Auguste Comte: Course of positive philosophy 1 - Solving equations

In a way that sounds rather old-fashioned these days, Comte defines mathematics as a science of measuring magnitudes. Of course, back in his days, mathematics was mostly about numbers, so the definition makes more sense. Furthermore, Comte instantly qualifies his statement, by noting that immediate measuring of, say, length with a ruler or temperature with thermometer is not yet mathematics. Instead, mathematics is all about indirect methods of measuring unknown magnitudes through their relations to others, known magnitudes. In other words, mathematics has to do with solving equations between magnitudes.

Comte is convinced that mathematics is a universal science that applies in principle to anything. His conviction means that everything should be in principle quantifiable. Comte mentions that this is in direct opposition to Kant’s table of categories, where qualities are kept strictly separate from quantities (this seems a rather peculiar way to understand Kant’s division of categories, but I’ll let it pass now). Even organic and social phenomena should be quantifiable, although their complexity might prevent us ever giving a full quantification of them, Comte hastens to add.

Comte divides mathematics into two parts. One part or concrete mathematics deals with measuring magnitudes in empirical and phenomenal matters, such as geometry and mechanics - I shall leave these disciplines to a later post. The other part deals with measuring magnitudes in abstract fashion, or as Comte puts it, with the logic of mathematics. More precisely, the topic of abstract mathematics consists of, Comte says, equations between abstract functions. By function Comte means some type of dependency, for instance, a sum of two magnitudes is their function, because the sum is dependent on what the magnitudes are. Now, abstract function is a dependency that can be understood only on the basis of bare magnitudes or numbers - sum of two magnitudes is the same, no matter whether the magnitudes are units of length, time, mass etc. On the contrary, concrete function expresses a dependency, understanding of which requires something more than mere numbers, such as geometric or mechanical properties.

Comte’s definition might still leave it unclear what to actually include in the abstract functions. He points out that we can at least enumerate some examples of pairs of simple abstract functions (pairs, because they consist of a function and its inverse). He adds that we can then know that any complex function that could be constructed from these simple functions is also an abstract function. These pairs would at least include, Comte recounts, addition and its inverse or subtraction, multiplication and division, raising to a power and roots, and exponential and logarithmic functions.

A more intricate question is whether to include among abstract simple functions also sinus and inverse sinus. The question is difficult, because if sinus is taken as a simple, unanalysable function, then it seems far from numerical, since it receives its meaning from a certain geometrical context (e.g. a unit circle). Then again, sinus can be defined in a purely numerical fashion, but then it is not anymore a simple function. Yet, just because of this dual nature Comte accepts the pair among simple functions, and instantly notes that other functions might also deserve to be included for the same reason, for instance, Jacobi’s theta-function.

Comte divides abstract mathematics into two disciplines, corresponding to two stages of solving equations. Firstly, one transforms or resolves functions given in the equation into other, more easily solvable forms - this is the task of algebra. Secondly, one finds the values of these easier functions - this is the task of arithmetic. Comte notes that this notion of arithmetic is more extensive than what is usually meant by it, since it includes also e.g. the use of logarithmic tables. He also points out that arithmetic is in a sense just a special case of algebra, since finding a value for a certain formula just means turning it into form (10^n)a + (10^(n-1))b + (10^(n-2))c + (10^(n-3))d + …, where a, b, c, d ... etc. are natural numbers smaller than 10. We might thus say that abstract mathematics is nothing but algebra.

Comte divides algebra further into a study of what he calls indirect functions - transcendent analysis or infinitesimal calculus - and study of direct functions - algebra in the proper sense. I shall concentrate in the rest of this post on the latter, leaving infinitesimals for the next one. Well, there’s not that much of philosophical interest in what Comte still has to say about algebra. He notes that the current state of algebra was far from complete, since general solutions had been found only for polynomial equations up to fourth degree of complexity - he was apparently unaware of the recently discovered fact that such general solutions could not be given for more complex polynomials. Still, this supposed incomplete state of algebra gives him an opportunity to mention that numerical methods of solving equations form a second part of algebra.

More interesting is Comte’s idea that because algebra abstracts from all the conditions for the meaningfulness of functions and equations, it instead aims for being as general as possible. Thus, in algebra all the functions or operations are defined so that they always will have results, no matter whether these results can be interpreted meaningfully. Hence, the notion of number is extended, first, to negative numbers smaller than zero, because all subtractions should produce some results, and finally even to seemingly impossible imaginary numbers, which are roots of negative numbers.

