Christian Hermann Weisse: System of aesthetics - The art of time

The modern ideal of beauty, Weisse suggests, is in a sense a point where beauty coincides with truth. That is, whereas by the idea of truth Weisse means the philosophical understanding of the whole world, both the physical nature and the human history, the modern ideal of beauty means experiencing again this whole world and all that has happened in it as beautiful. The difference is that the modern ideal does not merely passively observe this beauty, but tries to create an image of this living world with the help of some inert matter, which in itself is lifeless. This creation of images of beauty is the definition of art, Weisse says.

Like many German philosophers of the time, Weisse provides a hierarchy of arts. He does not mean to say that works from the arts of the lower level of hierarchy would be automatically less beautiful than works from the arts of the higher level. Indeed, works from any art can have an infinite variety in their degrees of beauty. Instead, the idea of the hierarchy is that the lower levels contain more abstract types of arts, which in themselves point to more concrete types of arts as embodying features that the lower levels lack.

It is then understandable why Weisse would begin from music as the most abstract type of art. Indeed, the place of music in Weisse’s hierarchy might appear surprising, when compared with thinkers like Hegel and Schopenhauer, who placed music much higher in their respective hierarchies of arts, Schopenhauer reserving for it even the highest position. Yet, while both Hegel and Schopenhauer move in their hierarchies from brute physicality toward more immaterial types of art, Weisse chooses to begin from an art that is closest to the mere modern ideal of beauty: while the ideal is nothing but consciousness of beauty, music embodies this beauty in the most fluid type of material or movement of sounds.

Following Hegelian philosophy, Weisse notes that sounds or what we can hear embodies especially time, while what we can see embodies space (sounds appear and vanish, following one another, while colours and figures abide longer and side by side one another). Yet, it is not all sounds (Klang) that music is concerned about, but notes (Ton), that is, purified sounds that are quantitatively related to a whole system of other notes.

A piece of music, Weisse continues, contains firstly a temporal succession of such notes - this is the melody of that piece. As Weisse has constantly emphasised, we can not give any surefire quantitative recipe for making beautiful objects and this is the case also with music: we cannot say beforehand what melody is beautiful, despite music being able to be represented through quantitative notions. A certain regularity is required, and this is provided by the rhythm of the music, although Weisse thinks rhythm is still subservient to the melody. A further, equally indeterminate demand is that a piece of music should let the melody be diversified into different and even dissonant modifications, although from a more higher perspective of the whole composition a harmony would prevail.

In its most abstract phase, Weisse says, music uses such sounds that do not occur in ordinary nature, but are made through mechanical instruments (we have to wonder what Weisse would have thought about synthesisers that take this type of music into its extreme). Because of this divorce from nature, Weisse suggests, instrumental music is closest to the pure consciousness of the modern ideal of beauty. Indeed, he continues, as the most abstract art, capable of expressing nothing more than raw emotions, like joy and pain, it has necessarily been the youngest art, because such a height of abstraction has required plenty of cultivation. Indeed, the modern ideal of art, which is even more abstract, being mere consciousness of beauty, could not have appeared before this most abstract of arts had appeared in history, Weisse concludes.

Although the sounds of instrumental music are artificially created, they are in a sense natural sounds, Weisse reminds us, since all sounds are part of the physical nature. Yet, the more they resemble naturally occurring sounds, such as raindrops, the further away the instrumental music moves from what is proper to its level, Weisse argues. Still, despite the fact that instrumental music imitating natural sounds is a corruption, in Weisse’s eyes, the tendency to make such experiments shows that music has a natural drive toward more natural sounds.

A more proper development of the music into more natural sounds is shown in singing, Weisse says, where instruments are replaced by human voice. Singing is not anymore just an expression of the pure modern ideal of beauty, but adds layers from the whole human life. In its simplest forms, Weisse notes, singing can be used to express all sorts of frivolous topics, which are far from true art (one suspects Weisse would categorise pop music here). This lower form of singing can be saved only by the help of poetry and an instrumental background.

Yet, Weisse continues, singing can have a higher purpose, namely, when it strengthens the relation of a finite observer to the divine. While worldly singing usually works with one or two singers, religious singing often requires a whole chorus of voices. A further difference is, Weisse concludes, that religious choral music does not require the help of instruments to become real art.

