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?

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.

James Mill, Analysis of the Phenomena of the Human Mind 2

Second volume of Mill’s work on philosophy of mind clearly divides into two different parts, and I shall treat the latter part in a writing of its own. The first part continues the discussion Mill started in the first volume, that is, the analysis of common concepts pertaining to human cognition.

The first of these concepts are relational terms. You might remember from my previous post that Mill appeared to accept only monadic predication, which seems, of course, problematic for understanding relations. Mill’s solution is to suggest that relational terms like “parent-child” are actually names for a single phenomenon, consisting of two things. The different names just emphasise different aspects of this same totality. Connected to this notion is Mill’s idea that difference and opposition is necessary at least for such conscious entities as us - we do not really sense or think anything, if there is no change in our sensations and ideas. Together these ideas suggest surprisingly close tendencies with broadly Hegelian philosophers, who view the world as an necessarily interconnected system of oppositions.

Mill goes into great details classifying the various possible relations: difference and sameness of sensations, sensations as preceding one another, spatial relations of objects, temporal relations of objects, quantitative equality and inequality of objects, qualitative likeness or unlikeness of objects, same or different composition of ideas and ideas as preceding one another. A clear fault in Mill’s schema is the ignorance of relations with more than two relata. The closest Mill comes to discussing them comes with spatial relations, as he admits that even the idea of position of an object involves relations to all other objects. Still, even here Mill tries to simplify matters and concentrates on juxtaposition of two objects and such pairs like high and low or front and back.

If we inspect Mill's account of spatial relations in more detail, we find him, in a manner reminiscent of Fichte, reducing the genesis of our experience of space into muscular exertions, by which we move ourselves or things around us. In another sense, Mill’s notion of space reminds one of Leibniz, because just like him, Mill says that space is nothing but an abstraction from all the concrete positions and spatial relations between things. While Kant would object that surely space is prior to positions in it, he might approve Mill’s admission of a subjective element in our notion of space - we always associate a place with a further place, thus making our notion of space infinite.

A further peculiar element in Mill’s notion of space is that when we think of empty space, he insists, we think of something positioned in that space and then the absence of this something. This is a feature of Mill’s general theory of privative statements - he insists that thinking of absence of something, we think of both that thing and its absence (e.g., when thinking of darkness, we think of light and its absence). In case of space, this would mean that by empty space, we always mean space empty of something. Indeed, he confirms, because our experience of space is intrinsically related to the idea of something resisting our efforts, we cannot think of any space empty of everything.

Mills account of time is similar enough to his account of space that we do not have to deal with it in great detail - like space is for Mill just an abstraction of all positions, time is nothing more than just an abstraction from all successions. Reminiscent of Kantian tradition, Mill connects numbers also with successions. In other words, he points out that although numbers are used to count the greatness of synchronously existing objects, the act of counting happens through some successive operation.

Mill also investigates the notion of personal identity, especially over time, and reflection. His basic solution to this conundrum is to point out that personal identity is just a special case of any identity. Identity as such, say, of other persons, we come to know by seeing a person and remembering that we saw her earlier. Similarly, Mill says, we are always conscious of ourselves, remember being conscious of ourselves and hear testimonies of others concerning us at times, which we don’t remember.

Mill’s explanation seems flawed in many ways. Firstly, one might point out that he hasn’t ever really explained what it means of being conscious of oneself, because saying that consciousness in general is just a general name for sensations and ideas explains nothing about self-consciousness - what are we conscious about when we conscious of ourselves? Secondly, Mill’s explanation would at most tell what evidence we use to justify our assertions of self-, or indeed, any identity, but not what it means that something is identical with itself. Finally, while it is undoubtedly obvious that our assertions and experiences of self-identity are particular examples of assertions and experiences of identity, one might wonder, in the manner of Fichte, whether e.g. our experience of self-identity underlies our experience of all other identities (that is, whether we must be able to identify ourselves, before being able to identify anything else).

James Mill: Analysis of the Phenomena of the Human Mind 1 (1829)

In my previous post on James Mill, I noted that he emphasised the need for the study of the human mind as a precondition for planning education. Mill tackles this very topic in his two-volumed Analysis. First volume, which I will be studying now, deals roughly with the theoretical side of the human mind, while the consideration of the practical side is left for the second volume.

