Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Intellectualised appearances

Hegel noted disparagingly in his Wissenschaft der Logik how most of the earlier books on logic did nothing to justify why a certain ordering was followed, but merely started with concepts, continued to judgements and so on. Indeed, the traditional books on logic – and Wolff's logical works are no exception – followed closely the order of Aristotelian Organon: first, Categories, a book classifying the basic concepts, then, On interpretation, a book telling how to combine concepts into statements or judgements, Prior analytics, a book telling how one could use syllogisms to deduce judgements from previously known judgements, and finally, Posterior analytics, a book applying syllogistic to scientific methodology.

I already remarked in the previous text how peculiar the inclusion and even the centrality of methodology might be to a modern reader, who is used to logic as a study of certain formal languages and at most a canon of reasoning. Furthermore, the order followed would be strange to this reader, because the order of the modern truncated logic is in a sense reversed, when the propositional logic already deals with deductions and only the more complex predicate logic introduces the more fine-grained structure of the propositions.

But perhaps most alien to the modern notion of logic is the doctrine of concepts. In Aristotle's Organon, Categories has a clear ontological import, as it e.g. defines individual substances as the primary entities. Modern formal logic, on the contrary, has become more and more removed from any ontological presuppositions. Thus, when Russell stated in Principia mathematica that logic presupposes the existence of at least one entity, modern free logic remains valid even if there should be nothing at all. This is perhaps the ultimate in deontologisation of logic, which strives to be an ontologically neutral science.

Wolff's theory of concepts is not so much an ontological, but a psychological study, which might be even more of a blasphemy to the modern logic that began with the Fregean critique of psychologism. Yet, Wolff is not alone in this, but instead almost all logicians of the time defined concepts in psychological terms – or at least this is what Hegel accuses them of doing. I shall leave the discussion of Wolff's theory of concepts to the next time, but for now I wish to speak of two mental capacities that lie on the basis of concepts in Wolff's account, that is, Empfinden, which we might translate here as perceiving, and Gedancken or thinking.

The reason why I want to discuss these two terms is Kant's analysis of where his predecessors went wrong when they discussed human theoretical faculties. Kant roughly divided the previous philosophers into two camps. The followers of Locke emphasised the status of perceptions, and indeed, Kant suggests, tried to make thoughts into a sort of perceptions. The followers of Leibniz, on the other hand, tried to do just the opposite, that is, they tried to intellectualise all appearances. Both sides erred, says Kant, because thoughts and perceptions or concepts and intuitions are two very different processes.

Wolff is usually seen as a follower of Leibniz, so one would assume Kant would count him as one of the intellectualisers. And indeed, Wolff boldly states that perceptions are a species of thoughts. But here we have a good example of philosopher's speaking right past each other. Two botanists won't probably have any difficulty to understand one another, when they both speak of potatoes, but mixing concepts of two philosophers is bound to produce confusion.

We shall have to decide later on – in about decade or so – what Kant meant by concepts and intuitions, but now is the time to define Wolffian terminology. Those who know German have most likely been raising objections in their mind to me translating Empfinden as perceiving. But Wolff's definition of the term is so broad that perception appears to me a more suitable translation in this context than e.g. sensation. We perceive something, Wolff says, when we are conscious (bewusst) of it as being present to us. Although Wolff's actual examples then consist of mere sensations like pain or light, the definition itself could also apply to an awareness of e.g. a red ball or a sugar cube.

Now, the link to consciousness is the important element in Wolff's definition, because through consciousness Wolff also defines thoughts. Thoughts, for Wolff, are effects of the soul, by which we are conscious of ourselves. Wolff does not clearly state why perceptions then would be thoughts, but the connection seems clear and goes back to Descartes. When we are conscious of something, we at the same time must be conscious that we are conscious and thus we must be conscious of ourselves. In other words, consciousness presupposes self-consciousness. With these particular definitions Wolff must arrive at the conclusion that perceptions are also thoughts. Indeed, the connection of the two terms in their Wolffian meaning is so natural that it does not imply any intellectualisation of perceptions that Kant is worried about.

That's all for now. Next time we shall see how perfect Wolff's concepts are when compared with the concepts of his competitors.

Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth (1712)

I have already mentioned how Christian Wolff had discovered how to make profit in academia by writing text books. Wolff had also apparently invented how to make a succesful brand of his creations, a long time before the notion of brand had been formulated. And indeed, if one nowadays knows that most of the hard cash paid for a Harry Potter -book goes to the pockets of J. K. Rowling, the 18th century Germans knew that a book with a title beginning with the words ”Vernünftige Gedancken von...” would be the handiwork of the philosophical Jack-of-all-trades Christian Wolff, no matter what the issue investigated was. Unfortunately Wolff did not have the chance to sell the filming rights to his series.