In choral music, finite human life is immediately unified with the ideal consciousness of beauty, through which the humans try to connect to the divine they yearn to find, Weisse says. Yet, finite life can also be put into opposition to the ideal consciousness, as something lowly and unworthy of serious interest. Yet, just like in the earlier stage of comical, this frivolous finite life can also help to show the ideal, by revealing its own unworthiness and contradictoriness.

This revelation is achieved by the final phase of music, Weisse notes, that is, opera, where music is used to embody the drama of life. Opera in a very literal sense combines the previous types of music, by connecting instrumental music to singing, thus proving Weisse’s earlier point that if singing of lowly matters is to become an art, it must be helped by musical instruments. So, opera can be taken as the final truth of music.

Yet, opera shows also the very reason why music is ultimately not the most satisfying vehicle of art, Weisse emphasises. In order to express the ordinary human life, music requires a more robust shape of embodiment. Indeed, opera has to rely on the help of other arts - it uses visual arts to provide the setting and poetry to create something to be sung. Even more so, Weisse remarks, opera must be embodied by living people, who express the beauty of life through gestures and facial expressions - or even by activities like dancing and pantomime. In other words, in opera music wants to become visible, which is also the key to the next stage in Weisse’s hierarchy of arts.

Christian Hermann Weisse: System of aesthetics - History of an ideal beauty

The development of beauty, with Weisse, had to contend with immediate beauty not necessarily conforming to the sublime infinity beyond it and being thus ugly. Now, while we can see what an ideal beauty corresponding to the sublime should be like, he continues, originally the discovery of ideal beauty required the effort of generations, and indeed, several historical cultures.

Weisse connects the notion of ideal beauty especially to the notion of mythology. Mythologies at their best are for Weisse not just arbitrary creations of fantasy, but reflect the life and thoughts of a people. A mythology sums up the hopes and fears of a culture in a symbolic shape of mythical personalities. These symbols, Weisse says, resemble the previous aesthetic shape of comical humour, in that they both rise above the ordinary life of finite, momentary and decaying shapes, the difference being that while humour merely notes and rises above this decay, mythological symbols try to grasp something stable from this play of finite entities.

Mythology as such is not yet beauty. It is no surprise that Weisse would follow the general trend in thinking that the ancient Greek were the people responsible for transforming mythology into an ideal beauty, because Hellenic mythology was a particularly natural creation of the spirit of Greek life. Hellenic mythology, as Weisse envisions it, had two different types of mythological shapes. Firstly, the mythology tells about heroes living in the distant past, who express the essence of what the Greek thought being a Greek meant. Secondly, the heroes interact with the beautiful gods, who represent various aspects of the superhuman realm beyond human history.This mythology is then embodied in the Hellenic cult, which represents the relations of humans to this supernatural realm.

Just like Hellenic ideal was a result of historical development, it was also subject to further development, Weisse notes. While in Hellenic mythology gods were characterised by their beauty, such an external appearance was revealed to be frivolous compared to god being a self-conscious entity: beauty is replaced by something more valuable. This does not mean that beauty was completely ignored after this historical transformation, but its place in the hierarchy of values was just lowered.

With this transformation, Weisse explains, the ancient ideal of beauty turned into a romantic ideal, with its own mythology. While Hellenic gods were present as beautiful shapes, romantic God is something beyond the mythological or legendary figures - an unreachable infinity. This doesn’t mean that romantic God would never be thought to appear in the finite world, Weisse admits, but the relation of the divine and the finite was just reversed. While Hellenic gods were embodied divinities, romantic God could divinise a body - an obvious reference to the notion of incarnation. Furthermore, Weisse continues, unlike with Hellenic gods, this appearance of the romantic God in the world of finity was meant to be just temporary and God returned to the realm beyond.

The divination or at least spiritualisation of the corporeal world in the romantic mythology happened also in an opposite fashion from incarnation, Weisse notes. The finite world or at least some part of it still appeared to be severed from God - in terms of earlier concepts, it was an ugly world. Now, this inherent ugliness of the human world, Weisse continues, was presented in a non-bodily form as evil spirits opposing God. Indeed, often the beautiful gods of antiquity were now interpreted as these evil spirits or demons. Human world was then seen as a battleground between the spiritual forces of good and evil.