Mill’s viewpoint is clearly empiricist and he begins with an assumption of Humean undertones that everything in our experience is reducible to sensations (what Hume would call impressions), ideas, formed on basis of sensations, and groups and trains of ideas, generated from sensations and ideas through association. Starting from sensations, he enumerates the usual smell, hearing, taste and sight, divides sensations of touch into sensations of (moderate) temperatures, sensations of tactile contact, sensations of disorganisation, such as pain or itch, and sensations involving muscular exertion, and adds the sensations belonging to alimentary canal and possibly also to other intestines.

Mill’s portrayal of these various types vary from ordinary to perplexing, latter being exemplified by his insistence that the effects of opium are sensed in the alimentary canal and just strengthened by associated ideas. Indeed, one might argue that Mill should have added a separate class for clearly pleasurable sensations, such as is occasioned by stimulation of various substances and sex. In any case, Mill makes an important remark that when speaking of sensations, at least English has a deplorable lack of distinct words for a) sensation of one kind (say, a smell), b) whatever causes this sensation (also known as smell) and c) the organ for this type of sensation (yet again, smell).

Ideas, Mill continues, are then traces of sensations. While we have some say on what we sense e.g. by changing our position in relation to our environment, he affirms, we cannot control what ideas we will have or think about, because they are generated by sensations and other ideas through association. This association is governed, again in a Humean manner, by habit - if we are used to sensing various sensations or ideas together, either simultaneously or successively, we are bound to think one part of this group when another part is sensed or thought.

Although the account is rather simple, there are some questions that Mill leaves unanswered. For instance, Mill notes that in case of succession we are used to following the train of thoughts to the direction of the process involved: thus, a farmer is used to thinking of first planting the seed and then the seed sprouting. Problem is that we often think such processes in reverse and particularly do so, when looking for explanations, for instance, if we want to explain why plants grow, we think first of their growing and only then the idea of someone planting the seeds. Another problem is that Mill does not point out that there are associations, which appear to be a necessary product of the way we sense - for instance, a combination of colour with extension seems not just coincidental nor even one based on the nature of things sensed, but on a fact that we cannot sense coloured things, which are not extended.

Such groups and trains of ideas can, then, form a single idea out of many individual ideas, Mill says. Thus, since we are accustomed to sensing, say, a distinct rosy smell with a certain kind of sight, we tend to think of them as a single unit (a rose). Similarly, temporal trains of ideas should form complex ideas of processes.

The purpose of language, Mill suggests, is merely to communicate to others or to our future selves these various sensations and ideas, either individually or as complex trains of thought. Later on, he adds another purpose for the language, that is, making it possible to think of such otherwise difficult ideas like a polygon with hundred sides. This seems a very limited understanding of the purposes of language. Particularly, this appears not to note that sentences of a language are meant to indicate e.g. that something happens or happened, not just that we have sensed or thought something. Mill himself appears to be at least unconsciously aware of this, since he notes that all verbs carry with them the notion of existence. Yet, Mill uses “existence” here in a very strong sense, which makes him instantly think of this immediate connection as a detriment to language - if we speak of e.g. what chance is, Mill suggests, we immediately assume that chance is something existent, namely an individual object. It seems obvious that we need not go so far, in other words, when we say that chance is so and so, we truly affirm something about chance without assuming that chance is an individual existent.

The trains of thought we express in language indicate, Mill suggests, causal relations, relations of resemblance or different names for same things. The third option here is the most perplexing, since Mill includes in it such mathematical statements as one plus one equals two. One might accept this classification in such simple cases, but whenever the calculations become complex enough, it is doubtful that we have any concrete idea to name in both sides of the equation. For instance, can we really say we have a distinct idea of what 1000^1000 is meant to say? What this expression indicates is more like a certain task of calculation (multiply 1000 with itself 1000 times), and the sentence connecting this task to its result indicates a certain relation between numbers.

The remaining part of the first volume is dedicated to an analysis of certain central concepts of human cognition in terms of concepts introduced thus far. For instance, Mill suggests that consciousness means no feeling distinct from sensations and ideas, but is simply a general name for both of them: if I am conscious of something, I either sense or have an idea of it. Although one might agree with Mill that consciousness is nothing distinct from sensations and ideas, one might still ask in the manner of Leibniz whether consciousness - or better, conscious sensations and thoughts - could form a subspecies of all sensations and ideas. Indeed, we might assume that different sensations and ideas have various degrees of consciousness and by moving our attention to one aspect of a sensation or an idea we might become more aware of its components.

