maanantai 13. maaliskuuta 2017

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


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.

perjantai 3. maaliskuuta 2017

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


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.