I have translated ”Vernünftige Gedancken” as reasonable thoughts, but one could also speak of reasoned thoughts, because what supposedly makes Wolff's thoughts so reasonable or makes them accord with reason is that these thoughts are presented in the already familiar axiomatic form of reasoning from evident basic truths and definitions to further truths. Thus, it is just logical that Wolff would present this form and in general the methodology of philosophy in the first published work of the series, Vernünftige Gedancken von den Kräfften des menschlichen Verstandes und ihrem richtigen Gebrauche in Erkäntniss der Wahrheit.

From modern perspective it probably appears rather strange that Wolff would actually call a book on methodology logical. This perplexity is instigated by the Kantian tradition where logic is seen as a mere canon for rejecting the obviously impossible statements. But before this Kantian novelty, logic had been usually described as the organon of or a tool for a philosophical or scientific investigation, and indeed, one of Aristotle's logical works, the Posterior analytics, was similarly a study of the scientific method.

I shall investigate the Wolffian methodology in more detail in later texts. Now I am interested in the moment of philosophical self-reflection at the beginning of the book, where Wolff defines what philosophy is and then divides it into different disciplines. More precisely, Wolff does not speak of philosophy, but uses his own invention, Weltweisheit, that is, worldy and not divine wisdom.I shall reverse Wolff's own schema and start from the individual parts of philosophy and thwn progress towards the definition of the whole philosophy.

Philosophers have been very keen to classify different philosophical disciplines in a rigorous manner. Perhaps the most influential has been the Stoic division of philosophy into logic, physics and ethics: similarly Descartes compared logic and metaphysics with the roots of a tree, physics with its trunk and ethics as one of its fruits, while Hegelian classification of philosophy into logic, philosophy of nature and philosophy of spirit clearly resembles its Stoic predecessor.

Compared with this classical division, Wolff's classification appears rather haphazard. Wolff begins with the discipline he is studying in the current book, that is, logic or ”the art of reason”. Wolff clearly means it to be the first part of philosophy, because one must first learn the method before using it. The second part of philosophy considers then the supposed ultimate explanation of all other things or creatures, namely God, who is investigated in natural theology or study of God.

The third part, which considers the creatures or things other than God, Wolff leaves unnamed, but divides it further into pneumatology or study of spirits and physics or study of nature. Fourthly, Wolff notes that in addition to reason or understanding, human soul has also the faculty of will, which is an issue of yet another unnamed discipline that contains such subdisciplines as the natural right, ethics and politics. Finally, Wolff notes that we must also have a discipline investigating the common characteristics of all things, that is, ontology or fundamental science.

Wolff's sixfold division seems a failure from a systematising viewpoint, because it is not obvious why philosophy is to be divided into these particular subdisciplines. Especially unsatisfactory is that some of Wolff's suggested disciplines have no name, but consist of even further subdisciplines. The unclarity is even furthered by Wolff's remark that the natural theology, study of creatures and ontology together form yet another discipline called metaphysics. One might think that Wolff should have divided philosophy just into three parts – logic, metaphysics and the nameless discipline considering human will – or then raise ethics, politics and other subdisciplines of the philosophy of will into the status of independent disciplines.

Somewhat more intruiging is then Wolff's definition of what philosophy in general is: it is a science of all the possible things, considering how and why they are possible. The mention of possibilities is particularly interesting. Indeed, philosophers have always had a knack for discovering hitherto unimagined possibilities. This knack is extremely evident in the habit of using thought experiments as crucial links in arguments. This is a procedure common especially in modern analytic philosophy – witness for instance Gettier's problems of epistemology or the Twin Earth thought experiment.

Yet, this knack or art of finding possibilities is probably a quite recent candidate in the competition for the essence of philosophising. Wolff himself speaks of philosophy as a science instead of a mere art. By science Wolff means a capacity to deduce things from incontrovertible grounds, and he explicitly states that scientific knowledge is more certain than mere art where particular examples might be confused with general rules. In effect, Wolff is still committed to the old idea of philosophy as an axiomatic system.

