Carl Friedrich Göschel: Monism of thought (1832)

(1784–1861)

Right after Hegel’s death in 1831, the interpreters dashed like vultures to feed on the carcass of Hegel’s system – for what was left of Hegel’s philosophy after his death was certainly not anymore a living, developing organism, but a mere dead corpse that could be used by other thinkers as nutriments for developing their own philosophies. After a while, two broad extremes of Hegel-interpretation rose: the so-called Young or Left Hegelians and the opposing Old or Right Hegelians, the dividing line being roughly determined by the political stances

An example of the Right Hegelianism is provided by Göschel’s Der Monismus des Gedankens, which is a perfect example of an interpreter reading more into a philosopher’s texts than could be gleaned from the text itself. The occasion for Göschel’s text is provided by Christian Hermann Weisse, whose Hegel-inspired book on aesthetics we have considered. Weisse did not even present himself as a pure Hegelian, but wanted to improve Hegel’s philosophy, while Göschel then suggests he wants to defend Hegel against this perceived attack.

Since we have dealt only with a part of Weisse’s suggested improvement, I will go through it in a summarised fashion. Firstly, he had said that he didn’t want to remove anything from Hegel’s logic, but thought it had to be supplemented in a fashion. Logic, Weisse had said, was a science of necessity, so all necessities should be handled in it. Thus, he continued, the final chapter of the logic had to still contain another trivision: logical method, as an undivided continuum of all the previous structures of logic, should be transformed into a differentiated space, which would then be transformed further, through connecting these differentiations into a new process or time.

Now, speaking as a Hegel-scholar, I see nothing to object in the process of developing space and time out of the logical method, as presented by Weisse, since this is in fact how Hegel himself does it. What is against Hegel’s intentions is only the grouping of space and time within logic, on the pretext that logic is somehow a science of necessities, compared with other parts of his philosophy, which should be sciences of freedom, according to Weisse. In other words, what Weisse is saying is that while logic describes what must be, the later parts of Hegel’s philosophy should describe only what exists, because of the free choice of God.

The problem is that Weisse’s notion of necessity is very different from the manner in which Hegel understands the word: of course, Weisse is really using the ordinary notion of necessity, so the confusion is understandable. For Hegel, necessity means in a sense the necessity of connections, in the sense that it is necessarily possible to move from what is described in an earlier part of his philosophy to what is described in a later part. Thus, for instance, once one has discovered the logical method, one can then divide the moments of the method and so form a sort of mental space of them. Of course, since such a necessarily existing possible connection need not still be used – we can freely choose whether to think space after the logical method – this necessity does not in any manner hinder our freedom. Then again, this duality of necessity and freedom occurs in all parts of Hegel’s philosophy and even in logic. What is novel in the further parts of Hegel’s philosophy, as opposed to his logic, is that one can then also find dedicated empirical counterparts for such logical models or concepts discovered through these connections – for example, the experienced, geometrical space.

What Weisse had considered as a necessary outcome of his emphasis on the difference of necessity and freedom is that the next part of Hegel’s philosophy – the philosophy of nature, should have begun with matter, which is something that does not necessarily exist, but is created by God. As something freely created, Weisse had added, matter is something of more worth than mere thinking, and in addition, something thinking can never completely penetrate. Of course, the truly Hegelian counterpoint would be to point out that all Hegel needs is to show that a model of space and time is necessarily connected to a model of matter – something moving within space and time – in the sense that we can, as it were, construct the latter model from the former. This connection of models is then in a sense reinforced by Hegel’s conviction that empirical space and time are nothing but ephemeral sets of relations that could not empirically exist without anything having those relations (that is, matter we can empirically observe). Furthermore, there would then be no question of not understanding what matter is, since the construction of its model just would be this required understanding.

From matter, Weisse had jumped straight to spirit, noting that like nature, spirit must also be something that is not describable by mere logical categories, but also something freely created by God. We probably need not repeat the basic fault in Weisse’s reading, but just in case let’s do it: jump from nature to spirit does not say that spirit would be somehow necessary, but only that a) given a model of nature (a realm of disparate entities), we can move to a model of spirit (a self-regulating process using these entities consciously for its own purposes) and b) empirically natural processes can in some measure be controlled by such a conscious self-regulating process.