Of the concept of conception Mill notes that it simply refers to complex ideas. This is undoubtedly true description of how some thinkers have used the word or similar expressions in other language (say, Begriff in German). Yet, other thinkers seem to use concept and conception only in relation to linguistically expressed notions (thus, a mental image of a bicycle would not yet be a concept, whereas a linguistic expression “bicycle”, perhaps also with a definition of the expression, would certainly be). Related to this is the question whether conception is somehow active in comparison with mere sensation and “having an idea”. Mill suggests that the supposed activeness of conceiving is just a misdirection of the active form “I conceive” and that conceiving is as passive an event as dreaming or dying, which can also be expressed in an active form. While Mill’s point is well made, it could be that when philosophers have spoken of the activity of conception, they might have actually meant the relative independence of conceptions from external objects. After all, we can conceive things that do not exist, especially if conception is understood in a linguistic manner.

In case of imagination, Mill goes even farther in his redefinition of words and suggests that imagination is just a new term for having a train of ideas. This seems to go against all ways the word has been used. In an early modern discussion, imagination was closer to what we would nowadays call memory, meaning thinking of anything that isn’t currently present, even if we have perceived it earlier. Nowadays, we are mostly using imagination in the sense that Mill disparages as Dugald Stewart’s redefinition of the word: that is, imagination is a train of new thoughts. “It is implied in every wish of the child to fly”, Mill says, trying to be sarcastic, but we indeed speak of imaginative kids as having such inventive thoughts, often divorced from reality.

Especially confusing is Mill’s discussion of belief. He at first divides belief into three distinct classes: belief in events and real existences, belief in testimony and belief in the truth of propositions. Later he makes a reclassification, saying that belief in testimony is just a special case of a belief in events (those told by other people). Nowadays, many philosophers might want to classify all beliefs as concerning propositions, because the content of belief can be expressed as a proposition. On a closer look, Mill himself might agree, because he thinks that one part of beliefs in propositions, those concerning individual objects, are essentially beliefs in events and real existences (he would just have to make the assumption that all such beliefs can be expressed in propositions). This still leaves the problem that Mill’s account of propositions is too simple, allowing in principle only monadic predication - a problem we already saw in the case of his idea of mathematics.

An even bigger problem lies in Mill’s attempt to describe belief merely in terms of sensations, memories, associations and abstractions. For instance, he says that in the case of sensations and memories, belief means simply having them - if we sense something red or have a memory of it, we believe it. These cases already seem problematic. If Mill means just that having a sensation or having a memory is such an experience we can never doubt, we might accept that this is true, but this is, for the most part, not a very characteristic way to speak of belief (one might even say that only philosophers say things like “I believe I have a sensation of redness”, whereas usually people use the concept of belief only in cases where there is at least a possibility of things not being so). If, on the other hand, Mill would want to use the notion of belief here in a more substantial manner - say, as a belief that there is something that causes this sensation of redness or that there was something that I now remember as red - then we immediately see that belief is definitely not identical with sensation or memory. True, sensation and memory might well be evidence and occasion for such a belief, but there might still be doubt e.g. whether what we remember really was red and not blue.

Mill’s mistake is even clearer in his supposed examples of belief. If we spin around very fast and stop, we get the feeling of the earth spinning beneath us. Mill suggests that we also believe that earth is really spinning, which seems far-fetched. Even more, Mill makes a quick throwaway line that fearing something implies also believing in it - hence, he says, anyone who is afraid of ghosts must believe in them. A quick consideration makes Mill’s statement seem unfounded, even if he thought it so self-evident to need no justification. Picture a person hearing a strange noise in her house during the night. She might be afraid that there is a burglar in her house, but she might still be uncertain whether there is any burglar and thus not really believe in the existence of this burglar.

Indeed, Mill appears to make no distinction between modalities of actuality and possibility, belief being intimately related to the former - we might consider e.g. a sensation or memory, without being sure of its object being real. In the more complex cases, he says that habitual associations just are beliefs - if we tend to think of the sun rising after it has set, we believe that it will rise again after setting. Again, this regular or habitual association might be a ground for believing something, but there are cases where we might still have doubts, whether e.g. the sun will rise everyday for the whole infinity of time.