What is then the distinguishing characteristic of a science of possibilities, and even more importantly, from what other sciences philosophy is to be differentiated from? There surely cannot be any separate science of impossibilities, or more precisely, it would be identical with the science of possibilities: as Aristotle already said, one and the same science deals with opposite concepts, because in order to know what is e.g. impossible and what is not, you must also know what is possible and what is not.

I suppose that the proper point of comparison for philosophy as a science of possibilities would be a science of actualities, that is, a system based on incontrovertible grounds that would tell what there is and why there are such things. Now, it seems evident, and Wolff would probably agree, that such a science of actualities is beyond human capacities. Humans can usually at most observe what is actual, that is, when it comes to actualities, they are capable only of history in the original sense of the word. From human perspective, then, philosophy just equals science.

One last question then is how a philosopher should go about proving that something is possible. The answer is already familiar from the study of what real definitions mean for Wolff. We know when something is possible, according to Wolff, when we know how this thing is generated and especially when we can ourselves generate such a thing. In effect, science at its best becomes almost like technology or handicraft.

Thus, if reason is defined as an insight into the relations between truths, the possibility of reason is shown when we demonstrate how a person can have such an insight through his or her own capacities. This is the task of the rest of Wolff's book, and we shall see how well he handles it in later blog texts.

Christian Wolff: Elements of all mathematical sciences - Is beauty in the eye of the beholder?

The issue of my final posting on Wolff's Anfangs-Gründe is not mathematical, but aesthetical. Quite early in a chapter dealing with architecture or art of construction, a little bit after defining what a building is, Wolff tells us how to understand beauty: ”beauty is the perfection or a necessary appearance of it, in so far as both the one and the other are perceived, and generates in us a liking”.

In simplistic terms, we can discern in the various philosophical theories of beauty two distinct classes: one could say, firstly, that beauty is an objective characteristic of things – i.e. that beautiful objects would be beautiful, no matter who perceived them – and secondly, that beauty is a subjective characteristic dependent ont the observer of the so-called beautiful thing. The objective interpretation of beauty had for a long time been the philosophical norm, but at least from Kant's Critique of judgement onwards the subjective interpretation has probably been more influential.

(Coincidentally, when I was writing these words, the major Finnish newspaper published a news that neuropsychologists had found out what part of the brain produced an experience of beauty. An overenthusiastic science reporter declared that this would finally solve the question whether beauty was truly in the eye of the beholder. Well, that's it then: close down all the philosophy departments and let the neuroscientists deal with the difficult questions.)

Now, Wolff's definition of beauty appears to fall mostly to the objective side of the equation. True, there are some elements in the definition requiring the interaction of the objects with a subject in order to make the notion of beauty meaningful. Thus, beauty can be also an apperance of perfection and at least requires that someone observes this perfection, that is, it is ridiculuous to call something unobservable beautiful.

Despite this connection to the observation, beauty is for Wolff primarily defined through perfection of things, and even the appearance of perfection or the liking in the observer is necessarily caused by the beautiful thing. Furthermore, Wolff defines perfection in teleological terms as usefulness to someone: beautiful thing is then something that we like, because it seems to fulfill our needs and also does so.



One could easily raise some serious doubts towards Wolff's definition. For instance, how would Wolff explain cases where people have different opinions on what is beautiful? Still, Wolff's definition has some merits especially in case of architecture. For instance, consider some of the houses designed by Gaudi. I have never been inside such a house, but for the sake of an argument, let's assume that they are not very comfortable, and indeed, fail to satisfy even the lowest living requirements. Then we might justifiably deny that the houses by Gaudi were truly beautiful. In fact, they would scarcely be houses at all. In other words, to be beautiful as a house, a building must first satisfy the purpose of accomodating people.



Does this mean then that we could not say that a Gaudi house is beautiful? Not at all, it would just not be beautiful as a house. Yet, it might perfectly satisfy some other purpose, such a being a mock-house – and indeed, what else would it be, but a mock house, that is, a building designed to fool the observer to think it is a house. Thus, it would be a truly beautiful mock house.



With these thoughts I end the consideration of Wolff's Anfangs-Gründe. Next time, I will be studying a philosophically more rewarding work, when I embarge on the first reasonable thoughts of Wolff.

Christian Wolff: Elements of all mathematical sciences - Infinitesimals = grains of sand?

I shall probably have to apologise to my potential readers who are not very fond of mathematics. The book I am currently going through – Christian Wolff's Anfangs-Gründe – is a textbook on mathematics and hence the investigation of it must almost inevitably deal with mathematical questions. I promise that I shall try to be less mathematically intimidating in my next posting, which also ends my investigation of this book.

