Carl Gustav Jacob Jacobi: Fundaments of a new theory of elliptic functions (1829)

(1804-1851)

The development of mathematics has often been one of generalisation: concrete problems have demanded development of very abstract tools that can then be applied to various other fields. One clear example is integration. Originally developed as a tool for calculating lengths, areas and volumes, it could then be used in various other contexts where calculating limits of infinite sums with high precision was required.

Another aspect of the development of mathematics, then, has been that these very abstract tools themselves provide new problems and topics for discussion. For instance, going back to the example of integration, if you progress beyond Calculus 101, you soon learn that not all integrals can be solved through those neat formulas given in the textbook, but in the worst case scenario you have to go to the definition of integral and approximate it through various finite sums. Of course, mathematicians have found various new tools for simplifying this numerical process in individual cases. A particularly interesting case concerns the so-called elliptic integrals.

The very concept of an elliptic integral belies an origin in quite concrete geometric problems: measuring the length of pieces of a curve called ellipsis (picture an elongated version of a circle). In truth, this example is just one version of elliptic integrals, the unifying elements being certain simplicity in the formal characteristics of the base function integrated (to put it very briefly, they involve nothing more complex than fractions with denominator a square root of polynomial of third degree). These kinds of integrals are already harder than those in elementary textbooks and their exact values can often be just approximated. The question is how to simplify this process of approximation.

First step in this simplification was provided by Adrien-Marie Legendre, who showed that all the various elliptic integrals could be reduced to three paradigmatic cases - already a huge improvement. Another important step was to note that these elliptic functions could be expressed in terms of two parameters: an angle called an amplitude of the integral and a number called module. In the particular case of ellipsis, the amplitude describes the angle the x-axis makes with a line joining the origin and the tip of its particular arc, while the module describes how elongated the ellipsis is (0 being the case of a proper circle).

Jacobi’s Fundamenta nova theoriae functionum ellipticarum took the simplification a few steps further. The basic problem Jacobi set out to solve was to show that an elliptic integral with a seemingly more complex structure could be reduced to an elliptic integral of a less complex kind, provided some relation of them, expressible through relatively simple algebraic means, was shown to hold between them. In other words, by knowing that such a relation existed between the two integrals, one could calculate the value of the more complex on the basis of the simpler one. Jacobi calls this transformation of the elliptic integral.

Now, the question was how to determine this relatively simple transformation. Jacobi showed that this question was essentially the same as determining a certain type of relation between the modules of the two integrals (what he called modular equation). In principle, this modular equation could be calculated through algebraic means, but in practice, the more complex the equation changes, the more cumbersome this calculation becomes. Jacobi’s solution is to go a bit further in the level of abstraction and to construct a more general rule picking a series of suitable modules that can be linked with such transformations.

Jacobi’s derivation of this rule is based on the second parameter, the amplitude, and the so-called elliptic functions, which can be defined on the basis of the amplitudes. To give a rough idea of these elliptic functions, we can compare them with simpler trigonometric functions. It is a well-known fact that trigonometric functions can be described in terms of a unit circle and angles set up on its centre. Elliptic functions can also be described in terms of an angle set up on a centre of an ellipse.

It was just inevitable that this new tool - elliptic functions - became a topic interesting in itself. Thus, the second half of Jacobi’s work is dedicated to the study of elliptic functions, which, just like trigonometric functions, are a source of many beautiful equations. A particular question Jacobi dealt with was how to express these functions as infinite series. In effect, this was yet again a way to find more and more good approximations for elliptic functions. Finding these infinite series required the introduction of yet another tool: the so-called theta functions, which are a certain type of series of complex numbers - and the development of the mathematics continued.

James Mill, Analysis of the Phenomena of the Human Mind 2 - Yawns and hiccups

The latter part of the second volume of Mill’s book concerns the active or volitional part of the human mind. Mill notes that the cognitive and volitional sides of human mind share the same structure - for instance, volitional side also deals with sensations and ideas. The difference is, Mill suggests, that while sensations of the theoretical part are, in a sense, indifferent to the mind, the sensations of the volitional part are not, being either pleasant or such that the mind would want prolong, or painful or such that the mind would want to stop them. Mill makes it sound like the trivision of sensations would be hard and fast, while one might object that e.g. a sensation pleasant now might become indifferent and even painful, if it has to be endured too long.