This battle was not supposed to be never ending, Weisse notes, but it was assumed to end with the overcoming of the forces of evil. Yet, this final victory was not really thought of as occurring at some definite point of time, but only in the hazy future - or, one might say, it had already been won, since nothing could hinder God's plans. This victory or salvation of the finite world was wrapped in the notion of divine love of the finite world.

While Hegel had ended the development of aesthetics with romanticism, which made his contemporaries assume he had assumed the death of aesthetics, Weisse continues further. Indeed, this very next step he assumes is inherent in the notion of divine love of finity - like the romantic God was supposed to do in the future or have done in an atemporal manner, we humans have again come to appreciate the beauty of the world around us. When this change has happened, yet another form of ideal has appeared - the modern ideal.

While many German romantics had supposed that a new ideal would require a new mythology, Weisse comments that all we really need now is the science of beauty itself or speculative aesthetics. Indeed, we need not even a complete aesthetical theory, but just a certainty that beauty is something equally eternal as truth and God are. This certainty is then accompanied with the historical appreciation of the former shapes of ideal beauty and with the expectation of further beauties of innumerable measure.

The historical development of the aesthetic ideals has stopped now, Weisse emphasises, but this does not mean that no further beauties would not be found. Instead, quite the opposite has happened, since by understanding beauty as such, we have liberated it from any necessary connection to further mythologies. We have thus learned to appreciate beautiful objects, which each in their unique manner express the modern ideal of beauty. In other words, Weisse implies, we now appreciate art for its own sake, not just as an expression of religious notions.

Christian Hermann Weisse: System of aesthetics - From beauty to non-beauty and back again

Last time, Weisse had just finished showing the reader a canon that ties individual beautiful objects into a unified whole - a macrocosm behind microcosms. This macrocosm limits the activity of fantasy in finding individual beautiful objects: not everything can be beautiful. This limit, Weisse says, is not expressible through mere numbers. Indeed, he continues, the limit is something beyond what we can understand - it is a force behind, but also beyond what appears beautiful.

This macrocosm or force behind individual beautiful objects Weisse calls the sublime. In other words, we experience sublime, when we experience finite beautiful objects being swallowed into something beyond our comprehension. The sublime can also be called beautiful, but it is a beauty different from the beauty of finite objects. An important difference, Weisse insists, is that the macrocosm can never be experienced as a completed whole, but it can only be approximated. This means also, Weisse says, that the sublime can not be described fully in aesthetic terms. Instead, it must be understood through religious and ethical concepts, like divine and good.

Now that the sublime macrocosm has appeared to us, individual beautiful objects should get their beauty from their relation to this macrocosm. Like the sublime itself, the beautiful objects, as related to it, receive a religious and ethical flavour - they appear graceful or dignified, Weisse says. There still exist beautiful objects without any relation to this divinity, or more precisely, beautiful objects denying their relation to the sublime. Paradoxically, they now seem in comparison with the sublime just plain ugly.

Ugliness, Weisse continues, is thus not just a lack of beauty, but beauty turned upside down. Because the basis of beauty was truth, ugliness is then defined by deception, and while the experience of beauty was one of blessedness, experience of ugliness is one of damnation. Indeed, Weisse says, this experience of ugliness makes us imagine a whole hell full of horrendous ghosts. Even concrete objects we deem ugly seem thus uncanny and frighten us.

Weisse’s notion of ugliness has thus also a link to ethics: what is permanently ugly is that which is evil or turned against the divine. Then again, Weisse says, if we separate ugliness from evil and related ethical notions, it becomes mere instability - seemingly ugly, temporal realities, not sustained by an evil will, vanish like all finite things.

The person observing this disappearance of individual ugliness feels itself as being constant throughout these changes and even as actively cancelling these ugly objects. In other words, this person feels its own power in the play of forces that causes the instability of everything it senses: things external to the observing subject appear frivolous and even ridiculous. As a sudden outburst, such a feeling becomes an experience of comical, Weisse says, and this experience is bodily felt as a trembling of one's whole body in laughter.

In comedic experience, a person recovers itself from a detour through creation and destruction of ordinary finite entities. In other words, Weisse says, a bit of wit or a whiff of irony helps us to isolate ourselves from the world of finity, while the objects of this world appear of lower stature to what is sublime. Yet, we can also have the original experience of beauty, when looking at them, but now this beauty seems childlike or naive. The result of this experience of humour or irony is to notice that these seemingly naive objects can take up their place in the flow of sublimity, which creates an experience of an ideal beauty, which shall be the topic of my next post.