This posting is then still about mathematics and more precisely about the so-called infinitesimal calculus. One might wonder why I have chosen to deal with such a non-philosophical question. Yet, the choice is not without precedents. For instance, George Berkeley criticised the inconsistencies implicit in the Newtonian understanding of the differential calculus. Even more importantly, Hegel dedicated a long passage in his Wissenschaft der Logik to the study and criticism of earlier opinions on infinitesimal calculus.

What then was so problematic in infinitesimal calculus? While regular algebra dealt with quantities in the usual sense, or as Wolff says, with finite quantities, infinitesimal calculus in its Leibnizian form suggested the idea of infinitesimal or infinitely small quantities. Although it may not be evident at first sight, the notion of infinitesimal quantity, as used by mathematicians of the time, was context-dependent. Thus, compared to lines, points were infinitesimals, but compared to planes, lines were infinitesimals: that is, a point could be regarded as an infinitely small line and line as an intinfinitely small plane.

Even the notion of inifnitesimals is somewhat suspicious, but even more suspicious is that infinitesimal calculus appeared to calculate with them as with ordinary quantities. For instance, if one takes two points in a curve described by some equation, the relation of the differences of the respective coordinates of the points describes the direction of the line connecting the two points. But then one assumed that the points were infinitesimally near one another and proceeded to make calculation as with ordinary quantities. To add an insult to an injury, the infinitesimals were finally just discarded – something that cannot surely be done with ordinary quantities. The result of this peculiar operation happened to be the direction of a tangent of the curve in that point, that is, a line touching, but not cutting the curve at that point, and one assumed that it was also the instantaneous direction of the curve at that point.

It was this dual role of infinitesimals that worried philosophers like Berkeley and Hegel. Nowadays mathematicians have constructed strict formal systems in which infinitesimals can be used, but such formal rigour was far from the undisciplined use of infinitesimals in 18th century. Furthermore, the original uses of the infinitesimal calculus were far removed from such abstract systems, and indeed, it is hard to see what physical sense one could make of such infinitesimals (e.g. points do not really have directions).

The first viable solution for the problems of infinitesimal calculus actually discarded the whole notion of infinitesimals. Instead, one spoke of limits. For instance, the forementioned ”direction of a point” could be taken as a limit for the direction of lines connecting the point to some point near it. Even Newton had used the notion of limit, but in the 19th century this notion was represented in a truly mathematical fashion. In effect, a limit for some operation in a given point is such that we can always choose an environment near the point where the results of the operation all fall within some arbitraty parameters. Thus, for instance, ”the direction of a point in a curve” is just a quantity such that we can find a part of the curve around this point where all the points connect to the reference point through straight line with direction that is arbitrarily close to this ”direction of a point”.

Well, how did Wolff then fare in dealing with the seeming incompatibility of the two ways to handle infinitesimals? We have seen that Wolff's dissertation handled differential calculus, but only on a very superficial level. Furthermore, the pragmatic tone of Anfangs-Gründe would suggest that Wolff would not really bother himself with difficult questions concerning the philosophy of mathematics. And indeed, Wolff merely presents the rather awkward analogy between infinitesimals and grains of sand: measuring a mountain has not failed, even if we have left one grain of sand unmeasured, because the quantity of the grain is so insignificant in comparison. Here Wolff confuses the insignificancy of small quantities with the complete immeasurability of infinitesimals. Thus, a grain of sand has some definite, albeit small quantity, while e.g. a point has no quantity. No wonder then that Hegel finds Wolff an example of the worst sort of muddle, when it comes to differential calculus.

The study of infinitesimal calculus ends Wolff's Anfangs-Gründe, but I still want to dedicate one blog text to this book. Still, one might rejoice that no mathematical questions are considered anymore. Next time we shall see what beauty means for Wolff.

Christian Wolff: Elements of all mathematical sciences - The axiomatic method in Wolff

We have seen how Wolff differentiates between nominal definitions, which merely explain what a word means, and real definitions which tell how to generate a thing corresponding to a definition. In the actual text of the Anfangs-Gründe, all the Erklärungen are mere nominal definitions, while no explicit real definition is mentioned. We shall see in a while that despite this the work does contain many real definitions.

From mere nominal definitions one cannot really deduce anything substantial and no philosopher has ever denied this. Well, Aristotle appears to say in his Posterior analytics that one could literally deduce all theorems from mere definitions, but Aristotelian definitions are not supposed to be nominal, but reveal the essence of something, so this might not actually be an exception. In any case, we do need some basic propositions (Grundsätze, as Wolff says in German) and from these basic propositions together with nominal definitions we can then derive other propositions, which Wolff calls Lehrsätze.

