Science of Logic,p138)

Yet Woods represents Hegel’s outlook in the following way:

In the section on Quantity in the first volume of The Science of Logic, Hegel points out that, while the introduction of the mathematical infinite opened up new horizons for mathematics, and led to important results, it remained unexplained, because it clashed with the existing tradition and methods. (p355)

This is misleading. Firstly, one should start with the section Infinity, in the Science of Logic, if one wants to know Hegel’s thoughts on the infinite directly. But Hegel is no less forthright in the section on Quantity to which Woods refers – the stanza of poetry above is from this section. Here Hegel examines Kant’s antinomy as to whether the world is finite or infinite. Woods mentions this antinomy, and comments “It fell to the great dialectician Hegel to resolve the contradiction in The Science of Logic.” (p 146) So he did. But Woods fails to mention how Hegel resolves it. Hegel concludes his discussion with following reference to the dialectics of ancient Greece:

But the so-called world… is never and nowhere without contradiction, but it is unable to endure it and is, therefore, subject to coming-to-be and ceasing-to-be. (Science of Logic, p238)

Hegel, in other words, does not embrace a universe which is infinite in time and space in the Newtonian sense, but instead argues that the universe has a birth and a death.

Nevertheless, it is true, as Woods infers, that in a remark on The Specific Nature of the Notion of the Mathematical Infinite, Hegel begins by recognising that the mathematical infinite led to “important results”. (Science of Logic, p240)  But let us be clear. Hegel here calls the infinities which mathematics uses, whether infinitely large or infinitely small, “pictorial conceptions which, when looked at more closely, turn out to be nebulous shadowy nullities”. (Science of Logic, p238)

In calculus (which Hegel is discussing here), a series of numbers gets smaller, appearing to be an infinite series. But this series never reaches infinity. Instead, a new quality emerges from the result of the calculus. (Science of Logic, pp244-5) Hegel sees the dialectic at work here, where a new quality emerges from a quantitative process. For this reason, among others, Hegel praises the Newtonian method of calculus. (Science of Logic, p257) Hegel is normally in the habit of sharply criticising Newton in the Science of Logic.

Hegelian infinity: the negation of the negation

In his section Infinity, Hegel discusses the dialectic of a simple infinite series (a series where, for instance, you can always add one more to whatever number you arrive at). Each step in the series appears to be a step towards infinity, Hegel says, only to be negated, because this step takes you no nearer infinity at all. There is no point at which infinity is nearer, no matter how many numbers one counts, as the stanza of poetry which Hegel quotes demonstrates. Thus infinity can be said to be ‘negated’ by the finite.

But, yet, the counting has not stopped, and there is no conceivable point at which it will stop. So the finite is once again negated. In this way, Hegel introduces his famous “negation of the negation”, because this second negation can be called the negation of the first negation, or the “negation of the negation”. Once familiar, this concept is not complex. The finite is negated by the infinite, and then this negation of infinity is itself negated with the next finite step in the infinite series, which again raises the hope of achieving infinity. “The infinite is the negation of the negation”, Hegel states. (Science of Logic,p137)

The infinite, concludes Hegel, always contains the finite within it, and is comprised of the finite:

The finite reappears in the infinite itself as its other, because it is only in its connection with its other, the finite, that the infinite is. (Science of Logic,p142)

The infinite, Hegel says, again referring back to ancient Greece, is properly understood “essentially only as a becoming” (Science of Logic, p148), something that is in a process of further determination. Hegel’s dialectic is a way of explaining why we consider, or can call, an infinite series ‘infinite’, although it never reaches infinity. Hegel emphasises that there is no “progress”towards infinity, since it is always negated and never gets nearer; there is only an infinite “process” which never leaves the finite.

We should note that Hegel is conscious of the concept of the Christian god as infinite, a matter we cannot go into here. Suffice to say that Hegel’s approach can strike one as remarkably secular in some passages, however mystical it is as a whole. Indeed, not long after the publication of Science of Logic, Hegel had to defend himself against accusations that he was an atheist.

Woods asks: “How can the universe be finite, and yet have no boundaries?” (p218) Hegel supplies the essence of the answer to this question, precisely one century before Einstein’s general theory of relativity allowed that the curvature of space-time might cause the universe to curl round on itself: “the image of true infinity, bent back onto itself, becomes the circle.” (Science of Logic,p149)  To put it another way, the surface of any sphere is finite yet has no boundaries – an ant can crawl over a football forever, never coming to an end point, a boundary marking the end of the sphere.

Hegel and “bad infinity”

Woods says:

In mathematics, it is possible to have an infinite series of numbers which starts with one. But, in practice, the idea of infinity cannot begin with one, or any other number. Infinity is not a mathematical concept. It cannot be counted. This one-sided “infinity” is what Hegel calls bad infinity. (p218)

This is incorrect. It is, in fact, the opposite of what Hegel meant. Hegel called “bad infinity” the consideration of the infinite as something separate from the finite, as we will go on to show. But first on the modern translation of the term “bad” infinity used by Woods. “Bad” has been replaced by the word “spurious” in the modern translation of Hegel’s Science of Logic. The Moscow (Progress Press, Lawrence and Wishart) translations of Engels’ Anti-Dühring use the word “bad”, but the highly regarded 1969 translation of Hegel’s Science of Logic by A.V. Miller – marking the beginning of a revival of interest in Hegel – translates term to which Engels was referring as “spurious” rather than “bad”.

It should already be apparent from the foregoing discussion that Woods has misunderstood what Hegel believed. Hegel did not accept the actual existence of an infinity which exists apart from processes which can indeed be counted, whether mathematical or historical, except in his somewhat pantheistic concept of divinity, and he called it the “bad” or spurious infinite.

So whereas Woods argues that the finite and the infinite are qualitatively distinct, Hegel says:

The infinite as thus posited over against the finite, in a relation wherein they are as qualitatively distinct others, is to be called the spurious infinite.(Science of Logic, p139)

Hegel’s spurious or bad infinity is the complete opposite of Woods’ description of it, and it is precisely Hegel’s spurious or bad infinity which Woods embraces – the infinity which cannot be counted, which stands apart from the finite, an infinite which “really and actually exists” as Bayle had said. Hegel says that: “such an infinite must be seen as a falsity”. (Science of Logic, p149)

Hegel correctly associates this spurious infinity with the divine. When Woods repeats a favourite phrase of Ted Grant’s (quoted in full below) that the infinite universe contains, “only galaxies and more galaxies stretching out to infinity”, (Authors’ Preface to the second Spanish Edition of Reason in Revolt, italics in the original) that is Hegel’s bad infinity. Grant and Woods completely reverse the position that Hegel takes, and Engels correctly champions Hegel’s position, if somewhat obscurely.

Commenting on the Hubble telescope, the telescope which was launched into space and has captured many stunning images, Woods says:

For our part, we welcome these epoch-making investigations, because they take the debate about the Big Bang out of the realm of abstract theorising and mathematical models, and into the field of practical observation.

We will predict now that they will see new surprises: not the Big Bang, but only galaxies and more galaxies stretching out to infinity.

Reason in Revolt, Preface to the 2001 Spanish edition, emphasis in original

The Big Bang had long been taken out of abstract theorising, while on the other hand, infinity will not be seen through the Hubble telescope. When astronomers turn their telescopes to view galaxies whose light has travelled to us over millions and billions of years, they are looking back in time in the sense that what they see actually took place millions or even billions of years ago, but its light has only just reached us. And what they see when they look back in time, in general, are galaxies in an earlier stage of formation and development. They see, for instance, galaxies in which the stars have not had time to manufacture as many of the elements that over millions and billions of years are products of the fusion process that powers the stars.

Woods omits to acknowledge that this is the overall picture. This process of the development of galaxies from the remnants of the Big Bang will be glanced at later. But what is not seen through telescopes is a mixture of older and younger galaxies irrespective of distance. There are no objects which challenge the widest range of ages given the universe, from ten to twenty billons years. The concept of an infinite universe containing “galaxies and more galaxies stretching out to infinity”is in conflict with the evidence, and has been for a very long time.

But only a year after his comments on the Hubble telescope in 2001, writing in the preface to the 2002 USA edition of Reason in Revolt,Woods endorsed a version of the 70-year-old cyclical Big Bang theory that interprets space as finite but time as infinite. Presumably, Woods was therefore prepared to accept the fallacy of the confident prediction of the previous year, of “only galaxies and more galaxies stretching out to infinity”, a prediction which, after all, is not and cannot be based on practical observation at all, only on abstract theorising. However far one can see, one can never see infinity through a telescope.

The idealist philosopher Hegel supports the materialist view that infinity is an abstraction that is never realised, except, that is, in god – Hegel is still an idealist. Woods takes the idealist position, which Hegel calls “spurious” or “bad” infinity. It is an undialectical position, Hegel says:

The falsification of the finite and infinite by the understanding which holds fast to a qualitatively distinct relation between them and asserts that each in its own nature is separate from the other, comes from forgetting what the Notion [i.e. the dialectic] of these movements is. (Science of Logic,p145)

Hegel uses the term “the Notion” in his Science of Logic to denominate his own system of dialectical thought.

Hegel has firmer dialectical reasons for rejecting the concept of a progression towards infinity. It is not just that infinity could never be reached or brought any nearer. For Hegel no apparently infinite process will go on indefinitely. Hegel understood that each additional quantity added to an infinite series could – and at some point in the concrete, material world, will – lead to a qualitative leap, and the whole process will be transformed into something else. Nothing stays the same. Everything comes into being and passes away. In the same way, those processes that we imagine could continue forever are mere figments of our imagination. When infinities appear in equations, physicists invariably work on the assumption that these infinities only mark out a point of qualitative transformation, or phase change. The idea of infinite space, stretching on without limit is undialectical because it is an idea of quantitative accumulation without a qualitative change. The same would apply to infinite Big Bangs.

Hegel explains that bad or spurious infinity “the spurious infinite… is commonly held to be something sublime and a kind of divine worship”. (Science of Logic,p228) He clearly rejects the infinite universe which Woods supports at the beginning of the Encyclopaedia (also called his Shorter Logic), with these words:

A second question in these metaphysical systems was: Is the world finite or infinite? The very terms of the question assume that the finite is a permanent contradictory [i.e. in permanent contradiction] to the infinite…

Dogmatism consists in the tenacity which draws a hard and fast line between certain terms and others opposite to them. We may see this clearly in the strict ‘either – or’: for instance, The world is either finite or infinite; but one of these two it must be. The contrary of this rigidity is the characteristic of all Speculative [i.e. dialectical] truth.

Hegel, Encyclopaedia, paragraph 28 (remark), paragraph 32

Hegel uses the term “Speculative” in the Encyclopaedia to identify his own system of dialectical thought. In modern terms, we may interpret this to say that if the universe expands forever, as seems likely at present, then we see the infinite within the finite, since the universe will always remain finite in extent and in time, yet is expanding endlessly. Hegel sees this as a dialectical unity of opposites, the finite and the infinite, whereas those like Woods who insist the universe must be infinite are dogmatists.

We must, however, add a caveat. Engels explains that the Hegelian system presented itself in such a way that, in the final pages of his Science of Logic: “the whole dogmatic content of the Hegelian system is declared to be absolute truth, in contradiction to his dialectical method, which dissolves all dogmatism”. (Feuerbach and the End of Classical German Philosophy, in Marx and Engels Selected Works, p589)

Hegel incorporated Kant’s support for Newtonian absolute space (not to be confused with infinite space) into his philosophy. In the closing pages of Science of Logic, Hegel appears to mystically link absolute space and time with what he terms the Absolute Idea, a kind of mystical godhead.

Hegel writes that the Absolute Idea takes on the form of the “externality of space and time existing absolutely on its own account without the moment of subjectivity”. (Science of Logic, p843) In a sense, Hegel is suggesting that once the Absolute Idea is reached in a great mystical cycle of the dialectical development of all things towards godhead, it returns, albeit at a higher level, to “nature”, “the end being wound back into the beginning, the simple ground”, (Science of Logic, pp842-3) as Engels points out.

