On Hegel's 'Philosophy of Nature' : A Free Reflex of Spirit - part nineteen.

'There are other planets'

by Konstantin Dmitrievich Balmont  (1867 - 1942)

There are other planets. The skies are clear and completely calm there,

the mimosa blossoms are softer, and sweet grasses grow higher.

The clarity that plays there, it is less changeable than here,

we cherish it always and can always smile.

There are other planets for another existence. 

We will return there, but later, but much later,

when a day we have lost cannot be returned to us unchanged,

when we don't like anything in this world where the herbs grow grey

and without fragrance, funereal herbs.

The sweet grass trembles sadly under the stars,

seeking peace in the mournful places,

and pushes on our tombs,

so calmly, so calmly, so sad and calm,

under the serenity of the moon.

'Есть другие планеты'

Есть другие планеты, где ветры певучие тише,

Где небо бледнее, травы тоньше и выше,

Где прерывисто льются переменные светы,

Но своей переменой только ласкают, смеются.

Есть иные планеты, где мы были когда-то,

Где мы будем потом, не теперь, не теперь,

А когда, потеряв, себя потеряв без возврата,

Мы будем любить истомленные стебли седых

Шелестящих трав без аромата.

Тонких высоких, как звёзды печальных,

Любящих сонный покой мест погребальных,

Над нашей могилою спящих

И тихо, так тихо, так сумрачно тихо

Под луной шелестящих.

====

'The Lightning' ('Il fulmine'), 1932, Alessandro Bruschetti

Georg Wilhelm Friedrich Hegel, (1770 - 1831). 'The Philosophy of Nature'.

Hegel and Kepler.

Scientism. The view that science and the scientific method are the best or indeed the only way to render truth about the world and reality.

Scientism. A word to confine to the dustbin, in my opinion. An empty charge the purpose of which is to close down debate on the grounds that your interlocutor is deluded by prejudice and ideology whereas in fact he or she is drawing upon the enormous success enjoyed by science in answering questions that belong to, well, science. As you will have just seen if you were willing to subject yourself to and able to endure the above preacher prattle, it is hard to argue against what someone is saying when they are saying nothing at all. 'They don't understand what serious Biblical people mean when they use the word God. The one thing God is not is one being or one cause among many. Thomas Aquinas refers to God not as ens summum the highest being but rather ipsum esse which means the sheer act of to be itself .. God is rather the answer to: why is there something rather than nothing? What explains contingent reality? Prattle ... prattle ... matter is characterised by potentiality ... he is cribbing of Aristotle there ... matter is characterised by potentiality .. is that saying anything at all?

(On Thomas Aquinas, (1225 – 1274), see my articles 'A Bellowing Ox and a Roaring Lion', parts one to six).

'And I John saw the holy city, new Jerusalem, coming down from God out of heaven, prepared as a bride adorned for her husband. And I heard a great voice out of heaven saying, Behold, the tabernacle of God is with men, and he will dwell with them, and they shall be his people, and God himself shall be with them, and be their God. And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away. And he that sat upon the throne said, Behold, I make all things new. And he said unto me, Write: for these words are true and faithful. And he said unto me, It is done. I am Alpha and Omega, the beginning and the end. I will give unto him that is athirst of the fountain of the water of life freely. He that overcometh shall inherit all things; and I will be his God, and he shall be my son. But the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death'.

- 'Revelation' 21:2-8

Such is the God of the New Testament that Bishop Barron believes in. Does that sound like God as ipsum esse whatever that is supposed to mean? So which is it? A God of justice, wrath etc that interacts with the world? Or God the sheer act of to be itself (the God of 'serious' Biblical people)? Make your choice because you cannot have both.

Side note: as for that oft told tale concerning Pierre-Simon, Marquis de Laplace, (1749 - 1827). 'Sire, je n'ai pas eu besoin de cette hypothèse'. 'No, Sire, I had no need of that hypothesis'. Reputed reply to Emperor Napoleon I who had asked why he hadn't mentioned God in his discourse on secular variations of the orbits of Saturn and Jupiter. 'Mais où est Dieu dans tout cela?'. 'But where is God in all this?'. Did that happen? According to astronomer Hervé Auguste Étienne Albans Faye, (1814 – 1902):

'In fact Laplace never said that. Here, I believe, is what truly happened. Newton, believing that the secular perturbations which he had sketched out in his theory would in the long run end up destroying the Solar System, says somewhere that God was obliged to intervene from time to time to remedy the evil and somehow keep the system working properly. This, however, was a pure supposition suggested to Newton by an incomplete view of the conditions of the stability of our little world. Science was not yet advanced enough at that time to bring these conditions into full view. But Laplace, who had discovered them by a deep analysis, would have replied to the First Consul that Newton had wrongly invoked the intervention of God to adjust from time to time the machine of the world (la machine du monde) and that he, Laplace, had no need of such an assumption. It was not God, therefore, that Laplace treated as a hypothesis, but his intervention in a certain place'.

- 'Sur l'origine du monde: théories cosmogoniques des anciens et des modernes'

I don't expect you to sit through this comical 'debate' between William Lane '2 PhDs and we know it is 2 because it is always being mentioned' Craig, (1949 - ), the philosopher and Sean Carroll, (1966 - ), theoretical physicist, who at one point even says that this isn't a debate. Just the first 10 or 15 minutes will do for me to make my point.

Craig, the philosopher, straight in with nonsense philosophical arguments. The Kalam cosmological argument. Everything that begins to exist has a cause. The universe began to exist. Therefore, the universe has a cause. Therefore God. And Carroll, straight in with the science. 'If the universe began to exist it had a transcendent cause.. the problem with this premise is that it is false. There is no explanation or justification given for this premise ... but there is a bigger problem with it in that it is not even false. The real problem is that these are not the right vocabulary words to be using when we discuss fundamental physics and cosmology. This kind of Aristotelian analysis of causation was cutting edge stuff 2500 years ago. Today we know better. Our metaphysics must follow our physics. That is what the word metaphysics means. And modern physics, you open a quantum field textbook or a general relativity textbook, you will not find the words transcendent cause anywhere. What you find are differential equations. This reflects the fact that the way physics is known to work these days is in terms of patterns, unbreakable rules, laws of nature .... given the world at any one time we will tell you what happens next, there is no need for any extra metaphysical baggage like transcendent causes on top of that, it is precisely the wrong way to think about how the fundamental reality works ... ' And then he continues on about mathematical models matching on to reality ... And then back comes Craig with his philosophy ... is God's existence more probable given current cosmology ... the universe begins to exist ... prattle and more prattle ... No advancement is made.

What Craig the philosopher should have seized upon was Carroll the physicist (although according to Wikipedia he is also a philosopher but I see no evidence of that) pointing out that open up a modern physics textbook and you find differential equations, and physics being concerned with finding mathematical models matching on to reality .. now there you can have a proper debate between a physicist and a philosopher. See my article part twelve in this series. Peter J. Lewis in 'Quantum Ontology: A Guide to the Metaphysics of Quantum Mechanics' writes: Quantum mechanics 'is a theory in which we have no idea what we are talking about, because we have no idea what (if anything) the basic mathematical structures of the theory represent'.