Christian Hermann Weisse: System of aesthetics (1830)

(1801-1866)

As a specialist on Hegelian philosophy I’ve come to notice that in studying works engaging with Hegel I must distinguish the picture of Hegel presented in these works from what Hegel really said (or more precisely, what is my picture of what Hegel really said, since I cannot be sure I have deciphered Hegel's meaning correctly). Thus, when reading Weisse’s System der Ästhetik, a work expressly indicated as a development and even an improvement of Hegelian philosophy, I had to constantly remind myself that Weisse is speaking not so much of Hegelianism in itself, but of the view he had of it. For instance, Weisse takes as Hegel’s definition of the topic of philosophy, the idea, that it is the eternal and necessary form of all that exists. True, Hegel does say that idea is eternal and necessary, but one might wonder if this is just a description of this “idea” and leaves out some crucial and defining characteristics.

Weisse’s understanding of Hegelian idea is in line with his understanding of the apex of Hegelian philosophy, where all the previous developments are summed up in the very notion of philosophy itself. For Weisse, this is ultimately just a passive, theoretical move, where the philosopher merely grasps the truth of what has been previously exposed, but does not actively engage or do anything with it. One might argue that the Hegelian philosopher as a living, conscious individual is constantly doing just that, but Weisse thinks that there is rift between this viewpoint of a finite individual and the viewpoint of the Hegelian philosopher supposedly knowing the essence of all there is to know.

Weisse insists that this rift must be bridged. Hegel himself might say that the whole of his philosophy is dedicated to bridging this rift and showing how anyone could become a philosopher, but this is another discussion. Weisse, on the other hand, thinks the bridging should be done through something completely different - an idea of beauty surpassing the idea of truth embodied in the Hegelian philosophy. Weisse thinks he is here following Hegel’s dialectic-speculative method and surpassing Hegelianism by its own means.

What is lacking in Weisse’s account is the crucial idea that the outcome of such a move should be based on its very starting point. That is, we shouldn’t just find a supposed inadequacy in the starting point and suggest a way to amend it, but we should be able to transform the very starting point by its own means to the end point. In other words, we should be able to start with a philosopher recognising the truth of philosophy and showing how this very experience can become or at least give rise to an aesthetic experience of beauty. This is a path that one could very well take, since a recognition of a deep and meaningful truth could be described as aesthetic. Still, it is not the path that Weisse takes. On the contrary, he thinks that such a recognition of truth is by itself not capable of being an aesthetic experience, which for him requires something surpassing mere truth.

Beauty is then for Weisse, in terms taken from Hegel’s philosophy, an Aufhebung of truth, that is, it contains truth, but only as one aspect in a greater whole. What this more is that is added to the truth is that beauty appears in the viewpoint of finite individuals, that is, in the ordinary human life. Still, Weisse insists, beauty also presupposes truth, which means for Weisse that aesthetic theories not understanding Hegelian viewpoint will not express the truth of beauty.

Like truth, Weisse says, beauty is still something connected to Geist or life of conscious beings. Here Weisse points to the notion of fantasy (Phantasie), as used by Fichte and romantics. Fantasy, Weisse says, is something different from mere imagination (Einbildungskraft), which means simply our ability to have mental images of things we have perceived, but are not currently perceiving. Fantasy, on the other hand, should be a higher capacity of representing things pertaining to the very nature of consciousness. In other words, fantasy should be a capacity by which we can think of things like beauty as embodied.

Beauty, as fantasised, is then not just a word we understand, but something that we can perceive and feel, Weisse says. In other words, in fantasy we experience beauty as something that is distinct from our consciousness and that makes us feel as being part of something more universal than us as individuals. In comparison, our finite selves feel utterly insignificant. We are here witnessing another dialectical movement, in which consciousness has an experience of beauty and then ascribes this feeling to an object different from itself as the subject. In other words, we have moved from beauty as a subjective experience to a beautiful object conceived as causing this experience.

While the truth of philosophy is something that is infinite in the sense that in it the philosopher knows that philosophy could potentially understand anything coming across, in the experience of beauty this potentiality is in a sense actualised. This does not mean, Weisse explains, that we would experience an infinite amount of beautiful objects all at once, but that we experience more and more beautiful objects, without any limit. Furthermore, it means that no individual beautiful object or any finite sum of them manages to completely satisfy our notion of beauty, but shows only some limited aspects of it.