Auguste Comte: Course of positive philosophy 1 - Classification of sciences

Classification of disciplines has been a staple of philosophy since the time of Aristotle, and Comte’s studies make no exception. Of course, there have been plenty of classifications presented as the correct one, so it is reasonable to ask what is so special in Comte’s. Comte himself has a clear answer: earlier classifications derive from a time when all fields of science had not reached the status of positivism.

Before introducing his classification of sciences, Comte considers the question of what he should actually be classifying. He first distinguishes theoretical sciences from practical arts and delineates between these two extremes the field of engineering, which aims at applying results of theoretical sciences to practical questions. He notes that arts and engineering are essentially dependent on theoretical sciences. Indeed, he adds, one art can depend on various sciences, for instance, agriculture requires theoretical knowledge of plants, of chemicals and even of sun, moon and stars. Thus, he concludes that the basic classification should be made at the level of theoretical sciences, not on the level of their practical applications.

Another distinction Comte makes is that between general or abstract sciences and particular or concrete sciences. Abstract sciences, he explains, deal with what is possible, for instance, according to known physical and chemical laws. Concrete sciences then deal with actual instances of such laws: examples include natural history and mineralogy. Comte also points out that just like practical arts depend on theoretical sciences, concrete sciences depend on abstract sciences, being their specifications. Thus, the disciplines good for classification are abstract theoretical sciences.

Comte goes on to speak about the method one should use in the classification. He points out that while we strive for what could be called a natural classification, we can approach such classification only through various artificial classifications. Indeed, he notes, we often begin classification of a new science historically, that is, by noting new ideas and discoveries in the order in which they were found. The more a science is developed, Comte notes, less and less possible it becomes to use the historical approach, because the number of theorems involved becomes too unwieldy. Historical classification is then replaced by a dogmatic approach, where the ideas in question should form a systematic whole. This dogmatic approach abbreviates the historical approach. In the particular case of classifying all sciences, there is the further link that the most abstract disciplines, from which the dogmatic approach begins, are also historically the earliest to reach a more complete stage.

Comte’s classification is meant to be a basis for a complete reform of the educational system. His idea is simple: if we can organise general theoretical sciences into a hierarchical system, in accordance with the dogmatic approach, we then know the ideal order of science education - the education must always start with the most abstract science and move towards more concrete ones. That way, researches working with problems of the more concrete sort would have the necessary tools for understanding more abstract field, on which the more concrete questions depend.

The actual classification Comte suggests seems somewhat problematic - this is just to be expected, since the development of science has been explosive in the last two centuries. Comte’s main dividing line between sciences of inorganic and organic nature seems acceptable, but subdivisions of the two feel less successful. For instance, Comte divides study of inorganic nature into study of stellar phenomena or astronomy and study of terrestrial phenomena, which he then divides into study of more mechanical phenomena or physics and chemistry. Considering that Comte clearly states that zoology and botany are not divisions of the abstract study of organic nature, being more like concrete applications of the study of organisms for two actual species of organisms, one might protest that astronomy is also just application of the same physical laws into stellar objects. Indeed, this is even more evident nowadays, when we know that chemistry could also be used to describe elements of the stellar objects.

Comte’s division of the study of organisms is also problematic. He suggests dividing this whole into a study of individual organisms or physiology and a study of interactions of organisms or sociology. One has to wonder if this is just a circumspect way to distinguish study of humans from study of plants and animals, trying to avoid the same criticism that Comte himself leveled against distinguishing zoology and botany, or if he truly will accept study of all populations of organisms as application of sociology. Even if the latter would be true, it is still doubtful whether we really can meaningfully separate study of an individual organism from study of the interactions of organisms in the same species.

There’s one very conspicuous absence in Comte’s classification - mathematics. This is simply because mathematics plays a very special role for Comte. In a sense it is a part of the classification, the most abstract science there is. In another sense it is for Comte a general methodology for all concrete sciences. Because of this role, mathematics is tightly linked to things it is used for, although it also has a part that is pure of all applications - but this is a discussion I’ll be entering later.