Within Hegel’s philosophy of spirit, Weisse had ignored the so-called subjective spirit, mostly because he wanted to concentrate on both objective and absolute spirit, instead. Weisse had envisioned the realm of objective spirit to be divided into parts on language – which Hegel studied in the division on subjective spirit – on state – which Weisse apparently thought to contain almost all of what Hegel had studied in his philosophy of right – and finally, on world history – which for Hegel was only a final part of a study on state. From this short summary it is difficult to judge what Weisse really meant with this division, and the analogy he gives is not that much more helpful: language being like mechanism (perhaps because speaking persons are still independent of one another?), state like chemism (perhaps because persons unite into communities?), and finally, world history like teleology (perhaps because it is supposed to be a process with an end?).

Weisse’s theory of absolute spirit we are already very familiar with. He had begun with truth or science as life of spirit, then proceeded to beauty and art as spirit’s cognition of the external world and finally ended with divinity or religion as the most absolute phase of spirit. I have discussed this part of Weisse’s system (or more particularly, the second division of beauty) in more detail earlier, so I’ll overlook it here.

As for Göschel's critique of Weisse, he clearly knows the Hegelian liturgy, since he can at the right places state some key phrases loved by many followers of Hegel. Thus, when Weisse criticises Hegel of formalism, Göschel can at once point out that Hegel's philosophy is just the opposite of formalism, avoiding mere application of all external schemas. He connects this point at once with the idea that the universal beginning of Hegel’s philosophy particularises itself, thus becoming more determined with its development. Weisse’s supposed improvement, Göschel says, is on the contrary dualist, because it assumes something beyond logical method – content – which the method has to grapple with. All this liturgy has a proper Hegelian meaning – thinking is, for Hegel, an activity that can create content even when nothing else is given and can thus be used to model anything in the empirical world - but it is unclear how Göschel understood these words.

This unclarity becomes clear when we see how Göschel wants to solve the dilemma of necessity and freedom that led Weisse to improve the Hegelian philosophy. Göschel says explicitly that the thinking we follow throughout the Hegelian philosophy is not human, but divine thought – or at least it would be that, if this system could repeat exactly all the intricacies of the world around us. Thus, he says, the world around us is the creation of divine thought and therefore necessary, but there is much that seems contingent for us as mere humans. This is, of course, completely against the spirit of Hegelian philosophy, where the idea of necessary contingency is explicitly endorsed and no divine viewpoint outside the human viewpoint is accepted.

Göschel, on the other hand, is clearly committed to the Christian notion of God. Hence, while he recognises that Hegelian philosophy just begins, from nothing, he at once adds that beginning differs from beginner, thus, that it is God doing this beginning. This does not mean, he explains, that we should begin from God, since then we could never move away from God, which would drive us to pantheism. Similarly, he adds, we should not start from unconscious nature, consciousness or both together, since these beginnings would lead us, respectively, to realism, idealism and dualism.

A further example of Göschel’s tendencies is his choice of example for the supposed cyclical nature of Hegel’s philosophy, where the beginning and end should coincide. This example is the incarnated God that, Göschel says, is both the basis of true human cognition – because according to him, we would know nothing without Christ – but also the end of all human cognition. Thus, moving toward Christ, Göschel says, a person moves, on the one hand, to their beginning, but also to their end.

Göschel moves on to consider the specifics of Weisse’s reimagination of the Hegelian system. Beginning from Weisse’s derivation of method, space and time, Göschel notes quickly – and quite correctly – that actually these three are implicitly present much earlier in the development of Hegelian philosophy than Weisse thinks. Indeed, method is, of course, used through Hegel’s logic, and furthermore, abstract spaces and “times” or processes have appeared almost from the beginning of the logic (i.e. when Hegel starts by pointing out that thinking nothing begins a process – or becoming, as he calls it – which results in a construction of a space of qualities), and the later philosophy of nature really adds just the discovery of empirical counterparts for both of them.

Just like space and time, Göschel says, so is matter to be found already in Hegel’s logic, even if Weisse thought it was something unique to philosophy of nature. Indeed, the move from space and time to something that is spatial and temporal is similar to many moves in Hegel’s logic, where he moves from the thinking of e.g. existence to the thinking of something existent. As Göschel points out, we do a similar movement in our ordinary thought, when we introduce the notion of matter – we perceive something and then decide something or matter must be the basis of these perceptions. In fact, he continues, matter just is a hypothesis of thought and not something thought couldn’t understand. Göschel also suggests that while Weisse saw matter and even whole nature as inimical to thought, nature in its innermost essence really is at least an analogy to spirit – in other words, it is not just a passive receptacle, but an active striving toward self-organisation and even self-knowing. Therefore, it is just natural that matter will next give rise to motion.