Hegel took the side of Gottfried Leibniz, (1646 – 1716), view in opposition to that of Isaac Newton’s, (1643 - 1727), interpretation of Johannes Kepler's, (1571 – 1630), Laws, though in a more systematic way objected to the misrepresentation by Newton of Kepler’s laws and in particular and the mis-redirection of Newtonian orientation of physics and cosmology in general towards the unreflected view of understanding as opposed to the view of reason or dialectics, a distinction between unreflected thinking (denkende) and comprehending consideration of Nature (begreifende Naturbetrachtung) yet Newtonian physics and British empiricism won out in the end in natural science and yet developments in quantum phenomena are nudging theoretical physics further into the domain of metaphysics, mathematical idealism and theology.

Not my area of expertise but Peter J. Lewis in 'Quantum Ontology: A Guide to the Metaphysics of Quantum Mechanics' writes: Quantum mechanics 'is a theory in which we have no idea what we are talking about, because we have no idea what (if anything) the basic mathematical structures of the theory represent'. Mathematical models of the universe are no explanation of reality unless you accept a particular metaphysical assumption .. mathematical idealism. Isaac Newton's programme of mathematization of physics was brought to its logical conclusion at the turn of twentieth century with Albert Einstein's, (1879 – 1955),  mathematical idealism and his theories of relativity demonstrating conceptual physics which in fact reflected Hegelian dialectic for any truth being extended beyond certain limits either turns to its opposite or becomes an absurdity and so the discovery of quantum phenomena, indeed biological evolution too, subverts the mathematical idealism that began with Newton thereby vindicating Hegel’s view of reason or dialectics. That would have been an interesting debate, mathematical idealism, what would Carroll have to say about his supposed method for explaining, (or modelling as he puts it, modelling what? and see below about explanation in general), reality being grounded upon a metaphysical assumption that needs to be defended and argued for if you are claiming to be explaining reality.

Hegel is generally understood as one whose philosophical interests were principally oriented around the topics of history, politics, art and religion while the contents of the Encyclopaedia’s Part II, 'The Philosophy of Nature', have rarely been treated as having the same importance and even those generally responsive Hegelianism this is usually understood as so dependent upon the state of empirical science of his time as to have little relevance today and critics not so responsive and who are unaware of the depths of Hegel's vast learning dismiss it entirely as a the mere interference of an amateur out of his depth, a rear-guard endeavour to preserve a prior philosophical/theological worldview that was being more and more challenged by the great discoveries of science. While defenders of Hegel like to characterise his stance towards scientific explanation by characterising it as directed more against the promotion of an exclusively scientific worldview (scientism) than science in itself yet his critics include those that one would expect to appreciative of a supposed anti-scientistic stance. William Whewell, (1794 –1866),  English polymath, scientist, Anglican priest, philosopher, theologian, and historian of science, whom Thomas Posch (1974 - 2019, for more of whom see below), cites as having declared Hegel’s account of the solar system as non-sensical. Whewell, an Anglican minister, opposed Darwin's theory of evolution because of its conflicts with Christian doctrine Darwin’s theory of evolution. One of his works is called 'Indications of the Creator', 1845, but of course Darwin was after Hegel's time, it was the attack on science’s idol Isaac Newton that ruffled the feathers of Whewell the Christian science advocate. It is worthwhile quoting Whewell here because this is rather amusing, he is being ironic (or sarcastic) of course when he refers to celebrated metaphysicians:

'A similar absence of distinct mechanical thought appears in some of the most celebrated metaphysicians of Germany. I have elsewhere noted the opinion expressed by Hegel, that the glory which belongs to Kepler has been unjustly transferred to Newton; and I have suggested, as the explanation of this mode of thinking, that Hegel himself, in the knowledge of mechanical truth, had not advanced beyond Kepler's point of view. Persons who possess conceptions of space and number, but who have not learnt to deal with ideas of force and causation, may see more value in the discoveries of Kepler than in those of Newton. Another exemplification of this state of mind may be found in Mr. Schelling's speculations; for instance, in his Lectures on the Method of Academical Study. In the twelfth Lecture, on the Study of Physics and Chemistry, he says, 'What the mathematical natural philosophy has done for the knowledge of the laws of the universe since the time that they were discovered by his (Kepler's) godlike genius, is, as is well known, this: it has attempted a construction of those laws which, according to its foundations, is altogether empirical. We may assume it as a general rule, that in any proposed construction, that which is not a pure general form cannot have any scientific import or truth. The foundation from which the centrifugal motion of the bodies of the world is derived, is no necessary form, it is an empirical fact. The Newtonian attractive force, even if it be a necessary assumption for a merely reflective view of the subject, is still of no significance for the Reason, which recognizes only absolute relations. The grounds of the Keplerian laws can be derived, without any empirical appendage, purely from the doctrine of Ideas, and of the two Unities, which are in themselves one Unity, and in virtue of which each being, while it is absolute in itself, is at the same time in the absolute, and reciprocally'. It will be observed, that in this passage our mechanical laws are objected to because they are not necessary results of our ideas ; which, however, as we have seen, according to the opinion of some eminent mechanical philosophers, they are..... '

- 'The Philosophy of the Inductive Sciences, Founded upon their History'

'Acrobazie aeree sulla campagna Umbra', 1934, Alessandro Bruschetti

While Hegel’s social and political philosophy has continued to be the most prominent focus of interest within those re-assessments of his philosophy that have developed over the last fifty years an informed understanding of his philosophy of nature is somewhat lacking and what there is is often accompanied by a general interpretation emphasizing the degree to which Hegel had relinquished the pretension to a metaphysics of pure reason with which Immanuel Kant, (1724 - 1804), had broken, some type of substantive, a priori theoretical knowledge of the world that might be seen as being in competition with modern science and such defensive strategies depend upon lierating Hegel from those shortcomings of Kantian transcendental idealism of which he had been an astute critic. Recent examples of such a metaphysically modest interpretative attitude extended to Hegel’s 'Philosophy of Nature' can be found in the accounts of Michael Wolff, (1942 - ), Sebastian Rand and Christian Martin, Wolff for instance conends that 'one can only understand Hegel’s procedure [in relation to the philosophy of nature] appropriately if one conceives it as a modification of Kant’s transcendental philosophy' with this attitude manifest in the Logic from which the 'Philosophy of Nature' issues in particular in relation to the status of its central metaphysical category of the absolute idea. This notion, Wolff assers, refers to 'nothing other than our own logical, or more precisely: dialectical thinking—our thinking by which we know the contradictory character of the categories'. Martin similarly writes that the Logic 'deals with the thinking of us thinkers' which 'requires spatial and temporal embodiment'. Nonetheless for methodological reasons it does so 'in abstraction from such embodiment'. This initial abstraction can give the false impression that the Logic 'treats of a different kind of thinking from ours', that of a divine thinker, for instance and given that such humility at a metaphysical level would deflate even the intelligibility of a thinkable transcendent domain beyond empirical reach, the domain of Kant’s unknowable things in themselves, a certain scientific realism might now be thought to possibly coexist with Hegel’s project.