This is not to say Mill’s book is completely without merits. His account of abstraction is particularly interesting. Mill’s basic point is the difference between what calls notation and connotation. A word like “red” notes a certain property of individual objects, but it also connotes something, namely, those very objects that can be called red. In more modern terms, we might say Mill is here talking about the difference of intension and extension. Abstraction, then, means that we ignore the connotation of the word and think only of its notation: we think of redness without any reference to red things. Mill’s point is that abstractions are then nothing truly existent, but always dependent on the ignored connotation - there is no Platonic redness floating around without red objects.

Günther, Anthon: Preschool to speculative theology of positive Christianity. Second part: Theory of incarnation (1829)

 The first part of Anthon Günther’s work began with the theoretical problem of describing the nature of God and his relation with nature and finite spirits. Günther opposed modern pantheism, which tried to put God on the same level with finite entities. He insisted that divine self-consciousness is on a completely different level from human self-consciousness, where the former involved God positing its own substance as an object (Logos) different from God as subject and noticing the identity of itself as object and itself as subject as a third entity (divine Spirit), while human self-consciousness could note its own dependence on this interconnected trinity of entities. At the end of the first part, Günther hinted that we could still learn more about divinity through our conscience, which he suggested was a direct pronunciation of divine existence to humans.

The second part of the book engages this very topic of human conscience. The setting of the first part - a theology student exchanging letters with his uncle, a mouthpiece of Günther - continues in the second part, by nephew presenting to his uncle the problem of evil. Nephew notes that because modern pantheism denies freedom of human will, it has to accept that evil is just a necessary aspect of human consciousness, which could be overcome only by returning humans to the unconscious existence of animals. In effect, nephew insists, this would mean either that the devil is equally strong as God or that God himself is also the source of evil.

To make his point, nephew picks out two recent philosophers who had discussed the question of evil. First of these is I. H. Fichte, whom we met in the first part of the book and who was the son of the more famous philosopher, J. G. Fichte. Fichte’s ideas on evil begin from the Kantian notion that humans have an innate and inexplicable predisposition toward evil, which hinders the freedom of our will. Fichte tries to explicate this notion further by suggesting that the universality of evil requires that we must assume an inexplicable, supernatural, destructive principle as the source of all evil - that is, Fichte endorses dualism. Furthermore, Fichte assumes that humans have had to decide between the creative principle or God and the destructive principle, temporal existence in separation from God being the result of a wrong choice. Nephew points out that the idea of a pre-temporal choice releases temporal human consciousness of all responsibility for their life choices. In addition, he notes that this idea of temporality as a result of corruption is in contradiction with another theory of Fichte’s, namely, that of God revealing himself through human spirits.

As his second example, nephew chooses Bernard Heinrich Blasche, better known as a pedagogue. In his more philosophical works, Blasche endorses a panentheist position, where God is the point of unity of the world, finite entities being just necessary self-particularisation and differentiation of this unity. Evil means for Blasche simply separation from original unity, being thus a mere intrinsic development of a possibility inherent in God himself. In effect, nephew concludes, Blasche simply identifies God with the devil.

Uncle agrees with nephew that the two philosophers fail to explain the true origin of evil. He sees Fichte as ultimately being committed to pantheism, because of his notion that individuals are just revelation of God, which means that they can never truly be separate from God and therefore cannot be evil. Blasche, on the other hand, ultimately follows the trend set by Schelling, whereby freedom is swallowed by necessity. In an interesting side remark, uncle notes that his contemporaries do the very same thing, when they explain bad decisions of individuals by differences of temperament and bodily influences.

With the options of the eternal dualism and the interpretation of evil as a modification of good ruled out, it is up to the nephew to come up with a third option. The only solution left, he thinks, is that evil arises from the level of finite entities, being thus something that is not a necessary part of existence. For instance, he concludes, in the original state, as intended by God, humans were not evil, but they became evil, when they opposed divine will and made their own interest into their highest law. Yet, this solution begs the question: how could a human, made perfect by God, fall into evil?

Nephew tries to find the answer from the works of a noted theologian, August Tholuck. Tholuck had insisted, firstly, that God couldn’t have given human a predisposition toward evil. Yet, Tholuck had noted, humans had an ability for evil in the sense that being evil wasn’t in contradiction with the essence of humanity, but only with the end appointed by God to humans. No true reason could persuade a perfect human to turn evil, Tholuck admitted, but they could be instigated to a blind, reasonless act, if deceived by deceptive reasons. Tholuck’s conclusion had been that the allure of the devil had been persuasive enough.