Although Göschel at first appears to understand the introduction of matter in a manner faithful to Hegel, he then adds the un-Hegelian point that we should also investigate the genesis of matter or of nature in general, or in the more poetic terms he uses, how the water of thought solidifies into the ice of matter. Hegel really isn’t interested in the question of how things have temporally begun, especially in the case of nature – philosophy, for him, deals with what is present. Göschel, on the other hand, explicitly answers this question by saying that it must have been absolute spirit – in essence, God – that has created matter.

From matter, Göschel jumps straight to the end of Hegel’s philosophy of nature, that is, to living organisms, which are the most spiritlike forms nature can produce. In other words, while the realm of nature, Göschel says, differentiates and solidifies the moments of the fluid process of thought, its forms become more and more unified, in the sense that in the more complex forms and especially in organisms their parts are less and less independent of one another. Yet, even organisms are still no spirits. Göschel, following Hegel, suggests that we could thus call the whole nature sick in the sense that it always dissatisfies the demand of becoming spirit.

Since Weisse ignored the first part of Hegel’s philosophy of spirit – the so-called subjective spirit – Göschel does not say much about it, but he does reprimand Weisse for not taking it seriously. Subjective spirit should be the first step away from nature and still partially linked with nature, and because this necessary first step is missing, Göschel insists, Weisse’s account of objective spirit appears to lack objectivity, which it then has to find beyond thought. It is for this reason, Göschel suggests, that Weisse considers human community or state to be mere means to an end – the world history. On the other hand, Göschel, following Hegel, thinks that world history is just the highest shape of the human community – the court of the whole world.

More sympathetically Göschel relates to Weisse’s idea that language is part of the objective spirit, while Hegel had restricted it to the subjective spirit. In fact, Göschel thinks that logic already as pure thought is some type of language, as the etymology suggests (logos meaning word). Subjective spirit, then, is the place where language is generated, Göschel continues, but objective spirit does have its own language, and the deepest shape of language can be found in absolute spirit, and more particularly, in art.

It is no wonder that Göschel finds Weisse’s account of absolute spirit very wanting, since it is based on the very idea that philosophy and thought is not enough and that there’s something more to be found beyond it. Göschel himself, on the other hand, suggests another way to amend the end of Hegel’s philosophy. In a rather formalistic manner – and in a manner followed by many Hegel-interpreters after him – Göschel takes a bit too seriously the trivisions of Hegelian philosophy and tries to correct Hegel by suggesting a better trivision, thinking he has thus made considerable progress. Thus, while logic is for Göschel the natural first point of the system, the second point should not be the philosophy of nature, but the whole philosophy of creation, dealing with nature (passive beginning), soul (separation of something active from nature) and human spirit (free activity). The third point, then, would be theology, as the study of God. Although he does not explicate this idea more closely, it clearly appears opposed to basic ideas of Hegel’s philosophy – God is not just something completely removed from creation and especially from human life, but merely one form of human existence.

Thomas Carlyle: Sartor resartus (1831)

(1795–1881)

At first glance, Carlyle’s Sartor resartus is a parody of philosophical treatises, especially those from Germans of late 18th and early 19th century. This is suggested by the name of the imagined thinker, Teufelsdröckh (something like “Devil’s dirt”), and his supposed hometown, Weissnichtwo (“Know not where”). Add to this that Teufelsdröckh is meant to be a professor in Allerley (“all sorts of things”) and that his main work – the theme of Sartor resartus – studies philosophy of clothes, and the reader is bound to expect that Carlyle is not taking any of this seriously.

Yet, when Carlyle finally gets to the fictional work itself, an air of depth and seriousness appears. He does make mockery of a scholarly tendency to get bogged down in insignificant details in his description of the historical-descriptive part of the work, where Teufelsdröckh, among other things, spends a whole chapter on the topic of aprons. But the tone of the work changes, when Carlyle turns to the second, philosophical-speculative part of Clothes-philosophy.

Clearly influenced by Kantian thinkers, Teufelsdröckh begins by rejecting the absoluteness of time and space and comes to a conclusion reminiscent of Fichte – there’s nothing as real to be discovered in the world as me. Yet, he immediately continues, it is not as obvious what this me is. His first suggestion is to note that underneath all the clothing, we humans are nothing more than naked animals. Still, Teufelsdröckh at once adds, humans are also spiritual beings, and this especially through their clothes: a judge radiates a certain authority just because of certain clothing. Indeed, without clothes, civilisation as we know it would be quite impossible: for instance, where to carry one’s purse, if not in one’s pockets?