Similar deflationary elements can be discerned in Rand’s interpretation of Hegel’s philosophy of nature since as Rand has it as conceived philosophically by Hegel nature is interpreted not so much as such but from a particular point of view, that oriented by project of the realization of human freedom structuring his philosophy as a whole, and while the idea of nature as construed internally to a goal of practical reason and action has a Kantian flavour about it Rand emphasises that what separates Hegel from Kant is the different ways in which the concepts freedom and nature are related in their respective philosophies. For Kant the ethical freedom of a human being is primarily conceived a type of freedom from what constitutes him or her as a natural being while in contrast Hegel had recognised nature in the broader sense as needing to be included in the domain for which freedom is to be attained.

Connecting the freedom of any particular human to that of all others becomes now widened out to a picture in which the freedom of humans as a whole is somehow similarly bound up with realms of nature beyond them and evidently this is a point of view possibly more appertenant in the face of the present ecological crisis threatening humanity. Construing Hegel’s 'Philosophy of Nature' within the context of such metaphysically humle accounts of Hegel’s Logic is not the only way in which his 'Philosophy of Nature' has sought to be rescued with more robustly metaphysical interpretations being pursued by others, for instance upon the basis of Hegel’s Logic James Kreines argues for the substantive rationalist metaphysical claims of the objective existence of reason in the world independently of human reasoners connected to a non-empiricist understanding of natural laws.

Whether a deflationary actualist understanding of Hegel’s metaphysical claims is desirable or not I put to one side, anyway I don't go for anything deflationary when interpreting Hegel, that is totally against the spirit of Hegelian philosophy, I just make he point that such a humble strategy is more likely to address the types of concerns typical of Whewell’s working scientists likely to be suspicious of any substantive claims about the world arrived at in some a prioristic manner rather than empirically but then who cares about satisfying Whewell who denies the science when it grates against his Christian beliefs?. From the perspective of Wolff’s understanding of Hegel’s ultimate metaphysical content of the Logic a response to the working scientist’s scepticism might start with mot much more than a request that she or he reflects upon the self-correcting logical processes he or she shares with her scientific colleague, effectively what Wolff’s version of Hegel’s absolute idea might amount to, and to consider whether the epistemic success of those processes could be explained in terms of the same causal explanations of the world that those processes have accompanied, and this could make possible to ensue a spirit of harmony between Hegel or at least some more recent application of his purported approach and the scientists.

Whatever metaphysically humble interpretation you propose it must align with Hegel’s endorsement of the ancient model of philosophy and of Plato. Olivier Depré emphasises this dimension of Hegel’s position in 1801 whereby indications of such an engagement can be discovered in Hegel’s first venture into the Philosophy of Nature his 1801 'Dissertation on the Orbits of the Planets'. Hegel contended that Kepler rather than Newton had been able to formulate realistically understood laws of planetary motion on the basis of empirical findings and the criteria to which he appeals would speak to the modern scientist nonetheless for Hegel Kepler’s success had been connected to his pursuit of what appears the antithesis of the modern empirical approach, Plato’s music of the spheres approach to cosmology. Cinzia Ferrini has summarised some of the negative reception of Hegel’s dissertation as including the accusations of 'incongruities, inconsistencies, absurdities, obscurity, external formalism and powers of imagination, crudest empirical ignorance, ridiculous errors and lack of scientific knowledge, ill-grounded a priori logical deduction of empirical reality, introduction of real content hidden within the formalism of the logical movement, the inability of the dialectical method to produce any actual progress at a cognitive level, and so on'.

The Dissertation which manifests clear connections with principal themes of his later Philosophy of Nature adopts and adapts a sequence of seven numbers from Plato’s 'Timaeus', 1, 2, 3, 4, 9, 16, in an apparent endeavour to explain the comparative distances of the then known seven planets from the sun an assertion that he never retracted and it has been claimed that Hegel was thereby dismissing on a priori grounds the possibility that an eighth planet might be discovered and indeed doing so in the very year in which a minor planet or asteroid was discovered and were this to be the case Hegel would be engaging in the type of a prioristic reasoning to features of the actual world that he seems to deny in the 'Philosophy of Nature' with the contention that it is not only that philosophy must accord with the experience nature gives rise to, in its formation and in its development philosophic science presupposes and is conditioned by empirical physics.

'The relationship of philosophy to what is empirical was discussed in the general introduction. It is not only that philosophy must accord with the experience nature gives rise to; in its formation and in its development, philosophic science presupposes and is conditioned by empirical physics. The procedure involved in the formation and preliminaries of a science is not the same as the science itself however, for in this latter case it is no longer experience, but rather the necessity of the Notion, which must emerge as the foundation. It has already been pointed out that in the procedure of philosophic cognition, the object has not only to be presented in its Notional determination, the empirical appearance corresponding to this determination also has to be specified, and it has to be shown that the appearance does in fact correspond to its Notion. This is not however an appeal to experience in regard to the necessity of the content, and an appeal to what has been called intuition, which was usually nothing more than a purveyance of random concepts by means of fanciful and even fantastic analogies, is even less admissable here. These analogies may have a certain value, but they can only impose determinations and schemata on the objects in an external manner'.

- 'The Philosophy of Nature'

Several defenders of Hegel have responded to the gross exaggerations found in standard accusations of this a prioristic approach to cosmology including Posch and Hegel they note had been commenting on a similarly formulated series of numbers that had been offered by the astronomers Johann Elert Bode, (1747 –1826), and Johann Daniel Titius, (1729 – 1796), Titius had advanced the idea in 1766 with Bode taking it up in 1768, on the basis of which Bode had urged astronomers to search for an eighth planet between Mars and Jupiter, as predicted by the series. Hegel had simply suggested that were Plato’s series to be accepted rather than the Titius-Bode series then no thesis of a missing planet would be required and such defenders in addition point out that the greater part of Hegel’s dissertation had been devoted to a topic much more expected of a modern philosopher, a critique of the idea that Newton’s laws could be said to explain the laws of planetary motion that Kepler had arrived at empirically. From this point of view in his defence of Kepler over Newton Hegel’s strategy as Cinzia Ferrini has contended, 'leaves room for an absolute principle of empirical reality'. Brigitte Falkenburg also stresses the role played by regularized empirical observations in science on Hegel’s account and while the phenomena of empirical science for Hegel are theory-dependent and not the bare givens typically postulated by empiricists nonetheless 'the empiricist criticism of unobservables such as forces and atoms, was completely in the spirit of Hegel’s own phenomenological attitude towards physics itself'.

In his support of Kepler Hegel was on this reading supporting an eminently empirically based approach to astronomy. On the basis of his extensive observational data of the movement of Mars accumulated over decades, together with data acquired from his employer in Prague, Tycho Brahe, (1546 – 1601), Kepler had come to propose three laws of planetary motion, a planet orbits its sun in an ellipse that has the sun standing at one of its two foci, next, a line segment from the planet to the sun, its radius vector, sweeps out equal areas in equal units of time, entailing the planet speed up as it approaches the sun and slow down as it moves away, and finally the square of a planet’s orbit is proportional to the cube of the length of the shorter diameter of its elliptic orbit.

