Jean-Robert Argand: Essay on a manner of representing imaginary quantities through geometric constructions (1806)

(1768-1822)

Originally, one meant by a number something made out of several units - thus, unit itself was not yet a number, but the first number was 2. It goes without saying that all these numbers were what nowadays are called natural numbers, and in addition to 1, the important 0 was also missing from the list.

In addition to numbers, one also spoke of magnitudes or quantities. While numbers were sets of discrete units, quantities were more like continuous wholes, which divided always into further quantities. Still, there appeared to be a clear relation between numbers and quantities. One could express the relations between quantities through relations between number - or so it was thought at first. It became soon apparent that some quantities - such as a side and diagonal of a square - had a relation that could not be expressed through any numbers.

This development took European mathematics from natural numbers to fractions and then to certain irrationals. The set of irrationals known increased when mathematicians started to wonder about the relations between the circumference and diameter of a circle, but the inclusion of zero to the list of numbers or quantities had to wait until Europeans came in contact with this Indian invention through Arabs.

Although negative numbers are usually presented nowadays as the first increment to natural numbers, their introduction to mathematics in Europe happened rather late, during 15th century. Even then, it took centuries before they were fully accepted by mathematicians. Negative numbers did have obvious uses, for instance, when speaking of debts, but they seemed still problematic, because one apparently could not apply same operations to them as with positive numbers.

Particularly, taking roots of negative numbers appeared an absurdity. Yet, the calculations appeared to work in the sense that one could do calculations with these roots or imaginary numbers and results were still consistent. The only question was what meaning to give to these quantities.

An answer to this problem was given by a dilettante mathematician Argand in his work Essai sur une manière de représenter les quantités imaginaires par des constructions géométriques. Argands’s idea was to return the question back to geometry, but to add a new element to the equation, that is, direction.

Picture, Argand says, a positive quantity - say, 1 - as a line heading to a certain direction from some point of origin. Then, we could represent the corresponding negative number (-1) as a line of same size, but moving from the same point of origin to opposite direction. How to account then for the roots of -1?

Argand’s idea is simply to note that number one has the same relation to the square root of -1 as this root has to to -1, that is, if we use the i to represent the square root of -1, then i/1 = -1/i. In layman’s terms, i is is an sense just in the middle of -1 and 1. In terms of size, i is then represented by a line of same size as 1 and -1, but its direction, beginning from the same starting point, goes perpendicular to -1 and 1.

This simple move to two-dimensions and the inclusion of direction to the notion of quantity allows Argand then to represent imaginary numbers and their combinations with ordinary numbers of quantities - in effect, these are then lines moving to various diagonal directions.

The majority of Argand’s essay is spent on justification of this innovation - he attempts to show that e.g. some interesting trigonometric truths can be derived from this assumption. Yet, the major result of the essay is not any of these rather bland proofs, but the innovation itself. This is the first time, when an actual meaning is given to such a seemingly fictional kind of mathematics. We might take this as a beginning of period, when seemingly more and more useless and abstruse mathematical notions find their applications. It is also the first time when a mathematician has in sense proven that a consistent set of truths actually has a point of reference - a semantics is combined to mathematical syntax.

Maine de Biran: Considerations on the influence of habit (1800 and 1802)

1766-1824

Marie-François-Pierre-Gonthier Maine de Biran - or as he is usually called, Maine de Biran - counted himself in the so-called ideological tradition of philosophy, continuing and building upon the works of Condillac and Cabanis - at least in his first major work, Mémoires sur l'influence de l'habitude, published in two parts. The word “ideology” does not refer to any political set of opinions, but to the Lockean habit of calling the basic elements of human consciousness ideas.

Although Maine de Biran identifies his work with the tradition beginning with Locke and Condillac, he is also extremely critical of the somewhat speculative manner in which the two gentlemen carried out their project. Instead, he is eager to follow Cabanis in his attempt to ground human mentality in concrete physiological studies. Thus, his first task is to differentiate between sensations and perceptions in physiological terms and not just by defining perceptions as a clearer form of perceptions. In fact, Maine de Biran appears at times even to suggest that when sensations have more clarity, perceptions are less clear, and vice versa.

In the first book, Maine de Biran suggests a quite concrete physiological difference between sensations and perceptions, sensation being an affection of sensory organ, while perception is formed by brain acting upon such affections. In the second book, the difference is laid out in more abstract terms - while sensations are passive, perceptions are active, just like nervous system consists of affective and motive nerves. In effect, Maine de Biran is giving a new physiological twist on the old idea of the difference between cognition and volition.

