Günther, Anthon: Preschool to speculative theology of positive Christianity. First part: Theory of creation (1828)

 

1783 -1863

A controversy surrounding pantheism (or atheism and Spinozism, as it was first called) has been a mainstay of German philosophy since at least the time of Christian Wolff, and ever new generations of philosophers have faced the condemnation of having breached the lines of orthodox Christianity. In Anton Günther’s Vorschule zur speculativen Theologie des positiven Christenthums it is the generation of Schelling and Hegel that faces now the same accusation.

Günther’s aim is to defend Christianity and to show how the new philosophies fail to understand its basic tenets. At the same time as Günther is criticising them, he is also clearly influenced by some of the ideas of the other German philosophers, in ways that resemble Coleridge’s new reading of the idea of trinity.

Günther’s book is structured as an exchange of letters between a theology student and his uncle, who is afraid that the nephew has fallen to pantheist thinking. The evidence for this suspicion lies in certain theses reminiscent of ideas common to many of the idealist thinkers of the time. In these theses, philosophy is regarded as a single entity, developing throughout history through various relatively true philosophical systems. Furthermore, philosophy is here regarded as having essentially the same mission as religion, and in addition, as bridging two different phases of religion: an undeveloped awareness of divine in feeling would have been supplanted by Greek realistic and polytheistic philosophy and Greek philosophy would have been superseded by Christian idealistic philosophy, which would finally lead to an age of reasoned consciousness of God, where philosophy would not be needed anymore and wise humanity would rule over all nature and create a new heaven.

Nephew then defends himself that despite his conviction that philosophy will lead to a new understanding of God, he does not think that this would necessarily imply supplanting Christianity with pantheism. Indeed, he insists that theism is an equally valid way to do philosophy.

Nephew outlines his position in more detail by dividing all possible philosophical systems into two kinds: critical philosophy begins with the faculty of cognition and takes care of not overstepping its limits, while transcendent philosophy does precisely what critical warns against and starts from what is absolute. He specifies this rather Kantian division by noting that transcendent philosophy describes the relation between absolute and relative being in terms of substance and its accidental modifications. Critical philosophy, on the other hand, tries to - again in Kantian terms - leave room for faith in matters pertaining to things outside cognition.

The student makes an even more intricate division of philosophical systems, although only in a footnote. Uncle is not that impressed with it, noting that all philosophical systems are in a sense critical, because they all must begin with some facts of consciousness, and at the same time also transcendent, because absolute or God is always one of these facts of consciousness.

His own understanding of philosophy is built around several oppositions. One of them is the primary opposition between absolute and relative being, by which uncle means, effectively, God and the world. Another opposition lies within the relative being, that is, opposition of nature and spirit. A third opposition cuts across the second opposition: in both nature and spirit we can differentiate between accidental or phenomenal appearances and their substance or noumenal ground. Furthermore, he finds yet another opposition between the phenomena multiplicity and the noumenal unity: both natural and spiritual appearances can be reduced to two opposed forces. One of these forces is always passive - in case of nature, contraction, in case of spirit, receptivity - while the other is active - respectively, expansion and spontaneity.

All of these opposed concepts are familiar from Kantian and post-Kantian discussions. Uncle - the probable voice of Günther himself - notes that they are also among the facts of consciousness, from which philosophy should begin. For instance, uncle notes that in our self-consciousness we are conscious of ourselves as spirit, but also of something distinct from us and spirit, that is, nature. By studying both spiritual and natural phenomena, uncle suggests, we can find the basic forces behind them. Uncle points out that this already leads us from accidental variability to more stable and substantial level; even more apparent this substance behind phenomena is to us in our experience of ourselves as a stable unity behind individual experiences. Finally, we are aware of ourselves being conditioned, which then requires the assumption of something that can exist just by itself.

Basic question of philosophy, uncle suggests, has always been to explain how these various components of experience are related to one another, and different systems have a different answer. Some of them concentrate on the relation between nature and spirit, like naturalism, which take substance of spirit be just an effect and a product of substance of nature, spiritualism, where the roles are reversed and the substance of nature emanates from the substance of spirit, hylozoism, where the two are identified, and dualism, where the two substances differ, but interact. Some philosophical systems try to settle the relation of natural and spiritual appearances, like sensualism, which takes all spiritual phenomena to be caused by natural phenomena and therefore raises the status of senses, or intellectualism, which takes the opposite stand and states that the world of experience must be a product of our representations.