'Forme negli spazisiderali', 1973, Alessandro Bruschetti

With his universal laws of gravitational attraction Newton is standardly understood as having been able to explain Kepler’s laws because while Kepler’s had said something about the behaviour of planets Newton’s were able to account for the behaviour of many other things besides, for instance Newton’s laws simultaneously explained Galileo’s Law of Fall for terrestrial objects, that the velocity of a falling objects increases proportionally with the square of time, and this kind of subsuming generalization is standardly understood as exemplifying progress in science and Newton acknowledging standing on the shoulders of Kepler to see further ... Kepler had come from the area around Stuttgart, Hegel’s place of birth and and like Hegel had studied at the Tubingen seminary so as to the question as to why Newton’s success be denied the answer is that in line with the modern scientists’ insistence on the role of empirical evidence only Kepler’s laws were empirically based while Newton’s relied upon postulates that themselves could not be empirically justified.

Concerns with problems with the type of explanations employed by Newton were still extant within the generally scientific community at the turn of the nineteenth century and Hegel contended that Newton’s reduction of Kepler’s laws involved an unacceptable reliance upon infinitesimally small magnitudes and such an appeal to infinitesimals is standardly associated with the framework of differential and integral calculus of which Newton along with Leibniz had been co-developer and in Europe Newton’s mechanics had been developed throughout the eighteenth century with the aid of the calculus but that had not been the method employed by Newton himself in 'Principia Mathematica' nonetheless both Newton’s and calculus-based approaches to mechanics had relied upon the idea of infinitesimal magnitudes that Bishop Berkeley, (1685 – 1753), would later ridicule as the 'ghosts of departed quantities' for according to Berkeley the physicists were as much committed to entities as mysterious as those they condemned in the accounts of the theologians and one did not have to endorse Berkeley’s rival spiritualistic metaphysics to feel the weight of this critique and concern about the use of infinitesimals in science still persisted around the turn of the nineteenth century and in the Logic Hegel devotes considerable discussion to this problem engaging with books that were state of the art at that time. John Bell has described two French works appearing 1797, Joseph-Louis Lagrange’s 'Théorie des fonctions analytique', and Lazare Carnot’s 'Reflexi, ons sur la Metaphysique du Calcul Infinitesimal', as 'the last efforts of the eighteenth-century mathematicians to demystify infinitesimals and banish the persistent doubts concerning the soundness of the calculus'. Hegel owned both of these works and employs them in the Logic in discussing the problem of infinitesimals.

Indeed the infinitesimal had been just one of a number of bothersome mathematical magnitudes that had come to be employed in science since the seventeenth century, these impossible numbers, as Thomas Nagel, (1937 - ), calls them included the number zero, the negative numbers, and the imaginary number that when multiplied by itself gave the result of ̶ 1. These magnitudes had been introduced into the sciences as a consequence of the adoption of algebra from the late sixteenth century and had been demonstrated to be indispensable for the solution of the types of equations the emerging sciences employed yet pragmatic accomplishments alone did not appease critics who frequently pointed to the apparent absurdities involved in their employment. Kant had addressed the problem of negative numbers in a pre-critical essay in 1763 and their problem engendering status is said to have been particularly live around the turn of the nineteenth century when Hegel was writing his Dissertation and as with infinitesimals justification for such magnitudes could not merely be answered in empiricist terms but neither could they be justified in terms of the sorts of criteria that had been inherited from Greek mathematics and the axiomatic method in which theorems are deduced from axioms and definitions had provided the main model of justification found in Greek geometry and the Greeks had neither a well-developed system of algebra nor zero or negative numbers that would be required for such algebra. Simple justification by the success enabled by their employment was really all that remained but many at the time would have concurred as indeed many still would with Hegel’s declaration in the Dissertation that a principle must not be evaluated simply 'on the basis of its use and consequences'. One might afford these mysterious entities some role in the prediction of phenomena but one wonders why extend this to belief in the reality of those structures described in the theories that employed them.

Hegel thereby raises the question of the ontological status of mathematical objects within the coupling of physics and mathematics and issues a caveat against confusing 'purely mathematical relations with the physical ones; rashly taking the lines used by geometry to construct demonstrations of its theorems for forces or directions of forces'. Nonetheless this itself was not to deny that mathematical entities in themselves could have an empirical reality for Hegel contends that Kepler’s ellipses are instances of precisely this. Kepler’s ellipses are real figures to be observed over time as carved out by the actual movements of the planets in their paths around the sun and this is to ignore those imperfections in orbits caused by the gravitational effects of other planets, so-called perturbations and while Hegel recognises that it is a strength of Newton’s approach that it is able to account for these he does not view this as contradicting his main criticism, on the contrary his criticism is directed at Newton’s directed line-segments or vectors to which such elliptical shapes are reduced in his explanations of the elliptical orbits because of the role of infinitesimals in this reduction, and Hegel’s criticism here is directed towards that of the empirical reality of Newton’s vectorial analysans.

It is the reality of Newton’s posited rather than observed centripetal and centrifugal components into which Kepler’s curves (although personally I'd be more interested in Mrs. Kepler's) are resolved along with the forces defined in terms of them that are at issue here and justification for the resolution of a curve into rectilinear components is not to be found in pure geometry which 'does not modify the true form of the circle' and in contrast the non-pure geometry in question is one 'that endeavours to subject the circle to calculation and to express numerically the relation of circumference to the radius' and that does so by seeking refuge 'in the hypothesis of an infinitely-sided regular polygon' but which 'does this, however, in such a way that in the same move it suppresses this very polygon and the straight lines by means of the concepts of the infinite and of last ratio'. Two types of non-pure geometry can be seen as implicit in this passage, ancient and modern.

Hegel knew that in ancient times the mathematician Archimedes, (c. 287 – c. 212 BC),had employed a method for estimating the ratio between the circumference and the radius of a circle later designated the number π and a German translation from 1798 of Archimedes’ works 'On the Sphere and Cylinder and Measurement of a Circle' was in Hegel’s library and this was representative of Hegel’s 'heavy investment in standard works on the calculus and mechanics' as evidenced by his library as André Mense says, the publication dates of which seeming to indicate their having been ought around the time of the dissertation.

In this method of exclusion Archimedes had constructed a regular polygon inside a circle such that the polygon’s vertices touched the circle’s circumference and while the length of an arc of the circle could not be measured the lengths of the corresponding side of the polygon could be and a similar polygon could now be constructed outside the circle such that its sides touched the circumference of the circle and if P denotes the combined lengths of the sides of the inner polygon and P' the combined length of the sides of the outer polygon it can be observed that the circumference of the circle will be greater than P but less than P'. By increasing the number of sides of both polygons the difference between P and P' steadily decreases allowing the circumference of the circle to be estimated with greater and greater accuracy, Archimedes had achieved a value of pi as lying between 3 1/7 and 3 10/71. In the Principia Newton would employ effectively the same technique into order to measure the area under a curve but for Hegel there were crucial differences in the ways the Greeks and Newton had employed this mathematical procedure.