But this does not mean that Hegel endorses Newton’s concept of an infinite universe. Space and time are absolute, in his view, but not infinite. Even in the closing paragraphs of Science of Logic, which contain an exposition of his dialectic – so that one might easily suppose that the “Absolute Idea” is nothing other than Hegel’s dialectic – Hegel argues that the infinite is not in fixed opposition to the finite, as something “beyond”.

Next: Engels on materialism, the infinite and cosmology

In the eighteenth century, Immanuel Kant speculated on the nature of the universe. His ideas had a remarkable influence and he is still cited today, for instance, as one of the first to suggest that there are galaxies other than our Milky Way, in his book, Universal Natural History and the Theory of Heavens published in 1755.

In 1781, turning to philosophy, Kant discussed contrasting theories of the universe which were held at that time, such as: Is the universe finite or infinite? He termed such contrasting theories, cosmological ‘antinomies’. In Critique of Pure Reason, and in the Prolegomena, Kant gives the first of his cosmological antinomies, or contradictions, as follows:

Thesis: The world has a beginning in time and space (a limit).
Antithesis: The world is spatially and temporally infinite.
(Prolegomena, Section 51)

Kant’s cosmological antinomies, which began by counter-posing the concepts of a finite and an infinite universe, were the announcement of the conscious reintroduction of dialectics into philosophy.

Following Kant’s reintroduction and re-interpretation of the dialectics which originated in ancient Greece, Hegel immersed himself in a study of ancient Greek philosophy as a student. Later Hegel recognised that there were not just four cosmological antinomies , or contradictions, as Kant supposed, but opposing tendencies in everything. Engels terms this the “interpenetration of opposites”.

Kant’s theory of the evolution of the solar system from a “nebulous” state, a gaseous cloud, is still credited in science today for revolutionising our understanding of the solar system’s formation.

Kant began his career by resolving the stable Solar system of Newton and its eternal duration, after the famous initial impulse had once been given, into the result of a historical process, the formation of the Sun and all the planets out of a rotating, nebulous mass. From this, he at the same time drew the conclusion that, given this origin of the Solar system, its future death followed of necessity. His theory, half a century later, was established mathematically by Laplace, and half a century after that, the spectroscope proved the existence in space of such incandescent masses of gas in various stages of condensation. (Engels, Socialism, Utopian and Scientific, Selected Works,p408)

Is this quoted anywhere in Reason in Revolt? If so, we must have missed it. Engels appears to suggest a universe with a history in time – a beginning and an end. Woods mentions Kant’s theory twice, but fails to draw from it the conclusions that Engels does. Engels calls Kant’s insight “the greatest advance made by astronomy since Copernicus. For the first time the conception that nature had no history in time began to be shaken.” (Anti-Dühring, p72)

Here Engels explains that previously the universe appeared to people only “as an incessant repetition of the same processes”. After Kant, this could no longer be so easily asserted. For Engels, the birth and death of our solar system must for the same reason apply to all the solar systems in all the galaxies (or ‘island universes’ as Kant termed them). Engels repeats the idea briefly in Ludwig Feuerbach and the Outcome of Classical German Philosophy and develops it in greater detail in the introduction to Dialectics of Nature.

We should add that, before turning to philosophy, Kant participated in the dispute originally between Newton and Leibniz mentioned above, in a treatise defending Newton’s concept of absolute space. Leibniz more correctly argued that space was relative. Engels says simply that Kant “didn’t see clearly into the matter” (Dialectics of Nature, The measure of motion – Work, p118). As we have noted in the chapter, Galileo and the relativity of space (under the subhead What is space?), Engels correctly recognised that space is relative.

Copernicus’ theory led to fresh speculation about the nature of the universe. The modern concept of an infinite universe first began to emerge here, linked to religious expressions of an infinite god.

Newton did not prove but merely asserted that the world was infinite. The idea of an infinite universe was undoubtedly extrapolated at least in part from the belief that to the vast quantities of stars and space that Galileo discovered through his telescope, there must be added vast quantities more, without end, to the glory of god.

But Newton also realised, as we discuss in this chapter, that his concept of infinite space brought serious problems — contradictions that were only resolved when science discovered, to everyone’s astonishment, that the universe was expanding.

A Cardinal – and the cyclical Big Bang theory

The universe, Woods writes,

was rapidly ‘expanded’ – in the minds of men – and… is now thought to measure tens of billions of light years across, and time will show that even the present calculations are nowhere near big enough. For the universe, as Nicolas of Cusa and others thought, is infinite. (p184)

Recent calculations suggest that the universe is at least 156 billion light-years wide. The German cardinal, Nicolas of Cusa (1401-1464), anticipated Copernicus (1473-1543) by nearly a century, proposing that the earth rotated and, as Woods rightly points out, argued that the universe was infinite.

In 2002, Woods appeared to have changed his estimation of the width of the universe. In his preface to the 2002 USA edition of Reason in Revolt, Woods offers his support to a mainstream re-working of the old speculative cyclical Big Bang theory. The idea that the universe goes through cycles consisting of a Big Bang followed billions of years later by a Big Crunch had been first suggested in the 1930s, soon after observation suggested our universe had a hot dense origin a few billion years ago.

Woods embraces the theory advanced in 2002 by prominent physicists Paul Steinhardt and Neil Turok, or more correctly, Woods embraces the theory as reported in the popular media. Woods asserts that Steinhardt and Turok’s theory is fully compliant with dialectical materialism, subject to experimental proof, purely because these two physicists, at least in their popular presentation, talked about a universe infinite in time.

Steinhardt and Turok’s cyclical big bang theory proposes that the universe goes through a perpetual motion of Big Bangs followed by what they term “big splats” as the universe reaches the end of the cycle. Woods, in 2002, thus continues to defend an infinity of time, but not an infinity of space, which expands and contracts perpetually with each cycle, according to the model. Are we to conclude that Woods now concedes that dialectical materialism does not prescribe to the universe an infinity of space, as he originally asserted in 1995 when Reason in Revolt was published?

Opposition develops to the Aristotelian view of a finite universe

Preceding Newton were a number of prominent religious thinkers who argued that the universe was infinite in space.

Nicolas of Cusa argued that the universe is infinite because god is infinite. Today the concept of an infinite god in infinite space is a commonplace concept. Nicolas of Cusa developed this concept from the ideas originating with the ancient Greek idealist philosopher Plato and the school of philosophy which Plato established. Cusa argued against the existing scholasticism based on the Aristotelian model of the finite universe, which the Catholic church embraced at that time.

Thomas Diggs (1546-95), an early supporter of Copernicus, was the first modern European astronomer to argue that the universe was infinite. He said it reflected the greatness of god, although the church at the time objected that an infinite universe left no room for heaven. This theme, that god was infinite and that the universe should reflect this, began to be adopted by the most far-sighted ‘theorists’ of the day, who had broken from Aristotle’s influence, although it was not until the end of the nineteenth century that it became an uncontested commonplace viewpoint.

William Shakespeare, a family friend of Diggs, reflected the conflicting views of the universe in many allusions (some quite obscure) in Hamlet. John Barrow cites, in The Infinite Book, a mention of the concept of infinite space in one of Hamlet’s declamations: “I could be bounded in a nutshell, and count myself a king of infinite space.”  (Hamlet, Act II, scene ii)

Speculation about an infinite universe lacked a basis in fact, but was linked to abstract religious considerations. The old concept that the universe consisted of concentric spheres began to break down. Giordano Bruno (1548-1600) was burned at the stake after refusing to recant his belief in an infinite universe. He claimed that there was no limit to the power of god and that god could have created an infinite universe.

Thus is the excellence of God magnified and the greatness of his kingdom made manifest; he is glorified not in one, but in countless suns; not in a single earth, a single world, but in a thousand thousand, I say in an infinity of worlds. (On the Infinite Universe and Worlds, 1584)

Bruno introduced many ideas that became commonplace in the later centuries, such as that space with its infinite worlds extends without limit in all directions and it has no central point. Bruno also suggested that life exists on other planets.

Bruno was a Neo-Platonist mystic with no understanding of astronomy, and today the concept of an infinite universe is often credited to Copernicus. What is called the Copernican principle – that the sun-centred solar system occupies no special place in the universe, and that the sun is one of many stars – is perhaps better credited to Bruno.

Among Newton’s influences was his contemporary, Henry More (1614–87). More was one of the leading philosophers of the influential group of philosophical ‘divines’, now known as the Cambridge Platonists, who broke with Aristotelian tradition. More believed that space was infinite, since infinite, immaterial space is analogous to god, who was an infinitely extended spirit.

In 1654, just a few years before Newton observed an apple fall and wondered whether the same attraction of the apple to the earth might keep the moon in tow, William Charleton wrote, in opposition to Aristotle, that time “flow[s] on eternally in the same calm and equal tenor” and is distinct from any measure of it. (Stanford University’s Stanford Encyclopaedia of Philosophy website: Newton’s Views on Space, Time and Motion) These views on time, once again, can be attributed to the influence of the thoroughly idealist philosopher Plato.

Newton’s infinite, absolute space and time

But it was Newton who most certainly set in motion what became our ‘common sense’ ideas about the universe, until the advent of Einstein’s theories and then the Big Bang cosmology. Newton’s general views on infinite time and space were essentially the same as these contemporaries. In the closing discussion in his Principia, Newton explains why he regards space and time to be infinite and absolute:

by existing always and every where, [god] constitutes duration and space. Since every particle of space is always, and every indivisible moment of duration is every where, certainly the Maker and Lord of all things cannot be never and no where. (Principia, book three, General Scholium, p1,158)

For Newton, infinite absolute space was a meaningful concept for these reasons. Gottfried Leibniz, the German philosopher and mathematician and one of the most prominent of Newton’s scientific contemporaries in Europe, opposed this view. There was a long-running, bitter dispute between the two. Newton must take most of the blame for the bitterness, but the debate extended over a wide range of issues and continued for decades among the most prominent scientists of Europe.

Woods cannot, in fact, distinguish between Newton and Einstein on these questions. Woods argues that “the greatness of Einstein” was to reveal the relative character of “the ‘absolute truths’ of classical Newtonian mechanics”, but adds that the “relative aspect of time, was, however, not new. It was thoroughly analysed by Hegel”. (p147) This view cannot be supported. The examples given have no bearing on the meaning of Einstein’s relativity in relation to time, the origin of which will be touched on very briefly in its proper historical context. Woods later gives examples such as: “A year on earth is not the same as a year on Jupiter” (p158), and so forth, in a series of irrelevant commonplaces, which never escape from Newtonian physics.

Contradictions in Newton’s beliefs: absolute space

In fact, Newton grasped these issues more profoundly, since he also understood Galileo’s principle that we experience space as relative and admitted that he had failed to provide evidence of his belief in absolute space.

In the opening Scholium, or discussion, of his Principia, Newton asserts: “Absolute space, in its own nature, without regard to anything external, remains always similar and immovable.” He also discusses relative space, which he assumes takes place in absolute unmovable space. “Relative space is some movable dimension or measure of absolute space.” But Newton admits that absolute space cannot be detected: “… the parts of that immovable [absolute] space, in which these motions are performed, do by no means come under the observation of our senses.” Newton ruminates that “the thing is not altogether desperate” and provides a range of arguments and suggests experiments that might detect absolute space.

But Newton’s absolute space is undetectable because the Newtonian concept of absolute space is false. It is relative space on which Newton’s laws of motion are based.

Scientist	Motion		Universe		Infinity
	Space	Time	Space	Time
Aristotle	Absolute	Absolute	Finite	Infinite	Denied actual infinite
Galileo	Relative	Relative	Finite (assumed)	Infinite (assumed)	Showed paradoxes of infinite
Newton	Absolute	Absolute	Infinite	Infinite	God as infinite

Aristotle: Denied the existence of space & time outside universe (= relative)

Newton: Newton’s Laws of motion were based on relative space & time; he assumed absolute space & time were real but undetected

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Problems with Newton’s universal gravity

Newton admitted he had no idea what formed the basis of the mysterious “action at a distance” by which his universal gravity binds tiny planets in the vastness of space in their orbit round the sun. In his concluding General Scholium of his Principia, he famously says:

But hitherto I have not been able to discover the causes of those properties of gravity from the phenomena, and I frame no hypotheses.

