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.

Augustin-Louis Cauchy: Lessons on applications of infinitesimal calculus (1826)

While the previous book of Cauchy I discussed remained mostly on the level of pure mathematics, the very title of this book, Leçons sur les applications de calcul infinitésimal, promises to deal with applied mathematics. Of course, even applying can happen at different levels, and Cauchy is here dealing not with, say, application of mathematics to other sciences, but with application of one part of mathematics within another part mathematics, more precisely, in geometry.

What is applied in geometry is infinitesimal calculus, which consists of differential and integral calculus, two methods which in a sense are counterparts to one another. Yet, in a sense this is not enough, since Cauchy is actually using a variety of mathematical tools. For instance, the book begins with a long introductory section on trigonometric functions. These functions express various relations between lines and angles and can thus be used in simplifying the formulas dealt in calculus.

Another example of mathematical tools used by Cauchy is provided by polar coordinates. Unlike the regular xy-coordinates, polar coordinates express all positions through a distance between the position and the origin and the angle that the line expressing the distance forms with one of the axes. As Cauchy notes, some geometric shapes are easier to express with the polar coordinates. This is particularly true of spirals. Spirals circle in a regular fashion around a centre, which we can think as located at the origin. While the distance between the centre and a point in spiral grows, the direction of the point from the centre changes in a regular fashion.

Differential and integral calculus are still the primary methods used in the book. Originally both methods have been justified through the idea of infinitesimals or infinitely small quantities - hence, their common name, that is, infinitesimal calculus. In my previous discussion of Cauchy, I noted that by an infinitesimal calculus he meant actually a variable which was thought to be diminishing into nothing. Here, he returns to the more relaxed notion of true infinitesimal quantities, perhaps because such deep theoretical questions need not be addressed in a more applied context.

A good example of Cauchy’s tendency to ignore the theoretical questions is his rather free use of the notion of different levels of infinitesimals. It is undoubtedly difficult to understand how one infinitely small quantity can be of a different level from another infinitely small quantity, that is, larger or smaller than it. Cauchy’s theoretical account of infinitesimals is rather enlightening. Different levels of infinitesimals could be defined through different rates at which quantities approach nothing. For example, if a quantity is approaching zero, its square will also approach it, except that the square will approach the goal in a quicker fashion than the original quantity.

Of the two methods, integral calculus seems simpler in the sense that it has less areas of application - and indeed, this part of Cauchy’s book is just a fraction compared to the part dealing with differentiation. Integration in its original sense was thought to consist of dividing something (curve, area or body) into infinitely many infinitely small parts and then, as it were, adding them up. In more modern terms, integration means finding more and more fine grained divisions of curve, area and body and sums of these divisions, and examining whether these sums approach some definite limit. The result of integration is then, simply, the magnitude of the curve, area or body. In fact, the result is at worst only an assumption that the curve, area or body will have some magnitude, and for actually defining this magnitude, information on other relations between the geometrical entities is often needed.

Basic use of differential calculus, on the other hand, is to find out relations between differentials, or again in terms Cauchy used in the previous book, to find out whether relations between variable, diminishing quantities approach some definite limit. An obvious application for this is the relation between variable x- and y-coordinates of points of a curve. One begins by looking at cords connecting ever closer points of the curve and the relationship of their coordinates, which determines the direction or inclination of the cord. The limit of this approach is then a direction or inclination of a tangent moving through this one point of the curve - or, one might even say, the inclination of the curve at that point. That is, in case the tangent even exists.

The last sentence is important. Cauchy already understands that differentiation is not a universal method, but fails at some points. The curve might have a sudden change in its direction, or it might loop back and touch itself, making it impossible to say what is its inclination at this point. It is an important change in self-understanding of mathematicians to accept mathematics as not interested merely of general rules, but also of particular exceptions to these rules.

Inclination of a curve is only one of its characteristics. Indeed, a curve and its tangent have the same inclination at one point and still look quite different around that point. One would like to say that the line has less of a curve than the curve, but it needs a more precise definition to do this. Cauchy asks us to think of a circle - the bigger the circle the more the circle looks like a straight line from a given point and less curved it seems. Hence, since the circle grows with its ray, Cauchy defines the curvature of this circle as inverse of its ray.

Now, while inclinations are represented by tangents of a curve, curvature of a curve can be explained by circles touching the curve - or more precisely, the inverse of their ray - which are known as osculating circles. This is still not a proper definition of curvature, since it is not clear what circle touching the curve one is to take as the osculating circle.