Clothing, Teufelsdröckh says, is something through which a spirit converses with other spirits. In a mystical fashion, he suggests looking through all these raiments of clothing, into the innermost essence of human beings. Here is revealed a mythical conclusion that the whole material universe can be seen as clothing for eternal.

Carlyle interjects at this point a fragmentary biography of Teufelsdröckh, supposedly composed from a collection of his random notes. This biography has its share of fantastical elements – Teufelsdröckh being supposedly a foundling with a possible noble heritage, raised by a pair of commoners as their own. Still, for the most part, the beginning of the story is a quite realistic, if somewhat parodied tale of a German would-be-scholar trying to make a living by finding a patronage for himself.

The biography takes again a more fabulous turn, after Teufelsdröckh experiences a failed romance and starts to travel, apparently meeting world historic individuals, like Napoleon. Yet, Carlyle leaves these external details mostly untold and concentrates on the spiritual development of Teufelsdröckh, who begins from a state of scepticism and desperation (everlasting no), but finds inner strength to resist the fear of a world without meaning. Then, after renouncing himself, Teufelsdröckh discovers the divine or the everlasting yea in the world and has now a purpose for which to produce things.

Returning to the Clothes Philosophy, Carlyle shows Teufelsdröckh praising the simple, self-made leather clothing of Quaker George Fox as the most important point in modern history. In comparison, the official church seems to Teufelsdröckh full of sham priests, whose clothes have become shallow masks with no spirit within. This does not mean that Teufelsdröckh would disregard all religious symbolism in favour of simple adornments. Quite the contrary, he praises symbols in general, which at the same time stay silent or conceal and speak or reveal things, and especially religious symbols, in which the artist behind the symbol has also become a prophet. These religious symbols just need to have behind them the divine or infinite, which uses the symbols as signifying some deep truths.

Two kinds of persons Teufelsdröckh is said to honour: an honest worker, toiling for their daily bread, and a spiritual worker, toiling for the Bread of Life. Best it is, he says, when both are combined in the form of a peasant saint. Although this admission might seem like a praise of a worker, it has its darker side: Teufelsdröckh does not take seriously the Malthusian idea that workers could starve to death if overpopulation continues. Instead, he is more worried that in the current time, when even priests are mere empty cloaks, workers will not be able to find their spiritual bread. The result of this non-existence of religion, Teufelsdröckh says, is that everyone will turn against everyone else, caring for nothing but their own independence, poor perishing from hunger and overwork, and rich perishing from satiety and idleness.

Teufelsdröckh is still optimistic. Even if the dying society is empty like disregarded clothes, at least clothes will carry the memory of the person who used to wear them. He also hopes that the society will at some point be reborn like phoenix, although it is now burning itself up. Indeed, he even suggests that like a snake is already growing itself a new skin, when it discards its old one, something new is already building up in our society - radicals of today already carry the seeds for the rebuilders of tomorrow.

From the current politics, Teufelsdröckh returns to the metaphysical, or as Carlyle says, transcendental level. Teufelsdröckh begins by comparing the notions of miraculous and lawlike. At first noting that science has progressed away from finding divine in miracles and searching it in the regularity of the cosmos and its laws, he continues by insisting that science has still only penetrated a small portion of the infinity of nature. Furthermore, he suggests, if nature is nothing but clothing for the divine - or to take another simile, a book written by God - why should it be just a cookbook full of recipes (i.e. laws of nature) and nothing more significant? Instead of regularities, Teufelsdröckh again wants us to return to appreciating the miraculous in our study of nature. Isn’t a miracle still a miracle, if it appears regularly? And isn't a miracle still a miracle, even if it is caused by internal workings of our mind (e.g. when we appear to see devils and witches)?

Teufelsdröckh goes even so far in his disdain of regularities that he, taking a cue from Kant, denies the validity of space and time, when it comes to metaphysics. Space and time may have to be taken for granted in our every-day life, but when it comes to the divine and even to the capacities of the mind, he insists, space and time are of no relevance: indeed, we can instantly think of things far from us or past or future. While living in this spatio-temporal world, he concludes, we are like ghosts appearing in a physical form, and after our seeming death, we return to the eternal from which we arrived.

Carlyle ends his summarisation of Teufelsdröckh with the consideration of two classes peculiarly engaged with clothes: dandies and tailors. Dandies, he defines, are people whose lives are dedicated to clothing and who thus, in a sense, have an inborn understanding of the philosophy of clothes: a poet making a work of art of themselves. In fact, dandies form, Teufelsdröch suggests, a religious cult with their own dogmas of what to wear and especially what not to wear: a new form of the old superstition of self-worship. As a contrast to dandies, Teufelsdröckh points out the so-called poor-slaves or drudges, whom he somewhat ironically describes as resembling monastic orders in having taken an oath of poverty. If the dandies worshipped themselves, he adds, the drudges worship earth, since all they think is where to get the next round of potatoes. In fact, these two so-called sects are nothing but the two classes of rich and poor, whom Teufelsdröckh earlier envisioned as tearing down the civilisation in their quarrel with one another.