Auguste Comte: Course of positive philosophy 1 (1830) - Toward the age of positivism

(1798-1857)

The style and topic of Comte’s Cours de philosophie positive feel strikingly modern. Indeed, it is one of the first examples of philosophy of science, field of study that became increasingly dominant in the philosophy texts of the 20th century, and its main thrust still feels topical today.

Of course, there are clear signs of Comte being a child of his time, especially in his idea of the three stages of human thought: theological, metaphysical and positive. Similar notions of progressive stages of humanity, leading from crude religious thoughts to scientific outlook had been a staple of French thought. One need mention just Saint-Simon, whose work indeed had an influence on Comtee. Compared to the systems of his predecessors, who might have distinguished a dozen stages, Comte’s version seems much more streamlined, which shows his attempt to transform such crude historical schemes into a real law of human progression.

A more important novelty is Comte’s idea that the three stages are not really distinct, but more like abstractions from a concrete continuum. Thus, he notes that we really cannot determine a specific spot where e.g. the positive stage began, because it has progressed in different manners in different sciences, and while physical sciences have already managed to eradicate theological and metaphysical notions, they still abound in human sciences.

While it is quite easy to understand what Comte means by the theological stage, where everything is explained by actions of divinities, the notion of metaphysical stage is not so simple to understand. Metaphysics should supposedly replace gods with abstract forces that act as explanatory causes. This description could fit a number of theories, for instance, Neoplatonic hierarchies of abstractions, but when we see Comte suggesting that metaphysics ultimately strives toward unifying all these causes in the notion of Nature, it appears that he is especially referring to various materialist philosophies that try to explain phenomena through some ultimate group of material existents.

Now, an obvious question such historical schemes suggest is whether they are meant to be just a very general description of past events or whether they imply that such progression has been necessary. Comte at least seems to take the latter route. He states that the three stages can be found even in the development of individual human beings, at least in the sense that we all must begin as theologians, searching for purposeful actors behind everything we experience. Indeed, Comte says, at the dawn of humanity such behaviour was quite rational, since there simply was not enough information to tell how e.g. the stars moved.

Even if the beginning of human history is necessarily theological, it seems to require more justification to state that it is necessary to move from this stage toward the so-called positive stage. Comte does not delineate any argument for this stance, at least in the first chapter of his Course, but he appears to have the idea that when we gather more and more information about the phenomena around us, we firstly notice that we really cannot find the ultimate causes of them, and secondly, also notice that we actually need no such explanations. In other words, moving toward the positive stage means rejecting all theological assumptions, but not by assuming other, materialistic assumptions. Thus, the positive stage is one of agnosticism and skepticism about ultimate causes, that is, in it we merely describe the laws or regularities of phenomena, but do not try to explain them.

The final justification of this historical scheme should apparently be given in the science of human societies, the foundation of which is one task Comte sets for himself in the course. This scientific study of humanity should replace psychology of his times, which he thinks to be still filled with theological and metaphysical assumptions. Comte especially criticises the use of self-observation as a method of psychology. In physical sciences, he notes, we have already learned that human observation might fake us, suggesting e.g. sun to be a much smaller object than it actually is. Why should we assume that observation of our own actions would be more trustworthy?

The task of creating a social science is in Comte’s eyes intricately linked to a second task, that of forming a system out of all individual sciences. Only through social science, Comte says, can we recognise logic, that is, scientific methodology, and so it helps us to understand relations between all sciences. Comte notes that creation of such a system or hierarchy of sciences satisfies our need for unification, evident in the earlier replacement of polytheism with monotheism and of pluralism of forces with monistic Nature. Comte speculates that this unification has its limits and that we can probably never reduce laws of different sciences into one law. Still, he notes, there’s at least a homogenous method combining all sciences into a unity.

In addition to these two tasks, Comte suggests several benefits his undertaking might have. Firstly, delineating the relations between scientific disciplines might suggest fruitful interdisciplinary studies. Comte mentions as examples Cartesian application of algebra into geometry and recent studies in organic chemistry. Another benefit lies in the reorganisation of education, where mere literary studies could be replaced by a curriculum designed around the system of sciences and beginning with their general methodology.