Each of these beautiful objects, Weisse says, although agreeing with others by awakening the experience of beauty in us, is still unique in its own way. Weisse clarifies that this uniqueness does not mean just that each beautiful object exists at its own specific time and place, but also that beautiful objects differ in their characteristics. Because beautiful objects still are Aufhebung of truth, while truth already contains the essence of everything studied at previous levels of philosophy, they in their turn contain in a sense the whole world in them. In other words, beautiful objects are microcosms. How a whole world could be expressed in a singular object is a mystery that can never be fully explained by philosophy, Weisse thinks.

Each of these beautiful objects must be a real object in our world and not just a figment of imagination, Weisse says. Beauty must be embodied, but this is as far as necessity goes, he continues. In other words, there’s a degree of arbitrariness and almost of a free choice in picking out what we actually experience as beautiful. This picking out involves then a new dialectical movement. The beautiful object is also something more than just beautiful or it has a number of other characteristics. Beauty is then relegated into a level of a property among other properties.

As comparable with other properties of the object, Weisse explains, beauty becomes a quantitative characteristic: a thing can be more or less beautiful, depending on its other characteristics. Still, Weisse adds, beauty differs from other characteristics of a thing by expressing all of the thing and not just a single feature. Weisse describes this relation also by saying that beauty is not part of the appearance of the thing, but its whole appearance. In other words, beauty indicates a complex relation among the other features of the thing - it is their rule, canon or measure for measures. Weisse also emphasises that beauty as a rule or measure cannot be expressed through a simple numeric equation - beauty lies not just in e.g. symmetry. In this sense beauty could also be called irrational, just like irrational numbers cannot be expressed through mere multiplications and divisions.

The canon of beauty ties the disparate realm of beautiful objects into a unified whole, Weisse says. At the same time, it creates some limits to what can be called beautiful, even if these limits cannot be clearly expressed. In other words, by changing the characteristics of beautiful things, we could slowly turn them into something that is not beautiful. I shall study the outcome of this dialectical transition in the next post.

Auguste Comte: Course of positive philosophy 1 - Mechanics

Comte’s first volume of his positive philosophy ends with a study of mechanics. Although mechanics is for Comte even more concrete than geometry, it still falls within mathematics. Indeed, Comte is against all readings of mechanics, where some elements of analysis are interpreted as real forces, although they would be nothing but means for making calculations. What forces are real can be decided only by observation and experience. Furthermore, Comte adds, mechanics does not investigate what is the nature of these forces, but merely the movements caused by them.

The abstract nature of mechanics, characteristic to mathematics in general, Comte notes, is seen in the fact that mechanical calculations are simplified by assuming bodies to be passive or inert, although in reality they are in many ways active. This idealisation of bodies as inert in mechanics, Comte explains, is not to be confused with inertia in the sense expressed in one of the basic laws of mechanics - the fact that bodies tend to move in straight lines and retain their state of movement. Together with the other two basic laws of mechanics - one being Newton’s law of action and reaction, the other being Galilei’s discovery that forces are independent of one another and can thus be composed with the parallelogram law - the law of inertia is, according to Comte, based on observation, not on any a priori deduction. Particularly, Comte adds, the law of inertia cannot be deduced from the law of sufficient reason.

Comte divides mechanics, expectedly, into statics dealing with instantaneous forces and uniform movement or equilibrium arising from them and dynamics dealing with continuous forces and varied movement arising from them. Within both statics and dynamics, he then differentiates a part examining solid bodies from a more complex part examining fluids. An important problem in this classification in Comte’s opinion concerns the relative status of statics and dynamics. Statics is clearly the older discipline, studied by ancient mathematicians long before dynamical questions. But as Comte has said earlier, the historical order of disciplines does not necessarily correspond to the order of the disciplines in a completed science. Indeed, when dynamics was finally introduced at the start of the Modern Age, statics was regarded as a mere abstract limit case of dynamics.

Yet, as Comte’s favourite mathematician, Lagrange, had argued, the notion of virtual displacement - an application of his calculus of variations to mechanics - could be used to reduce all of dynamics to statics. In effect, Comte is referring to the central idea of the so-called d’Alembert principle that all apparently dynamical systems can be regarded as being in equilibrium. Comte further links Lagrange’s idea with Poinsot’s notion of a force couple, which Comte thinks is a modification of a notion of force from translation to rotation.