The more important question in Günther’s schema concerns the question about the relationship between absolute and relative worlds, and it is here especially where the character of uncle finds evidence for his pantheism accusations. Some systems regard nature and spirit as mere accidents of divine substance: following tradition, uncle calls this acosmic pantheism, because this system denies the existence of any world distinct from God. In a sense opposed to this is what the uncle calls cosmic pantheism, where God is just the goal to be actualised through world history, which is then a process of God’s self-revelation. Further forms of pantheism the uncle mentions hold God or absolute to be spatiality (monadistic pantheism, where all becoming is mere illusion), temporality (compared to previous, all stability is here illusion), substantially identical with nature (pantheistic naturalism, where nature is revelation of divinity) or substantially identical with human spirit (here human spirit is just a limited mode of God and nature this limitation, and God is revealed through the system of finite spirits). With all of this evidence before him, uncle concludes that pantheism is a necessary result, when philosophy attempts to describe the relation of God to nature and spirit.

Nephew objects that there is still some room for a non-pantheist philosophy, where God is described as a creator of the world, creation being an external relation between God and nature. He does concede that creation cannot be handled theoretically, because it is based on God’s unconditioned, free causality, which cannot be conceptualised. Creation is still justified on practical grounds, because it is the only possible basis of ethical freedom.

The basic idea of creation, nephew says, presupposes that there is a world distinct from God and God distinct from the world. World can, he admits, be said to be in God, but only in the sense that God thinks even possible worlds. In addition to this, world still needs to receive existence, which is based on a free act of God. At the same time, nephew says, God reveals himself as the most perfect entity through the world to other, less perfect entities outside him, who are capable of knowing and loving God and so taking part in his blessedness. These less perfect entities or spirits God has created as his likeness, but not as equal to him.

Nephew also makes a Kantian distinction between the sense world and the intelligible world. The sense world is only the manner in which the intelligible world appears to finite spirits, thus, God has no real relation to the sense world. Because spatiality is the property of the sense world, God is not in any spatial relation to the real world. Still, nephew insists, God is outside the world in the sense that he is not finite.

Uncle notes that nephew’s model of creation resembles the system presented in Vorschule der Theologie (note the similarity with the name of this book), written by I. H. Fichte, son of the more famous J. G. Fichte. In Fichte’s system, as presented by Günther, absolute realises itself by giving independent existence to a difference implicit in it and forming a sort of body to itself through its self-division. Absolute retains and knows its unity in and through its division and is thus a self-conscious God only because of creation. The divisions are in the fashion of Leibnizian monadology themselves relative absolutes or self-contained universes and therefore also self-conscious personalities.

The concept of relative absolute reminds the nephew of another recent philosophical work, by Georg Friedrich Daumer. Daumer’s starting point, nephew tells, is an idea found in Schelling’s essay on freedom that God himself had a basis or ground in an ultimate, featureless indifference, on top of which divine self-consciousness was built. For Daumer, nephew says, God is always self-conscious and this indifference is more like something divinity rips out of itself - the relative absolute - and makes into a basis of finite world. In effect, this Daumerian relative absolute is a blind generating force that searches for its own self-consciousness and eventual reconciliation with God.

Nephew uses the Daumerian idea to solve a fault he finds in Fichte’s system, where there appears to be no place for nature outside self-conscious subjects. Nephew notes that most post-Kantian philosophers have taken nature as a dynamic whole and thus something real. If Fichte were right, nature would then also form a self-conscious subject and would be on the same level as human spirits, which the nephew cannot accept. His own idea was to take nature as a mere shadow of true reality, but now Daumer has given him a new idea - if nature is just this blind relative absolute, humanity is still on a higher level, because it does not just strive, but can also reach back to God. The only worry nephew now has is that it seems to restrict God’s omnipotence, if he can never create something truly different from himself, but only modifications of divinity - when God creates, he should not be just positing his own essence.

It is time for the uncle to lead the student away from the quagmire he has got himself into. The first target of uncle’s criticism is nephew’s Kantianism, evident especially in latter’s insistence that spatial sense world was nothing real and that God had nothing to do with it. Günther goes into more detail with Kant’s idea of space and time as a priori forms of experience in an appendix, where he balances Kantianism with Aristotle’s objective notion of space and time and both with Augustine’s manner of grounding space and time to God. In the main book, uncle merely notes that while it was right to say that worldly and therefore spatial relations do not concern God, it is wrong to insist that no relation holds between God and the spatial world. He also adds that Kant’s attempt to make room for faith actually managed to just set faith aside somewhere where it could be forgotten.

