Science and Marxism

Hegel on the dialectics of infinity

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

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