Like indifferent sensations, pleasant and painful - or interesting - sensations can also be revived in ideas, Mill says, and such ideas of pleasant and painful sensations are called desires and aversions. This choice of nomenclature seems peculiar. Mill himself points out one possible point of contention: we also say that we desire things like cake, which are not sensations. In fact, Mill says, in all such cases we ultimately desire some sensation, e.g. the taste of the cake. Yet, there is a more important point of contention, because we can think of a pleasure we’ve had without desiring it. Mill offers the explanation that desires are actually only ideas of future pleasures. This explanation seems insufficient, because we could think of future pleasures and still not yet desire them. Indeed, it seems far more plausible to take desire as a primitive concept, especially as Mill’s explanation of pleasures already implicitly referred to it (pleasure is a sensation we would want or desire to continue).

In case of pleasant and painful sensations, Mill continues, we also make associations to their causes. In fact, thinking about such causes might affect us more than thinking of the pleasures and pains themselves. For instance, we might think of a past stomach ache with indifference, but still avoid the food that we think caused this sickness at the time. Mill appears to think that such affections are not just caused, but also defined merely by such associations, which seems again insufficient - surely they also contain the aspect of us wanting to gain or avoid something.

Mill notes that we often have stronger affections toward more remote causes of pleasures and pains. For instance, people often desire wealth, power and dignity more than, say, food. Mill has a quite convincing explanation for this peculiarity: remote causes, like money, allow us to gain more and a larger variety of pleasures than mere immediate causes.

An important subset of affection toward remote causes concerns other persons. In some cases, Mill notes, such affections are awakened by shared experiences and common interests. Then again, he adds, we also have a general compassion toward all humans, because the similarities with them make us associate their pleasures and pains with our own pleasures and pains. The more similarities we have toward some group of people, the more affection we have for them, for instance, in case of people from the same class (in a pre-Marxist fashion, Mill insists that only privileged classes could have such a class consciousness).

Mill touches also aesthetic affections, but unfortunately, not in any great detail. His main point is that we call sensations beautiful and sublime not because of themselves, but because of associations they convey in us. Thus, the humming of a beetle is found beautiful, not because of any intrinsic feature, but because it awakens in us the notion of summer. Because such associations may vary from one person to another, there is no universal criterion of beauty, Mill concludes: black seems ugly in a culture, where it is associated with death, beautiful in a culture, where it is associated with festivities. What is especially lacking in Mill’s account is the explanation what kind of associations are required for calling something beautiful or sublime.

When we have an idea of ourselves as the only possible cause for gaining some future pleasure or for averting some future pain (or some of their causes), we have a motive, Mill defines. Clearly, this definition works only if we assume, like him, that all thoughts of future pleasures involve desire. In any case, motives can work against one another, and indeed, Mill insists, only another motive can prevent us putting one into action. Different people are affected in different measures by same motives, and a particular affinity to some motive is called disposition. Mill notes that in common parlance we often confuse affections, motives and dispositions: thus, we may speak of lust when speaking of a positive affection toward sex (idea of sex as causing desirable sensations), of a motive for engaging in sexual relations (idea of ourselves as instigators of sex) or of a disposition to engage in sexual relations.

In a Humean manner, Mill considers cause and effect to be nothing more than a name for a regular association of certain events. Thus, he sees no problem in saying that sensations cause certain bodily actions - we can say that a pungent odour makes us sneeze, while a certain sensation in our stomach makes us hiccup. These examples might make Mill’s analysis of causation seem suspect, since it seems more likely that in such cases the sensation is not really the cause of the bodily movement , but merely shares with it a common cause (some bodily process).

In any case, because Mill thinks sensations can produce bodily movements, he sees no difficulty in ideas causing them also. In fact, he points out laughter caused by humour or weeping caused by sadness as examples of this kind of causation. Of course, Mill adds, such uncontrolled weeping is still not voluntary action. What is still required is the presence of a desire: when motives make us act, they are called will.

Mill does not then believe in any motiveless will. He insists also that will cannot awaken its own motives - this would be like baron Munchhausen lifting himself from his own hair. This rather plausible suggestion makes Mill go even so far as to suggest that it has no effect on the train of ideas, being just a process of translating ideas into action. Mill considers two possible objections. First is the notion that we often seem to will to recollect something. Mill’s answer is that actually we always only desire to do so. Similarly, to the second objection that we often will to attend to some sensations or ideas, Mill answers that attending means just that we find these sensations or ideas interesting. Mill’s answers seem just verbal confusions, since he admits that the very element that makes acts into acts of will (desire or interest) is involved also in these two cases. One might even rephrase the objections in a manner suggesting that there is a choice involved. Suppose we are attending to something complex, like a bicycle. The idea we have of it has different aspects, and we may then choose to attend to one of these aspects, say, one of the tires. Couldn’t we then say that we willed or wanted to attend to the tire?