Uncle also notes that Fichte’s system is essentially another form of pantheism, because - as the nephew had feared - in it God just multiplies itself or posits other absolutes similar to itself. He also notes, making fun of the idea mentioned in the beginning about apparently different philosophies being just phases in the development of one philosophy, that pantheism, as exemplified by Fichte’s system, and Kant-inspired criticism, as exemplified by nephew’s system, are more like eternal rivals, neither of which can be seen as development of the other. Ultimately, the decision between the two systems boils down to the question whether facts of self-consciousness should weigh more than demands of reason. Human self-consciousness is inevitably bound to a consciousness of something else limiting us - nature - and thus to an inevitable dualism between finite or relative and infinite or absolute being. Then again, our reason strives to reduce everything into a unified whole and to envision the relative world as a revelation of absolute. Although not yet spelling it out, uncle hints of a possibility of a third option more satisfying than either criticism or pantheism.

Uncle also makes short shrift of Daumer’s position. At first, he quickly dismisses Schelling’s essay of freedom, which influences Daumer, by noting that Schelling introduces without any justification dualism within absolute, thus, making it into relative being. Daumer himself, uncle continues, makes again the pantheistic assumption that God can only posit something similar to itself (relative absolute) and that God needs the world for its own self-consciousness.

We then get an interlude with nephew’s letter, where he notes the similarity of his time with the time of Neoplatonists and Church Fathers. He suggests that just like Neoplatonism was a syncretist hodgepodge of various mystery religions and mythologies covered with magical superstition, modern idealism is also a similar confusion of oriental pantheism with mysticism of intellectual intuition. Furthermore, nephew insists that whereas Church Fathers appropriated from Neoplatonism all that was worthwhile (especially the notion of supersensuous world above the world of senses) and added solid historical background in the form of Bible, the task now ahead is to take what is useful in modern pantheistic idealism and give it again a solid grounding in Christianity.

Uncle takes up this challenge by picking as the important result of modern idealism the analysis of self-consciousness. Consciousness as such unites two distinct moments: what is represented and what represents. Self-consciousness is a form of consciousness, in which representing subject makes itself its own object. Immediately subject can know only its activities or interactions with some objects, while it can know its substance or essence only through the mediation of these activities: in the words of a common analogy, eye can see or be aware of itself only through the seeing of other things.

God, the uncle continues, should also have self-consciousness, although it is one of a peculiar kind, because it is absolute. Still, even God must have something as its object, before being capable of self-consciousness. Unlike human consciousness, God can simply posit its own essence as its own object. At the same time, it must posit a common substance for God as subject and God as object. In a manner very reminiscent of Coleridge uncle states that this threefold structure of divine self-consciousness is what Christianity has called trinity. In other words, God necessarily emanates three persons from himself in order to know itself.

Self-revelation of absolute, which reason required for completing its demands, happens then already at the level where only God exists, because God posits its own essence as a new object. Question is why God would still create something other beyond himself and his persons. Uncle concludes that God’s goal must have been to reveal himself to someone else who is not God.

Within the created realm, uncle continues, nature and spirit share similarities, but on a more substantial level they differ and oppose one another. Thus, both nature and spirit are based on a pair of opposed forces and their underlying unity, but with nature these two forces are necessarily related (action and reaction), while with spirit spontaneity need not correspond with the receptive side of human mind, and thus, spirit can freely rise above what it senses. Spirit and nature cannot then be reduced to one another and also not to any indifferent substance underlying them. Their opposition is also not based on their combination, which is just as finite and relative as both of them, namely, humanity as a combination of spirit and body. Their existence, uncle concludes, can then be based only on something third above them, which has created them.

God is then something completely different from both nature and spirit, uncle says, and completely incommensurable with them. In a sense, God is even opposed to finite creatures. God has a unified essence and substance, which appears in three essential forms. World, on the other hand, contains three different kinds of substances (nature, spirit and their combination or humanity), but they all share the same form of revealing their substance through a dualism of basic forces. Still, uncle points out, both the world as a whole and individual created beings can be regarded as God’s images - individual creatures are relative absolutes, since they have life principles that work as a foundation of a sum of appearances, while the world as a whole is a trinity of substances.