Since sensations and perceptions are two quite different and even opposed states of human mind, it is quite easy to see why making one stronger will weaken the other - if we are overwhelmed by strong sensory effects, we are far more likely to lose our ability to perceive the objects around us clearly. Then again, when the force of new sensations subsides, for instance, when we become accustomed to them, it is far easier to e.g. distinguish objects around us.

The example above shows us also an example of the supposed main topic of the book - the influence of habit. Indeed, it is just this habit of having sensations that allows us to have more control over our perceptions. This habit makes it easier for us to decide what aspects of our sensory field do we want to concentrate on, just like practicing any activity will make that activity easier for us.

Yet, Maine de Biran continues, habit is not just a positive force. This is especially the case with our imagination fathoming combinations between two quite distinct things: for instance, whenever we’ve seen one event following another, we easily conclude that one is the cause of the other, although there wouldn’t actually be any connection between them.

If imagination is involuntary and passive association of our ideas, the corresponding active movement from one idea to another Maine de Biran calls memory. Memory, he says, is essentially based on taking some sounds or inscriptions as signs of our ideas, and through their means actively recollecting the ideas associated with them. The recollection might stop at the level of mere signs, and then we are dealing with mechanic memory, that is, with mere repetition of a string of signs or words, known by heart. Then again, words might recollect, not ideas, but mere unclear feelings, and then we are dealing with sensitive memory - this is the faculty of poets and mystic philosophers.

The perfect mode of memory or representative memory works in many cases in close cooperation with the two other modes of memory. For instance, Maine de Biran points out that when we use words from ethics, such as good or virtue, we do attach to these words clear representations, but also certain affective emotions. Further complications arise from the fact that the representative memory has to sometimes deal with abstractions like time and space, which refer to a number of ideas.

A further level of habituation generates judgements - that is, a judgement is born when we frequently associate some ideas and signs together. If this association is made quite passively, the judgement is mechanical - like when we repeat the multiplication table by rote - but when we actually follow the train of ideas and signs we are dealing with a reflective judgement. Finally, the effect of habituation on judgements might again be either passive and harmful or active and beneficial. We might associate judgements which have nothing to do with one another, which leads to faulty reasoning. It is then no wonder that Maine de Biran concludes with the thought of representing all judgements in a Leibnizian-style universal calcul, which would allow perfect reasoning, like in mathematics.

François-René de Chateaubriand: Genius of Christianity - Cult

The topic of the last book in Chateaubriand’s apology is supposed to concern the cult of Christianity. In essence, the book divides into two different strands of argument. Firstly, Chateaubriand continues the theme he touched already in the previous book, namely, the experiential side of religion. While in the last book he mentioned the ability of churches to provide human beings with a feeling of infinity, here he emphasizes burial rituals as a way to face the impending reality of death.

This part of Chateaubriand’s argument is probably the most important. Religions interest people, because they offer vivid emotions – the lure of hope, love and self-certainty is rather understandable. Even more interesting are the mystical experiences, like the inexplicable feeling of being filled with divinity. These experiences are certainly not felt by all members of a religious community, but they could be quite important for people who are most ardent followers of some religion.

Even if these strong experiences and emotions are the core of religion, one must still wonder why they should be coupled with any strict dogmas. Even if a religious mystic would describe her experiences through certain concepts that would seem to commit her to some clear dogmas – such as the existence of divinities – there literally would be no need to take these concepts as referring to anything. Chateaubriand’s strongest argument then leads at once to a quite different outcome from what he envisioned – to a notion of religion without any dogmas, that is, to a sort of embracing the mysterious without any need for explaining it with some theories.

Chateaubriand’s second argument concerns more the supposed benefits Christianity has provided in the development of society. One might be a bit skeptical about the beneficence of some of the things Chateaubriand mentions, such as the founding of militaristic religious orders, but some developments, such as the disapproval of infanticide and the abolition of slavery, are clearly positive (although Bible has also been used for justifying the enslavement of Africans). In a manner fitting to a romantic age, he even speculates of the possibility that the spiritual being of humanity would directly cause changes in the material state of both humanity itself and nature in general – barbaric paganism, with the massacre of innocent Christians, caused the destruction of Roman Empire and the spread of atheism might as well destroy Europe. This speculation is, of course, quite unbelievable, but also against the very dogmas of Catholic Christianity – no lesser authority than Augustine stated already that the coming and goings of earthly empires have nothing to do with the fate of the heavenly kingdom.