Nikolai Lobachevsky: Geometrical researches on the theory of parallels (1840)

 

(1792-1856)

Euclid’s book on geometry has for ages been seen as an ideal of an axiomatic theory, in which everything is based on a solid basis of definitions and evidently certain axioms and proven through strict demonstrations, making the results presented appear indubitable. No wonder many works of philosophy tried to imitate Euclid’s style, to make their theories seem as indubitable and necessary, usually failing miserably to be as convincing as Euclid.

If you know your Euclid by heart, you know that he had not really achieved the ideal many want to see in his book. There are sometimes slight hidden assumptions in his proofs - and isn’t it a bit too empirical to carry around triangles and put them on top of one another (Euclid, I.4)?

The most glaring fault in Euclid’s work is, of course, the infamous parallel postulate. When compared with other postulates of Euclid, it appears complex and far from self-evident: “if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles”. The postulate can be made a bit clearer with some rewording and use of more modern phraseology - if line A cuts two other lines on the same plane, B and C, and the sum of interior angles on one side equals pi (or 180, if you are more into degrees), B and C eventually cut one another on that side and are therefore not parallel. Even with this rewording, it seems like a theorem we should prove, not a postulate to be just assumed.

Many professional geometers and even more geometry dilettantes were equally unimpressed by this postulate and tried to demonstrate it from other postulates. Their efforts led at most to finding other postulates that could replace Euclid’s. Most famous of them is the so-called Playfair’s axiom: given a line and a point, we can draw through the point, at the same plane as the line and the point, only one line that does not cut the first one. This does sound simpler, but still lacks the self-evidency of the other postulates.

Dissatisfaction with Euclid’s postulates and its alternatives continued, but no solution was forthcoming. All of this was changed by Lobachevsky’s seminal paper, Geometrische Untersuchungen zur Theorie der Parallellinien. Well, to be truthful, he had already written papers on the topic in his native language, Russian, in 1820s, but these did not circulate very widely (and in addition, I cannot read Russian).

Lobachevsky’s starting point is the Playfair’s axiom, but instead of just assuming it, he asks what would happen, if there were more than one line we could draw through the point - lines which would not cut the given line. He notes that even then we could find a single particularly interesting one among those non-cutting lines, namely, the limit between lines that do and those that do not, and suggests calling this the parallel line. Then, by tying Playfair’s axiom back to the original framing of Euclid’s postulate (C being the original line, B being parallel to it, A cutting them both and A and B meeting at the given point) and by making the assumption that A cuts C perpendicularly, he notes that in this peculiar setting B and A form an angle less than half the pi (or 90 degrees), thus contradicting Euclid’s postulate. The angle formed by A and B (and dependent on the distance of the given point from the line C), Lobachevsky calls the angle of parallelism.

Although Lobachevsky’s new geometry - later dubbed hyperbolic geometry, although he himself called it imaginary - has clearly different properties from Euclidean geometry, a significant portion of Lobachevsky’s paper is committed to show similarities to Euclidean geometry. The simplest similarity is that the new definition of parallel lines works similarly enough to the Euclidean notion, for instance, parallelism is a symmetrical and transitive relation.

A more intricate similarity Lobachevsky finds through notions of oricycle and orisphere. By oricycle Lobachevsky means such a curve in hyperbolic geometry, all perpendiculars or axes of which are parallel to each other. Furthermore, oricycle is also a sort of limit for circles - by enlarging the ray of the circle indefinitely, in hyperbolic geometry, the curve tends toward the oricycle. In a figurative way, we could say that an oricycle is an infinite circle. Interestingly, the same concept in Euclidean geometry means ordinary straight line.

The notion of oricycle taken into three dimensions forms, then, an orisphere. Technically, an orisphere can be formed from an oricycle by turning it around one of its axes. What is interesting is that oricycles on orisphere work like straight lines on a plane in Euclidean geometry, for instance, in a “triangle” formed of segments of three different oricycles, the sum of the angles equals pi.In effect, two-dimensional Euclidean geometry can be ingrained within three-dimensional hyperbolic geometry.

Although the aforementioned pseudotriangles in hyperbolic geometry do follow same rules as regular triangles in Euclidean geometry, regular triangles in hyperbolic geometry do not. Yet, Lobachevsky points out, when sides of the triangles in hyperbolic geometry decrease indefinitely, the more they start to resemble the triangles in Euclidean geometry, for instance, the sum of their angles approaches pi.An interesting consequence of this is that the bigger the triangles in question are, the more apparent the difference of the two geometries becomes. It becomes then an empirical problem to decide whether we live in a space with a Euclidean or a hyperbolic geometry - just make astronomical measurements of distances between stars and you might notice signs of non-Euclidean properties.