Nephew’s final letter starts by recounting traditional properties of God, such as his omnipotence, omniscience and love, all of which are essential reasons why God created the world and how he could do it. He notes that some philosophers deem such properties too anthropomorphic, while only accepting spirit as God's characterisation. Nephew points out that it would be actually better to distinguish spirit as a finite entity from God as absolute - although God is not necessary, like nature, he is also not free in the sense of finite spirit.

He then turns to the question, whether human reason can by itself know that God exists. He notes that philosophers have had two positions on the topic. First position insists that reason has in its own self-consciousness also an immediate mystical awareness of divinity. Nephew cannot accept this position, because it is an obvious fallback to the pantheist position, where humans and God are identified. The second position, on the other hand, accepts that we have only mediate knowledge of God, but suggests that this is possible only if God comes and directly interacts with us. Nephew is skeptical, whether knowledge requires such a direct interaction with its object - we can deduce the existence of absolute even from the awareness of ourselves as conditioned entities.

A related problem concerns the role of faith in getting to know absolute. The prominent opinion, as the nephew points out, was that faith had at least something to do with it. One party, endorsing the idea of reason having a immediate connection to God, identified this immediate connection with faith and insisted that only this immediate cognition or faith was required, while understanding as mediate cognition was dependent on immediate cognition and could only be of use in the multiplicity of the sense world. Understanding and its traditional proofs of God’s existence could still enliven faith in us, even if they really couldn’t prove anything. The opposing party insisted that understanding has something to do with cognition of divine: faith was created in us by God’s interaction with us and understanding was required for refining it into cognition through traditional proofs of God’s existence. Nephew points out that both parties have tried to find an immediate connection with God, one through its own self-consciousness, other through God’s interaction with us. Yet, both must admit that understanding and its mediate, conceptual thinking is required for knowing God, since immediately we can know only our own states, but not what produces them.

In the concluding letter of the book, uncle congratulates the nephew for endorsing two controversial ideas. First of these is the idea that God is properly speaking not a spirit. True, God has similarities to spirits: both are non-material and have self-consciousness. Still, they both have widely different essences, just like animals and humans do not share their essence, although both have the capacities to sense and imagine.

The second controversial position is that we can know God even without direct connection to him. The opponents of this idea insist that only similarities between the knower and the known make knowing possible. Uncle points out that this would make it impossible for us to ever know things of nature, which have a different essence from us. He points out the possible objection that humans can know nature through their bodily side, but rejects it instantly by noting that it ignores the unity of human self-consciousness - it is not the eye that sees, but the self-conscious human using the eye.

Uncle notes that the principle of similars knowing similars has a true kernel: we must know something similar to ourselves, that is, we must be conscious of ourselves, before we can know anything else. Still, this does not mean that we have to turn in something else in order to know its existence. Furthermore, the knowledge of God provided by our knowing the conditionedness of ourselves is only theoretical knowledge of absolute. If we truly want to know God as God or his personality, uncle concludes, we must know him also through hearing him speak to us through our conscience.

James Mill

(1773-1836)

Some thinkers have had the misfortune of being overshadowed by both their predecessors and their successors. This is true e.g. of Karl Reinhold, a philosopher known often just as the link from Kant to Fichte. A similar fate has been experienced by James Mill, who is often regarded just as the person to transfer Bentham’s utilitarianism to his more famed son, John Stuart Mill.

Even so, James Mill was an interesting philosopher, with a wide range of interests. Indeed, this variety calls for some selection. I will not look at his role as a historian of India, although this was a work close to his professional interests as an administrator in East India Company. I will also ignore his economic writings, mainly because Mill's main achievement was to inspire David Ricardo to complete his book on economy. Instead, I shall focus on Mill’s political essays, the most famous of which is undoubtedly his essay on government.

Mill begins his look on the best form of government with a conundrum. Clearly, pure democracy is impossible, he says, because every time a governmental decision is required, we cannot go around asking everyone what to do. Then again, aristocracies and monarchies can - and even more strongly, Mill appears to say - tend to devolve into dictatorships where rulers follow their own interests. Sometimes solution was seen to be in mixing democracy with monarchy and/or aristocracy. Mill noted that this would mean merely that the government would not be unified, but a battlefield of the different interests of the nobles and the commons.

Mill’s simple answer is representation - the decision must be made by people representing the real owners of the political power. Evidently, representation is pointless in case of monarchies and aristocracies and does not remove the threat of tyranny. One might naturally ask whether the representational democracies don’t also face the threat of tyranny, in case representatives gain too much power. Mill’s answer is that if the representatives are regularly changed then they will have an incentive to rule for the best of the whole community - after all, they will soon be just regular citizens.