  More frankly, in a letter to Richard Bentley in 1693, Newton writes that action at a distance is “so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it”. (Quoted in Newton: Philosophical Writings, Cambridge University Press, p102) Woods falls into it.

Newton’s rival, Leibniz, argued that Newton’s universal gravityhad an “occult quality”. “The fundamental principle of reasoning”, Leibniz emphasised, “is, nothing is without cause,” yet Newton, “is admitting that no cause underlies the truth that a stone falls towards the Earth.” (Quoted by James Gleick, Issac Newton, p156)

In a damning letter Leibniz wrote:

“Thus the ancients and the moderns, who own that gravity is an occult quality, are in the right, if they mean by it that there is a certain mechanism unknown to them, whereby all bodies tend towards the centre of the earth.”

Referring specifically to Newton, Leibniz continues, “if they mean that the thing is performed without any mechanism by a simple primitive quality, or by a law of God, who produces that effect without using any intelligible means, it is an unreasonable occult quality, and so very occult, that it is impossible that it should ever be clear, though an angel, or God himself, should undertake to explain it. (Leibniz, Philosophical Writings, 112, quoted by Andrew Janiak in Newton as Philosopher, p7, footnote 14)

Newton did not necessarily disagree on this point, however vehemently he and his followers disputed with Leibniz. It is now widely recognised that Newton spent a great deal of time on what would now be classed as the occult, particularly alchemy. The economist John Maynard Keynes, who acquired many of Newton’s writings on alchemy, stated: “Newton was not the first of the age of reason: he was the last of the magicians.” (The Collected Writings of John Maynard Keynes, Volume X, pp 363-4) 

The reason he frames no hypotheses, Newton says, in the above quoted passage from his Principia, is because hypotheses, “whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy”. Science had not yet fully separated itself from alchemy, astrology and the occult. Newton rules out neither mechanical nor occult qualities to explain the action of gravity at a distance but he does rule out hypotheses. He was searching for proof, not hypotheses, and he was not able to discover any explanation for his universal gravitation, despite a considerable amount of investigation into the occult. Nevertheless, Keynes is not entirely correct. The Principia, more than any other work of the era, was defining the new ground of experimental physics and mathematical proof and, in addition, replacing an interventionist god with a god that designed the physical universe along rational and universal principles only at the moment of creation.

The mechanists of the period were “labouring to banish occult influences – mysterious action without contact,” James Gleick points out in his biography of Newton. Yet “Action at a distance, across the void, smacked of magic. Occult explanations were supposed to be forbidden.” (Issac Newton, p96, p142)  How did gravity mysteriously act on bodies completely remote from them, with no intervening substance? Hegel chides Newton for not developing laws which go beyond a mere description of the actual mechanics of gravity’s effects. “Even Newton’s proofs,” he says, somewhat stretching the point, are “nothing more than mere jugglery and window-dressing” (Science of Logic, p273), especially those which merely gave mathematical expression to the motion of the planets which Johannes Kepler had already discovered.

It was Einstein’s general theory of relativity that eventually resolved this paradox, by showing how space and time are bent (“warped”) by mass and energy. It is this warped path in space-time that the planets follow. There is no force acting at a distance through the void on the planets. The planets do not depart from Newton’s first law, which says that no object will depart from a straight path unless a force compels it to change direction. No force is acting on the planets to make them move from a straight path, but the space-time they inhabit is itself bent, as viewed from the perspective of the solar system, and all paths bend with it. In the dark vastness of space, the planets follow curved paths because space and time are bent by the sun’s gravitational effect. It is a stunning discovery, both mathematically and experimentally proven.

No longer could space and time, mass and energy, be treated as absolutely independent of one another. The sun’s great mass dimples the space-time around it so that the planets ploughing through space-time naturally follow the curvature of space-time around the sun. Thus Newton’s occult force which acts at a distance is replaced with a material effect. In scientific terminology, the Newtonian term ‘gravitational force’ is replaced with the term ‘gravitational effect’. Yet Woods disparages Einstein’s general theory of relativity and wants to return to Newton. To save science from mysticism, he wants to deliver it to the occult.

The near century that lies between Galileo’s discovery of the moons of Jupiter and Newton’s publication of the Principia is a remarkable period. Galileo demolished Aristotle and showed that there was corruption in the spheres – the universe must have had a beginning and an end. He further showed that space was relative and that the earth went round the sun, and stood on trial before the Inquisition. Eight decades later Newton reasserted that space and time were absolute and re-established a universe that was infinite in space and time, so long as god was the Prime Mover.

Problems of the infinite: starlight

Not all scientists in Newton’s time, however, accepted an infinity of space and time. Newton’s contemporary, Edmund Halley, who was the first to calculate the orbit of a comet using Newton’s laws, attempted to refute “the ancient notion, some have of late entertained, of the eternity of all things”. (Quoted in Stephen Jay Gould, Eight Little Piggies, p175)

This did not mean, as has been supposed, that Halley was a creationist. On the contrary, Halley refused to take The Bible literally. John Flamsteed, the Astronomer Royal of that time, opposed Halley’s appointment to a post at Oxford University, saying he would “corrupt the youth of the university”.

Halley required evidence, and there was neither evidence for the biblical creation, nor for an infinite universe. There was evidence against in both cases, however. Halley noted a serious contradiction in the concept of an infinite universe:

I have heard it urged that if the number of fixed stars were more than finite, the whole superficies of their apparent sphere would be luminous. (Quoted by John Barrow, The Infinite Book,p151)

In other words, if the universe was infinite and therefore populated with an infinity of stars, the night sky should be brilliantly illuminated, as if it were day. This contradiction was rediscovered by Wilhelm Olbers (1758-1840), and became known as Olbers’ paradox. Despite many attempts, no explanation of this paradox (such as interstellar dust, distance, etc), in the context of a universe infinite in space and time, has been successful.

Suppose you are in a deep forest with an infinite amount of trees. Every line of sight soon ends up at a tree. But if you are in a small wood with a finite amount of scattered trees, every line of sight does not end up at a tree. We live in a universe that has a finite amount of scattered stars and galaxies, with great voids where there are no stars or galaxies.

Problems of the infinite: gravitational collapse

Woods contrasts his version of infinite space with that of Einstein’s, which he says was “closed” and “static”. This is not true. Einstein’s theory allows for both an open and a closed universe, and makes no claims that the universe is static. It was Newton who developed a view of an infinite universe in a “static or a permanent state of equilibrium”, as Woods puts it.

If space is finite, Newton correctly argues, gravity would make stars move “towards all the matter on the inside and by consequence fall down to the middle of the whole space and there compose one great spherical mass.”

But, Newton reasons, in infinite space it might be possible to position each star so precisely that it is equally attracted by all on all sides. Then, argued Newton, the stars would not be able to fall into one another. But only a divine power could position the stars so exactly, as Newton explains:

but that there should be a Central particle so accurately placed in the middle as to be always equally attracted on all sides and thereby continue without motion, seems to me a supposition fully as hard as to make the sharpest needle stand upright on its point upon a looking glass. For if the very mathematical centre of the central particle be not accurately in the very mathematical centre of the attractive power of the whole mass, the particle will not be attracted equally on all sides…

Yet I grant it possible, at least by a divine power… they would continue in that posture without motion for ever, unless put into new motion by the same power. (Letters to Richard Bentley, 1692-3)

No scientific solution (as opposed to a spiritual one which invoked the hand of god to initially set things in motion) could be found to this apparent contradiction between the theory of gravity, Newton’s greatest scientific discovery, and an infinite universe, Newton’s unshakable belief.

Newton saw that the problem of gravitational collapse is posed for any system, finite or infinite, however dynamic or static, so long as it contains matter.

It makes no difference if there is a “continual process of movement and change, which involves periodic explosions, expansion and contraction, life and death”. (p189) Or whether, “Long periods of apparent equilibrium are interrupted by violent explosions.” (p215) Whichever scenario you choose, continual movement or interrupted equilibrium, neither scenario avoids the issue. Gravitational attraction between places of equilibrium, or expansion and contraction, would pull them together over a period of time that would be a blink of an eye compared to an infinity of time, as Newton foresaw.

What is the answer to the conundrum of gravitational collapse? Why has all the matter in space not collapsed in on itself in the universe?

Expansion of space

The definitive answer came in 1929-31 with Edwin Hubble’s earth-shaking discoveries. Hubble provided the first evidence that the universe is expanding. Using powerful telescopes, Hubble showed that galaxies are generally receding from one another and from us, not simply moving this way and that. Hubble also noticed another remarkable fact that was far more significant.

It appeared that space itself was expanding. Hubble’s results showed a universe expanding in such a way that clusters of galaxies move away from ours at a speed that increases with distance. Galaxies are not all receding from us at around 700 miles per second – 2.5 million miles per hour – as Woods nonchalantly says. (p155) In general, at 100 million light-years away, galaxy clusters are moving away from us at 5.5 million miles per hour, while those at 200 million light years are moving away at twice as fast, at 11 million miles an hour, and at 300 million lights years away, they are moving away three times as fast. (Brian Greene, The Fabric of the Cosmos, p229)

Why is this? If a “great explosion”, as Woods calls the Big Bang (p189), had torn apart some pre-existing primordial mass – the equivalent on a much larger scale of a star going supernova – then the speeds of the different objects observed would tend to be related to their masses, with the lightest pieces being thrown further with the greatest motion, compared to the heaviest. There would at least be a great variation in speeds. Hubble did not find this.

Instead, Hubble saw the universal orchestration of an orderly expansion. Hubble recognised that this could only be explained if what was expanding was space itself.

In the same way, when a cake stuffed with raisins rises in the oven, the raisins (the equivalent of galaxies) move apart from one another in a simple relationship determined by the surrounding cake mixture – in particular, the amount of self-raising flour in the mixture. If the cake explodes, one sees quite a different dynamic.

It is this expansion of space which is such a significant indication of a hot, dense origin of the universe and of space-time itself. The evidence is not consistent with what Woods calls a “great explosion” taking place in infinite space and time. (The term ‘Big Bang’ is a mischievous misnomer, which amuses astrophysicists but trips up Woods. Ironically, it was first coined by Fred Hoyle, who believed to the end of his life that the universe was infinite in space and time but was forced to admit that the Big Bang was the only existing satisfactory explanation for astronomical experimental data. Hoyle used the term derisively, and was perfectly aware of how misleading it was.)

Where gravity is strong enough to counteract it, it is thought that this expansion of space is halted. Within galaxies and some clusters of galaxies, for instance, this expansion of space is overcome and gravity has taken over. Nevertheless, as a whole, the universe is expanding and gravity has been unable to overcome this expansion, and thus has been unable to cause the entire universe to collapse into “one great spherical mass”.

By the end of the nineteenth century, so ingrained in common sense was the concept of an infinite universe (whether containing within it regions of expansion and contraction or equilibrium), that even Einstein, who seemed to question every common sense conception, did not at first question it, and tried to solve the problems which Newton pondered. Only hard scientific evidence provided by Hubble and reinforced by countless observations since, caused Einstein to abandon the concept of an infinite universe.

Woods does not seem to understand the nature of the problem: “The Achilles’ heel of Einstein’s static, closed universe is that it would inevitably collapse in on itself because of the force of gravity.” (p204) Woods does not seem to realise that this is also the Achilles’ heel of his universe where “Long periods of apparent equilibrium are interrupted by violent explosions.”

In his 1917 paper, Cosmological Considerations on the General Theory of Relativity, Einstein considers the problem, and ruminates that “if we really have to regard the universe as being of infinite spatial extent”, then, “It seems hardly possible to surmount these difficulties on the basis of the Newtonian theory.”  This is because Newton’s infinite universe suffers the same Achilles’ heel.

Einstein suggests that, “if it were possible to regard the universe as a continuum which is finite (closed) with respect to spatial dimensions”, a solution can be found, but only if there was a repulsive force, which he termed a cosmological constant, which could counteract gravity.