Techniques like that of Archimedes had allowed the Greeks to apply geometry to the world and estimate astronomical distances with great accuracy and Archimedes’ friend Eratosthenes for instance made accurate calculations of the circumference of the earth yet while the Greeks were satisfied with values of π that were accurate enough for their purposes Newton imagined this method as being carried to infinity such that what had been originally lines were ultimately reduced to points and the Greeks were in general not positively disposed to the notion of the infinite as conveyed by Alexandre Koyré’s contrast of the 'closed world' of the Greeks to the 'infinite universe' of the moderns. Hegel says that with this modern mathematics 'conceals this identity of incommensurables with the word ‘infinite'' alluding to the fact that the Greeks had treated curves and linear magnitudes as incommensurable magnitudes, they were different kinds of number. The number pi was not proved to be irrational until the eighteenth century but the Greeks were very aware of the incommensurability between curved and linear numbers as testified to by the ancient problem of squaring the circle. Newton hides this difference, he 'suppresses this very polygon and the straight lines' from which it calculated his areas. When Hegel refers to 'the geometry called higher geometry' that 'reduces the plane to the line, and both to the infinitely small, that is, to the point' in this way he evidently has in mind the modern analytic geometry that had been introduced in the seventeenth century by René Descartes, (1596 – 1650), and Pierre Fermat, (1607 - 1665), and upon which Newton had drawn in his mechanics.

'Occhio magico', 1951, Alessandro Bruschetti

In his 'Geometrie' of 1637 Descartes had proposed a type of algebraized geometry by introducing what is now taken as standard x and y orthogonal coordinates within which geometric figures could be situated and as each point on a two-dimensional figure can be linked to a pair of numbers via the coordinates such a figure could be represented by an algebraic equation as when for instance a circle centred on the origin of the coordinates and of radius a units is represented by the equation x2 + y2 = a2 and analytic geometry had rapidly caught on in virtue of the great advantages offered to developing sciences, Newton et al were concerned that the use of algebra did not conform to the strict axiomatic method found in ancient that is to say synthetic geometry and with the dependence upon infinitesimals Newton’s methods were still able to be understood at the end of the eighteenth century as far from being free from the sorts of metaphysical fictions that operative contemporary scientists might denounce in Hegel and alternatively Kepler’s ellipses were for Hegel genuinely empirical objects and he had emphasised Kepler’s empiricist attitude in his rejection of Bruno’s conception of an infinite universe, nonetheless the empirical scientist was only one aspect of Kepler’s character and Hegel’s reference to Plato’s number series is simply presented as a hypothetical alternative to an existing Titius-Bode law, and yet there is an alternative Kepler, the Platonic cosmologist, and Hegel knew that Kepler’s empirical researches coexisted with a commitment to the ancient thesis of the music of the spheres as expressed in Plato’s 'Timaeus' and in a move similar to Hegel’s employment of Plato’s number series Kepler in his 'Mysterium Cosmographicum' of 1596 had modelled the solar system by embedding the orbits of the planets within a nested structure made up of the five Platonic solids each of the solids fitting between the spheres of the orbits of the then-known six planets (from outer to inner: Saturn - (cube) - Jupiter -(tetrahedron) - Mars - (dodecahedron) - Earth - (icoshedron) - Venus -(octahedron) - Mercury) so Kepler’s enthusiasm had persisted and while the first two of his laws had been published in one of the major works of modern astronomy, his 'Astronomica Nova' of 1609, the third law would be published ten years later in 'Harmonies Mundi' devoted mostly to the music of the spheres thesis.

Kepler was still in part grounded in the ancient cosmos that was in the process of being transformed into the modern universe and granted the relatively informed level of the discussion of contemporary issues that could be found in earlier parts of the Dissertation one wonders about Hegel's point with regard to his allusions to Platonic cosmology, well Hegel’s attitude to the Platonic musical cosmology correctly understood discloses fundamental features of Hegel’s logic that can be seen to complement his non-atomistic empiricism in matters of science. In brief the importance of Plato’s cosmology was not so much cosmological as logical, and Ferrini has argued for the logical infrastructure of Hegel’s assertion of the empirical realism of Kepler’s cosmology.

Plato’s cosmos gave expression to the logical structure implicit in the pursuit of a modern scientific cosmology, as Hegel understood it. In 1794, during his time with Hegel at the Tübingen Seminary Friedrich Schelling, (1775 – 1854), had written a commentary on Plato’s 'Timaeus' that would find its way into the philosophy of nature that he would continue to pursue during his time at the University of Jena and his collaboration with Hegel and Schelling’s subsequent 1798 work 'On the World-Soul' had caught the attention of the scientifically trained romantic Naturphilosoph Franz von Baader (1765 – 1841),who in the same year published 'On the Pythagorean Tetrad in Nature, or The Four Regions of the World' critical of Schelling’s adoption of Kant’s model of opposed attractive and repulsive forces in nature found in his 'Metaphysical Foundations of Natural Science' and such opposed forces Baader contended should be seen as expressions of a single underlying force, gravity or heaviness (Schwere), which was to be considered as 'the immediate expression of the individual inhering in and individualizing itself (sich individualizierenden) in all single (einzelnen) or moveable bodies'. Baader’s influence is evident here in the Dissertation in terms of ideas that would be continued in Hegel’s 'Philosophy of Nature'. Hegel begins Part II of the Dissertation with Baader’s idea: Gravity so constitutes matter that matter is objective gravity.

'God’s actions are not external nor mechanical nor arbitrary nor coincidental. One thing must be clear: the forces they claim God put into matter truly dwell therein; indeed, they constitute the essence of matter in the principle of opposed forces, internal and immanent to it. Mechanics avoids this concept with its claim that inertial matter is always moved by an external impulse or, what amounts to the same thing, by forces alien to matter. It recognises neither God nor true force effectively, nor that which is internal and necessary. Mechanics only accepts external causes and does not comprehend nature rationally, so it is incapable of advancing to the principle of an identity that asserts difference within itself. Once restored to us, this principle went on to revivify philosophy, separated mechanics from physics, which, distinguished from mechanics by more than the name ‘dynamics,’ it finally gave back to philosophy. We shall now present the elements of the planetary system and develop them briefly'.

'Gravity constitutes matter such that matter is objective gravity. It is one and the same matter dividing itself into poles and thereby creating a line of cohesion, generating diverse shapes in a series of evolutions with different relations between the factors. This is gravity’s real difference, from which we distinguish the other, ideal difference, that of the potentials of time and space. One double thus implies another: one of poles, the other of potentials; and that makes four regions. Let us first consider the cohesion line. Gravity draws this line by asserting itself at all points, each of which is distinct in itself due to the reciprocal relations of factors, producing a series of nodes and centres for itself. In each of these points there is no lack of that multiplicity of relations to the others, now drawn together under the law and organisation of each, bundled by the power of its own principle. The solar system draws a line so much greater than the rest, which makes it so much more powerful, for where the cohesion line is broken, the body at that point carries its centre of gravity within itself; not with an absolute power certainly, but with greater force than that of the other bodies. No body, no matter that it is a whole in itself, is completely independent of the others and each is part and organ of the larger system. Still, the heavenly bodies enjoy, if not perfect, surely the greatest possible freedom and independence from gravity. The planets were not wandering aimlessly through infinite space on rectilinear paths, when they just happened to be flying in the neighbourhood of the sun and were forced under its law onto their orbits. And that hypothetical centrifugal force is not what holds them back from the sun. Rather, because they form an original system with the sun, the true cohesion force holds them firmly in place and keeps them apart'.