Mill’s account of representation has some notable gaps. First of all, it is a bit unclear whether Mill really suggests that all the representatives will be changed at appointed time, without any chance of same representatives being re-elected. A more important problem is Mill’s somewhat naive assumption that the interests of an individual representative will in every case coincide with the supposed interest of the whole community. Indeed, Mill seems to leave quite out of consideration the possibility that representatives might make decisions benefiting just some factions within the people. Mill does try to prevent this possibility by rejecting all systems where representatives are voted to represent certain professions or class. Yet, even if one would not officially be a representative of certain class, she might still feel the interests of this particular class as aligning with one’s own and make decisions based on it.

Despite his rather republican conclusions, Mill at once takes back his statement that mixed governments could not work and notes that a king and a House of Lords could still be maintained, if the representative government would find some use for them in administration. Mill’s suggestion might have been just a political move, meant to allay concerns that he would be deposing British royalty. Even so, he makes the further conservative move that voting could have an income limit. He doesn’t want the limit to be too great, which would make the system practically aristocratic, but allows a small limit, because, he says, people with little money can take care of the interests of those with no money. Furthermore, although Mill does not want rich people to buy votes for themselves and therefore endorses a secret ballot, he casually notes that representatives will most likely be rich and aristocratic in the good sense of the term, that is, capable men.

My use of the word “men” is quite deliberate. Mill states in clear terms that women are to be treated like children: they do not take part in voting, neither as voters not as candidates, because, Mill says, their interests are taken care of by adult men. Nowadays, Mill’s reasoning seems just preposterous. Even if one would believe that fathers and husbands always have the interests of their children and wives in their mind, the case of orphaned, unmarried women would still need some further solution.

Mill’s system of government has, then, an elitist streak. Indeed, he notes that while even a nation of mere uneducated people would benefit from representational democracy, its true worth is shown in nations with a large, educated middle class. Then again, Mill’s middle class is not a closed club, but people can become part of it by educating themselves. Indeed, Mill endorsed the idea that all people could and should receive education, with the exception of those that have some clear syndrome making learning difficult. This is the part where Mill’s philosophy most closely touches what was classically meant by philosophy, as Mill notes that finding best means of education requires first determining what human mind is like and what makes it happy. We’ll see in a later post what to make of this side of Mill’s philosophy.

In addition to representative government, Mill also spoke for freedom of press. According to him, there are only two cases where it should be restricted. Firstly, press should refrain from publishing slanderous lies told of individual persons. They could still make headlines about true scandals, except in cases where scandals are not true scandals, although irrationally held to be so because of outdated morals. Secondly, press should not hinder the proper activities of government. This does not mean that government could not be criticized, and in fact, Mill takes it to be the duty of free press, because people need to no when a time for a revolution is at hand.

August Ferdinand Möbius - Barycentric calculus: new aid for analytical consideration of geometry (1827)

1790-1868

Concrete sciences have often been a spur on the development of new mathematical methods. Take as an example Archimedes, who used the idea of balancing a mass of parabola with a mass of a triangle and thus finding out the size of an area limited by the parabola. The exact method Archimedes used in determining this area was a rudimentary form of integration, with Archimedes, as it were, adding infinitely small pieces of the area together.


The idea of mass and weight was, coincidentally, a spur for Möbius’ text called Der barycentrische Calcül: ein neues Hilfsmittel zur analytischen Behandlung der Geometrie. Möbius starts with the notion of the center of mass, determined by mass of three objects situated in a triangle. He then quickly finds out that in fact any point within the triangle can be expressed as such a center, by just changing the masses assigned to the apex. Indeed, one need not be confined to points in this triangle, but allowing that in this case some of the masses could have a negative mass.

The end result is then the expression of points - and ultimately lines and figures - through three points of an arbitrary triangle. One can expand this meth to three-dimensional case by switching triangle to a pyramid. One can even express various simple functions in these terms - e.g.express curves like parabola, hyperbole and ellipse in terms of a three points of a triangle - and translate some concepts of infinitesimal calculus in this “barycentric” form.