Uncle is not convinced of Tholuck’s account. He notes that it hinges on a formal notion of possibility, which outlines only what is thinkable. A more substantial notion of possibility is linked to actuality in the substantial sense, which the uncle defines as just another name for the system of causal relations. Possibility and necessity, he suggests, are then just different ways to reflect on this nexus of causality: we speak of necessity, when causal relations are certain and uniform, and of possibility or contingency, when the relations are uncertain. Furthermore, nature is especially characterised by necessity and spirits by possibility, because latter have a freedom to manipulate causal chains that the former do not, and indeed, uncle insists, with spirits disposition cannot really be distinguished from possibility or ability.

Because ability and disposition to evil cannot be separated, uncle notes, the seeming contradiction still remains - why would God allow humans to choose evil? The reason, uncle remarks, cannot be that finite creatures as such could not exist without this possibility, because natural things as necessary cannot be evil. Uncle thus denies the Augustinian idea that evil derives ultimately from finite entities being created out of nothing and thus having intrinsic lack in their essence - he notes that Augustine mistakenly treated nothing if it was something substantial. Augustine’s mistake led then to the same error pervading also Tholuck’s theory, uncle says - both philosophers thought that evil was caused by acting blindly and without reason, although it is quite the opposite, that is, evil and sinful action causes blindness.

Uncle returns then to the original state of human beings. God, he begins, created other beings, firstly, because he wanted to reveal himself to other beings, and secondly, because he lovingly desired that other beings would share in his blessedness. The original human being, uncle describes, had a spirit created immediately by God and a body fashioned by organic forces created by God. This original human being knew God as the source of its existence, itself as a subject of this knowledge of God and nature as something in close connection with itself. Now, uncle continues, God revealed also to human being the end fashioned to it and commanded the human to form itself according to that end by taking part in the blessedness of God. In a sense, then, the original human being, as imagined by the uncle, was not yet perfect, but could perfect itself through its own action, making human then a sort of image of God as the creator. The original human being became evil, uncle explains, because it chose to not subjugate to divine will. By so doing, it did not cancel its own existence and certainly not the divine existence, but merely contradicted the divine command to remain in connection with God.

Nephew accepts at once uncle’s account and adds quickly that other, non-embodied spirits or angels must have also faced a similar choice - essence of the sin of the devil is the same as the sin of humans, that is, choosing oneself over God. The consequence of this sinful choice, he notes, is that the sinful creature cannot anymore be called the image of God, that is, it is not in connection with divine substance - by trying to become positive in itself, a creature distanced itself from God, just like two positive poles of magnets repel one another. In case of humans, nephew explains, this distancing means especially the loss of God-given ability of human spirit to control its animal body effectively. In effect, then, nephew suggests, sin of spirit of trying to rule by its own force leads to flesh rebelling against the spirit. Concrete effects of this rebellion, he points out, are the eventual succumbing of human bodies to forces of nature in death and the necessity of humans to reproduce sexually. But it is not just nature that opposes the sinful human being, but also divine will, nephew adds, and the human hears this opposition as a voice of its own conscience declaring guilt for the sin.

Uncle is ready to add a few important details to the nephew’s account. Firstly, he points out that the sin of fallen angels differs from human sin, because the former do not have bodies and therefore do not reproduce and have further generations of angels or a history in the proper sense of the word. Thus, uncle deduces, disembodied spirits could not really experience any salvation, but they are eternally condemned to an existence without God. How can such eternal damnation be justified? Uncle’s answer is that this is simply what the fallen angels have wanted and always will want - their choice was to live without God, who out of his love lets the fallen spirits live without God. Indeed, he continues, this is the very purpose of the existence of free creatures - to let them decide whether they want to live in the light of God or to shroud themselves in a shadowy existence of a sinner.

Furthermore, uncle shows another sense, in which the image of God has been destroyed in the sinful humanity. What is given to a human being when he becomes an image of God is not the full divinity, which a finite creature could not obtain. Instead, it has to be something that could be shared by humans and God, namely, self-consciousness. Now, even sinful humans are conscious of themselves in a sense, but their self-consciousness is not as clear as when they understand their relation to God. Instead, sinful humans can confuse themselves with God or nature or confuse God with nature. In other words, various forms of non-Christian ideologies, like pantheism, are another consequence of human sin.