After learning that the universe was expanding so that gravity is currently being overcome by the expansion, Einstein called the addition of the cosmological constant to his general theory of relativity his “greatest mistake”. A cosmological constant could not in any case keep the universe in equilibrium, it was found. Recently, in a further twist, it has transpired that a repulsive force, or something akin to it, from an unknown source, appears to exist in the universe, since the universe’s expansion seems to be accelerating under its influence.

Why does Woods suggest that the problem of gravitational collapse affects only a closed universe? Is he obscuring from Reason in Revolt’s readers this significant and widely known contradiction: that a universe infinite in time and space would inevitably collapse under its own weight? Or is he simply unfamiliar with the science?

Next: Kant’s cosmology and Engels’ commentary

Some elementary scientific errors appear in Reason in Revolt, which quite undermine the authority of the author.

In this chapter we discuss the ideas of Copernicus, Galileo and – a little – Einstein. Those of a non-scientific background are welcome to skip this lengthy chapter – but it does attempt to show in familiar language the historical development of our ideas about the universe, and get to the core of an number of scientific issues related to the Big Bang.

One of the elementary scientific errors in Reason in Revolt is an inability to grasp Galileo’s ‘principle of relativity’, confusing it with Einstein’s theory of relativity and disparaging it. In this chapter we will attempt to disentangle this confusion by discussing this principle of relativity as presented by Copernicus, Galileo, Newton – and more profoundly by Einstein.

Another error Woods makes is to proceed from this failure to the conclusion that modern physics is saturated with “subjective idealism” since, Woods argues, Einstein was always concerned with “the observer” rather than the physical processes taking place independently of any observer in what he believes to be absolute space and time.

Woods misunderstands the term “the observer”, as if Einstein means the particular, subjective viewpoint of an individual, rather than a location from which an objective measurement can be taken.

Copernicus’ revolution

Aristotle’s heavens appeared incorruptible to western astronomers until the seventeenth century when, in 1609, Galileo constructed his telescope. Galileo discovered that there was change in the spheres, because moons orbited Jupiter, and the Earth’s moon was pitted with craters and not perfect. This gave rise to a new consciousness among the most advanced thinkers in the Renaissance.

Stars seemed to some to stretch to infinity. But Galileo did not see infinity through his telescope, nor has it ever been seen – only inferred by extrapolation, by the quantitative extension of finite observations.

Galileo defended the Copernican system, which redrew the map of the universe by rearranging the ‘perfect spheres’ so that the sun rather than the earth was at the centre of the solar system.

Nicolaus Copernicus’ famous On the Revolution of the Heavenly Spheres, published in 1543, is thought to have invested the word ‘revolution’ with its ‘revolutionary’ connotations, in the sense of a complete change or an overthrowing of the old conditions.

In his opening remarks, Copernicus briefly contrasted his revolutionary alternative of an immobile universe in which the earth rotates with the old Aristotelian idea of immense but finite heavens which rotate daily round an immobile earth. Ptolemy of Alexandria (87-150 CE), Copernicus tells us, had asserted that if the earth rotated, “this movement, which would traverse the total circuit of the earth in twenty-four hours, would necessarily be very headlong and of an unsurpassable velocity.” This rotational velocity, Ptolemy says, would have violently torn apart the earth long ago and all the pieces would have “passed beyond the heavens, as is certainly ridiculous”. (On the Revolution of the Heavenly Spheres, book one, Introduction, point eight, pp22-3)

Ptolemy assumes that if the earth is rotating we would certainly experience it. In fact a point on the equator of the earth is moving, in round numbers, at an astonishing 1000 miles an hour as part of our 24 hour daily rotation. Yet we do not experience this motion. Ptolemy finds the idea ridiculous.

Copernicus replies: “Why didn’t [Ptolemy] feel anxiety about the world instead, whose movement must necessarily be of greater velocity?” In other words,  Copernicus argues, in this seminal work: Would not the much greater rotational movement of the heavenly spheres around the earth, as defended by Ptolemy, have torn them apart and thrown them outwards? But this objection was not meaningful to the ancient philosophers, since they believed the “heavenly spheres” to be “incorruptible” and unchanging, incapable of any disturbance.

Copernicus then asks the reader a quite remarkable question: Have “the heavens become so immense, because an unspeakably vehement motion has pulled them away from the centre, and because the heavens would fall if they came to rest anywhere else?”

This remarkable question bears comparison with two aspects of the Big Bang theory. Copernicus asks whether the universe is so large because it has expanded from the centre and whether this expansion has prevented the collapse, under the pull of gravity, of the heavens, which if they came to rest would be bound to fall back to the centre. These tentative suggestions remarkably anticipate modern theory. Copernicus seems to present in one brief remark essentially an explanation to the problem of why gravity has not caused the supposedly static universe (the model of the universe which Woods defends) to collapse in on itself – because it is expanding outwards rapidly. This problem of gravitational collapse is a problem for supporters of the static universe which Woods had not fully grasped 450 years later. Needless to say, this problem is naturally resolved by the Big Bang theory. It will be touched on further in the next chapter. 

Copernicus ponders: “[I]f this reasoning were tenable, the magnitude of the heavens would extend infinitely.” Perhaps Copernicus has in mind Aristotle’s potential infinity – that in this scenario, the heavens would go on expanding indefinitely. But if Copernicus means “infinitely” in the sense Woods presents it (Aristotle’s actual infinity), this could only be true if the universe had existed for an infinite period of time. Yet if the universe is expanding “from the centre”, it must have began at some finite point in the past.

Galileo’s defence of Copernicus

What were Galileo’s views on an infinite universe? Just nine years before Galileo began looking through his telescope, in 1600, Giordano Bruno was burned at the stake by the Inquisition, after being tortured for nine years for preaching that the universe was infinite. He refused to recant. So, perhaps not surprisingly, Galileo did not express, or at least did not publish, any views on whether the universe was infinite or not. But he did discover additional contradictions which the idea of infinity leads to, (anticipating some of the concepts of the late nineteenth century mathematician George Cantor, who we discuss very briefly below) and elsewhere notes that infinities “transcend our finite understanding… In spite of this, men cannot refrain from discussing them… we attempt, with our finite minds, to discuss the infinite, assigning to it properties which we give to the finite and limited; but I think this is wrong…” (Galileo, Dialogues Concerning Two Sciences, First day, p418)

The ‘corruptible heavens’ which Galileo viewed through his telescope implied – by Aristotle’s own admission – a beginning and end to the universe, and this was bound to disturb all accepted dogmas.

Motion is relative, not absolute

Galileo set out to defend the theory of Copernicus that the sun, not the earth, was the centre of the solar system. This meant that, in addition to its “violent” rotational motion, the earth must move with incredible speed round the sun. To prove this, Galileo needed to demonstrate the rather surprising fact that steady motion in a straight line, such as the earth’s motion approximated to, cannot be readily detected by someone experiencing this motion (unless by reference to external evidence). 

Now the bumps and jolts we often experience when in motion are little accelerations – changes in motion from one speed to another. Acceleration is not steady motion. This is critical to our understanding of Galileo’s argument, since our most common experience of motion will consist of increasing and decreasing acceleration, and this misleads us, as it did in Galileo’s time, that we can tell, without external reference, if we are in motion. 

Steady motion in a straight line cannot be intrinsically experienced – such motion is not an “absolute” property of an object, as Aristotle had previously thought. In the next section of this chapter below we’ll see how Galileo does this. 

In the section, Problem Not Resolved, Woods says:

Einstein was determined to re-write the laws of physics in such a way that the predictions would always be correct, irrespective of the motions of different bodies, or the ‘points of view’ which derive from them. From the standpoint of relativity, steady motion on a straight line is indistinguishable from being at rest. (p161)

As mentioned earlier, this was not a ‘Problem Not Resolved’, but the standpoint of the entire scientific community since Galileo argued that the earth moved – that despite this “violent motion”, roughly 67,000 miles per hour, people on earth experience no motion at all – that, in this particular sense, we find our “state of rest” indistinguishable from being in motion. Einstein applied it rigorously to motions that were very fast, that is, approaching the speed of light, with astonishing results.

Today we might imagine a person floating in a spacecraft with its thrusters turned off. Despite travelling at enormous speed, once the thrusters which accelerate the spaceship are turned off and the rocket is in steady motion, the objects and the astronaut experience weightlessness rather than any impression of the speed of the craft. A person on the earth experiences gravity rather than weightlessness, but no sense of the 30km per second speed of  the earth through space. We will return to this shortly.

Woods seeks to undermine Galileo’s arguments all the time believing it to be a facet of Einstein’s relativity, asserting instead the existence of absolute space and time – meaning, that if an object is in motion, that motion is a property intrinsic to itself – and this means that it “experiences” that motion in the way that Aristotle conceived. Thus a person on earth should experience the earth’s 30km per second motion.

We must untangle this in what follows. This error leads Woods to take essentially the same position as the supporters of Aristotle in their dispute with Galileo in his defence of Copernicus. Copernicus, we should add in passing, briefly anticipates Galileo’s arguments. (On the Revolution of the Heavenly Spheres, book one, Introduction, point five, p23)

What is this fundamental law of physics, the principle of relativity, which formed the basis of Newton’s first law of motion, and from which Einstein took the name ‘relativity’? Let us take a glimpse at what Galileo and Einstein said.

Galileo’s thought experiment

The followers of Aristotle’s orthodoxy in the early seventeenth century thought that if the earth was travelling round the sun, or rotating, this would cause many very visible effects:

How would birds find their nest again after they had flown from them? Why does a stone thrown up come straight down if the earth underneath it is rotating rapidly to the east? (The Galileo Project, http://galileo.rice.edu/sci/theories/copernican_system.html)

Aristotle himself provided arguments against the notion that the earth moved, since one school of ancient Greek philosophers, led by Pythagoras, proposed that the earth did move. For instance, Aristotle asks, why do “heavy bodies forcibly thrown quite straight upward return to the point from which they started” if the earth has moved in the meantime? (On The Heavens, book II, chapter 14)

In the same passage Aristotle also argued that if the earth moved, one would surely see the stars pass by: “… there would have to be passings and turnings of the fixed stars. Yet no such thing is observed. The same stars always rise and set in the same parts of the earth.” This was a very powerful argument, not experimentally refuted until stellar parallax was measured by powerful telescopes in the 1800s. (Stellar parallax is the apparent movement of a star caused by viewing it from different positions, for instance, when the earth has moved a sufficient distance in its orbit round the sun.) Galileo could only suppose (correctly) that the stars were at too great a distance for parallax to be observed with the naked eye.

Galileo suggested experiments to prove the followers of Aristotle wrong. Adopting a popular, accessible style and writing in the native language rather than the scholars’ Latin, Galileo begins by asking his audience to imagine they are shut up “with some friend in the main cabin below decks on some large ship”. While the ship is stationary, Galileo suggests conducting a number of experiments designed to test motion in space, such as throwing and jumping, and setting a bottle to drip into a jar below. Then, he suggests:

… have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.  (Dialogue Concerning the Two Chief World Systems)

The ship, of course, represents the earth. Galileo is at pains to show that motion is “indistinguishable” from rest under these conditions, the accusation Woods levels at Einstein. 

The water still drips directly into the jar below – it does not fall behind the jar as the ship moves forward steadily. Many people have been below decks on a ship or car ferry, where you cannot see out, and experienced something similar: you cannot be sure if the ferry is moving or not, so long as it is going at a constant velocity. Galileo’s point is that a scientist, conducting experiments, could not determine by any experiment whether the ferry or, of course, the earth, was in constant motion or stationary either.

Einstein calls this Galileo’s principle of relativity.

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Scientist	Motion		Universe
	Space	Time	Space	Time
Aristotle	Absolute	Absolute	Finite	Infinite
Galileo	Relative	Relative	Finite (assumed)	Infinite (assumed)

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Einstein’s railway carriage

Thus far, the matter is quite clear. Einstein deepens this understanding, by discussing the trajectory of an object as seen from the two different standpoints – one at rest, and one in motion. And he truly deepens our grasp of precisely what it means to see the equivalence of these two.