Chateaubriand’s laudations are probably far from impartial – he ignores all the havoc caused in the name of Christianity and is silent about any positive effects of other religions or Enlightenment. Still, it is true that religions might at least have the potential for good effects – religious experiences might e.g. make people more charitable toward one another. Yet again, we might well ask, firstly, why this religion need be Christianity in particular. Secondly, we might also question whether these beneficent effects should be connected with any religious dogmas at all. Indeed, a religion without dogmas could satisfy the need for more humane society a lot better than any religion with strict dogmas that might always leave out some human beings outside their influence.

François-René de Chateaubriand: Genius of Christianity – Art and literature

When it comes to art, Chateaubriand's outlook is clearly literary – he has almost nothing to say about fine arts, when compared to his extensive take on poetry. Furthermore, even here we could as well repeat most of the criticism we launched against Chateaubriand's take on poetry. He is especially careful to point out that Christianity provides e.g. good topics for paintings. This appears rather pathetic defense of Christianity, because surely the worth of a painting lies in something else than merely their topic, even if a good topic does make it easier to make an aesthetic impression. Most original is probably Chateaubriand's take on architecture, where he points out how e.g. Gothic churches provide a more concrete experience of infinity than Greek temples.

Chateaubriand's take on literature is not that much better. He divides his topic intro three classes: history, rhetoric and philosophy or science. Of these, we might be quick with the rhetoric. Chateaubriand is especially keen to show that people like Church Fathers were good writers. This is again part of Chateaubriand's general strategy to show that Christianity is especially good religion for producing beauty.

History and science are a different matter. Chateaubriand does have a somewhat antiquated notion of history – it is more a literary effort of describing certain series of actual events in an aesthetic fashion than a research into past events. His defense of Christianity as a source of good histories is rather halfhearted. Chateaubriand merely states that Christian historians have been as proficient as Pagan historians and points out in a familiar manner how the Christian nations have had interesting fates worthy of a historian.

Chateaubriand's plea for the worth of Christianity in case of science and philosophy is the least convincing. In this case, religion either seems like a detriment to the work of scientists and philosophers or then at best apparently contributes nothing to their progress. Chateaubriand relies on name dropping, mentioning all the famous mathematicians, scientists, philosophers etc. who were clearly at least religious in general or even Christian in particular. In case of philosophy, he is especially keen to point out that times of heightened Christianity were those with great philosophers. One is somewhat perplexed that Chateaubriand has then nothing to say about the philosophy of Middle Ages, since that was the time with most Christian and even Catholic philosophers, but is satisfied with such modern names as Leibniz and Clarke. The basic problem is, of course, that Chateaubriand is already evaluating the worth of a philosophic work with lenses of a Catholic Christian.

The most convincing part of the third book of the Genius of Christianity is then the one not mentioned in the title of the book. This part has as its topic what Chateaubriand calls harmonies. In practice, these include, on the one hand, harmonies between religious buildings and nature, and secondly, between religious feasts and human life. The first continues Chateaubriand's interest on architecture. Places of Christian worship are not just capable of connecting us to divinity, he says, but they also exist in harmony with the surrounding nature, and even when they are becoming ruins, they show the power of divinity over the nature. The second topic opens up the discussion to the fourth book, which covers the cult of Christianity. Together, they contains ingredients of most importance in Chateaubriand's book – it is the practice and personal experience that has the greatest ability to make people Christian. This is a topic that I shall deal in more detail in the next post.

François-René de Chateaubriand: Genius of Christianity – Christian poetry

In the second part of Genius of Christianity Chateaubriand begins the tactic that will have most repercussions in the history of ideas. Ironically, this inheritance has little to do with philosophy of religion and more with aesthetics, which is in line with the topic of the second part – poetry. While attempting to uphold Christianity, Chateaubriand influenced the romantic movement of literature.

Chateaubriand's tactic plays on many levels, but his main line of offence is simple – Christian poetry simply is better than pagan poetry. Of course, this strategy also fails on many levels. For instance, Chateaubriand fails to take into account all poetry that falls outside either Greek or Jewish influence – there's no mention of e.g. Indian or Chinese poetry.