Finally, uncle points out that guilt is not in a proper sense of the word a consequence of the sin. Sin, he explains, is a relative concept, in which someone changes his relation toward God. Now, a necessary counterpart or the other side of this sinful action is a reciprocal action of God severing his connection to the sinner. Feeling of guilt, uncle says, is then the necessary form in which sinful persons experience their own sinful action. Difference is that the sinful action cancels itself immediately - by severing its ties to God, sinner also destroys the abilities, by which it could again make sinful choices. On the contrary, the feeling of guilt lingers on as a thought of God, which is now as good as dead for the sinner.

The account so far has shown how a human being in their original state could have sinned. The next problem is why humans now appear to be born already in a sinful state. Traditional answer was the notion of an original, hereditary sin, which was a middle position between traducianism - the idea that spirits of humans were born out of the spirits of their parents - and the Origenean theory that human spirits had existed before the birth of their bodies and had been condemned to an earthly life because of a choice in that earlier, pretemporal state. The official catholic position was that human spirits were generated by God, at the same time as their bodies were generated, thus making the two other positions untenable. The mediating position indicated that original sin of the first was somehow transmitted to their descendants, even if the spirits of these descendants were made directly by God. Question is how this transmission was possible.

Nephew considers first the solution offered by Philipp Marheineke, a theologian of Hegelian school. Marheineke suggested that all humans shared the original sin, because they all were part of the human genus. This general sin, instigated by first humans, had to still become our personal sin through our individual choice, Marheineke said, because without personal choice we could not be responsible for the sin. Marheineke also suggested that this choice was somehow inevitable, so that no individual was in the end exempt from guilt. Nephew notes that Marheineke’s solution is no real solution: if the choice of evil is necessary for humans, then they are not really responsible for the original sin and are like mere animals.

Nephew’s answer to Marheineke is that original, hereditary sin must be something essentialy different from personal sin based on our bad choices. Furthermore, he differentiates between a complete personality of an actual individual, created by free choices, and spirit as a complex of unactualised potentialities. Nephew points out that even before becoming a full personality, individual spirit is active, for instance, in differentiation of various psychological capacities. These prepersonal activities of spirit, nephew concludes, are still unfree or instinctual and could be said to be affected by the original sin, if they were hindered by separation from God.

Nephew’s answer still leaves open the question why God continues to create human spirits that remain in separation from God. His solution lies in the dual nature of humans. Beyond spirits, he says, humans also have bodies, which happen to be naturally generated and connected to divinely created spirits. This means, nephew insists, that humans form no mere abstract class, but a genus or family, where the first person represents the whole of humanity. Because new humans belong to such an organic whole, their spirits must be affected by the corruption of the first humans. In effect, this means that the new humans are also divorced from God, coming into contact only with the voice of conscience declaring human guilt. Even more, nephew insists, the activities of new humans are tainted by a tendency to pridefully choose oneself over God and a fixation with nature and its delights.

Nephew admits that God could still create spirits linked to divinity, but these would form a completely new species, unrelated to the descendants of the first human. He also admits that God could simply destroy the family of humans, but won’t do it, because of a possibility to eventually save some of the lost humans.

Uncle still presents some clarifications to the hereditary nature of original sin. His main point lies in the notion that humanity is connected by a sort of Platonic idea (although he calls it Kantian). This idea is not just an abstract concept, he explains, but a true organic unity, through which we humans can feel a connection with one another. In effect, it is also a blueprint for generating humans, which was perverted by the sin of the original humans and thus resulted in the creation of perverted humans.

Uncle also explains that the sinful disposition of the humans does not lie in their spirits as such, which still have the ability to choose God. Instead, the perversity or sinfulness lies in their being disconnected from God and at the same time connected with an animal body. The animal body as such knows nothing about God, because it feels only appearances and lives only for sustaining itself through its descendants. Spirit, on the other hand, starts to revolve around itself, when not sustained by divine light. Because the human spirit is organically connected with the body, it is also tempted by its sensuous concerns. Still, uncle notes, there is something helping spirit to gain control over the body and turn again toward God, namely the voice of conscience.

The next letter of the nephew concerns the various ways humans could relate to the call of conscience. The relation to conscience is, in fact, the way in which the nephew classifies various religions. He wants, thus, to present an alternative to various philosophies of religion of German idealists, where religions were usually put in a developmental order. As an example, nephew gives an account of Johann Erdmann’s philosophy of the history of religion, which seems like a slight modification of the Hegelian philosophy of religion. Nephew’s judgement is clear: human beings have an innate tendency for such abstract conceptualisations that have little to no basis in empirical facts.