- 'Dissertation on the Orbits of the Planets'

In the Philosophy of Nature he continues to describe gravity (die Gravitation) in the same way as the true and determinate notion of material corporeality ... Universal corporeality divides itself essentially into particular bodies (besondere Körper), and links itself together in the moment of singularity or subjectivity (Einzelheit oder Subjectivität), as determinate being appearing in motion; this, in its immediacy, is thus a system of many bodies.

'Gravitation is the true and determinate Notion of material corporeality realized as the Idea. Universal corporeality divides itself essentially into particular bodies, and links itself together in the moment of individuality or subjectivity, as determinate being appearing in motion; this, in its immediacy, is thus a system of many bodies'.

- 'Philosophy of Nature'

In the following paragraph Hegel describes 'the concept of gravity/heaviness (Begriff der Schwere) as that which realizes its full freedom it these bodies. The moments of gravity 'suffer the fate of being grasped as distinct forces corresponding to the forces of attraction and repulsion' such that the 'thought of universal gravity is annulled by this'. In the Dissertation Hegel also linked this Baaderian view to Kepler, who 'knew that gravitation was a quality universal to bodies'. In the 'Philosophy of Nature' Kepler’s views are defended whereby the solar system had been described in ways stressing the ideas of singularity and externality, notions central to his concept of nature as such hence the sun, exemplifying the category of abstract universality, stands at an extreme to lunar and cometary bodies which express the maximally external form of singularity (Einzelheit])while planets 'simultaneously stand as much in the determinations of self-externality, as they do in that of being-in-itself; they are in themselves centres and find their essential unity through relating themselves to the universal centre. This stress on externality and singularity links Hegel by way of Baader’s heterodox neo-Platonic understanding of the Christian Trinitary doctrine back to Plato’s 'Timaeus' and its Pythagorean features and from there to a criticism of Aristotle’s formal syllogism. The 'Pythagoraische Quadrat' in the title of Baader’s book had referred to a figurative number employed by ancient Pythagorean mathematicians called the tetraktys consisting of an array of 10 elements arranged like the pins in ten-pin bowling, that is, in four rows of 1, 2, 3 and 4 units respectively. Baader, a devout catholic, had linked the tetraktys to a triangular representation of the Holy Trinity via a diagram containing an equilateral triangle, the Catholic symbol of the Holy Trinity, with a central point. It was intended to somehow connect Catholic trinitarian theology to the world of the ancient Pythagoreans in a way which stressed the this-worldly nature of God’s incarnated son while at the same time avoiding pantheism. Baader also appears to have been responsible for having introduced Hegel and others around this time to the Silesian mystic Jakob Böhme.

Hegel alludes to Baader’s four regions of the world in the Dissertation in a discussion featuring Schellingian concepts but Hegel’s interest in Baader’s Pythagorean ideas is also evident in a now lost diagram dating from around the time of the Dissertation and found after Hegel’s death by his biographer Karl Rosenkranz, depicting a triangle of triangles showing the inverted embedding of one equilateral triangle within another which manifestly relates to both the ancient tetraktys and Baader’s Quadrat and according to Rosenkranz the diagram had been an endeavour to represent what Plato had described in the 'Timaeus' as a complex double-relation, a ratio of ratios, responsible for the unification of the various parts of the cosmic animal. In the 'Lectures on the History of Philosophy' Hegel refers to this beautiful bond as a syllogism and in this way Plato’s cosmology could be interpreted as a presentation of the type of logic which Hegel was pursuing in Part I of the Encyclopaedia and that was to provide the logical presuppositions for the 'Philosophy of Nature' and the 'Philosophy of Spirit'.

'A fondo', 1933, Alessandro Bruschetti

'The Four (τετράς) is the triad but more developed, and hence with the Pythagoreans it held a high position. That the tetrad should be considered to be thus complete, reminds one of the four elements, the physical and the chemical, the four continents, &c. In nature four is found to be present everywhere, and hence this number is even now equally esteemed in natural philosophy. As the square of two, the fourfold is the perfection of the two-fold in as far as it—only having itself as determination, i.e. being multiplied with itself—returns into identity with itself. But in the triad the tetrad is in so far contained, as that the former is the unity, the other-being, and the union of both these moments, and thus, since the difference, as posited, is a double, if we count it, four moments result. To make this clearer, the tetrad is comprehended as the τετρακτύς, the efficient, active four (from τέτταρα and ἄγω); and afterwards this is by the Pythagoreans made the most notable number. In the fragments of a poem of Empedocles, who originally was a Pythagorean, it is shown in what high regard this tetraktus, as represented by Pythagoras, was held:

'If thou dost this,

It will lead thee in the path of holy piety. I swear it

By the one who to our spirit has given the Tetraktus,

Which has in it eternal nature’s source and root'.

'From this the Pythagoreans proceed to the ten, another form of this tetrad. As the four is the perfect form of three, this fourfold, thus perfected and developed so that all its moments shall be accepted as real differences, is the number ten (δεκάς), the real tetrad. Sextus (adv. Math. IV. 3; VII. 94, 95) says: 'Tetraktus means the  number which, comprising within itself the four first numbers, forms the most perfect number, that is the number ten; for one and two and three and four make ten. When we come to ten, we again consider it as a unity and begin once more from the beginning. The tetraktus, it is said, has the source and root of eternal nature within itself, because it is the Logos of the universe, of the spiritual and of the corporeal'. It is an important work of thought to show the moments not merely to be four units, but complete numbers; but the reality in which the determinations are laid hold of, is here, however, only the external and superficial one of number; there is no Notion present although the tetraktus does not mean number so much as idea. One of the later philosophers, Proclus, (in Timæum, p. 269) says, in a Pythagorean hymn:—

'The divine number goes on,'...

“Till from the still unprofaned sanctuary of the Monad

It reaches to the holy Tetrad, which creates the mother of all that is;

Which received all within itself, or formed the ancient bounds of all,

Incapable of turning or of wearying; men call it the holy Dekad'.

'What we find about the progression of the other numbers is more indefinite and unsatisfying, and the Notion loses itself in them. Up to five there may certainly be a kind of thought in numbers, but from six onwards they are merely arbitrary determinations'.