Just like with other parts of mathematics, Comte is especially interested in the applications of mechanics. Thus, he points out that statics is used for finding mass centres of bodies, while in dynamics we are trying to calculate movement of a particle from forces affecting it, or the other way around, to find forces creating a known movement. We need not go in great detail to theorems that Comte lists as consequences of the three basic mechanical principles. I will just point out that Comte speaks against interpreting Maupertuis’ principle of least action in a theological or metaphysical manner suggesting that bodies would somehow choose to move in accordance with the principle.

Auguste Comte: Course of positive philosophy 1 - Geometry

While many philosophers had considered it an important problem to put geometry on secure foundations, Comte finds such attempts mere unfounded metaphysics. For him, geometry is simply a natural science with an empirical basis. Of course, it is the most abstract natural science, dealing only with static spatial properties of things, in abstraction from any movement. Still, Comte feels no need to prove the basic axioms of geometry, since he can just assume them as bare facts. On the other hand, he also feels no need to consider the possibility of other geometries with other axioms, since experience appears to agree with the ordinary Euclidean geometry.

Comte supposes that this empirical science has a very practical purpose, namely, that of measuring spatial features of things. Of course, he adds, not all measuring is geometry, for instance, if we fill an oddly shaped container with water and then measure the volume of the water, this is still not geometry. Instead, geometry, like all mathematics, is an art of finding out quantities indirectly, through calculations.

When we measure spatial features of things, Comte continues, we can measure all the dimensions of it or then only some of them. When a geometer speaks of planes or lines, he adds, they are considering just such abstractions, that is, they are ignoring some of the dimensions the thing has. Thus, lines in reality always are wide and thick, we just concentrate on their length. We can even ignore all the dimensions of a thing and consider only its position in relation to other things - this is the origin of the notion of point.

While the essence of geometry is indirect measuring of spatial features of things, a geometer must assume some ways to directly measure these spatial features. This implicitly assumed form of measuring, Comte suggests, is the measurement of straight lines or lengths by comparing them with a length of some other thing, like a ruler. Other geometrical figures (curves, areas and volumes) are then to be measured with the help of these straight lines.

Comte admits that geometry is full of other things beyond mere measuring, that is, full of propositions about spatial properties of things. Still, he insists, even these properties are ultimately studied, because they could help with measuring (perhaps in some more concrete science). Because one cannot know beforehand what properties help with measurements, Comte advocates studying as many such properties as possible.

Comte mentions the old distinction between synthetic and analytic methods in geometry, but seems to have no idea what these terms meant originally and what their difference was supposed to be. Instead, Comte suggests that the distinction is just a roundabout way to distinguish ancient from modern geometry. Ancient geometry, Comte suggests, was mainly dealing with concrete and individual figures, taking one type of figure and finding all its characteristics. Because of the uniqueness of the chosen figure, none of these characteristics could be assumed to hold for other entities. Modern geometry, on the other hand, deals with abstract geometrical problems, which can then be applied to many different contexts.

Comte has little patience with ancient geometry, and he especially dislikes the common habit of starting to teach geometry from works like Euclid’s Elements - as he has noticed earlier, history of a discipline is usually not the most convenient way to teach it. Comte is especially critical of the proofs of early propositions in Elements, where Euclid simply places figures on top of each other, to show their similarity - this is just as futile as an attempt to prove the parallel axiom would be.

Although Comte is very critical of ancient geometry, he does admit that Greek mathematicians did make some progress. Particularly, they perfected the study of the simplest kind of figures, namely straight lines and polygons and polyhedras. Furthermore, while Comte ridicules the Greek use of diagrams as proofs of proposition, he does suggest it to be a sort of precursor for modern projective geometry. Another field developed by ancient geometers to perfection, Comte concludes, is trigonometry, where they used certain lines (sines,tangents etc.) to represent angles and their relations and so simplified calculations.

Comte associates the birth of modern geometry with Descartes. While it had been long known that geometric forms could be described in terms of spatial situations of their limiting points etc., it was the invention of Descartes, Comte says, to reduce the talk of situations to talk of lengths and other magnitudes through the notion of a coordinate system. In effect, Comte concludes, Descartes was able to transform at first sight qualitative properties like geometric figures into quantitative properties.