In addition to this division of basic and derived propositions, Wolff dstinguishes propositions merely describing a state of affairs from propositions stating a method or a need of a method for doing something. This division resembles the division of definitions to nominal definitions describing how to use a word and real definitions telling how to manufacture a thing of a certain kind. Similarly, basic propositions can be divided into axioms describing certain fundamental states of affairs from which proofs can begin and into postulates describing certain fundamental methods for doing things. Finally, derived propositions can be divided into theorems that need to be proved and into problems that need first to be solved and then these proposed solutions need to be proved to solve the problem. We might thus classify the constituents of the axiomatic method in a following form:



Unfortunately, this classification oversimplifies Wolff's thought. Firstly, while nominal definitions can be separated from axioms and theorems, real definitions are not a type of statements different from postulates and problems. Indeed, real definitions tell us how to find or manufacture certain things and thus are included either in postulates or in problems.

Secondly, Wolff notes that we can actually do without the problems altogether, because they could be turned into theorems. Theorems, Wolff says, are always of the form ”if p, then q”, whereas problems could always be translated into this form in the following manner: if the solution is followed, the problem is solved.

The interesting question undoubtedly is how Wolff suggests the axioms to be justified, that is, what is the ultimate basis of his demonstrations. Now, the Anfangs-Gründe investigates also handicrafts like architecture or artillery and Wolff admits that he must use information from experience as a justification of some theorems in these sciences. Yet, in case of pure mathematics, Wolff suggests that just by looking at the (nominal) definitions we can immediately see that certain basic propositions are valid. In Kantian terms, Wolff thnks that axioms are analytic.

Wolff's suggestion appears rather outdated for anyone who is acquainted with how axiomatics are dealt in modern mathematics: axioms are chosen not because we would be somehow immediately certain of their truth, but because we are interested in finding out what propositions follow from these axioms. Indeed, when Wolff suggests as an axiom that between two points we can draw only one straight line, it is difficult to see how this could be decided merely from Wolff's definition of straight line: straight line is a line where a part of the line is similar to the whole.

Actually one might criticise even Wolff's definition: how can he be sure that only the supposed straight lines are such that their parts are similar to the whole? Indeed, we might well imagine a sort of fractal line where the details of the whole would be repeated ad infinitum in smaller and smaller pieces of the whole. Yet, we should not be too harsh on Wolff, who surely could not have been aware of the exotic geometric shapes in modern mathematics. Furthermore, Wolff's definition is probably derived from tradition, because it appears to be an improved modification of Euclid's definition of straight line as a line which lies evenly with the points on itself.

Even if we accepted Wolff's definition, it would appear that his axiom presupposes characteristics of space that are not included in the definition of straight line. In Kantian terms we would say that Wolff's axiom is actually synthetic, and indeed, this is what Kant said of a related proposition that a straight line is the shortest line connecting two points. In other words, geometry requires according to Kant something more than mere conceptual analysis.

Although Kant's viewpoint on the axioms of geometry seems a clear improvement on Wolff's, it is interesting to note that on Hegel's opinion the very axiom on straight lines that Kant had considered is actually analytic. This does not have to mean that Hegel would have returned to Wolffian idea of axiomatics. To see this, we must first consider why Kant would have called the axiom synthetic. He probably had an independent notion of a straight line, perhaps something similar to Wolff's definition.Then Kant just needed to notice that this definition did not imply the axiom, and the syntheticity of the axiom became evident.

Hegel, on the other hand, apparently does not begin from an independent definition of a straight line. True, he begins his justification of Kant's axiom from the idea that a straight line is the simplest of all lines, but here this simplicity indicates not so much a characteristic, but a search for a characteristic. The most basic type of line is to be called a straight line and we need to choose what sort of line are we to take as the most basic. Kant's supposed axiom is then just the required definition of a straight line: the minimality of the length of a line is chosen as a criterion for its simplicity.

Understood as a definition, Kant's supposed axiom is undoubtedly analytic: being a shortest line between two points is just what being a straight line means. The analyticity of the axiom would still not undermine the syntheticity of geometry. We would still have to take as an axiom, firstly, that there is something corresponding to this definition – that there is no infinite series of shorter and shorter lines connecting two points – and secondly, that a line determined by two points is unique – that there are not more than one lines between two points equalling the lower limit of all lines connecting the two points (note that elliptical geometry does not satisfy the latter condition: for instance, there are infinite numbers of equally short routes from the North to South Pole on the surface of the Earth).