- 'Lectures on the Philosophy of History'

See below for Hegel’s Triangle of Triangles, the tetraktys, and Baader’s Quadrat. During the earlier years of his stay in Frankfurt from 1797 to 1800 Hegel had been attracted to such mystical or theosophical elements within medieval Christianity like Baader’s yet nonetheless soon moved beyond these to a more philosophical stance and according to Helmut Schneider 'the ‘triangle fragment’ does not rest on mystical experience. It is about rational construction and geometrical logic. The account of the syllogism in Hegel’s Logic clearly has something to do with the syllogism Hegel finds implicit in Plato’s cosmology in the 'Timaeus' and of which he regards Aristotle’s formal syllogism as both an elaboration and reductive distortion. For the technical vocabulary of his syllogistic, Aristotle had drawn upon the terminology used in Pythagorean music theory, that is, the same sources as Plato’s cosmology in the 'Timaeus'. While Aristotle seems to have adhered to much of Pythagorean science he did not accept the application of music theory to cosmology.

For Hegel the superiority of Kepler’s approach to science stems from the fact that it instantiates the type of logic implicit in Plato’s cosmology in contrast to the otherwise dominant Verstandeslogik implicit in Newton’s which had its ultimate source in Aristotle’s formal logic and a key difference in Hegel’s account of the Platonic syllogism in comparison to Aristotle’s is that Plato’s has an explicit four-part structure in contrast to the three-part structure of Aristotle’s formal syllogism, and, as with Baader’s triune God acquiring a type of Pythagorean four-fold quality Plato’s concrete four-fold version of Aristotle’s trinary syllogism has to do with its being applied to the three-dimensional world and Hegel’s way of characterizing this is to describe the middle term of Aristotle’s formal syllogism as being divided, split or broken in Plato’s syllogism.

Hegel’s Interpretation of Plato’s 'Timaeus'. In Plato’s 'Timaeus' there is a complex of contemporary arithmetic, geometry and Pythagorean musical theory making up the framework of Timaeus’ mythical account of how the craftsman or demiurge had brought order to the cosmos out of its initial state of disorder whereby the demiurge had wanted everything to be good and nothing bad and brought order to an out of tune (plemmelos) disorderly state shaping the cosmos into a single living animal of which all other living things formed parts both individually and as kinds. How are the parts combined?

'[I]t isn’t possible to combine two things well all by themselves, without a third; there has to be some bond between the two that unites them. Now the best bond is one that really and truly makes a unity of itself together with the things bonded by it, and this in the nature of things is best accomplished by proportion [analogia]. For whenever of three numbers which are either solids or squares the middle term between any two of them is such that what the first is to it, it is to the last, and, conversely, what the last term is to the middle, it is to the first, then, since the middle term turns out to be both first and last, and the last and the first likewise both turn out to be middle terms, they will all of necessity turn out to have the same relationship to each other, and, given this, will all be unified'.

-'Timaeus'

In his discussion of this and related passages in his 'Lectures on the History of Philosophy' and in accord with Baader’s linking of the three-part trinity structure to the Pythagorean tetraktys Hegel focuses upon how this apparently trinary structure becomes complicated by the idea that not one but two middle terms are needed to unify its extremes and to understand this we need to understand something about how the tetraktys functioned in Pythagorean thought,the details of the Pythagorean music theory and cosmology by which Plato had been influenced. Hegel had been influenced by the heavily Pythagorean account of Plato’s philosophy promoted by late neo-Platonists like Proclus, (412 – 485).

The Pythagoreans had an arithmetic worldview, the principle of all number being the monas which, besides the unit used in counting was also meant to play the role of a spatial point, conceived as a monadic unit in position. According to Richard D. McKirahan Jr. the Pythagorean conception of a point as a unit in position 'skips over the facts that geometrical points are different to arithmetical units, and straight lines are determined by two points in a different way from that in which the number 2 is composed of two units'. In this way a line could then be conceived as composed of such units, two-dimensional figures such as squares and rectangles could be conceived as compounded of arrays of two-dimensional lines, and three dimensional solid figures as compounded of those planar ones. That is, what we know as square numbers, numbers multiplied by themselves, were, for the Pythagoreans, literally square, determinations of areas rather than lengths, and cubic numbers were similarly cubic. Heath discusses the way that such figured numbers as square, triangular and oblong numbers were used in Pythagorean calculations. This meant however that there was no sense to be given to the idea of a number raised to a power greater than 3. Around the time of Plato this reduction of geometric continuous magnitudes to arithmetic ones was being challenged by a new geometrical approach in which such continua were understood as is some sense primary a view reflected in Aristotle’s thought as well as in his logic yet Plato was still more aligned with Pythagorean mathematics within which the four rows of the tetraktys were meant to represent these four spatial dimensions of point, line, area and volume, the series of exponential powers associated with them, as well as the four fundamental elements, fire, air, water and earth, and like the kinds of the four substances with which they were associated numbers raised to different exponential powers were understood as different kinds of numbers with the idea of different and incommensurable kinds of magnitude being a deeply held belief within Greek mathematics after the discovery of incommensurable numbers sometime in the fifth century BC. This was the type of incommensurability holding between curves and straight lines to which Hegel had alluded in his critique of Newton’s reduction of curves to straight lines and ultimately points and observing that anything with bodily form must be both visible and tangible Plato has the demiurge compose the body of the cosmic animal out of the elements of visible fire and tangible earth, elements existing at the extremes of the tetraktic structure and hence having two intermediaries.

'Acrobazie tra monti e laghi', 1933, Alessandro Bruschetti

+++++

Hegel’s Triangle of Triangles, the tetraktys, and Baader’s Quadrat (yes I know my illustrative diagrams are not of the best quality but I do my best with Microsoft Paint. Perhaps I should invest in some more advanced software)

Along with this, Timaeus points out that were the cosmos planar rather than three-dimensional, there would be needed only a single middle term but as it actually is three-dimensional there is a need for two middle terms standing between the monadic elements themselves and the solids into which they were compounded and the demiurge had thus 'set water and air between fire and earth, and made them as proportionate to one another as was possible, so that what fire is to air, air is to water, and what air is to water, water is to earth. He then bound them together and thus he constructed the visible and tangible universe ... making it a symphony of proportion'. In relation to Plato’s rational syllogism with its Pythagorean structure Hegel portrays Aristotle’s formal syllogism as a type of corrupted form with only a single middle term. Aristotle seems to have been heavily influenced by contemporary geometers who were reacting against the Pythagoreans reduction of continuous to discrete magnitudes and as with continuous magnitudes the relations among its three terms, A, B and C, are understood comparatively according to the idea of the containment of the smaller in the larger, the transitivity of which means if C is contained in B and B in A, then C is contained in A and this is how we how we are to understand how the combination of two premises, A-B and B-C, can result in the conclusion, A-C. All of Aristotle’s terms must be general as each must be able to play the role of subject (container) or predicate (contained) in a judgment and singular terms such as proper names cannot be predicates hence there are no remnants of the Pythagorean monas (Hegel’s singularities), in Aristotle’s logic thus there is nothing in Aristotle’s logic to properly capture the absolute singularities of external nature as understood by Hegel.