Einstein discusses Galileo’s principle of relativity, which he calls “the fundamental law of the mechanics of [Galileo] Galilei-Newton” at the very beginning of his short popular primer, Relativity (first written in 1916 and still in print). He poses two questions:

I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices that the stone falls in a parabolic curve. I now ask: Do the ‘positions’ traversed by the stone lie ‘in reality’ on a straight line or on a parabola? Moreover, what is meant here by motion ‘in space’? (Einstein, Relativity, p9)

Dropped from the train, as seen from the footpath, the stone will continue moving forward as well as downwards, because before its release it is already set in motion forward by the forward motion of the train. Its downwards motion is accelerating under gravity. It will therefore cut out a curve (a parabola) in space as viewed from the footpath. You can plot this motion out on a piece of paper. But the train passenger, since he or she is also moving forward at the same speed as the stone, only observes the downward movement, so the stone appears to fall straight down to the passenger. If you find it difficult to visualise the effect of the stone falling without air resistance, on a fast moving train, imagine that the stone is dropped inside the carriage rather than out. It drops straight down. So it would outside the train, if it were not for air resistance, from the point of view of the train passenger.

Einstein reminds us that Galileo and Newton (in respect of his first law of motion) have shown that in reality both views, the one from the train, and the one from the footpath, are equally valid. Both views of the trajectory of the “misdeed” of dropping the stone, whether falling in a parabola or a straight line, furthermore, are objective descriptions of reality. This is quite a striking fact which we will explore further. But neither views are merely subjective impressions. Despite what Woods maintains, the subjective thoughts or impressions of individuals do not come into it.

Woods argues: “Einstein regretted his earlier subjective idealism, or ‘operationalism’, which demanded the presence of an observer to determine natural processes.” (p167) Einstein never demanded the presence of an observer to determine natural processes. It is a “complete misinterpretation of Einstein’s ideas” as Woods himself says slightly earlier (p163), without appearing to understand what he says. Einstein proceeds to re-phrase his own words this way:

The stone traverses a straight line relative to a system of coordinates rigidly attached to the carriage, but relative to a system of coordinates rigidly attached to the ground (embankment) it describes a parabola. (Relativity, p10)

It helps to consider the train to be a substitute for the earth. When we drop a stone on the earth it falls straight down despite the earth moving. What concerns us is the relative positions of the stone, as measured according to two different system of coordinates, or frames of reference, one moving and one stationary – the train and the footpath.

But does one frame of reference offer a correct description, while the other is merely secondary? No, they are both correct. At first, we may be tempted to say that the pedestrian on the footpath has the correct view or, to put the same thing another way, that the stone, as measured according to the frame of reference of the earth, is the correct measurement, because the pedestrian is the ‘stationary’ one, standing on the ‘stationary’ earth.

But the earth is not stationary. We must keep in mind that in the few seconds it took for the stone to fall, the earth and the stone have travelled perhaps sixty kilometres or more around the sun. Why take the earth as the correct or absolute reference point? In addition, the sun itself travels round our galaxy, and our galaxy is moving in a complex gravitational dance with our local cluster of galaxies. And all independent clusters of galaxies in the universe are moving away from us at great speed, in proportion to their distance from us.[1]

Whose space is the correct space? From the point of view of physics, each view, each measurement, whether from the railway carriage, the footpath, or the Andromeda galaxy, is equally valid. According to the Newtonian laws of motion which begin with Galileo’s insight (and which Newton acknowledged), the view from Andromeda is just as valid as the view from the train. Hence motion and space are relative to the observer (whether that observer is a person or a scientific instrument is irrelevant), meaning, relative to the frame of reference you choose to take – the train or the embankment in this case. 

Einstein’s universe

Now, according to classical Newtonian physics, we have no trouble at all translating the measurements of one frame of reference into that of another. They have a very simple, physical relationship. Suppose the train is moving at twenty miles an hour and a passenger is walking towards the front of the train at three miles an hour in the railway carriage. To put it another way, relative to the railway carriage frame of reference, the passenger is moving at three miles an hour. By simple addition, we say that the passenger is moving at twenty-three miles an hour relative to the footpath frame of reference – the speed of the train plus the speed of the passenger in the carriage.

We make this rather obvious point to make clear that no-one, whether Galileo, Newton or Einstein, is suggesting that, because the measurements are, in the common idiom, relative to the observer, these measurements are “subjective” in some way, or that physics has wallowed in subjective idealism ever since Galileo, which is the unintended essence of Woods’ claim.

However, Einstein realised that these calculations fail to take into consideration the speed of light. When we observed the train moving, we did so with the aid of light. But light does not travel instantaneously as our Newtonian calculation above assumed but at a definite speed. Furthermore, light has very unexpected properties. It is only once we have learnt about the strange qualities of light and have taken them into account that we can start to discuss Einstein’s universe. (In order to calculate the real transformation of the speed of the passenger relative to the carriage into his or her speed as measured from the platform, an equation associated with the physicist Hendrik Lorenz must be used, which takes the strange properties of light into account: the Lorenz transformation.)

What is space?

Asking what is meant by motion in space, Einstein says we “cannot form the slightest conception” of what ‘space’ means, since it seems to have two quite different values according to the person on the train and the pedestrian. Instead, he reconstructs the description of the stone’s trajectory in terms of two systems of coordinates – the moving train and the footpath. He concludes:

there is no such thing as an independently existing trajectory but only a trajectory relative to a particular body of reference. (Relativity, p10)

Therefore, in popular terminology, the motion of the stone dropped from the train must always be described according to some ‘observer’ – a particular body of reference – the earth, the train, the sun, etc, to have any meaning.

This is what Woods considers to be subjective idealism. But Woods cannot claim to have the authority of Marxism on this question (even if it were correct to use Marxism in this way). In fact, Engels also understood that there was no such thing as an independently existing trajectory. When Engels first conceived of writing about the dialectics of nature, in 1873, he began by noting the following:

1. The first, simplest form of motion is the mechanical form, pure change of place:

a) Motion of a single body does not exist – [it can be spoken of] only in a relative sense… (Dialectics of Nature, p329)

In words, Woods sometimes denies and sometimes echoes the idea that time and space are bound up with matter. But when he argues that, “Time and space are properties of matter, and cannot be conceived of separately from matter” (p146), it becomes clear from the context that Woods is not embracing Einstein’s theory of relativity, but essentially arguing that if a body is travelling at a certain speed, this motion through space and time is an inherent property of that body, without reference to any other body, in other words, not relative to it. In this sense, it is an expression of absolute space and time. And thus, as we will very shortly see, Woods expects that a person travelling very fast (such as someone sitting on the fast-moving earth) would experience that motion in “material damage” to their internal organs.

The earth’s motion must be judged in relation to other bodies, such as the sun. Taken as a single body, the earth’s motion “does not exist”, as Engels puts it. We may treat the earth, in accordance with our day-to-day earthbound experience, as stationary. The earth’s creatures do not experience its motion because space is relative.

This is the deeper understanding of the meaning of the statement that steady motion and rest are “indistinguishable”, as expressed by the fact that you do not experience the motion of the earth while sitting reading this page of text. Remember that all we are discussing here is Galileo’s principle of relativity and Einstein’s discussion of it. But Woods rejects this, thinking he is rejecting Einstein’s “subjective idealism”.

Clocks, twins and time

Despite calling Einstein’s special relativity “one of the greatest achievements of science” (p160), Woods proceeds, sometimes insidiously and sometimes openly, to attempt to denigrate Einstein’s relativity, particularly in the sections Problem Not Resolved, and Idealist Interpretations. (Einstein’s ‘special relativity’ deals only with the special case of motion unaffected by gravity or acceleration. His ‘general relativity’ includes gravity.)

Woods discusses the commonly used ‘twins’ example of the effect of motion according to the theory of relativity. Here, one twin goes on a high-speed intergalactic journey and returns. The effect is that she will have aged less than her earthbound twin – the amount depending on what fraction of the speed of light she travelled. Woods’ treatment is impeded by his failure to grasp Galileo’s principle of relativity. Let us see how Galileo’s science contradicts Reason in Revolt.

Woods begins:

A controversial idea here is the prediction that a clock in motion will keep time more slowly than one that is stationary. However it is important to understand that this effect becomes noticeable at extraordinarily high speeds, approaching the speed of light. (p163)

There is much that is wrong here but, above all, the effect of motion on the timekeeping of clocks is not “controversial”, it is incontrovertibly proved (as Woods admits elsewhere). For instance, navigation systems using the Global Positioning System (GPS) constantly make use Einstein’s special and general theory of relativity in about a dozen distinct types of calculations, in order to ensure the accuracy of their results, twenty-four hours a day. It is quite misleading for Woods to witheringly assert: “Unlike special relativity, experimental tests which have been carried out on [the general theory of relativity] are not very many.” (p172) Fifty years ago Woods’ assertions were true. The reader may have noticed already that Reason in Revolt is trapped in a kind of fifty-year-old time warp. (This is true for the second half of the book also, which we do not discuss in this review.)

In Einstein’s Universe, Nigel Calder describes the definitive experiment on this question, carried out in 1971 using four robust atomic clocks, which were placed aboard regularly scheduled commercial passenger jet aircrafts which took them right around the world. “One circumnavigation was made eastwards and one westwards, both journeys taking about three days. The result of the experiment was that the clocks no longer agreed about the time of day.” The clocks were compared to similarly highly accurate atomic clocks which remained at the US Naval Observatory in Washington DC. (Einstein’s Universe, p60)

The two experimenters, JC Hafele and Richard Keating, had predicted a loss of 40 nanoseconds eastbound, and the clocks did indeed lose time, although it was slightly larger, at 59 nanoseconds. Westbound the experimenters predicted a gain of 275 nanoseconds and the clocks gained 273 nanoseconds, a very close agreement indeed.

“In Newton’s universe, there would be no accounting for the discrepancies in such highly reliable instruments,” Calder remarks. Since then, subsequent experiments have tested the theory to far greater precision.

Woods proceeds to admit that this ‘time dilation’ effect, as it is called, has indeed been observed, and now objects: “The whole question hinges upon whether the changes, observed in rates of atomic clocks, also apply to the rate of life itself.” (p164) Woods’ line of argument could only arise if he has not grasped Galileo’s principle of relativity, since it does not matter in the least what is moving – living organisms or mechanical clocks – the point is their steady motion is measured relative to a stationary observer (another frame of reference, such as the earthbound twin). It is only relative to earthbound clocks that the clocks on the spaceship run slow.

In the section Idealist interpretations, Woods says, “it is not easy to see” how “the process of aging” of the astronaut twin can be “fundamentally affected either by velocity or gravitation, except that extremes of either can cause material damage to living organisms.” (Page 165, My emphasis)

He continues:

If it were possible to slow down the rate of metabolism in the way predicted, so that, for example, the heart-beat would slow to one every twenty minutes, the process of aging would presumably be correspondingly slower. It is, in fact, possible to slow down the metabolism, for example, by freezing. Whether this would be the effect of travelling at very high speeds, without killing the organism, is open to doubt. (p165)

Relativity, of course, makes no prediction about slowing down a person’s metabolism. It is not a biological science. But can extremes of velocity “cause material damage to living organisms” as Woods appears to believe? The followers of Aristotle’s orthodoxy in the early seventeenth century thought that if the earth was “travelling at very high speed” it would cause very visible effects, and ridiculed the idea mercilessly. Yet the entire earth’s population is going round the sun at roughly thirty kilometres a second, or one ten-thousandth of the speed of light.

Woods feels that the time dilation effect on “life itself” is “open to doubt” because he is convinced that travelling at very high speeds is injurious to life. Would not this very high speed “kill the organism” or at least cause some “material damage to living organisms” just as Woods ponders it might? Does our metabolism slow down? It does not, no matter how fast we travel at a steady velocity, because space is relative, as Galileo explained.