The very last fragment Carlyle has selected to introduce from Teufelsdröckh’s work – before announcing that Teufelsdröckh has vanished from the face of the earth – is reminiscent of the earlier fragment about George Fox as a producer of his own leather outfits. Here Teufelsdröckh consequently praises tailors as sub-creators. Indeed, he suggests, aren’t all creative people in some sense tailors of their own kind?

János Bolyai: The science absolute of space: independent of the truth or falsity of Euclid's axiom XI (which can never be decided a priori) (1832)

 

(1802–1860)

A good example of a remarkable coincidence of two persons having almost the same idea roughly simultaneously is the discovery of non-Euclidean geometries, and more precisely, the so-called hyperbolic geometry. We have already seen how Russian mathematician Lobachevsky approached the idea of not assuming Euclid’s parallel axiom, and we are now about to see the Hungarian János Bolyai do it in his own manner.

One might say that the interest in the parallel axiom ran in the Bolyai family, since János’s father, Farkas Bolyai had for a long time tried to deduce the axiom. Indeed, he had warned his son that the parallel axiom was something younger Bolyai should keep away from, since one could waste a lifetime thinking about it. To older Bolyai’s surprise, younger Bolyai sent his father a short paper dealing with the issue, which older Bolyai published as an appendix to his own textbook on mathematics.

The topic of Bolyai junior’s article is primarily the absolute geometry, that is, a geometry where neither Euclid’s parallel axiom nor its denial is assumed. Thus, just like Lobachevsky, Bolyai is interested firstly in the similarities between the Euclidean and the hyperbolic geometry. Again like Lobachevsky, Bolyai defines as a parallel line to a given line X as that precise line, which is a sort of limit of all the lines drawn through the same point y and not cutting the given line X, in the sense that any line falling from that point y more toward the given line X will cut this given line.

Bolyai also shows, like Lobachevsky before him, that parallelism, defined in this manner, is a transitive relation. Bolyai goes even further and shows that with the parallel lines, the congruence of line segments is also a transitive relation. He then notes that given a line segment AM, one can consider a collection of all such points B that if a line segment BN is parallel to AM, it is also congruent to it. Bolyai doesn’t really give any other name to this collection, but F. In addition to F, Bolyai considers the intersection of F with any plane containing AM, which he calls L, while AM he calls the axis of L. He also notes that F can be described by revolving L around AM. Because parallelism and congruence of line segments is transitive, every line segment BN starting from a point B in L and parallel and congruent to AM is also an axis of L.

In Euclidean geometry, Bolyai notes, this L is simply a line perpendicular to AM - and F, similarly, a plane perpendicular to AM. In hyperbolic geometry, on the other hand, L is not a straight line, but a curved line, and similarly F is a curved surface. Indeed, they are what Lobachevsky called respectively oricycle and orisphere. Like Lobachevsky, Bolyai notes that oricycles in an orisphere work like straight lines in a Euclidean plane.

Now, Bolyai pictures an axis AM move through its oricycle L, always staying at same angle to L. He saw that the other points of the axis AM described further oricycles, that is, taken C from AM would describe an oricycle, for which AC would be the axis. Furthermore, taking corresponding parts of two such oricycles, the relation X of their lengths is always a constant, which depends not on the length of the parts, but only of the distance x of the points on AM. Bolyai also notes that these oricycles are always congruent, although part of one appears to be multiple of the corresponding part of the other.

Bolyai further notes that whether one supposes Euclidean or hyperbolic geometry to hold, spherical trigonometry – that is, study of triangles, as it were, on the surface of a sphere – always follows the same rules. Then again, he adds, with the ordinary geometry the case is quite reversed. In Euclidean geometry, given the length of at least one side of a triangle and at least two other elements of the same triangle (whether angles or sides) are known, the other elements can be solved. In hyperbolic geometry, on the other hand, one has to also refer to some length x, of which the corresponding relation X is known, to make similar calculations. Bolyai suggests using length i, defined by having as the corresponding relation e - the basis of natural logarithms. This length i would then work as a sort of natural unit of length in the hyperbolic geometry.