A more interesting question is why Möbius uses such a peculiar form of expression instead of the more familiar method of expressing geometrical figures through two coordinate axes - that one can do seems a poor reason for actually doing so. The answer appears to be simply that some important relations are easier to express in this manner. Möbius is especially interested of what we nowadays might call equivalence relations between different figures. Two examples of such relations have been known since the beginning of mathematics - first is the equality of the sizes of the areas, no matter what their shape is, and second is the similarity of shapes, no matter what their size.

Beyond these two equivalence relations Möbius considers two others, first of which, affinity, had been introduced already by Euler. One might say that this affinity is a relaxed version of similarity, that is, a relation that could hold even between non-similar figures, although similar figures are always affine. Möbius introduced this relation by asking his reader to change the triangle used to determine figures with some completely different (not even similarly shaped) triangle. The figures defined by such change need not be similar - an example of such a pair is formed by a square and a rhombus. Still, they do have some resemblances, for instance, quantitative relations between respective parts (more precisely, parallel line segments) of these figures remain same.

A further form of equivalence relation Möbius discovers is what he calls collinear resemblance. Simply put, Möbius starts from affinity, takes away even further properties of the two figures and leaves only one point of resemblance - that all straight lines in one figure correspond to a straight line in the other figure. This definition loses even the identical ratios of lines, resulting in even more abstract resemblance. Some common quantitative properties still remain, namely, the so-called cross-ratios (a specific type of ratio for four points in the same line). Still, this is a very loose sort of relation - even circle and parabola have this sort of collinear resemblance.

Jean-Babtiste Joseph Fourier: Analysis of determined equations (1827)

One of the parts of mathematics most easily applicable to practice is the study of equations. You need to find out a certain quantity or certain quantities, and you know that it has or they have a certain relation to other quantities. Finding out what these unknown quantities are means solving the equation.

Although a layman might think that mathematics should always give exact solutions to such problems, it is quite obvious that whether and in what magnitude giving exact solutions is possible is a question that does not always have a clear cut answer. Take such a simple equation like x2 = 2. We know that the solution to this problem cannot be expressed as a ratio of whole numbers. Still, despite the objections Pythagoreans would have had, we are usually accustomed to say that solutions involving roots are precise - at least they tells us that the relationship that the searched for quantity has to known quantities, even if we can express the numeric value of the former only approximately.

It has been long known that for some, relatively simple equations, such relatively exact answers can be found, if there just is an answer to be found. Let’s take a case where we are searching for a single unknown quantity, with a relation to zero, describable in terms of such simple calculations as sum, multiplication and squaring:  x2 + ax + b = 0 (the so-called quadratic equation). There’s a simple enough formula for solving such equations, using again only very simple operations - addition, subtraction, multiplication, division, squaring and square roots.

We know that the formula for quadratic equation will give us two, one or none solutions - the last option occurs, when the formula would involve a square root of a negative number, something that is usually an impossibility, when applying mathematics in more concrete fields, although we can construct an abstract system with such square roots of negative numbers (the so-called imaginary numbers). We also know that the solutions revealed by the formula are all the solutions the equation could have, and we can even represent this geometrically: the equation describes a figure known as parabola, which can cut one of the axis of coordinate system twice, touch it at one point or then not cut or touch it at all.

The situation becomes somewhat more complicated when we allow exponents larger than 2 in the equations, that is, when we deal of general polynomial equations of the type xn + axn-1 + bxn-2  + … + rx + s = 0, where the highest exponent is called the degree of the polynomial. We do know something general about the solutions of such equations. If the n is odd, the figure described by the polynomial function xn + axn-1 + bxn-2  + … + rx + s is like a rising line: with very large, but negative values of x, the result of the polynomial is negative, while with very large, positive values of x, the result is positive. If n is even, the figure resembles parabola, where large values of x, whether positive or negative, produce positive results. The only difference is that with larger exponents, the figures might have more bends - the maximal number of bends in the figure is always n - 1, where n is the degree of the polynomial. This means that the maximal number of solutions for the equation is also the degree of polynomial - every new bend makes one more point of contact with the x-axis of coordinate system possible.

Although the maximum number of solutions of polynomial equation is known, we might not always be exactly sure what these solutions are. With polynomials of degree 3 or 4, a general solution of similar sort as with quadratic equations can be given. Then again, with polynomials of higher degree such a general solution does not - and even more, cannot - exist. We might be able to find the exact solutions sometimes, but there’s no guarantee we could do it always.