The final benefit Comte suggests links his undertaking again to the contemporary discussions in French philosophy. He notes that theological and metaphysical thinkers had disputed about the best possible form of governance: it is likely that he is referring here, on the one hand, to Catholic conservatives siding with absolute monarchy, and on the other hand, various materialist leaning thinkers speaking for more republican or at least constitutional state. Comte appears to be suggesting that this debate cannot be solved through philosophical disputations, but only through scientific description of how human societies work and how they could be organised most effectively - a task for the social sciences.

Louis-Gabriel-Ambroise vicomte de Bonald: Philosophical demonstration of the constitutive principle of society (1830)

In his book Démonstration Philosophique du Principe Constitutif de la Société, one of the leading stars of French conservative school of philosophy, de Bonald, returns to questions of political philosophy. Still, he also has something to say in the introduction of the work about different philosophical schools, especially as they had appeared in contemporary France. In a previous book, Recherches philosophiques, de Bonald had distinguished two separate schools of philosophy - Platonists, who lean toward innate ideas as the source of knowledge and uphold spiritualism and theism, and Aristotelians, who lean toward sensations as the source of knowledge and uphold materialism and atheism.

Now, de Bonald, distinguishes also a third school, eclecticism, represented by thinkers such as Maine de Biran, who were not part of the empirical tradition of Condillac, but who also were not attached to the very Catholic inspired school of de Bonald. De Bonald’s quick judgement is that such thinkers are simply inconsistent and that there really are only two possible philosophical positions to choose from - and in the end, only Catholicism is actually true.

In the book itself, de Bonald attempts to give a new justification to his political theory. He begins with a very traditional account of family: family consists of three roles, father, mother and child. Family as such is always monogamous, de Bonald says, because polygamous family would mean just combining many families together under same father (note how de Bonald conveniently forgets the possibility of polyandry). He goes even so far as to suggest that serial monogamy, based on the possibility of divorce, is just polygamy in disguise, although here the different families succeed one another and do not overlap in time.

De Bonald’s idea of family is not just heteronormative, but also patriarchal, as he insists that all the power in the family should reside with the father. Power of the father is absolute and independent of the mother and the child and divides into two parts: power to judge what is good for the family and power to combat any obstacles against the good of the family. Despite the power of family being concentrated to the father, the purpose of the family, de Bonald notes, is to take care that human species will continue through its individuals and especially children.

Between the power of father and the service of children lies the role of mother, who works as a sort of minister for the father, de Bonald tells. In a remarkably insulting statement de Bonald reveals that mother as a mediating element of the family resembles both men and children - being servant to one and controlling the other - and could thus be called a manchild.

Families tend to reproduce and thus multiply, and the aim of a state, de Bonald states, is to guarantee the continual regeneration of family life, just like families guarantee the continual regeneration of human individuals. Thus, he concludes, states should be like big families. If states have grown from a single family, this happens quite naturally, de Bonald insists, by central power remaining always in the hereditary line of succession. Then again, even in a case where a group of unrelated individuals and families combine into a state, there is usually some heroic person who acts as the central node in bringing all together, de Bonald assures the reader. He is explicitly criticising the idea of a social contract made in a state of a nature, where a group of unorganised individuals could invent a political structure to guide them.

De Bonald’s ideal of a state is thus monarchic, as we know from his previous writings. Central power must always be unified, absolute and independent of everything else, he says and adds that a king should have the final authority in deciding the ownership of the land and soil of the state. Just like in an ideal formation of a state, the monarchy should be perpetuated through heredity.

An ideal state, de Bonald continues, shouldn’t be just a two-rung hierarchy, with nothing mediating between the monarch and the subjects, like in the Ottoman empire, the favourite example of despotism for early modern thinkers. Instead, de Bonald argues that like domestic society or family had to have a mediating position of woman, political society or state should have a mediating position of the nobility. Like woman in the family, de Bonald notes, nobility should share the nature of extremes, being subject to the monarch, but being also like little kings, having absolute authority in their own piece of land.

While it seems that there is no place for the opinion of the subjects to be heard in Bonaldian state, he does allow a position for them in the form of General Estates, although it has only a consultative role in the state affairs. De Bonald dilutes this concession by noting that, as was the tradition, one third of the Estates was to be filled by nobility, who represented the political society, while the second third of the Estates was then supposed to represent church or the religious society. In his earlier works, de Bonald had thought that the final or the Third Estate was superfluous to the proceedings, but he now finds a justification for their inclusion: they represent the domestic societies or the ordinary families.