We must emphasis here another point that Woods fails to grasp. What is being discussed here is constant velocity or steady motion in a straight line. Woods also uses this term: “From the standpoint of relativity, steady motion on a straight line is indistinguishable from being at rest.” (p161) Einstein’s special theory of relativity, written in 1905, takes the special case of steady motion in a straight line (velocity), and excludes acceleration. Acceleration is quite different to steady motion. An accelerating jet fighter plane today can generate enough g-forces to kill the pilot if sustained for long enough. Einstein’s later general theory of relativity, published in 1916, deals with acceleration, and he showed that acceleration too can affect time and space.

But the entire point in the twins example is that the clocks and heart-beat of the space traveller moving at high speed are slow only relative to her twin on the earth. The motion of the spaceship is not an absolute motion, a spaceship which has the “property” of moving at high speed. Although it must have accelerated to its current speed, now it is cruising in steady motion it is only moving at its current high speed relative to the earth from which it departed. Relative to a frame of reference confined purely to the spaceship, the astronaut feels herself to be stationary, and her life processes are unaffected by her relative motion as she floats weightlessly inside her craft. She could “survive thousands of years into the future” (p164) but only as measured from the earth, only into the future of the earth, not as measured from the spaceship, where she will live a normal life span – disappointing as that may be. It is clear that Woods cannot consistently grasp Galileo’s principle of relativity here, let alone the ‘twins’ example itself in relation to Einstein’s relativity (which is more involved than can be adequately discussed here).

It is perhaps necessary to add that currently no spaceship can remotely approach the kind of speeds that would be necessary to observe a twin “time travel” years into the future in the way that Einstein’s theory of relatively revealed. These speeds must be a sizable fraction of the speed of light. Space flight currently renders astronauts imperceptible fractions of a second younger any earthbound twin, as measured by atomic clocks. Scott Kelly, who spent 520 days in orbit travelling at 28,000km an hour round the earth, is now 5 milliseconds younger than his twin Mark Kelly, Space.com reported (Einstein’s ‘Time Dilation’ Spread Age Gap For Astronaut Scott Kelly & his Twin, 13 July 2016).

The discussion of Einstein’s relativity in Reason in Revolt never grasps the seventeenth century scientific debate between Galileo and Aristotle’s supporters, and at no point clearly recognises the validity of Galileo’s arguments (as Engels certainly did) or of Newton’s first law of motion. Essentially, in this respect, Reason in Revolt sides with those who supported Aristotle’s views of a stationary earth, at the centre of the celestial spheres.

Woods makes a further error when, as discussed above, he asked whether “the changes, observed in rates of atomic clocks, also apply to the rate of life itself”. Woods tries to draw a distinction between processes taking place in humans or other living things and those in inanimate objects moving at high speeds. This is an unintentional departure from materialism, since it suggests humans or living things have a special, non-physical (and by implication therefore spiritual) essence which does not necessarily always obey the laws of physics by which material things are bound.

Criticising modern cosmological applications of Einstein’s relativity, Woods intones, “Here the study of philosophy becomes indispensable” (p216) but he has not grasped the problem, the most basic, elementary physics and, in fact, cannot escape from Aristotelian or Newtonian concepts of absolute space and time, on which his philosophical criticism of modern science is based. Philosophy is no use when you have no grasp of your subject.

Einstein applied the same relativity principle to time, but these considerations still do not yet depart, in essence, from classical Newtonian laws of motion. The issues that Einstein addressed which brought about an entirely new understanding of the universe will be briefly touched on later.[2]

Next: Newton: belief and contradiction

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[1] For those familiar with these concepts: according to the satellite COBE’s 1996 measurements, our solar system is moving at roughly one thousandth the speed of light (about 300 kilometres per second) in the direction of the constellation Leo, relative to the cosmic background radiation. http://arxiv.org/PS_cache/astro-ph/pdf/9601/9601151.pdf. Our local cluster of galaxies is travelling at twice this speed in the direction of the constellation Crater. http://www.arxiv.org/abs/astro-ph/0210165 (NB: Incidentally, unlike velocity, rotational movement can be determined by experiment.)

[2] But we are justified in considering so closely Galileo’s contribution since, as the physicist Hermann Bondi once said, “I always say that Einstein’s contribution has a name for being difficult, but it is quite wrong.  Einstein’s contribution is very easy to understand, but unfortunately it rests on the theories of Galileo and Newton which are very difficult to understand!” (Quoted by Gleik, Issac Newton, p 200)

“Aristotle would never have made the mistake of talking about a time before time existed,” claims Woods. (p209)

Woods quotes Aristotle on many occasions throughout Reason in Revolt. It is true that Aristotle believed that the universe was eternal. Yet it is not clear how familiar Woods is with Aristotle’s writing on time, space or infinity. In general, Aristotle takes a contrary position to that of Woods.

If one takes into account the limitations of his epoch, Aristotle achieved some quite remarkable insights into the nature of our universe. Even though Aristotle developed a logic that effectively usurped the ancient dialectics of Anaximander, Aristotle made occasional use of this ancient dialectics when discussing ‘the heavens’.

Woods presents Aristotle’s idea of time using this quote from his Metaphysics: “Movement can neither come into being nor cease to be: nor can time come into being, or cease to be.” Woods sneers: “How much wiser were the great thinkers of the Ancient World than those who now write about ‘the beginning of time’, and without even smiling!” (p145)

Woods is skating on thin ice. In chapter three of Physics, Aristotle discusses the infinite. Aristotle argues, as we have already noted, that there are two sorts of infinity, potential infinity and actual infinity. He concludes very firmly that actual infinity does not exist. It is true, Aristotle explains, that one can always imagine an infinite process, such as in addition. It is not hard to imagine infinity in this sense. This is Aristotle’s potential infinity, but it will always remain finite. Something extra can always be added. So:

there is potentially an infinite… For it will always be possible to take something as extra. Yet the sum of the parts taken will not exceed every determinate magnitude… (Physics, chapter 3, part 6)

Aristotle’s conclusion is spelt out very clearly: “… the infinite has a potential existence. But… There will not be an actual infinite.”

Curiously, Woods does say that Aristotle, “Polemicised against geometers who held that a line segment is composed of infinitely many fixed infinitesimals, or indivisibles.” (p354) This means, in simple language, that Aristotle opposed the idea that something can be infinitely divided, or is comprised of an infinite number of parts. Woods does not state this clearly, but rather in general allows the reader to draw the conclusion that Aristotle supports the same views as Woods.

Aristotle was the first western philosopher to clearly explain the materialist position that there is no such thing as the actual infinite, as a separately existing thing. Woods argues that the infinite does exist, and he makes it a central tenet to the philosophy of dialectical materialism. This introduces an element of idealism (in the philosophical rather than the common or popular sense of the term) into Reason in Revolt and its interpretation of dialectical materialism, because the infinite is a human idea without any proof of its material existence. It is “beyond all human experience” Woods admits, and yet argues that science should accept this idea as the basis of cosmology!

An idealist approach, in philosophical terms, is one which makes ideas primary and the material world secondary. Idealism explains developments primarily though ideas, and relegates experience to a secondary role, whereas Marxism makes the experience of the material world primary, from which ultimately arises, in a general sense, our ideas about the world. It is important that Woods’ interpretation of dialectical materialism on this question is corrected. The philosophical meaning of the terms ‘idealism’ and ‘materialism’ is discussed in Engels’ Ludwig Feuerbach and the Outcome of Classical German Philosophy.

Woods takes the position that: “The reason why infinity can be used, and must be used, in modern mathematics is because it corresponds to the existence of infinity in nature itself.” (p358) Infinity is not used in this way in mathematics – as Woods concedes later on the same page – because infinity does not correspond to nature itself.

Indeed, Aristotle explains that the type of infinities used in mathematics corresponds to his potential infinity. After explaining the illusion of the actual infinity, Aristotle goes on to say:

Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase… In point of fact they do not need the infinite and do not use it. They postulate only that the finite… may be produced as far as they wish… Hence, for the purposes of proof, it will make no difference to them to have such an infinite instead, while its existence will be in the sphere of real magnitudes. (Physics, chapter 3, part 6)

By real magnitudes, of course, Aristotle means concrete, measurable quantities that exist in the real world. This issue will be touched on very briefly in the short chapter, The infinite in mathematics. But more than mathematicians, scientists tend only to use Aristotle’s potential infinity, even among those who defended the idea of an infinite universe. When the Big Bang theory began to gain some adherents, the physicist Fred Hoyle attempted to develop an alternative theory which could explain current cosmological observations on the basis of an infinite universe. In a book popularising his ideas, written in 1955, he simply says: “the word ‘infinite’ should cause no conceptual difficulties. It simply means that however much of the [universe] we consider there is always more of it.” (Fred Hoyle, Frontiers of Astronomy, p277) The reader will perhaps recognise that Hoyle’s definition of an infinite universe avoids Woods’ ‘actual’ infinity and approximates to Aristotle’s potential infinity: it can be “produced as far as [you] wish”. Hoyle’s alternative theory, the Steady State theory, was not successful.

The infinite and the divine

Aristotle makes an exception of time and of the ‘divine’ in On the Heavens. The divine is taken to be infinite by definition. Aristotle explains that all philosophers agree that anything that changes has an origin in time and is finite, because change is a result of the dialectic of the interpenetration of opposites, of coming into being and passing away. If the heavens (by heavens Aristotle means the cosmos) are capable of change then, according to the dialectic of both Anaximander and Aristotle, they must have had a beginning, and will have an ending, and are finite.

But we do not see the heavens change, says Aristotle (apart from their fixed rotation). He says that the stars are “fixed” in the heavens, and that therefore the heavens must be incorruptible, imperishable, unchanging except for their rotational motion through the heavens, since “a thing whose present state had no beginning and which could not have been other than it was at any previous moment throughout its entire duration, cannot possibly be changed.” Whereas, “clearly whatever is generated or destructible is not eternal.”  (On the Heavens, book 1, part 10)

In other words, for Aristotle, dialectics teaches that if there is change in the heavens, the heavens must have an origin and an end. We now know, since Galileo, that there is change in the heavens – that is to say, in the universe. Galileo, incidentally, was convinced that with the evidence before him, Aristotle would have changed his views.

For Aristotle, the heavens’ eternity reflected the divine. The reason why Aristotle’s aether (the fifth element, the heavens) is eternal, “unaging and unalterable and unmodified”, should be clear to “all who believe in the existence of gods at all”. Aristotle maintains that the aether is the “seat of all that is divine”. (On the Heavens, book 1, parts 3 and 9) While the earth was corruptible, the heavens were not. (In the Renaissance and until the beginning of the twentieth century, the term aether, or luminiferous aether, was used to describe the medium through which scientists thought light propagated.)

Absolute and relative space and time

Woods might have been advised to hesitate before recruiting Aristotle to his cause. He also fails to mention that Aristotle thought the universe, although infinite in time, was a sphere of finite size. Aristotle discusses how concepts of time and space can be meaningful outside this rotating sphere of the heavens. This is fascinating because of its relevance to the concept of the universe that arose from Einstein’s relativity, and therefore from modern speculation about into what substratum the matter, time and space of the Big Bang universe might be expanding.

Firstly, we remember that Aristotle argued that within the heavenly spheres, material things fall down to the earth, because the earth happened to be “down”, at the centre of the universe. (Aristotle appears even to suggest that the earth is simply an aggregate of everything that has fallen “down”.) Fire, on the other hand, heads “up” to the heavens.

This meant that it could be said that for Aristotle time and space were absolute within the bounds of the universe. The earth was the absolute, stationary frame of reference for all motion. For instance, stones fell because the natural place of heavy bodies was the centre of the universe, and that was why the earth was there.

Later, Galileo attacked Aristotle’s views. He pointed out that there was not just one universal frame of reference for motion. For instance, if one was to imagine a game of table tennis below decks on an ocean liner, where the ship is travelling in steady, constant motion in calm seas, without rolling from side to side, one can readily predict that the players will not see any difference in the behaviour of the ball compared to a game on terra firma – all the laws of motion apply, relative to the frame of reference of the steadily moving boat, just as they do on the shore, relative to which the boat is moving.