Interestingly, if we accept Kant's axiom as the definition of straight line, as Hegel suggests, Wolff's axiom becomes in a sense also analytic: indeed, there can be only one unique shortest line connecting two points. That is, if there just is any shortest line: as we have seen, this is not guaranteed by the definition.

We are still not completely through comparing Wolff's idea of mathematics with the ideas of later philosophers. Next time, we shall see whether Hegel's criticism of Wolff's notion of differential calculus was justified.

Christian Wolff: Elements of all mathematical sciences (1710)

The mathematical magnum opus of Christian Wolff is undoubtedly his Der Anfangs-Gründe aller mathemathischen Wissenschaften, which covers in four volumes areas such as arithmetics, astronomy, hydraulics and algebra. The book was apparently heavily used in teaching, because several editions were printed: the one I read hails from 1738. In addition to the book itself, Wolff also wrote a supplement with trigonometric tables, a shortened version and a Latin translation. Apparently he had already realised that the true source of wealth for academics are good text books.

The book is also famous as the target of Hegel's criticism in his Wissenschaft der Logik, where Hegel makes fun of how Wolff tries to formalise mere handicrafts like constructing a window. Indeed, it is rather peculiar to find in the section on architecture definitions of windows and doors (window, for instance, is an opening in the wall from which light can fall into the building). Even more disturbing is the section on artillery, where Wolff defines such things as bombs, hand grenades and mines (e.g. hand grenade is a bomb small enough to be thrown).

As is already apparent, the book would nowadays be read by future engineers and not by future mathematicians. Even in the case of arithmetics and geometry Wolff is more interested of the practical capacities implicit in these sciences: arithmetics, for instance, Wolff says to be of use in counting money. Indeed, the sciences dealt with are as mathematical as the method of Spinoza in Ethics is geometrical. Nowadays we would say that both Spinoza and Wolff applied or at least attempted to apply the so-called axiomatic method. Wolff's book even contains a description of mathematical or axiomatic method, which is undoubtedly the most interesting part of the book for a modern philosopher.

The basic idea of the axiomatic method is that certain propositions are to be developed from other propositions of a more foundational level. Some of these propositions merely tell us what things are or define them, while others or axioms spell out some further characteristics of things. I shall now concentrate on what Wolff has to say about definitions and leave the axioms to my next writing.

It has been customary to separate mere verbal or nominal definitions, which merely explain how certain words are to be used, from real definitions describing the essence of the defined thing. Although the distinction seems clear, it is somewhat difficult to determine when we have truly defined the essence of a thing or even whether thungs have any essences.

Wolff accepts the traditional notion of a nominal definition, but his idea of a real definition seems more creative. Real definition, for Wolff, tells how a thing is generated. Although in one example – that of a solar eclipse – Wolff clearly refers to general causes of a phenomenon, in most cases the generation refers to an actual production of a thing. Thus, Wolff suggests that a real definition of a circle describes how a circle is drawn.

Wolff has replaced the question about essences of things with a more pragmatic aim of extending our abilities to do things. This is clearly in line with Wolff's rather pragmatic attitude towards sciences. Indeed, many of the theorems in Anfangs-Gründe are just such real definitions: we are first told the nominal definition of a window, but only later on do we learn how to build windows. Although German idealists work on a more refined level than at mere handicrafts, one might think that Wolff's idea is a sort of precursor of the later idea of the primacy of practical reason.

According to Wolff, sciences are then more about how to do things than about what things are, that is, sciences are intrinsically connected or even identifiable with technologies. The most interesting consequences of this connection arise in the realm of pure mathematics. Notably, Wolff defines arithmetics not as a science of numerical relationships, but of methods for discovering numbers: 5 + 7 = 12 does not represent a relation between certain objects or sets of objects, but tells that by adding five to seven one gets a new number, that is, twelve.

The difference between the two interpretations of arithmetics is not so obvious when we remain at the level of finite set of numbers, because within a finite set we could at least in principle check whether the set contains a number having a desired property. But in the case of infinite sets the difference becomes more obvious. For instance, it is not at all obvious that we can find solutions to some difficult equations and we cannot even go through all natural numbers to find out if they do have.