True principle of classifying religions, nephew insists, is whether followers of that religion have obeyed the call of conscience or not. The latter, he surmises, is the essence of paganism. By turning away from the call of the God, pagans cloud their understanding of the proper relation of humans to God and nature. Firstly, they might think humans as essentially similar to nature, falling to some sort of pantheism. The nephew supposes two possible extreme forms of pantheism, with various mediating positions between them. First type of pantheism supposes that spirits are emanations from the absolute and nature is then a yet further emanation from spirit. Although nephew does not identify this idealistic pantheism with any particular religion or philosophy, it bears a striking resemblance to Neoplatonism. The second, realistic form of pantheism reverses the roles of nature and spirit, suggesting that human spirit is just a development of its natural side. Nephew finds traces of this idea in Greek polytheism, in the sense that Greeks believed some humans could eventually evolve from mere natural state into real divinities.

A contrast to the pantheistic religions, nephew continues, is provided by dualisms, where the struggle between spirit and nature is explained through the existence of two equally powerful principles. He suggests that dualism has two extreme forms, corresponding to two forms of pantheism. Just like realistic pantheism speaks for the self-development of humans, an ethical dualism, exemplified by Persian religion, states that humans should become heroes fighting for the good principle. On the other hand, just like idealistic pantheism speaks instead of action, more of an occasional mystical retreat into the original unity, theoretical dualism, which nephew identifies with Buddhism, supposes that humans spirits will eternally be mere playthings of dualistic forces, reincarnating from a better fate to worse and back.

Even paganism wasn’t completely mired in error, nephew admits and points out various indication of theism in Hindu writings. Even more important to him are hints of a coming saviour in various pagan religions. Yet, nephew says, the proper answer to the call of conscience - obeying God - was shown by Jewish people. Because of the proper answer, nephew explains, they received a second revelation of God, not as a commanding lord, but as a loving father. When the rest of humankind was falling more and more toward error, the chosen people could keep the idea of loving father alive.

In addition to the new revelation, nephew adds, the original revelation of a commanding lord had to be kept alive also, which happened by God declaring in audible voice the divine law. Thus, God very literally ruled the chosen people, while pagan societies were based on mere human will. Nephew also notes another contrast: while paganism spread all over globe, the force of Judaism was not in extension, but in intension, making it understandable, why there could be only one chosen people. Despite being a positive contrast to paganism, nephew reminds the reader that Judaism had its weak points also.Jews confused symbols, like animal sacrifice, with their true ethical meaning. Eventually, this led to Jewish cult becoming a mechanical following of rites and rituals, while the prophecies quickly became mere signs of an awaited upcoming political upheaval.

When the juxtaposition of paganism and Judaism had developed to its extreme, nephew continues, a saviour had to arrive. The task of this saviour or Christ, nephew says, was to build anew the original relation between humans and God. This second Adam had to be, then, directly unified with God or one of the moments of the trinity. In fact, nephew adds, he had to be the moment that is involved in God becoming an object both to himself and to others, that is, the second person or Logos. Because Christ should save humanity, he should also have a body belonging to the human family. Finally, Christ should face a similar choice like the first humans did, that is, he had to renounce sin both externally, by living in poverty, and internally, by freely obeying the plan of God.

What puzzles nephew is the relation of the human and divine side of Christ. The official catholic dogma was another mediating position between extremes, one of which held that the divine nature swallowed immediately and completely the humanity of Christ, while the other extreme held that the human Jesus was only in his adult life attached to Logos during his baptism. The mediating position stated that Christ united two different natures (humanity and divinity) within a single unified person. Nephew thinks that this unification of persons makes the human side of Christ too subservient, since Christ then has no human self-consciousness distinct from the divine self-consciousness.

The lack of human self-consciousness is especially vital in relation to the question of salvation, nephew clarifies. How can we call Christ a representative of humanity, if he experiences everything like God would, for instance, knows everything that is about to happen to him? And is his choice of pure life really something worthy of applaud, since the choice is made by a divine person? Despite his doubts about the means of salvation, nephew does not want to follow Duns Scotus, who said that God had just arbitrarily decided to let Christ be crucified and then accept this as a salvation for humanity. Instead, nephew insists God must have had a reason for choosing the particular method of salvation he did.