This is obvious to us. Today, it is second nature to us that motion is relative, not absolute, even if we are not aware of the laws of physics. Yet to Aristotle’s way of thinking, if the boat was in motion (or, more significantly, if the earth was moving through space), the table tennis balls should behave differently, reflecting the forward motion of the ship (or earth), perhaps as though there was an invisible medium through which they and the boat passed, which affected their motion, sweeping the table tennis balls backwards off the table. This is a concept of absolute space and time.

As we shall show, however, although motion is relative, this does not mean that the motion of the table tennis balls is not an objective, measurable phenomenon – there is no lapse into subjective idealism, which is what Woods associates with the concept of relative space and time.

Before time, outside of space

But outside the ‘heavens’, the sphere of stars surrounding the Aristotelian universe, Aristotle concludes that there can be no space or time. Nothing exists outside the heavens, Aristotle says, therefore there can be no movement. But since “time is the number of movement” – time is the measure of change – there is no time outside the heavens:

But in the absence of natural body there is no movement, and outside the heaven, as we have shown, body neither exists nor can come to exist. It is clear then that there is neither place, nor void, nor time, outside the heavens. (On the Heavens, book 1, part 9)

Woods asks the defenders of the Big Bang theory: “So what was there before time? A time when there was no time! The self-contradictory nature of this idea is glaringly obvious.” (p210) At least since Aristotle, the answer to the question has not been quite so “obvious”. Since Einstein, the question has to be rephrased: What was there ‘before’ the space-time of our universe?

Woods, however, wishes to distinguish between the measurement of time, which must be relative to some process of change, and the “nature of time itself” (p161), and says: “… in cosmology, the confusion of measurement with the nature of the thing itself leads to disaster in practice.” (p158)

Table 1. Schematic summary of Aristotle’s views

Scientist	Motion		Universe
	Space	Time	Space	Time
Aristotle	Absolute	Absolute	Finite	Infinite

But what is this “time itself” or “thing itself” which is to be abstracted from the measurement of change? It is the old Newtonian concept of time. Newton says, in the Principia: “Time exists in and of itself and flows equably without reference to anything external.” It is truly as if there is somewhere a giant watch, held by a great timekeeper in the sky, keeping track of all celestial matter. We are jumping ahead a little, but for Newton, this timekeeper was god. Einstein did away with the timekeeper. Einstein’s universe, it is often said, is a “democratic universe” with no dictatorial authority laying down on each and every planet the strict time of the universe. Historically, in any case, the notion of a single time from which all clocks must be set arose in that same historical period which gave rise to Greenwich Mean Time, and should be placed in the context of the politics of Britain’s seafaring exploration and conquest of that period.

Woods does not elucidate exactly what “disaster in practice” has occurred as a result of science treating time, since Aristotle, as the measurement of change. Woods concedes, however, that according to this conception: “defining what time is presents a difficulty”, which he does not resolve. Time is indeed a complex phenomenon. But this “time itself” which Woods defends is Newtonian absolute time. On the facing page, Woods correctly asserts the opposing view. “It is impossible to regard [space and time] as ‘things in themselves’.” (p159)

Aristotle’s argument that space and time do not exist if there is no matter is, very roughly speaking, an acceptable hypothesis to science today. There is no meaning, in general, to time and space unless there is matter and energy. This is a materialist position, although it can be hard to grasp. But we will leave it to Galileo to demonstrate aspects of this important concept, which he did when he argued – against the followers of Aristotle – that the earth was in motion.

Next: Galileo and the relativity of space

Woods seeks to enlist two ancient Greek philosophers in his scheme of the infinite, so this historical survey must begin with them. These two philosophers are Anaximander, from the sixth century BCE, and Aristotle, from the fourth century BCE.

Anaximander – the dawn of dialectics

Both materialism and dialectical thought can be traced back to the sixth century BCE and a remarkable and powerful city-state called Miletus, in Ionia (now Turkey).

In those days Miletus was experiencing a period of revolutionary upheavals as a rising merchant class challenged the old ruling elite for power. These revolutionary upheavals must have been earth shattering, like the 1789 French revolution (in which King Louis XVI was guillotined), or the 1917 Russian revolution. In those revolutions the whole social order was turned upside down – everything in the social order which seemed eternal was proved to be ephemeral, like the so-called ‘divine right of kings’. Something similar happened in Miletus. The rising merchant class for a time took power from the old aristocracy in a series of revolutions.

This revolutionary period gave birth to the philosophy of dialectics. It is no coincidence that Kant and Hegel, who made dialectics central to their philosophy and greatly developed the ancient dialectics founded in Miletus, also lived in such revolutionary times – the period of the 1789 French revolution. It was in these momentous events, as one class clashed with another, that these philosophers came to believe that a clash of opposites leading to a sudden qualitative, fundamental change represented the true underlying nature of all things.

In ancient Ionia, this brought about the philosophical school thought to have been founded by Thales and Anaximander, and it is called the Ionian school of philosophy. It will hopefully become clear that our discussion of this ancient philosophy is very relevant to the claims made in Reason in Revolt about dialectical materialism and the universe.

No doubt reflecting the revolutionary times in Miletus, in which it must have seemed that nothing was permanent, Anaximander speculated that the entire universe had come into being from some unknown substratum and would eventually perish.

Woods says, “from the beginnings of philosophy, men speculated about infinity. Anaximander (610-547 BC) took it as the basis of his philosophy.” (p353) Woods’ assertion that Anaximander, who is said to be the first western philosopher to set down philosophical ideas in writing, took infinity “as the basis of his philosophy” gives a misleading impression of Anaximander’s views.

Far from suggesting that our universe was infinite, Anaximander said that our universe had come into being in a ball of fire and would pass away. What was revolutionary about the philosophy of Anaximander, particularly from the point of view of dialectical materialism, was precisely his challenge to those who, like Woods, intone: “Thus it has been. Thus it ever will be.” Anaximander describes how a “sphere of fire” grew from a “germ, pregnant with hot and cold, [which] separated off from the boundless”, forming several rings, from which arose the sun, moon and stars.

For Anaximander, the “heavens and the worlds within them” have a beginning. He also believed they have an end. Anaximander’s views, of course, remind us of the Big Bang picture of the birth of the universe, beginning in a hot dense state, a kind of “sphere of fire”.

Nevertheless, Anaximander postulated some sort of substratum from which the universe arose in its sphere of fire. “All the heavens and the worlds within them” have arisen from “some boundless nature”, Anaximander said. He seemed to use the term, ‘the boundless’, to describe this substratum. It appears that ‘the boundless’ represented some kind of inexhaustible source of the creation of matter. But the boundless does not by any means necessarily stand for an infinity of space and time – that interpretation might be no more than an anachronistic extrapolation based on a Newtonian outlook. Some modern translations use the phrase “boundless chaos”.

Even so, it is speculated that Anaximander’s concept of the boundless arose as an extension of the idea of the immortal Homeric gods. A typical viewpoint states: “Anaximander added two distinctive features to the concept of divinity: his Boundless is an impersonal something and it is not only immortal but also unborn.” (The Internet Encyclopaedia of Philosophy) So the origin of the boundless is likely to be associated with concepts of the divine. Thales is thought to have said: “What is the divine? That which has no origin and no end.”

Two centuries later Aristotle pointed out, in Physics (book one, part seven), that the ancient dialectics of coming into being always assumes what he called a “substratum”, and we have adopted his term in this review. So from what substratum might our universe have been brought about as if ‘from nothing’?

Dialectics and the quantum fluctuations in the vacuum of space

Woods discusses such a candidate substratum (indeed, one of the leading candidates) when he discusses the strange subatomic quantum fluctuations observed in the apparent vacuum of empty space, in which subatomic particles appear to come into existence fleetingly in opposing pairs, only to recombine and annihilate each other.

What is curious in this connection is that the philosophers of the school which Anaximander and others brought into being, the most ancient, Ionian school, were best known for their concept of coming into being and passing away. Today’s cosmology, in various ways, links this subatomic coming into being and passing away to the sudden coming into being of the universe in the Big Bang.

Towards the beginning of Reason in Revolt, Woods mentions in passing what he calls a “restless flux of swirling quantum waves” (p107), when attempting to discuss quantum mechanics. Towards the end of Reason in Revolt he correctly quotes from a passage in a now otherwise largely obsolete 1959 book by Banesh Hoffmann: “What we would think of as empty space is a teeming, fluctuating nothingness, with photons appearing from nowhere and vanishing almost as soon as they were born.” (Quoted in Reason in Revolt, p386)

Woods here correctly points to the curiously dialectical concept of the quantum fluctuations in the vacuum, and compares this with the dialectics of coming into being and passing away, as discussed by Engels in Anti-Dühring: “but everything moves, changes, comes into being and passes away.” (Anti-Dühring, p30)

The theory of quantum fluctuations has been a standard part of quantum mechanics (quantum field theory) for more than fifty years. Yet midway through Reason in Revolt, at the pinnacle of its mockery of modern science and the Big Bang, Woods devotes an entire section, Thoughts in a Vacuum, to ridiculing this very same idea. (p219) Woods’ contradictory positions suggest that he has not understood what he has written.

The interpenetration of opposites

Woods repeats in the abstract: “Everything that exists deserves to perish” but expresses his cosmology like this: “Time, space and motion are the mode of existence of matter, which can neither be created nor destroyed. The universe has existed for all time.” (p199) This is not the dialectical viewpoint of the ancient Greek philosophers from whom Hegel developed his dialectics. Woods is here defending Newtonian cosmology. Anaximander conceived that the universe had a beginning and an end but, as a materialist, he always assumed it emerged from some underlying substratum.

Ancient philosophers of the school of Thales and Anaximander were materialists and had a dialectical outlook. Anaximander said:

Whence things have their origin,

Thence also their destruction happens.

Anaximander

This means, according to Aristotle: “whatever comes into existence should have an end”. This is the origin of the quote from Johann Goethe’s Faust, which Engels uses and Woods fondly repeats a few times in Reason in Revolt: “Everything that exists deserves to perish” (p141), more correctly expressed as Engels renders it: “All that comes into being deserves to perish.”

Anaximander’s philosophy of coming into being and passing away reflected turbulent political times in the ancient city state of Miletus. Furthermore, these ‘opposites’ of coming into being and passing away, of birth and death, creation and destruction, were understood to interpenetrate everything, even our universe itself.

In other words, this ancient school of philosophers believed that the opposites of coming into being and passing away were two integral aspects of everything capable of change: for instance, a person is born and dies, and this mortality is a part of their being. This was the origin of the unity and interpenetration of opposites, which Engels summarised so clearly – and which Lenin, following Hegel, considered the central element of dialectics. These opposites, which dialectics says is found in everything which changes, attempt to negate the other, until one finally triumphs and there is a qualitative change – Louis XVI is guillotined, water boils, atoms decay, the living die. There is a passing away and, perhaps, another coming into being.This was called the dialectic of becoming.

Next: Aristotle on the ‘heavens’

What does Woods mean by infinity? In the section, Does the Infinite Exist? Woods suggests that:

The idea of the infinite seems difficult to grasp, because, at first sight, it is beyond all human experience. (p353)

An infinite universe would indeed be “beyond all human experience”. As the physicist Brian Greene says: “Experimenters never measure an infinite amount of anything. Dials never spin round to infinity. Meters never reach infinity. Calculators never register infinity.” (The Fabric of the Cosmos,p335)

But this raises precisely the question we will address in this survey: how did the universe come to be reckoned to be infinite?

After all, science, which is instinctively materialist, bases itself on human experience (including, of course, through the use of scientific instruments of all kinds), not on what is “beyond all human experience”. This raises a second question: how can Woods’ claim that the universe is infinite be a materialist claim?

We can all envisage an unending series of numbers, a series of numbers that continually grows greater in an infinite repetition of some additional amount. No matter how large the number gets, we can always add one more. A simple repetitive task, we imagine, at least in principle, can always be repeated in an infinite process that need never stop. In this sense, we cannot agree with Woods’ claim that the infinite seems “difficult to grasp”.