Just like de Bonald preferred monogamous family over polygamous, he also prefers monocratic state over polycratic or democratic state. In a democracy, he insists, all social roles are confused, everyone being both a ruler and a subject at the same time. He is certain that democracy works only in small communities, like Swiss cantons, or then in such backward and almost savage countries like America.

Unlike with the case of family, de Bonald notes that there are various middle positions between the ideal monarchy and democracy. Main one of them is aristocracy, which de Bonald calls also acephalous or headless monarchy. This need of a central power, he notes, often forces aristocracies to elect a figurative monarch, which still lacks the status of absolute ruler, because of its dependency on the nobility. Another type of middle position is provided by English representative monarchy, which de Bonald interprets as an unstable combination of three states (monarchy, aristocracy and democracy), which can only work in such an isolated country.

We already mentioned that beside domestic and political societies de Bonald speaks also of religious society. The place of the monarch or father lies naturally with God, whom de Bonald assumes everyone should be aware of through innate ideas and who accounts for the preservation of the whole world. Analogically to the domestic and the political society, the natural form of religious society is monotheistic, while the respective perverted form is polytheism, which de Bonald calls either idolatry (in a natural state) or paganism (in an organised state).

The problem with forming religious society, de Bonald notes, is the mediation between God and humans, since the gulf between the two extremes is infinitely wide. This problem is especially strong with the question of sacrifice, de Bonald thinks. For him, sacrifice is an essential element in all societies and particularly for all mediators, who according to de Bonald should sacrifice themselves in order to pay for the care that the rulers provide for their subjects. Thus, de Bonald insists, nobles are meant to put their lives on stake for their country. In an even more drastic and patriarchal fashion, de Bonald thinks that women should yield their whole life to the service of their husbands.

In a religious society, the role of mediator has at first been in the hands of fathers of the families, who have shown their gratitude for God through sacrifices - de Bonald refers to the story of Abraham, who was willing to sacrifice even his own son for God, who then accepted animals as substitutes. When families combined into states, this role of mediators was taken up by a priestly class, who continued the tradition of animal sacrifices. Still, the gulf remained, and de Bonald concludes, it could be bridged only by someone partaking both of the nature of God and human and sacrificing himself for the whole humanity - here’s a justification for main tenets of Christianity.

De Bonald’s account of the role of sacrifice in religion makes God sound like a mafia boss demanding payment for his protection. Furthermore, like his account of family and state, his account of church or the earthly representative of God and his mediator is quite hierarchical, de Bonald practically endorsing Catholicism.

All attempts to reform Catholic church, de Bonald stated, inevitably lead the church toward the equivalent of democracy, where all individuals by themselves mediate their relation to divinity. Indeed, de Bonald notes, reformist churches often endorse the ideals of democracy and even allow divorce, which he thought to be just another name for polygamy, analogue of democracy in family life. Like with state, de Bonald admits there are various middle positions between Catholicism and full reformism, such as Orthodox, Anglican and Lutheran churches, which at least admit the need for a priest class, even if they do not submit under the authority of Catholic church.

Sir James Mackintosh: Dissertation on the Progress of Ethical Philosophy (1830)

Mackintosh’s dissertation on the history of ethics is mainly famous through its connection to James Mill. In one small paragraph Mackintosh criticised Mill’s essay on government, and Mill retaliated with vitriol. Still, the topic of Mackintosh’s work wasn’t really even political philosophy, and the dissertation deserves to be read on its own. One might think a history of ethics would be of mere antiquarian interest. Yet, a historian of ethics can have something to say about ethics itself, by pointing out what to criticise and what to applaud.

The main point of criticism Mackintosh raises against many earlier philosophers is the same: they have failed to distinguish between a criterion of moral actions and their motivation. This criticism is especially poignant in case of utilitarians, whom Mackintosh characterises as having confused jurisdiction with ethics. In other words, Mackintosh explains, utilitarians are overtly interested with the question of what actions are good and what not, while they almost completely ignore the question of how to improve people’s characters so that they would tend to act well. Thus, Mackintosh does not say that utilitarians were wrong in defining good actions as those useful for most of humanity. Indeed, he appears to mostly agree with the utilitarian definition, although he also says that in practice, we humans cannot apply this criterion alone, due to the limitations of our understanding of what serves the whole humanity best. Yet, even greater fault of utilitarians, in his eyes, is that they do not then properly explain how people could be inspired to act toward such generally beneficial ends.