It is again up to the uncle to explain what the nephew had not understood. His main point is that Church Fathers and scholastics were interested in their account about the objective side of the entity called Christ - is this combination of divine and human sides a unified individual entity? This is evidenced by the Greek of the original, hypostasis, that was translated as person by the Latin church - it is not so much a personality, but an individual existence, compared to general nature or substance, that the word refers to. Subjective experience, uncle insists, became a philosophical topic only after Descartes, so it is not a wonder, if the phenomenology of the experience of Christ was not a problem older philosophers or theologians considered.

Thus, uncle finds it acceptable to assume that the human personality of Jesus exists as well as the consciousness of divine Logos in Christ. He notes that multilayered personality is not a rare thing, because even the divine trinity has a unified abstract personality built on top of three more concrete personalities. Other examples uncle mentions are a consciousness of the natural genus, built on top of experiences of individual animals, and human personality, which consists of an animal and spiritual consciousness. Thus, although Jesus was joined with Logos to form a single, unified entity or Christ, Jesus the human need not have the divine perspective of things, even if he was always aware of sharing his thoughts with Logos. At times Logos controlled Christ, but at other times it left the control to Jesus, who then was just as human and ignorant of the true essence of everything as we all are.

Nephew still wants to tackle the question of how the suffering of an innocent person like Christ can help the sinful humans. His point of reference is the theory of de Maistre, who suggests that guilt has something to do with our animal life, embodied especially in blood, so that it has to be paid in blood, no matter whose blood it is. The nephew has several things to criticise in this account. How can our unfree animal side be held accountable for the sin, which has been instigated by our spiritual side? And how can someone else’s blood pay for one’s own sin?

The true explanation, nephew says, is the essence of a human being as a synthesis of animal body and spirit. Just like persons of trinity, our animal and spiritual sides are so closely interconnected that even guilt of the spirit touches the animal body. Now, while spirits as such are independent individuals, animals form an organic whole - a genus - which also makes the inheritance of guilt possible for humans. Indeed, a sin of the representative of the whole genus - the first living person - can make the whole of humanity guilty in the eyes of God. Because humans have a dual nature, their genus can have two representatives - while Adam represented the animal side of humanity, Christ represents its spiritual side. And, nephew concludes, just like Adam’s sin can be inherited by the whole genus, so can Christ’s salvation work be inherited by all humans. Of course, he adds, it requires some effort from the part of the individual human to imitate Christ and become worthy of his salvation.

Salvation is not yet over with the sacrifice of Christ, uncle notes, because Christ merely removed human guilt, but did not yet reconnect humans with God. The work of reconnection, he explains, is left for that moment of divine trinity, which unifies the earlier ones, that is, the Spirit or the sense of community between Father and Logos. The work of the divine Spirit, uncle continues, parallels the work of Christ. Christ was, firstly, a priest who sacrificed himself for the whole of humanity - his followers experience this sacrifice again and again in sacraments, like Eucharist. Additionally, Christ was a prophet who taught the truth of God to his disciples - his followers continue telling this truth to further and further persons. Finally, Christ was a king who set out a hierarchy among his disciples .- the community of his followers are ruled by law and justice.

What uncle is describing is obviously the Christian church, in which humans try to imitate divine salvation through their faith and charity and gain for this service the sacraments, in which the divine Spirit reconnects them with God. This church is to divine Spirit in a sense what Jesus was to Logos, with the exception that church and Spirit do not form a unified entity. As a good Catholic, uncle and Günther with him, places most importance to priests, who represent the human side of salvation. Priests form a distinct class in the Church, just like Jesus was special among humans, but they are not direct descendants of him, just like Jesus was not born from copulation. In fact, it is the class that matters in priests, so the moral worth of the priest does not affect the worth of their position. In fact, uncle concludes, priests are just servants created by the community for the purpose of objectifying the life of love.

What then should we think about Günther’s grand theory? Express catholicism is not a common sight among philosophers nowadays, so many of his basic suppositions seem suspect and quite outlandish. Furthermore, Günther’s attempts at modernising the old faith appear to not have convinced his catholic contemporaries. Even more, his means for this modernisation - theories of consciousness in idealist theories - have been opposed and even supplanted by other trends, making it look quite quaint in the eyes of all except scholars of this period of German thought. Günther's book reads then like a curious relic, which in another possible chain of events might have led to a large scale rethinking of the Catholic faith.