The intriguing question is then, which of the two, Euclidean or hyperbolic geometry, is the one describing the world we live in. Bolyai notes that we cannot really determine this without any empirical facts to guide us, since neither of the two geometries has any intrinsic flaw in it. This statement goes against the common idea of Bolyai’s contemporaries that Euclidean geometry is somehow inherently intrinsic. As if to just spite such thinkers, Bolyai ends his short article by showing how one can construct in hyperbolic geometry a square equal in area to a circle – something that is impossible in Euclidean geometry.

George Peacock: Treatise on Algebra (1830)

 

(1791–1858)

On a surface level, Peacock’s Treatise on Algebra seems a rather humdrum textbook on algebra, with no particularly original innovations to be found in it. Yet, what is most important in the treatise are not any purely mathematical results, but the more philosophical considerations on the nature of algebra and its relation to other parts of mathematics and especially arithmetics.

What Peacock is particularly interested in is the question whether algebra should be somehow based on arithmetics, which is based on concrete groups of objects, like seven apples. Peacock’s answer is a resounding no: arithmetics is far more constrained, allowing e.g. no negative numbers, since you cannot produce a group of, say, minus seven apples, although negative numbers are a common occurrence in algebra.

Not restricted by characteristics of arithmetical numbers, what is to determine what rules are to be assumed in algebra? Peacock makes the bold suggestion that these rules are just assumed: algebra concerns only symbols, and we could in principle choose any rules to govern our calculations in it.

Still, Peacock does not yet reach the modern notion of algebra, where we can have many different systems of calculation with different rules. Instead, he thinks that algebra should be especially the most general system of calculation, where all sorts of calculation are possible. Thus, he accepts not just calculations leading to negative numbers, but also roots of negative numbers, which we nowadays call imaginary numbers. Peacock goes even so far as to suggest that such seemingly nonsensical symbols like 0/0 could be used in a meaningful manner (although he mentions this possibility only in passing, he is referring to certain readings of the infinitesimal calculus).

Peacock still wants that algebra would be of practical use. Here, he suggests that the rules of algebra should at least correspond to the rules of arithmetics at least in those cases where the calculations and their results make arithmetical sense. Then again, he continues, results of algebra could also be used in other mathematical sciences, where some of them might be of relevance. Thus, negative numbers do make sense, for instance, when speaking of debt or of movement to an opposite direction. Even imaginary numbers can have concrete meaning, as describing movement perpendicular to a line.

Algebra becomes then, for Peacock, like a general tool for mathematical problem solving. When a concrete problem is given, it is turned into algebraic symbols: what quantities are known and what are unknown that are supposed to be determined in terms of the known quantities? After doing the algebraic manipulation of the symbols, one must still interpret the results. Often the context of the problem restricts this interpretation, and although algebra would lead to a number of possible results, only some of them might make sense in the context. Sometimes no result suggested by algebra would make sense, and then the problem itself will be impossible.

Évariste Galois: Dissertation on the conditions of solvability of equations by radicals (unpublished)

I’ve already discussed Galois as an example of a mathematical wunderkind. Like with many wunderkinds, his life was not long, as he was killed in a duel, when he was just twenty years old. Galois had not published much before his untimely death, and indeed, the work he is mostly remembered fpr nowadays, Mémoire sur les conditions de résolubilité des équations par radicaux, was rejected by the Academy of Sciences, due to being incomprehensible in its present form. Indeed, the evaluation of the Academy was not completely without grounds, since Galois often passes over required steps in his proofs, if he considers them to be sufficiently obvious. Still, it seems clear that at least part of the rejection was due to Galois’ method being so original and unseen.

The topic of Galois’ paper concerns the possibility of solving polynomial equations, that is, of equations of the form a1xn + a2xn-1 + …. an = 0. Methods for finding the possible solutions of the equation, using only the operations of addition, subtraction, multiplication, division, raising to power and taking a root of the coefficients, had been known for all cases where n is at most 4 (or as the mathematicians would say, where the degree of the equation was at most 4). In mathematical terms, such methods were known as solving the equation by radicals (radical being the symbol for taking the root). It had also been shown that already when the degree of the equation was 5, there are cases where the equation could not be solved by radicals. What Galois added was a criterion by which all equations solvable by radicals could be recognised.

Galois’ method of finding this criterion was preceded by the idea of Lagrange to study the solutions, or to put it in mathematical parlance, the roots of the equation, before they were actually known. Lagrange had especially considered what happens, when in combinations of these roots, using only addition, subtraction, multiplication and division, the places of these roots were changed - permuted, as they say in mathematics. Now, Lagrange had shown that the structure of the equations had something to do with these permutations, although he had not yet managed to find the complete tale.

