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.