Even if a general method for finding exact solutions does not exist, we might still have a method for finding inexact solutions, that is, better and better approximations of the searched for solutions. Such a method of approximation can also be of mathematical nature, because we might have good mathematical reasons to say why a certain method works. A good example is the method invented by Isaac Newton. The basic idea behind Newton’s method is that at small distances a curve is similar to its tangent. Thus, if we have an estimate that is close to the final solution, we can use the tangent at the point of the estimate to get an even better estimate of the solution - just check where the tangent hits the x-axis and you get the new estimate.

The problem with this method is that if the first estimate is not close enough to the real solution, it might take many iterations to get even fairly good approximations. The problem thus becomes how to determine the regions where we should go looking for the A partial answer to this problem is provided by Fourier’s posthumously published work, Analyse des équations déterminées.

Fourier’s starting point is unexpected. He asks us to produce a derivative of the original polynomial, then a derivative of this derivative and so forth, until nothing else is left, but a constant function. The series beginning from the constant function and ending with the original function has n +1 members. What has this series of derivatives to do with the solutions of the original equation? Well, consider the results of the polynomial and the series of derivatives for very large negative numbers. The final constant function is always positive, the result of the next derivative in the series - a polynomial of degree 1 - is negative for very large negative values of x, while the next derivative - a polynomial of degree 2 - has with these values positive results. Generally, the polynomials of odd degree in this series have negative results for very large negative values of x, while polynomials of even degree have positive results. In other words, with these large negative values of x, a result of the function in the series is always of different sign than the result of its derivative, which means that the sign of the result changes throughout the series n times.

By itself, this result seems quite meager, but some further reflections show its importance. Firstly, checking what happens with very large positive values of x, we notice that the original polynomial and all the derivatives of the series have positive results, which means that the series has no sign variations. All the sign variations have vanished when moving from very large negative to very large positive numbers. Indeed, the only point when the number of sign variations may change is with those values of x, when the original polynomial or one of derivatives in the series produces zero - either the function producing zero cuts x-axis at that point and its sign changes when moving through, or then it just touches x-axis and the sign of its derivative changes.

A further important point is that the number of sign variations can never increase. If the function changes from positive to negative near a certain value of x, then the function is diminishing and its derivative must be negative near the same value of x, and if the function changes from negative to positive, then the function is growing and its derivative must be positive. Thus, supposing that the function changing the sign is also a derivative of another function of the series and thus in the middle of two other functions, the change of its sign can never increase the number of sign variations. For instance, if in one part of the series the signs are + for f’’(x), - for f’(x) and - for f(x) (with one sign variation between them), the change of the sign of the middle term changes the series into +, + and - (again with one sign variation between them). Then again, if the series was at first, +, - and + (with two sign variations), after the sign change it will be +, + and + (with no sign variation).

In effect, then, if we take two different values of x, at the smaller value the series of derivatives of the polynomial cannot have less sign variations than at the larger value. In fact, if we consider the difference between the sign variations at these two different places, this difference gives the maximum number of places between these values of x, at which the result of the polynomial will be 0 (roots of the polynomial, as they are called).

Of course, this method provides us only with a maximum number of possible roots of the polynomial between two values of x, and the interval might actually contain a lot less of roots. Still, with systematic division of such intervals - and few tricks Fourier uses to weed out intervals, which really contain no roots, despite the number of sign variations - it is possible to pick out certain intervals where the searched for solutions lie. The next step in Fourier’s method is then simply to use Newton’s method to approximate the solution found within a certain interval. The whole procedure is then strictly mathematical, although the result might never be truly exact - we can even count, Fourier notes, how close our approximations are to the real solution.

Augustin-Louis Cauchy: Lessons on applications of infinitesimal calculus (1826)

While the previous book of Cauchy I discussed remained mostly on the level of pure mathematics, the very title of this book, Leçons sur les applications de calcul infinitésimal, promises to deal with applied mathematics. Of course, even applying can happen at different levels, and Cauchy is here dealing not with, say, application of mathematics to other sciences, but with application of one part of mathematics within another part mathematics, more precisely, in geometry.

What is applied in geometry is infinitesimal calculus, which consists of differential and integral calculus, two methods which in a sense are counterparts to one another. Yet, in a sense this is not enough, since Cauchy is actually using a variety of mathematical tools. For instance, the book begins with a long introductory section on trigonometric functions. These functions express various relations between lines and angles and can thus be used in simplifying the formulas dealt in calculus.