But it is important to realise that this infinite process of addition or repetition will never reach actual infinity. The number of repetitions, however large, will be a finite number, and will remain finite.

This infinite process is not the kind of infinity that Woods is talking about. “Infinity, by its very nature cannot be counted or measured.” (p353) “The idea of infinity cannot begin with one, or any other number. Infinity is not a mathematical concept.” (p218)

Potential and actual infinity

So we begin to see that, contrary to Woods’ assumption, there are two contrasting concepts of infinity which may concern us here. (We will glance at George Cantor’s contribution to this subject later.) The first is the familiar one, which can, in fact, begin with one, or any other number, where we can always imagine adding one more in an infinite process.

Woods does not accept this, but this is what Aristotle, the Greek philosopher of the fourth century BCE, termed “potential” infinity. It is a process that never leaves the finite – you never reach infinity – yet, at any particular stage under consideration, it is an infinite process. It is the only type of infinity which science recognises (in the real world, as opposed to mathematical methods such as calculus). The dial never reaches infinity even if the process appears to be infinite.

In this way, as we shall see later, Engels at one point envisaged a universe rolling out indefinitely in time and space, in an infinite process comprised only of finites, and never becoming infinite (Anti-Dühring, Part V, p67). Elsewhere, Engels envisaged the death of the universe, pointing out that at a certain point all the stars must exhaust their fuels.

The second concept of infinity – the only one that Woods recognises – is “beyond all human experience”. Aristotle calls this second type of infinity “actual infinity”. Woods claims that the infinite is quite distinct from the finite. Yet Aristotle and many philosophers and scientists, through the ages to the present day, have explained that the ‘actual infinite’, the infinite that is “beyond all human experience” according to Woods, is an ideal with ‘potential’ but no ‘actual’ reality.

Engels’ views

Before we start our historical survey let us address directly the question: is the universe – or for that matter, the ‘multiverse’ – infinite?

We answer this question in the course of this discussion, but to jump ahead, it may be useful to take a glimpse at Engels’ remarkable insight on this question. We will argue that among these insights, guided by dialectical considerations, are some that approximate to the position of modern science today.

As we will show, Engels discusses the coming into being of our universe and says that there must have been a cause to this event even if, at present, we have no idea what it is. In today’s terminology, science assumes that there must be a cause to the Big Bang and is searching for it.

But if one was to ask whether there must be an infinity of previous causes to the cause of the Big Bang, at another point Engels replies: infinity is a contradiction, and is full of contradictions (Anti-Dühring, PartV, p66). Engels was well aware that Aristotle had shown that the actual infinite does not exist. It was common knowledge. It was also common knowledge that Aristotle discovered contradictions in the concept of actual infinity, and others, beginning with Galileo, have discovered many more. One such contradiction is called the infinite replication paradox, and simply follows from the fact that infinity can contain within it an infinite number of infinities.

Consider an infinity of people. With more than six billion people in the world, there are bound to be people who look very much like you. Famous people sometimes employ look-alikes to pretend to be them. Since antiquity it was understood that if the universe was infinite – a Newtonian universe or a multiverse of modern conception – there must be an infinite number of worlds (or universes) of every possible type, since even the most improbable worlds occur infinitely given an eternity of time and an infinity of space. Among them there will be an infinite number of worlds like ours, and even an infinite number of people like us living on these worlds – in fact, there must be an infinite number of people exactly like us, on these infinite worlds, doing exactly what you and I are doing right now.

As materialists, we must leave the actual infinite for what it is, a contradiction. As scientists emphasise, we have no material evidence for an infinite universe – just a sense that there must always be an endless chain of causes. Let us now place our discussion in its proper historical context.

Next: The dialectic of ‘becoming’ in ancient Greece

One of the major themes running throughout Reason in Revolt is the infinite. Woods repeats many times, claiming the support of dialectical materialism, that the universe is infinite in space and time: “Dialectical materialism conceives of the universe as infinite.” (p189)

“From the standpoint of dialectical materialism,” Woods intones, it is “arrant nonsense” to talk about the beginning of time or the creation of matter:

Time, space and motion are the mode of existence of matter, which can neither be created nor destroyed. The universe has existed for all time (pp198-9)

Is it true that dialectical materialism conceives of the universe as infinite in time and space? Is it a materialist claim? Is it a dialectical claim?

The view that the universe is infinite in time and space may strike many people as a perfectly natural one. This concept has developed over the last five hundred years and should be understood in its historical development. It is a view that arises from definite historical and social conditions.

The Big Bang theory may well seem contrary to common sense to many readers. If we start from the very beginning – with the ancient Greek philosophers from whom so much has been learnt, even by modern scientists – we will find the answer to why science has taken this plunge into what appears on the surface to be an assertion that something can come out of ‘nothing’: that the universe – all its matter and energy, time and space – can emerge from the Big Bang. We will also discover the real material basis on which science establishes the origins of our universe, and the ancient dialectical concepts which proved so perceptive.

But first, a few remarks on what is meant by ‘universe’ and ‘infinity’.

One universe or many?

Firstly, what does Woods mean by the ‘universe’? When we say “the world” we may mean one of two things. We may mean the entire universe, or we may be referring to the earth. But what precisely do we mean by the ‘entire universe’?

No one imagined galaxies beyond our own, let alone universes, until a remarkable eighteenth century German philosopher suggested that there were other “island universes”.

Immanuel Kant (1724-1804), son of a German craftsman, introduced dialectics into modern philosophy

This philosopher was Immanuel Kant, who was later to reintroduce the ancient Greek concept of dialectics into modern philosophy. In the late nineteenth century Engels enthusiastically praised Kant’s foresight and, in time, island universes were discovered by powerful telescopes, and termed ‘galaxies’. By the 1920s, the very great distances of some of these galaxies from our own galaxy had been measured.

After Einstein overturned Newtonian physics, especially with the advent of the Big Bang theory of the origins of the universe, it became possible to conceive of universes outside of our own, leading to various concepts of a multiverse or meta-universe – a set of universes which are speculated to arise in various ways. So now, when we say ‘the universe’ we may not mean everything that exists, but only ‘our universe’ as opposed to possible other universes. To most physicists the term ‘the universe’ tends to refer to our universe, the universe we can observe. The Astronomer Royal, Martin Rees, who adopts the term “our universe” in this way, writes:

What’s conventionally called ‘the universe’ could be just one member of an ensemble. Countless others may exist in which the laws [of physics] are different…

This new concept is, potentially, as drastic an enlargement of our cosmic perspective as the shift from pre-Copernican ideas to the realisation that the Earth is orbiting a typical star on the edge of the Milky Way, itself just one galaxy among countless others…

The big bang that triggered our entire universe is, in this grander perspective, an infinitesimal part of an elaborate structure that extends far beyond the range of any telescope. (Rees, Before the Beginning, Our universe and others, p3-4)

Our universe appears to have had a hot, dense origin popularly known as the Big Bang. It does not exclude the possibility of other universes beyond our own. Scientists speculate about a substratum, as we term it here, from which universes might naturally arise. For instance, some envisage universes budding off from a quantum substratum like bubbles budding off from foam. But in modern science neither our universe, nor a multiverse consisting of many universes, is compatible with the old Newtonian universe defended by Woods.

For many scientists today, one significant element of our universe is the special physical attributes of atomic particles and forces of which it is comprised: “The entire physical world,” says Rees, referring to our universe, “is essentially determined by a few basic ‘constants’: the masses of some so-called elementary particles, the strength of the forces – electric, nuclear and gravitational – that bind them together and govern their motions.” (Rees, Before the Beginning, p236)

But if these forces were only marginally different the universe that we know would be a physical impossibility. Yet we do not know whether these forces are the only possible combination of constants – maybe there are many other possible variations, producing many other types of universe, beyond our own, which are hardly conceivable to us today.

In our universe the known physical laws appear to apply universally, and the space, time, matter and energy of our universe are bound together. Scientists often use the term space-time, meaning, in a special sense, that time and space together can be treated as a single phenomenon. This discovery was based on Einstein’s theory of relatively, which also showed that mass and energy are linked. For instance, when an atomic bomb explodes a small amount of enriched uranium is converted into a massive amount of energy, a dreadful demonstration of the truth of Einstein’s theory.

In Newton’s universe, space and time have an absolute existence of their own, independent of each other and of matter. Einstein showed that if the mass of our universe exceeded a certain amount, the gravitational effect of all that mass would cause space-time to bend until the universe became ‘closed’ like a sphere (which has three dimensions), but in the four dimensions of space-time (which is not easily conceived by us). By closed, we roughly mean that anyone travelling in the universe in what appears to be a straight line could eventually find themselves back at their starting point, as if we were ants scurrying around the inside wall of a gigantic football.

Diagram: Space and time is bent around a massive object such as a star (shown by the dimple). To an observer from a distance, distances have been shortened, and time is also running a little slower.

Light (shown by the line) passing nearby is bent from the straight path indicated by the dashes.

We will discuss how Einstein revolutionised our concepts of time and space in the course of this survey. But to anticipate these arguments slightly, let us take a moment to consider what this remarkable concept means. A star, like our sun, bends space and time – something that has been routinely confirmed by observation since 1919.

Light travelling to earth from a star will be bent if it passes close to an intermediate star or galaxy. Space and time are bent by the great mass of this intermediate star or galaxy, and light passing through this bent space and time behaves just as if it was going though a gigantic lens. Today, this is routinely observed and quantified. It can give rise to gravitational lensing, an extremely useful tool in astronomy, in which a galaxy or other object in front of a distant object acts like a giant magnifying glass.

In the same way, the mass of all the stars in the universe collectively, together with other matter, have the effect of bending the space and time of the entire universe – and if there is enough mass, it could be bent right round back on itself in various ways. Current observations, however, suggest that there is not enough mass for this to happen.

We should point out that Woods calls this result of Einstein’s general theory of relativity a “regression to the mediaeval world outlook of a finite universe”, in a short passage particularly densely populated with false ideas. (pp382-3) But we should also point out that earlier in Reason in Revolt, Woods has already unintentionally endorsed the idea of space-time bending, not once but twice: “This was proved in 1919, when it was shown that light bends under the force of gravity.” (p106) Later, Woods presents both his viewpoints on the same page, first appearing to deny or at least denigrate Einstein’s theory and then going on to say that:

… [Einstein] predicted that a gravitational field would bend light rays… In 1919… Einstein’s brilliant theory was demonstrated in practice. (p154)

Woods seems to fail to grasp here that the 1919 experiment attempted to show that space – ‘empty space’ – is indeed distorted by the existence of a massive body and that the effect of gravity is a consequence of this distortion. Arthur Eddington’s famous 1919 observations, taken during an eclipse on the island of Principe off the West African coast, showed that light from a star that passed very close to the sun was indeed bent by the mass of the sun.

Eddington’s grand expedition was the first experimental test of Einstein’s general theory of relativity. His measurements were soon improved upon, and much more accurate measurements have confirmed his result – the confirmation of Einstein’s prediction that space and time is warped. Newton’s theory of gravity can also be used to suggest that light bends by a certain amount. But Einstein’s theory predicts that the gravitational effect on light should cause it to bend by roughly twice as much as predicted by Newtonian science – and light does, indeed, bend by the amount predicted by the general theory of relativity as it follows the curvature of space-time.

When scientists today speculate about other ‘island universes’ they may envisage universes governed by different laws which lie beyond the space-time of our universe and which, therefore, could not be measured in distances and times from our universe. Such universes might not be gravitationally attracted to one another or to the matter in our universe and may have none of the basic ‘constants’ as Rees calls them, of our universe – or even, some suggest, the same space-time dimensions. Science stands on the very first stepping-stone of a path to the possible discovery of other universes, in the same way that Kant anticipated a vast enlargement of our horizons when he speculated about other ‘island universes’. So the term ‘the universe’ today can either refer specifically to our universe or, more broadly, to our universe and anything that may lie beyond it. But Woods is defending the old Newtonian notion of an essentially unchanging universe comprised of infinite time and space with “galaxies and more galaxies stretching out to infinity”.

Next: What is infinity?