Still, we can use the circle as a clue for finding out what curvature is. Picture a tangent moving through the circle, always remaining a tangent and changing its inclination as it advances through an arc of the circle. Taking smaller and smaller arcs around a point, the relation of the inclination of the tangent to length of the arc might approach a certain constant, which happens to be the inverse of the ray of the circle or its curvature.

This notion of curvature is at once applicable to other curves, since they also involve the variables of the arc and the inclination. True, the concept of curvature does not work with all curves, at least not in all points. In particular, it requires the curve in question to be twice differentiable.

Cauchy’s book deals with many other important concepts. Worth mentioning is what happens when we move from a setting with two coordinates to a case with three conditions. I won’t go into details, but note merely that then mere curvature defined in the sense above is not enough, because this first curvature deals only with a two-dimensional issue. We also need then to take into account another quantity defining the curvedness of the figure when looking at the third dimension - this curvedness Cauchy calls second curvature.

Jean-Babtiste Joseph Fourier: Analytical theory of heat (1822)

(1768-1830)

The history of the interaction of mathematics and physics has not just been one of unidirectional influence. Certainly the development of mathematics has been of great importance to physics, by providing it new and improved tools for modeling natural phenomena. Yet, physics has also offered inspiration and spur for development of new mathematical tools. The tale of Fourier’s Théorie analytique de la chaleur is of the latter sort.

Fourier’s starting point was the revolutionary use of mathematics in understanding nature, instigated by the works of Descartes and completed, in a sense, in the works of Isaac Newton. What they did was to extend the use of mathematics from mere tool for studying of figures into a tool for studying the motions of bodies. The success of Descartes and Newton inspired others to investigate whether mathematics could be useful in studying other natural phenomena.

One obvious candidate was the propagation of heat through a substance. Whatever heat was thought to be - often it was considered a distinct caloric substance that permeated all objects - it certainly appeared to “flow” through these objects, touching at first only one spot of the object in question and gradually spreading through the object and finding a point of equilibrium. The physical model was simple enough, all that was needed was to express the movement of the heat mathematically.

Fourier noted, firstly, that the movement of heat in an object was dependent on three things specific to the object, its constitution and its relation to its environment: the capacity of the object to assimilate heat (heat capacity), the capacity of its parts or molecules to transmit heat to one another (thermal conductivity) and the capacity of the environment to transmit heat to the object in question. In practice, we can limit our attention to the first two, because they form the basis of his theory of heat flow and the question of one object transmitting heat to another merely complexifies the basic theory. For simplicity’s sake, Fourier regarded these two quantities as simple constants, although the heating of an object might in reality affect them.

While physical objects are, of course, three-dimensional, we can first concentrate on the simple case of one-dimensional transfer of heat, e.g. within a barlike object. Clearly, the more distant a point in the object is from the source of the heat, the colder the point is and the opposite end from the source remains coldest. Now, Fourier supposed that the temperature of a point, at a given time, is in a sense proportional to the distance from the heat source. To put it more precisely, if at some time the temperature at the heat source is a and temperature at the opposite end of the bar is b and e is a given unit of temperature, then at a given point of the bar, with a distance z from the heat source, the temperature at that point can be calculated by subtracting z(a-b)/e from a.

Now, the temperatures a and b do not remain same, since heat is continuously flowing from one end of the bar to another and a keeps decreasing while b increases. To make the situation easier to handle, Fourier supposes that temperatures a and b are artificially kept constant, e.g. through an external heat source warming a. This means, he continues, that all the temperatures between the extremes of the bar also remain constant, that is, the temperature v always decreases while we move away from the heat source, at a rate (dv/dz) opposite to (a-b)/e. At the same time, heat is constantly flowing through the bar, and this flow, Fourier argues, is at least partially expressed by the formula (a-b)/e, that is, the greater the difference between the ends of the bar, the more heat flows from the warmer to the colder end.

If we forget the assumption of a and b being constant, we might say that (a-b)/e or −(dv/dz) partially represents the flow of heat characteristic of a bar of certain substance at a certain point of time. This expression cannot be the whole truth of the notion of flow of heat, because different substances have different capacities for conducting heat through them. Then again, he concludes, this expression together with the constant K describing the thermal conductivity of the substance describes completely the flow F of heat through a bar: F = K(a-b)/e or in terms of an infinitesimal change of temperature dv through an infinitesimally long length of bar dz, F = −K(dv/dz).