The outcome of the Cartesian transformation of geometry, Comte explains, was that now in two-dimensional geometry lines could be expressed by equations and equations by lines (in three-dimensional case, Comte adds, equations express surfaces, while lines are expressed by pairs of equations). These equations do not just characterise some random properties of the lines, Comte notes, but explicate how they could be generated. He adds that when a geometer is looking for an equation to describe a line, there is no need to choose any specific coordinate system - sometimes the searched for equation might be easier to describe e.g. in terms of polar coordinates, which are especially convenient when describing rotations. Then again, he admits, when finding lines to describe equations, it is best to pick the rectilinear coordinate system, which is the most natural for us to decipher.

Although Comte at first appears to say that all lines correspond to an equation and all equations correspond to a line, he is well aware that analytic geometry of his time has imperfections in this department. Firstly, he notes, discontinuous lines cannot be expressed so well in terms of a single equation. Furthermore, he continues, equations with more than three variables or equations with imaginary solutions have no proper geometric model.

Like in the case of abstract mathematics, Comte is mostly interested in the uses geometry could be put to and the problems that could be solved by its help. Solving some of these problems relies on simple algebraic means, such as when we try to find the number of points that are required for determining the course of a curve. Still, most of these problems, Comte notes, rely on the help of differential and integral calculus. Differentiation is useful not just for finding tangents, but also for describing e.g. the curvature of curves. Then again, Comte concludes, integration is the most useful tool, because it helps us to fulfill the true task of geometry, that of measuring lengths, areas and volumes.

Auguste Comte: Course of positive philosophy 1 - Mathematical analysis

As the second part of abstract mathematics Comte distinguishes what he calls transcendent analytics, which he characterises as a mathematics of indirect functions. Comte’s point appears to be that while in algebra one is interested of and manipulates functions and quantities given in the task, in the transcendental analytics - in effect, infinitesimal calculus - one introduces and searches for auxiliary quantities and functions, which are related to the original quantities and functions in a more distant manner.

Although Comte does mention the method of exhaustion, known already to Euclid, as a precursor of infinitesimal calculus, he singles out Leibniz and Newton as its first true inventors. Leibnizian method is based on the notion of infinitesimals or infinitely small magnitudes, like infinitely small parts of a curve. These infinitesimals are then used as a way to express certain functions and quantities (e.g. tangents) with the help of relations of these infinitesimals. After some algebraic manipulation of these relations, the infinitesimals can then be discarded in the end, leaving only the searched for function described in terms of ordinary quantities. This method is in Comte’s eyes full of misleading metaphysics - what are these supposed infinitely small magnitudes?

Newtonian method agrees in its results with Leibnizian method, but Comte finds it much more viable. The basis of Newton's method is the notion of a limit - for instance, we can understand tangent of a curve as a limit of a series of secants of the same curve, where the series is formed by letting the endpoints of the secants approach one another. Despite the increased validity Newton’s method, Comte finds it otherwise more cumbersome to use than Leibnizian method. A common problem in both, Comte suggests, is that they both distinguish analysis too severely from algebra, because the main concepts of both methods - infinitesimals and limits - are not something that can be expressed in algebraic terms. Of course, a more algebraic explication of the notion of limit was in the air - Cauchy had already used the famous epsilon-delta -proofs to show what a limit of something was.  Yet, Cauchy's proofs were not widely known and they were not put in form of a clear definition before Bolzano, so it is no surprise Comte still is unaware of the algebraic explication of limit.

Instead, Comte has to rely on Lagrangian understanding of infinitesimal calculus. This is remarkable, because here Comte agrees with another philosopher wanting to provide an account of all sciences, namely, Hegel. Lagrange’s idea was to define the result of both Leibnizian and Newtonian method  - what Lagrange calls the derivative function - in terms of the so-called Taylor series. Mathematician Brook Taylor had shown that many functions could be expressed as an infinite sum consisting of factors made out of the derivative, its derivative etc. In effect, Lagrange reversed this process - he just assumed that a function could be expressed as a Taylor series and then picked the derivative as included in the first factor of the series. Further derivatives could then be just picked from the further factors of the series.