Hegel’s Logic and the Paradox of Singularity. Hegel’s Logic is intended as an entirely internal project in which thought determinations are generated from the operations of thought itself. In the Logic’s 'Subjective Logic' we find Hegel’s analogue of Aristotle’s syllogistic but modified such that Aristotle’s terms designated by A, B, and C are now identified with Hegel’s three fundamental conceptual determinations (moments of the concept), universality, particularity, and singularity. The distinction between singularity and particularity is defined by the different ways in which a thought can be related to an object, the former, as expressed in a singular term such as a demonstrative phrase or a proper name picking out something in its specificity, the latter as in a definite description that is “satisfied” by whatever worldly object or objects it is true of.

In the Dissertation there are endeavours to apply the idea of a split middle term to various aspects of Kepler’s model of the solar system, for instance, 'From that real difference in gravity, we distinguish the idea of difference, namely, that of the potencies of time and space; for when a two-foldness has been posited, a double two-foldness—one of the poles, the other of the potencies—or four regions, must be posited. While much of Hegel’s language in the Dissertation is that used by Schelling in his philosophy of nature Hegel’s use of the split-middle structure is clearly contrary to Schelling’s thought hence Hegel describes a 'line of cohesion' that seems to allude to Schelling’s 'constructed line' from his 1801 essay 'Presentation of My System of Philosophy' et while Schelling’s line has an extreme divided by one middle term, Hegel’s line of cohesion is divided by two. This is a distinction that had been used by Hegel’s logic teacher, Gottfried Ploucquet and it incorporates the innovation of earlier nominalist logicians (transmitted by way of Leibniz) so to give a type of extensional interpretation to judgments. Ploucquet describes the distinction as between exclusive and comprehensive forms of particularity. However the methodology of Hegel’s Logic gets in he way of our understanding the distinction in this way for given its entirely internal constitution as the self-articulation of pure thought there is no equivalent of any extensional semantics in it and from the purely logical point of view singularity is understood as different to particularity and universality while related to both albeit such extensional semantic considerations start to come into focus in the early paragraphs of the 'Philosophy of Nature'.

Introducing the concept of nature Hegel states that 'Nature has yielded itself as the Idea in the form of otherness ... nature is not merely external relative to this Idea (and to the subjective existence of the same, spirit), but is embodied as nature in the determination of externality' and this theme of the externality of nature is further elaborated upon: 'In this externality, the determinations of the concept have the appearance of an indifferent subsistence and isolation with regard to one another'. This combination of externality and singularity gives a dimension to every actual natural thing what Hegel later refers to as its 'infinite singularization or separation' (unendlichen Vereinzelung).

'The Idea, as nature, has: I. the determination of extrinsicality and of infinite individuation. Unity of form, as it is external to this, is of an ideal nature, and as it is simply implicit, is merely sought after. This constitutes matter and the ideal nature of the system of matter, i.e. mechanics. II. the determination of particularity, in which reality is posited with an immanent determinateness of form and its own existent differentiation. This is a relationship of reflection, the being-inself of which constitutes natural individuality, i.e. physics. III. the determination of subjectivity, in which the real differences of form are also brought back into a unity of an ideal nature, which has found itself and has being for itself, i.e. organics'.

- 'Philosophy of Nature'

From a logical point of view the difference between the conceptual determinations of singularity and particularity cannot be accounted for by the question of how they pick out worldly objects but now in the context of the 'Philosophy of Nature' this internally articulated system of thought faces a world with its own inherent logical structure including the element of infinite singularization or separation and this status possessed by a natural thing qua natural clearly resists the type of conceptually articulated unification upon which intelligibility relies and where something is grasped as a particular it is grasped in terms of what it has in common with other instances of a universal. This rose, grasped in the judgment this rose is red can be grasped in terms of what it has in common with other roses or other red things and yet when grasped as singular such connections must be put to one side for Hegel’s split middle term signals the paradox of the conceptual determinacy of singularity as when I grasp this rose in a judgment I abstract from its specificity and weave it as a rose into the fabric of my understanding and this comes at the cost of annulling its infinite singularization and Hegel refers to this as the contradiction at the heart of theoretical reason. I seek to understand the thing as it is but the more thought predominates in ordinary perceptiveness so much the more does the naturalness, individuality, and immediacy of things vanish away, as thoughts invade the limitless multiformity of nature its richness is impoverished and its spring times die and there is a fading in the play of its colours and that which in nature was noisy with life falls silent in the quietude of thought its warm abundance, which shaped itself into a thousand intriguing wonders, withers into acrid forms and shapeless generalities, which resemble a dull northern fog. This is the price paid by Newtonian thought in which abstractions come to completely replace the specific entities, Kepler’s observed elliptical orbits, for example, that had been concretely perceived and as with understanding the role of mathematics within science one might advise against confusing purely conceptual relations with the physical ones and the infinite singularization or separation of things in nature does not mean they are unconnected, if nature is noisy with life then clearly there are real physical relations at work in the world and what it suggests is that these physical relations cannot be simply reduced to the types of relations existing among the determinate concepts we bring to it, concepts mutually determined within the fixed oppositions of the logic of the understanding.

Because of such consequences, Goethe, Schelling, Baader and various other Naturphilosophes had sought forms of science that preserved the immediacy of living nature, but Hegel’s solution was different: 'what has been dismembered may be restored to simple universality through thought' rather than intuition and through our becoming more self-aware of what we are doing when employing concepts in our practical and theoretical engagements with the world we may return to a proper grasp of the living processes implicit in nature itself and in this way Hegel’s philosophy of nature does not aim to challenge the results of the working scientist’s investigations with an account of how the world they are investigating really is, on the contrary as based upon a correct understanding of the logic implicit in the empirical investigation of the natural world philosophy of nature will help the working scientist to avoid the pitfalls brought about by the contradictions implicit within the mind’s own theorizing, examples of such pitfalls can be recognized in Newton’s projections of concepts onto the world such as the concept of the infinitesimal that not reflecting empirically detectable existences in the world itself have to be understood as mere posits of the mind. And so Hegel’s defence of Kepler over Newton is premised upon the Platonic cosmology to which Kepler had been attracted and in it Plato had provided the first concrete expression to the logic implicit in the activity of empirical science itself.

'The Way of Truth and the Way of Opinion'

by Parmenides, (fl. late sixth or early fifth century BC), (excerpt)

Horses that bore me, impelled by their courage,

Brought me to the much-famed streets of the goddess

Who leads the wise man to every kind of knowledge.

Maidens point out the way.

The axle sings hot as the daughters of Helios quickly approach,

Leaving the dwelling of night, pressing on to the light,

With mighty hands raising the sheltering veil.

- Parmenides, 'The Way of Truth and the Way of Opinion'.

'Dynamism of Horses' ('Dinamismo di cavalli'), Alessandro Bruschetti

For my adorable One, my Muse, dreaming of your curves ...the curve of your lips as they smile and make me feel happy.

When all the land was dark

And you appeared in light

Then the darkness cried

We danced above the earth

Through the heaven above

Sunlight, moonlight smiled

Until the end of the world

We touched beside the sky

The sun through golden rays

Whispering our love

Until the end of the world

Your bright eyes fill my soul

Your kiss a sacred dream

The dream is one that lasts

Until the end of the world

Julee Cruise, 'Until the End of the World':-

Coming up next:

Astronomy.

To be continued...