Chateaubriand suggests, firstly, that Bible itself is somehow deeper and more beautiful poetry than anything produced by Greeks – for instance, Chateaubriand praises the simplicity of the beginning of Genesis story of creation, which with few phrases creates a fantastic image of a world coming into being. One might answer that Homer has his beautiful phrases as well, but the most telling objection to Chateaubriand's tactic is irrelevance – who cares if Bible is more beautiful than Iliad, if we want to know which is truer?

The worth of Chateaubriand's suggestion lies more in his general opinion that all Christian poetry is beautiful. Here he is emphasising especially the usefulness of Christianity – Christian religion has the characteristic of bringing forth more beauty in the world. This is, undoubtedly, more of a pragmatic justification of the practice of believing its tenets, but quite in line with the general stance of Chateaubriand's book.

One might really question the selection of poetic works Chateaubriand takes to be representative of Christian poetry. It is surprising that Dante's Divine Comedy is barely mentioned, while Milton's Paradise Lost is given ample attention. Equally mysterious from modern viewpoint seems the negligence of Shakespeare's literary work and the inclusion of so many French tragedies from authors barely remembered today.

A more interesting line of criticism lies in Chateaubriand's peculiar reason for upholding the greatness of Christian poetry. Chateaubriand appears to say that Christian poetry has a better mixture of characters and topics to use. For instance, Chateaubriand sets the Homeric heros against the knights of the chivalric poetry. While the former are merely self-serving barbarians, the latter fight for a greater cause, namely, the honour of God. Similarly, the Paradise is according to Chateaubriand a mythologically more moral place than Elysian Fields, since getting to former lies not so much in heroics, but in living a good life.

Chateaubriand's argumentation seems here equally shaky. Sure, one could say that Christian poetry deals with more moral topics and has morally better characters. Still, this is no guarantee that a work dealing with such moral matters would itself be aesthetically more worthwhile. We'll see if Chateaubriand's case is more convincing with other arts and literature.

François-René de Chateaubriand: The Genius of Christianity (1802)

Chateaubriand’s Génie de Christianisme falls into a genre of apologies or defenses of the worth of Christianity, with which the history of Christian philosophy began and which was popular throughout the Middle Ages. But the supposed enemy, from which Chateaubriand tries to defend Christianity, is not the pagan culture of imperial Rome nor is it the competing might of Islam, but a completely new force, namely, the Enlightenment and its criticism of religion. Thus, Chateaubriand appears to think, a completely new kind of apology is required.

Chateaubriand’s apology falls into four books with distinct themes, which I will deal with separately. The first book most resembles a traditional apologetic work in that it tries to defend the dogmas of Christianity. Yet, even here Chateaubriand has more of a pragmatic take on the question, for instance, when he tries to show that compared with other religions, only Christianity has ever had a truly working moral doctrine in Decalogue, or when he speaks in somewhat Kantian tone of a moral necessity of the belief in afterlife.

The true essence of Christianity lies, according to Chateaubriand, in its mysteries, such as Trinity and the story of fall and redemption. These mysteries reveal some deep truths about God and human beings. A common attack against these mystical doctrines had been that they were actually just influences from other religions – for instance, other religions had examples of Trinities, thus, they were not a unique element in Christianity. Chateaubriand’s tactic appears to be in a sense accept these accusations – Trinities occur all around religions, thus, there must be something important in the idea. Then again, he is also quick to point out that these other examples lack some important details which occur only in Christianity and which truly explain the importance of these doctrines.

Chateaubriand does also try his hand with traditional argumentation for the existence of God. Yet, he is quick to dismiss overtly theoretical proofs, or at least he gives one only in end notes. His main line of offence is traditional teleology – Chateaubriand points out the orderliness of the universe and especially the wondrous nature of living beings, which do things that apparently should require the aid of divine forethought (a favourite example of Chateaubriand appears to be the migration of birds and other animals). We know that Chateaubriand is here fighting a loosing battle, since these wonders of nature do have natural explanations. A similar losing battle is going on, when Chateaubriand tries to show that the chronology of the Bible is the most accurate representation of time and that the Biblical events are far more ancient than events in any other holy text.

While this lackluster start wouldn't really make the work into a classic, same cannot really be said about the next book, in which Chateaubriand tries a new tactic.