Lagrangian method was at least as cumbersome to use as Newtonian. Furthermore, it created an added problem that one has to prove why the factors of Taylor series had the important properties derivatives we supposed to have. For instance, Lagrange had to still show that a line determined by the derivative at a certain point is tangent to the curve at that same point, because it is the line closest to the curve determined by the original function. Despite this added difficulty, Comte says, Lagrangian notion has the benefit that it makes analysis into a mere new modification of algebra, that is, an algebra of Taylor series.

With this general idea of what analysis is, Comte can easily divide analysis into two parts: differential calculus, which tries to find derivative from an original function, and integration, which goes the opposite way, trying to find the original function related to a derivative. Comte notes that the division is not strict, since many problems require the use of both methods. Both differentiation and integration, Comte continues, can then be further divided into differentiation or integration of explicit functions - in other words, problems, where a function is given and its derivative or a function, to which it is derivative, is asked to be provided - and differentiation and integration of implicit functions - in other words, equations, the solving of which requires the use of differentiation or integration.

Comte notes other ways to classify problems of differentiation and integration. For instance, differentiation can involve formulas of only one variable or of many variables (in the so-called partial differentiation). In relation to integration, on the other hand, Comte notes that integration over integration forms a task of far greater complexity than mere simple integration. He also points out that this is not true of differentiation, because finding derivative of a derivative is as simple as finding the first derivative. This difference, in effect, makes integration a much more complex study than differentiation, Comte concludes.

The basis of all these more or less complex problems, Comte insists, should be the differentiation and integration of the ten basic algebraic functions introduced in his study of algebra.. Even here, Comte points out, differentiation is more complete than integration, where we often have no other way to solve even very simple problems, but to give an approximation of its result through numerical methods.

For Comte, it is important not just to structurise methods of mathematical analysis, but to explain what they are useful for or where they can be applied. The applications he suggests are not surprising - differentiation can be used e.g. to find minimal and maximal points of a curve, while integration can be used in determining quantities of areas. More important is the pragmatic principle that even such an abstract field as mathematics should provide some benefit for society.

Following Lagrange again, Comte includes within transcendent analysis also a sort of generalisation of the problem of finding maximal and minimal points of a curve. The earliest example of this general sort, Comte notes, is Newton’s problem of finding what function determines a solid, which would experience minimum resistance when moving through a homogenous fluid in constant velocity. Another early example pointed by Comte is Bernoulli’s problem that if a body moves from a higher point to a lower point, not directly under it, what curve between the points would describe the fastest descent. Such problems involve minimising or maximising certain integrals by choosing a suitable function as a derivative. Lagrange had the idea that just like in differential calculus we consider ever smaller variations of variables, we could also consider small variations of functions. This similarity of method is enough, Lagrange and Comte conclude, to take this calculus of variations as just another modification of analysis.

In contrast, Comte does not want to include within the field of analysis the so-called calculus of differences, invented by the forementioned Taylor. Taylor’s idea was to generalise differentiation and integration. While regular infinitesimal calculus was based on the notion of infinitesimally small differences, Taylor’s calculus would study finite differences: for instance, instead of finding functions to describe tangents of a curve, it would try to find general form for functions of secants of the same curve. In his calculus, Taylor developed many analogues to ordinary differentiation and integration. Indeed, Taylorian calculus can be used to approximate integrals, which cannot be calculated in any other manner, as Comte himself also notes.

Lagrange’s opinion, which Comte evidently accepts, was that Taylorian calculus was not truly in par with infinitesimal calculus, but a part of regular algebra. More particularly, Comte understands it to be a modification of the theory of series. Thus, Comte notes that Taylorian analogue of differentiation corresponds to finding a rule governing a series of numbers or functions, while Taylorian analogue of integration corresponds to calculating a sum of a series. Somewhat ironically, because Comte did not have the modern explication of limit in his use, he didn't know that differentiation and integration can be presented in terms of limits for series of functions, which would thus reduce the difference between Taylorian and infinitesimal calculus.

While Comte’s take on calculus of differences is heavily inspired by Lagrange, clearly his own invention is to connect it with Fourier’s notion of periodic functions: idea seems to be that the periodicity of functions is a generalisation of a regularity in the finite differences of a function. Fourier’s notion serves also as a good bridge to more concrete mathematics, as Comte ponders the possibility that Fourier’s work of applying periodic functions to thermal physics might form a new part of concrete mathematics. Yet, Comte doesn’t develop this suggestion further, so I’ll instead consider next time his more developed take on another part of concrete mathematics, that is, geometry.