This difference actually is what separates the so-called Platonist and constructivist interpretations of mathematics: the former focuses on the mathematical relations and supposes that these relations do exist, even if we cannot know anything about them, while the latter focuses on the actual methods of solving mathematical problems and regards it meaningless to discuss whether these problems would have solutions without any method of solving them. Wolff is unconsciously heading towards the constructivist option, which is interesting, because I think that some of the German idealists have affinities with a constructivist reading of mathematics.

So much for definitions,next time I shall be looking at what Wolff says about axioms, postulates, theorems and problems.

Christian Wolff: Albegraical dissertation on infinitesimal differential algorithms (1704)

As promised, I shall begin my blog with Christian Wolff, the major figure in the arena of German philosophy before Kant. I am sure that many of you have heard the phrase Leibniz-Wolffian philosophy, which would indicate that Wolff merely copied Leibniz and had nothing original to say. Furthermore, many critics, like Hegel, have stated also that Wolff is a pedantic and boring writer. Both of these statements are far too critical. Even before this project, I had read German-language versions of Wolff's logic and metaphysics and have nothing to complain about his style: his writings are at least as exciting and engaging as writings of an average analytic philosopher.

In addition, we should not think Wolff as a mere faithful disciple or even imitator of Leibniz. Wolff and Leiniz corresponded mostly about mathematcial questions and Wolff could not even have known Leibniz's philosophy in a deep manner: the majority of the works of Leibniz were published only after his death and after Wolff had begun his own philosophical career. Indeed, we might as well speak of a Wolff-Leibnizian philosophy: Wolff was the true beginner of the philosophical tradition in Germany, while Leibniz's works influenced this tradition only later.

I have chosen to take Wolff's dissertation Dissertatio algebraica de algorithmo infinitesimali differentiali as a suitable beginning for his career, although the work is not philosophically important. In fact, the dissertation is not about philosophy at all, but as the title says, studies mathematics and particularly differential calculus. Wolff began as a mathematician and what we would call a physicist and in a later book he even congratulates himself on having made important innovations in the field of aerometry or the study of measuring air pressure.

The subject matter of the dissertation is from a modern viewpoint rather elementary. Wolff derives some easy formulas of differential calculus, such as how to differentiate a product of two functions – nowadays a clever high school kid could do this. Although differential calculus was still somewhat of a novelty, I can find nothing in Wolff's dissertation that either Leibniz or Newton wouldn't have achieved already – Wolff even himself makes copious references to his predecessors in the thirty-odd pages. If this was a standard dissertation at the time, I must say that the criteria have been tightened from those days.

What I find most interesting in the whole dissertation are the short corollaries at the end of the work. After rather mathematical considerations Wolff merely enumerates certain further consequences such as ”matter is infinitely divisible” and ”God is to creatures as our mind is to entities of reason”, without offering any argument or explanation as to how these rather philosophical sentences related to the question of differential calculus. This might remind one of certain analytic philosophers who do a whole article or presentation full of clever logical tricks and then end up by telling that all of this has rather interesting philosophical consequences, but that this is so evident one hardly needs to spell the argument.

This is undoubtedly all we need to say about Wolff's dissertation, but we are still not yet through with his mathematical writings: the next few texts shall describe Wolff's mathematical masterpiece that purports to go through the elements of all mathematical sciences.

Why we are here

My interest in German idealism started with two books: Kant's Critique of pure reason and Hegel's Lectures on the history of philosophy. Of the two gentlemen I was finally drawn more to Hegel, and after reading Phenomenology of spirit, I began my decade-long project of translating Science of logic into Finnish. During this project I was for a short while engaged in translating Foundation of Wissenschaftslehre, from which my interest on Fichte. I could say that I have read a significant portion of Kant's and Hegel's works and a good deal of Fichte's. Schelling is the German idealist I have always found least attractive, although I have tried to introduce myself to his philosophy: so far this attempt has been met with a failure to understand him.

When both the project of translating Hegel and myPhD thesis were nearing their end, I found myself wondering what to do next. I had the idea that I might use my knowledge of German idealism to some use by starting a blog where I would go through the highlights of German idealism in detail. My plans have a bad habit of becoming more and more grandiose. Finally, I decided that in addition to the most famous German idealists I would try to investigate more carefully the whole German philosophy during the German idealism, that is, not just the followers, but also the critics of German idealism and generally the philosophy written in Germany at that time.

The natural ending point for my blog was then inevitably the death of the last great German idealist, Schelling, in 1854. The starting point was more difficult to determine. It is somewhat unclear whether Kant should be called a German idealist, and some Kantians resent the idea. Still, German idealism without Kant would be like a torso without the head, because the German idealism grew out of a certain reading of Kant's philosophy.