Galois notes that if a polynomial equation with degree n has n different roots (the maximal number it could have), one could always form a combination of these roots, where every permutation of the roots would change the value of the combination. With a characteristic leap of thought, Galois just states that some combination of the form Aa + Bb + Cc + … would suffice (a, b, c… being roots of the polynomial and A, B, C … being appropriate integers), without proving this statement or even giving any method how to find such combination (the statement is true, but let’s just take it on faith and not go in full detail how to prove it). Furthermore, he points out that Aa + Bb + Cc + … can be assumed to be irreducible, that is, not expressible as a product of other polynomials.

Another point that Galois makes rather quickly is that all of the individual roots can then be expressed as roots of a polynomial in terms of V, which is the numerical result of the combination of the form above, where the root expressed is, say, the first in the combination. In other words, if V = Aa + Bb + Cc + …, then we can find another polynomial F, such that F(V, a) = 0. Even more, Galois points out, just by permuting the roots a, b, c … in a suitable fashion in the combination, one can find for each root (say, b) a numerical result of a suitable permutation of the combination (say, V’), which can then be used to express the root with the same polynomial F (e.g. F(V’,b) = 0). Note that it is quite arbitrary, what is the order of the roots in the first V: it is the permutation of them to make V’ out of V that is of great importance. Indeed, even the exact values V, V’, V’’ … corresponding to the roots a, b, c … are not as important, Galois notes, as the way in which the roots have to be permuted in order to construct the values V, V’, V’’ … . This group of permutations, or group of the equation, as Galois calls it, has the interesting characteristic that whenever some arbitrary function F(a, b, c …) has a value expressible in the by now familiar terms of addition, subtraction, multiplication and division of the roots, then this function will have the exact same value, if the places of the roots are changed with a permutation in this group.

Galois seems to have thought of his group of the equation as a sort of matrix formed of roots, where one line describes one permutation of the roots. Next, Galois makes the interesting suggestion that while the Vs determining the group of the equation are defined by integer coefficients A, B, C …, we could, as he says, adjoin a new quantity, beside integers, that could take the place of the coefficients (this quantity could be, for instance, an irrational root of some integer). With this addition or adjoinment in place, it might become possible to express the polynomial Aa + Bb + Cc + … as a product of further polynomials. If so happens, the group of the equation, expressed in terms of possible values of Aa + Bb + Cc + …, is, as Galois says, partitioned or decomposed into smaller groups. All of these smaller groups happen to be of the same size and also of the same form in the sense that one subgroup can be turned into another with a rulelike permutation.

Now, if just suitable quantities to be adjoined can be found, the procedure can in principle be continued further. In this case, we can move to polynomials of smaller and smaller degrees and eventually hit the rock bottom, when the corresponding group contains nothing but one row. If this can be done, the original equation will be solvable by radicals.

While the aim of the paper is to find the method for recognising polynomial equations solvable by radicals, Galois himself seems to become more and more interested in the supposed mere means for this goal, namely, the group of the equation and the permutations involved. Indeed, this is the road that the development of mathematics and especially algebra was to take: it was not anymore just a method for solving equations, but a more intricate study of such abstract structures like groups of permutations.

Amos Bronson Alcott (1830): Observations on the Principles and Methods of Infant Instruction

 

(1799-1888)

Amos Bronson Alcott is undoubtedly less known than his daughter, Louisa May Alcott, the author of Little Women. Even in the field of philosophy, he is overshadowed by the fame of his associates, Emerson and Thoreau, who, together with Alcott, were part of the so-called Transcendentalist movement.

What we are looking at in this article is Alcott’s work as a reformer of education. Alcott himself worked as a school teacher, but we are especially interested in his theoretical work on the topic of education, and particularly on infant instruction, where the infants in question are meant to be under the age of primary school, from two to seven years. Alcott’s main principle is the time-honoured idea, familiar from Plato onwards, that education means not just external pouring of information in a person's mind, but development of propensities already nascent within the mind.

Underlying Alcott’s ideal of education is also a notion of what it means to be a human. Humans are for Alcott active beings with a twofold nature, animal and intellectual. While the aim of education is especially the development of the intellectual side of the human being, the animal side should not be ignored. This is especially true of small children, where animal nature is still stronger than intellectual nature. Like the whole human being, their animal nature is also active. Thus, Alcott concludes, the natural physical energy of children should not be bound down, but allowed to develop through playing. By letting children use their physical energy, the teacher pacifies their animal nature and creates room also for the development of the intellect.