We might say that Mackintosh has here hit upon a difference between contemporary and classic notion of ethics. If we read an article on ethics nowadays, it invariably tends to be concerned with questions of the first sort, while the second type of question is considered by almost no contemporary ethicists. On the contrary, the second type of question was the main topic found in classical works of ethics, while considerations of the first type of question were more often found in books on natural law.

Even though Mackintosh admits that earlier writers of ethics tried to tackle the second question, he also notes that they confused it with the first question. This confusion was often connected with two errors: making ethical motivation too intellectual and interpreting human actions as always self-centred. Although distinct errors, they were often connected in theories of prudential reason that would make humans act ethically, because ethical actions agreed usually with the interest of the individual. The intellectual error here lies in the idea that reason alone could make us do something. Indeed, Mackintosh notes, prudence has an emotive element (the so-called self-love), although it is so weak that when acting prudently, we might confuse it with acting from mere reason.

The weakness of prudence or self-love as a motive links directly to the second error Mackintosh perceived in many of his predecessors, who assumed all human actions are ultimately based on self-love. How could such a feeble motive make us really do anything? Often what moves us, Mackintosh notes, are much stronger desires, say, a desire to eat chocolate. This desire does not have self and its welfare as its object, as indeed, fulfilling it by eating lots of chocolate won’t do good to our physical constitution. Instead, what is desired is the chocolate itself and its taste, and fulfillment of this strong desire produces equally strong pleasure.

As the example of the desire for chocolate showed, it is quite possible to have desires that are not based on self-love - and indeed, many of them are stronger desires. Furthermore, Mackintosh continues, some of these desires are disinterested in the sense that they aim for the pleasure or happiness of other people. That is, we feel pleasure when we see others become happy, especially if they become happy because of our own actions, and we feel discomfort when we see others in pain. This is an empirical fact that we just cannot disregard, Mackintosh insists.

Despite this empirical fact, Mackintosh notes, some philosophers have tried to argue that even disinterested desires are fundamentally selfish. Their argument is that such disinterested desires and emotions have been generated from more self-centered pleasures and emotions through association. For instance, in our childhood we might have experienced fulfillment of our desires in the company of our family members, making us sympathise with them, and later these feelings of sympathy could be generalised to other people through further associations.

Mackintosh isn’t convinced that the argument works. He admits the premise that disinterested desires are generated on the basis of more primitive desires - indeed, this is even assumed in the notion of moral education of people. Yet, he at once notes, such generation does not mean that the generated emotions would be dependent on their roots nor even essentially similar to them. Consider avarice, for instance. It certainly is a generated emotion, since newborn babies do not yet desire money. Instead, the love of money is generated, because people come to associate money as an indispensable means for gaining things, e.g. for satisfying primary desires, like hunger. Even so, avarice is independent from hunger and other similar desires, which is clearly shown by the actions of a miser preferring to save money instead of spending it on food.

What is true of avarice, Mackintosh says, is equally true of disinterested emotions - although we learn to love and help others only later in life, these affections are still real. This is true also of the highest type of disinterested affection, that is, conscience. Conscience resembles prudence, Mackintosh says, in that it is quite a weak emotion compared to more heartfelt emotions. Indeed, like prudence, it is a more general emotion and even one of different level: its object are our moral dispositions and actions proceeding from those dispositions, that is, it aims to improve our first-order emotions so that we would be more inclined to choose what is known to be good.

We circle back to the difference between criterion and motivation and their confusion especially in utilitarianism. Utilitarians, Mackintosh says, often seem to try to motivate good and virtuous behaviour by pointing out its good consequences for the society and for the individual. Especially in the latter case, Mackintosh notes, they appear to forget the most important beneficial consequences, that is, those affecting the mental condition of the individual. Take as an example one classic virtue, courage. Sure, courageous actions can help us get external advantage through the brave actions we can do because of it. Yet, there is a far greater advantage to be got through the effect of courage on our own mind - courage makes us impervious to fear. Similar positive effects to our mental condition are connected to all virtues, Mackintosh concludes: with virtue it becomes a delight to do good things.

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?