Next step in Fourier's argument is generalization of this formula to three-dimensional flow of heat. He begins by considering a prism, with one corner having the highest temperature A and heat flowing from this corner to all the other directions. Just like with the case of the bar, the further one goes from the source of hear, the less is the temperature, although now we have to account for three dimensions, when counting temperature at a given point at a given time - that is, the formula for counting the temperature of point (x, y, z) of prism looks something like A − ax − by − cz. Again, just like with the bar, by keeping the temperature constant at the limiting surfaces of the prism the temperature remains constant at all the points within the prism. By restricting then the investigation to heat flows in one dimension, heat flows in lines crossing planes perpendicular to x-, y- and z-axes will then be respectively −K(dv/dx), −K(dv/dy) and −K(dv/dz).

The final ingredient to be added to a general theory of heat is time. We again remove the assumption that the limits of a solid would have constant temperatures. Then the temperature of points within solid change as time goes on and heat flows in some manner through the solid, that is, temperature becomes a function of spatial coordinates and time: v = f(x, y, z, t). Fourier’s next move is to restrict the attention to flow through an infinitesimally small circle o at an infinitesimal instant dt, where due to extreme smallness the condition of the precious paragraph then apply. If we suppose the circle to be situated perpendicularly to z-axis, the heat flow going through it should be, Fourier says, −K(dv/dz)odt.

The notion of infinitesimals is, undoubtedly, unclear gibberish according to more modern understanding of differential and integral calculus, but even greater gibberish is to follow. Supposing the ring o to have an infinitesimal thickness, dz, we can distinguish between the heat flow coming within o and heat flow leaving o. The former is, expectedly, −K(dv/dz)odzdt, while the latter is almost the same, differing from this already infinitesimal quantity by “an even smaller infinitesimal”. The difference of the two quantities - that is, the amount of heat left within the ring after the instantaneous flow of heat - is just this type of “second-grade” infinitesimal, namely, K(d2v/dz2)odzdt. Here the expression (d2v/dz2) and its relation to (dv/dz) - rate of change of temperature, when moving through z-axis at moment dt - can be understood through an analogy with the relation of acceleration and velocity. In effect, (d2v/dz2) describes how the value of (dv/dz) itself changes, when moving through z-axis at moment dt.

Now, Fourier notes that the shape of o is not important and that we might as well take instead an infinitely small rectangle dxdy, making the flow through that rectangle, at instant dt, −Kdxdy(dz/dv)dt. Consider then an infinitely small cube of size dxdydz. The heat flow forming within that cube, at instant dt, is the sum of heat flows left within the cube, when heat flows coming in and going out from and to all three directions have been accounted for, namely K(d2v/dx2 + d2v/dy2 + d2v/dz2)dxdydzdt.

Now that the quantity of heat accumulating within an infinitesimal point of a solid at an infinitesimal instant of time has been, in a sense, determined, we can answer the question how does the temperature of that point develop over a period of time. It is not just a matter of dropping dt out from the formula, since heat and temperature are not completely same thing. Instead, we finally need the notion of heat capacity C of substance, which Fourier defines as the relation how much heat is required for increasing temperature of an object of certain weight. In order to get the required quantity of weight, we also need to take into account density D, that is, the relation how much certain volume of this substance weighs. By putting all these ingredients together, we find out that the rate of change of temperature over time, dv/dt, equals (K/(CD))(d2v/dx2 + d2v/dy2 + d2v/dz2).

What Fourier’s complex argument has provided is a position, where we can continue with purely mathematical methods. It is still unclear what the function f(x, y, z, t) determining the temperature of a point within solid at a certain time should be. We do know that the equation Fourier has found could correspond to infinitely many functions, but that certain additional conditions might be enough for determining the function. What really interests us is the method Fourier uses in solving the function from given conditions. To put it shortly, Fourier starts with an assumption that the function in question can be expressed in terms of simpler functions. To be more precise, he assumes that the function can be expressed as a sum of a possibly infinite series of trigonometric functions. The assumption happens to make sense in the context of heat transmission, because this physical process is not too erratic. In other words, the changes in the transmission can be approximately described with sums of cosines and sines. All Fourier then needs is a systematic method for determining these constituent functions, which is a simple enough task.

The idea of using sums of trigonometric functions as a way to determine heat functions was not completely novel. Yet, Fourier was the first person to assume that this method could be used in so extensive manner. While trying to solve a physical problem - how to describe movement of heat - he launched a completely new area of mathematical studies, the so-called Fourier series.