Children seek primarily enjoyment, Alcott says. He thinks this is a perfectly natural desire, but the task of education is just to direct this desire to lasting sources of enjoyment. Still, because small children prefer very immediate forms of enjoyment, their instruction must  remain at a very concrete level and leave abstract reasoning for the higher levels of education. Route memorising is to be replaced by amusing stories, pictures, music and poetry.

Alcott also endorses the idea that human beings feel a natural sympathy toward one another and are thus by nature communal animals. This implies, Alcott suggests, that the teacher should encourage the use of this natural sympathy and especially use it to build up the conscience of the children. The best way to do this, Alcott think, is that the teachers themselves become an object of children’s affection and thus serve as their role models, showing toward them the same kindness they want to awaken in them. If this succeeds, he concludes, the children become, in a proper Kantian fashion like law to themselves.

Ludwig Feuerbach: Thoughts concerning death and immortality (1830)

(1804 - 1872)

 If at all, Feuerbach is today remembered most often as a mere link from Hegel to Marx. This is at least so far true that the very first published writings of Feuerbach do bear the distinct mark of Hegelianism. Thus, in his dissertation De ratione, una, universali, infinita, Feuerbach contrasts in quite a Hegelian fashion individuality of sense life, where everything is distinct and separate from one another, with the universality of reason and thinking, by which any human being is inherently connected with all the other humans in a mutual act of recognition.

His next major work, Gedanken über Tod und Unsterblichkeit, continues in the same vein, but connects this distinction especially to the question of personal immortality. Feuerbach considers this question rather modern: ancient Greeks and Romans believed only in current life, lived in and for a community, while mediaevals merely hoped for immortality in some distant future, where everyone would be resurrected and good deeds rewarded and evil punished. It is only the protestantism and the rationalist philosophy, he insists, that suggested an individual subject to be of such importance that it deserved immortality in itself.

Yet, Feuerbach leaps to a conclusion, such an emphasis on individuals has made us forget that there is something much more important and perfect than mere individual life. This more perfect or infinite Feuerbach now calls God or spirit (again, in quite a Hegelian fashion). Just like reason in Feuerbach’s dissertation, God unites individuals, although now Feuerbach likens it more to love: in loving one another, humans lose their distinct individuality and merge into a wider community of persons. Indeed, God exists just through such a process of mortals losing their own individuality, in other words, by constant death of new human beings - God is the unity behind everything variable.

Such a pantheistic notion of God does not then leave any room for personal immortality, Feuerbach insists. Only in this particular life exists this particular human being, and when that life ends, there is nothing sensuous left of this person (thoughts about this person do remain, but these are not individual). If a person would continue to exist as an individual, it would have to exist somewhere, but where could this place be?

Feuerbach does consider the possibility that life after death would continue on other planets and stars, but ultimately refuses it. Life must have its spatial limits, in addition to temporal limits, that is, Feuerbach insists, there can be no life beyond Earth. His line of argument is very flimsy: life on other planets couldn’t be equally perfect to ours - nature shouldn’t just repeat things - but there couldn’t be life less perfect than the least perfect life on Earth - then it wouldn’t even be life - nor life more perfect than the most perfect life on Earth - what could be more perfect than thinking? Still, one cannot but laugh at Feuerbach’s idea that making stars into homes of reborn humans would just turn space into a comfortable hotel. Instead, he suggests, stars are not creations of utility, but are dreams of a still youthful nature.

Another point speaking against personal immortality, Feuerbach continues, is that individual life is always embodied. Spirit or God does live without a body - it is the process, where bodies are generated and destroyed - but an individual person cannot. Indeed, Feuerbach points out, when we think of an individual's immortality, we think of the soul as a kind of body, trapped in another body. Yet, he continues, soul is not a body nor is it even an individual thing, but an activity of living, just like fire is an activity in a burning body. And like fire stops, when a thing is burned up, soul vanishes when the body dies.

Feuerbach also relies on the Hegelian idea that a limit of something does not just end something, but also makes it what it is. In other words, mortality just is what makes us human beings what we are. This is nothing to be afraid of, he adds, since when this limit of ours or death is present, we are not. What death adds to our life is meaning, just like a melody adds meaning to fleeting sounds.

The analogue to melody in human life is the common memory of humankind, Feuerbach suggests. The first years of our life is something that we learn of only from the memories of other persons. Our own individual personality is thus something made by humanity outside us. The whole of our life is then a steady creation of more memories, which could be transmitted to other persons and thus universalised. When the process has finished, we still exist in spirit, as these idealised memories.