Another example of mathematical tools used by Cauchy is provided by polar coordinates. Unlike the regular xy-coordinates, polar coordinates express all positions through a distance between the position and the origin and the angle that the line expressing the distance forms with one of the axes. As Cauchy notes, some geometric shapes are easier to express with the polar coordinates. This is particularly true of spirals. Spirals circle in a regular fashion around a centre, which we can think as located at the origin. While the distance between the centre and a point in spiral grows, the direction of the point from the centre changes in a regular fashion.

Differential and integral calculus are still the primary methods used in the book. Originally both methods have been justified through the idea of infinitesimals or infinitely small quantities - hence, their common name, that is, infinitesimal calculus. In my previous discussion of Cauchy, I noted that by an infinitesimal calculus he meant actually a variable which was thought to be diminishing into nothing. Here, he returns to the more relaxed notion of true infinitesimal quantities, perhaps because such deep theoretical questions need not be addressed in a more applied context.

A good example of Cauchy’s tendency to ignore the theoretical questions is his rather free use of the notion of different levels of infinitesimals. It is undoubtedly difficult to understand how one infinitely small quantity can be of a different level from another infinitely small quantity, that is, larger or smaller than it. Cauchy’s theoretical account of infinitesimals is rather enlightening. Different levels of infinitesimals could be defined through different rates at which quantities approach nothing. For example, if a quantity is approaching zero, its square will also approach it, except that the square will approach the goal in a quicker fashion than the original quantity.

Of the two methods, integral calculus seems simpler in the sense that it has less areas of application - and indeed, this part of Cauchy’s book is just a fraction compared to the part dealing with differentiation. Integration in its original sense was thought to consist of dividing something (curve, area or body) into infinitely many infinitely small parts and then, as it were, adding them up. In more modern terms, integration means finding more and more fine grained divisions of curve, area and body and sums of these divisions, and examining whether these sums approach some definite limit. The result of integration is then, simply, the magnitude of the curve, area or body. In fact, the result is at worst only an assumption that the curve, area or body will have some magnitude, and for actually defining this magnitude, information on other relations between the geometrical entities is often needed.

Basic use of differential calculus, on the other hand, is to find out relations between differentials, or again in terms Cauchy used in the previous book, to find out whether relations between variable, diminishing quantities approach some definite limit. An obvious application for this is the relation between variable x- and y-coordinates of points of a curve. One begins by looking at cords connecting ever closer points of the curve and the relationship of their coordinates, which determines the direction or inclination of the cord. The limit of this approach is then a direction or inclination of a tangent moving through this one point of the curve - or, one might even say, the inclination of the curve at that point. That is, in case the tangent even exists.

The last sentence is important. Cauchy already understands that differentiation is not a universal method, but fails at some points. The curve might have a sudden change in its direction, or it might loop back and touch itself, making it impossible to say what is its inclination at this point. It is an important change in self-understanding of mathematicians to accept mathematics as not interested merely of general rules, but also of particular exceptions to these rules.

Inclination of a curve is only one of its characteristics. Indeed, a curve and its tangent have the same inclination at one point and still look quite different around that point. One would like to say that the line has less of a curve than the curve, but it needs a more precise definition to do this. Cauchy asks us to think of a circle - the bigger the circle the more the circle looks like a straight line from a given point and less curved it seems. Hence, since the circle grows with its ray, Cauchy defines the curvature of this circle as inverse of its ray.

Now, while inclinations are represented by tangents of a curve, curvature of a curve can be explained by circles touching the curve - or more precisely, the inverse of their ray - which are known as osculating circles. This is still not a proper definition of curvature, since it is not clear what circle touching the curve one is to take as the osculating circle.

Still, we can use the circle as a clue for finding out what curvature is. Picture a tangent moving through the circle, always remaining a tangent and changing its inclination as it advances through an arc of the circle. Taking smaller and smaller arcs around a point, the relation of the inclination of the tangent to length of the arc might approach a certain constant, which happens to be the inverse of the ray of the circle or its curvature.

This notion of curvature is at once applicable to other curves, since they also involve the variables of the arc and the inclination. True, the concept of curvature does not work with all curves, at least not in all points. In particular, it requires the curve in question to be twice differentiable.

Cauchy’s book deals with many other important concepts. Worth mentioning is what happens when we move from a setting with two coordinates to a case with three conditions. I won’t go into details, but note merely that then mere curvature defined in the sense above is not enough, because this first curvature deals only with a two-dimensional issue. We also need then to take into account another quantity defining the curvedness of the figure when looking at the third dimension - this curvedness Cauchy calls second curvature.