But starting from Kant was not enough. The critical idealism of Kant was preceded by his pre-critical writings, which were clearly influenced by the earlier Leibnizian-Wolffian philosophy. If I then wanted to really see how the German idealism had risen, I thought I should take a look at the pre-Kantian metaphysics. Here I put the final limit to the philosophy of Christian Wolff. Wolff's philosophy was undoubtedly influenced e.g. by Leibniz, but Leibniz is still more of a European than just German phenomenon, on par with Descartes and Spinoza. Besides, Leibniz did not write in German, unlike Wolff.

The task is set then. I shall read and comment on various philosophical writings that have appeared in Germany between the first publications of Wolff and the death of Schelling. The order of my reading shall be somewhat loose. Works of Hegel won't come before works of Wolff, but I shan't be too strict on doing all the writings of one year before the writings of the next year. I might even do some backtracking if I find that I have ignored some important work.

Of course, I shall not pay the same amount of attention to every writing, because some important works deserve a number of comments. Indeed, in addition to the obvious places, I would like to take a look at the not so well known passages of the classic works, for instance, the idea of different types of nothingness in Kant's first Critique.

Other works I will probably just skim quickly and make up some clever anecdote. But because even such a philosophically meagre work as Wolff's Anfangs-Gründe has required four separate blog texts, I shall probably have to spend a whole decade to get even to the precritical works of Kant. I assume it takes twice as much time to get to the 19th century, and by the time I finally hit the late works of Schelling, I might be on the verge of senility. We shall see whether that helps me to understand better what he is saying.

Although the main theme of this blog is philosophy, I shall not restrict my interest to strictly philosophical works – indeed, none of the four greats did either. Thus, I might look into some political and cultural essays and scientific writings. Then again, I shall try to veer away from poems, plays and novels, firstly, because it is far more difficult to see the true philosophical relevance of such literary works, and secondly, because I would just make myself look like an idiot if I tried to analyse them. Hence, you will more likely find in my blog a text on Goethe's theory of colours than on the first part of Faust, although both undoubtedly had an influence on Schelling and Hegel. Then again, in some cases an important philosophical figure has published in nis lifetime nothing else, but poems and plays and I feel compelled to take a look at them. For instance, if I would avoid reading the play Hyperion, I would have almost nothing to say about Hölderlin, the friend of both Schelling and Hegel and an important figure in their development.

My choice of what I shall read is mostly determined by accessibility. Thus, I shall usually limit myself to material actually published at the time the German idealists were living, because most of it is easily available even in a digital format. Still, at least in case of the four great names, Kant, Fichte, Schelling and Hegel, I will read at least some of the unpublished manuscripts and lecture notes of their students, because often the development of their thoughts requires this. For instance, Fichte had decided after a scandal in Jena that he would never again publish anything on his views and therefore we must reconstruct later changes to his philosophy from unpublished manuscripts, while many parts of Hegelian system were fully developed only in his lectures. And if some reader happens to have access to a rare manuscript of an obscure German philosopher that I should read for some reason, please send me a copy (do not send the original or it will probably face some tragic fate or at least be stained by jam).

Whether I read different editions of the same work will depend on the number and imporantnce of changes from one edition to another. Thus, because in the second edition of his Grundlage Fichte had added only some footnotes and changed few phrases, I won't be reading the different editions at different times. Then again, the changes to both Kant's first Critique and the first book of Hegel's Logic are so drastic that a differentiation of the two editions is clearly in order. In the cases where I shall read only one edition of a work, I shall usually try to choose the latest edition published at the lifetime of the philosopher or just after his death.

Finally, some words on my method are in place. Often one sees essay collections on Kant's philosophy where all aspects of his philosophy appear to be contested and no consensus on the meaning of Kant's works is reached, yet, one detail is accepted by all writers: Hegel got it all wrong. Similarly, there are number of Hegel-scholars who disagree on what Hegel intended to say, but agree that he definitely knew it better than Kant. Such presuppositions obviously colour how one then interprets the supposedly worse philosopher. I shall try to avoid all such pre-evaluations and instead aim at finding some reasons why a philosopher would want to say what he has said, even if what he says sounds ridiculous. Furthermore, I shall refrain on making any decision as to who got it right. Indeed, often different philosophers were not even pursuing the same agenda, so the question is somewhat moot.

With these words the necessary prologue ends: onwards to Christian Wolff.