François-René de Chateaubriand: Essay on revolutions (1797)

1768-1848

Chateaubriand, especially famous of his influence on the literary movement of romanticism, wrote also some more broadly philosophical writings. The most famous of these, concerning the essence of Christianity, could be said to represent a romantic stance on the philosophy of religion, laying emphasis especially on the aesthetic nature of religious experience. Yet, before we take look at that book, I would like to consider a work Chateaubriand himself later regarded as a juvenile experiment: Essai sur les Révolutions.

The aim of Chateaubriand’s Essay is to understand the forces behind revolutions. We need not wonder the reason for such a work: it is obviously the on-going French Revolution Chateubriand wanted to understand. Yet, he spends relatively little time with the current revolution, since his method is historical – he looks at the great revolutions in ancient Greek and he plans to continue with the Roman and modern European pre-French revolutions, although he later abandoned this project.

Chateaubriand’s main idea is that while the political status of Europe during French Revolution resembles that of Mediterranean during the rebellion against traditional Greek monarchies, its moral status resembles the time of downfall of Greek politeias under the Macedonian rule. We may be quite quick with the first comparison, because even Chateaubriand himself in his more mature period noticed the deficiencies of this approach. The young Chateaubriand drew analogies between the relation of Greece and Persia on the one hand and the relation of France and Holy Roman Empire on the other hand and continued e.g. by comparing Carthage with England and Scythia with Switzerland. Needless to say, these analogies fall apart very quickly – the attempted conquest of Greek city states by Persia had in most likelihood little to do with the rise of democracies, while the war against revolutionary France had a clear ideological basis, not to mention that Greek city states formed no unitary nation unlike France.

Still, if we at least accept the broad analogy of a fledgling and apparently weak democracy pitted against a huge monarchic conglomeration of nations, we can still appreciate Chateaubriand’s questions – will the new French government be able to counter the monarchic backlash and whether it is preferable that it does so? Starting from the latter question, it is surprisingly difficult to find out Chateubriand’s true allegiances in the ideological conflict between democracy and monarchy. Indeed, he was a member of the counterrevolutionary army of emigrants, but this was perhaps more to do with his belief that revolutionary France was no true democracy. Here we can see influence of Rousseau – Chateaubriand holds a legislative assembly of all citizens to be a necessary feature of democracy, while so-called representative democracy is then just a monarchy in disguise or an alliance of smaller democratic republics.

Still, even direct democracy is not the ideal form of state in Chateaubriand’s eyes, paradoxically, because of its lack of freedom – he is apparently thinking of the tradition of ostracism in ancient Athens or the mob trials during the Reign of Terror. Politically, Chateaubriand clearly leans more on the tempered English version of monarchy and not the French absolutism. Yet, sometimes even the English monarchy seems too restrictive for his love of freedom, as he speaks eloquently of the American savages who live in a state of no states in small familial communities.

Chateaubriand does not make the mistake of projecting his ideals to his predictions on the outcome of the Revolutionary Wars. Instead, he bases his estimates again on the example of the battle between Greek city states and Persian empire. What Greeks had and what French revolutionaries lack is a certain moral constitution, which made the Greek men into reliable defenders of their way of life. This moral constitution, Chateaubriand suggest, could be gleamed especially from the philosophers of the period, who all upheld traditional ideals of the city state and polytheistic religion. Then again, when the morality of the people relaxed and philosophers started to speak against both the democratic form of constitution and polytheism (both tendencies might be seen in Plato), the Greek city states fell under the Macedonian rule.

Chateaubriand suggests that the French nation had for long time been in this state of moral degradation. His primary evidence is formed again by the philosophical tendencies of the time – the so-called Enlightened philosophers were all moving toward abolition of traditional Christian monotheism. In this juvenile work Chateaubriand does not suggest that French philosophy should turn its course and start to speak for Christianity. On the contrary, he supposes that the whole Christian clergy is mostly rotten and decadent and that Christianity itself would then fade away. Chateaubriand does suggest the possibility that some form of Enlightement could take the place of religion in the future, but he is more likely to believe that this is all just a premonition of an up-coming time of barbarism, from which a new civilization might then gradually rise again. In his mature phase, Chateaubriand rejects this conclusion with the forceful rejoinder that there are not that much non-civilized areas in the modern world, from which the current civilization could be assaulted. Furthermore, he is far more positive about the strengths of Christianity. We shall see next time how Chateaubriand defended Christianity in his later years.