Reasoned thoughts on the useful study and application of mathematical sciences (1747)

Surprisingly late in Wolff’s career appeared yet another part of his Vernünftige Gedancken -series. Yet, Vernünftige Gedancken von der nüzlichen Erlernung und Anwendung der mathematischen Wissenschaften was not a book specifically planned for the series, but actually a case of Wolff’s earlier book, originally written in Latin, translated to German by Balthasar Adolph von Steinwehr. I have not managed to ascertain what the original Latin work was called, so I am satisfied with reading this translation.

The book itself is not an independent treatise, but more like a study guide, meant to accompany Wolff’s earlier work on elements of mathematics. The most philosophically relevant part of the book is the first chapter, where Wolff outlines three different grades of knowledge. The first and lowest grade consists of understanding what is held to be true by others: Wolff calls this in some places also historical knowledge. This sort of knowledge requires first and foremost, in the case of mathematical sciences, studying definitions, theorems and solutions to problems, but does not hinge on proofs for the theorems or solutions.

Wolff points out many pedagogically important points for gaining historical knowledge of mathematics. Some of these points pertain to the order of study, for instance, that the definitions should be learned before theorems using those definitions, and within definitions, those required for understanding other definitions should be learned first. Wolff also notes that human understanding requires sensuous aid and thus points out the importance of examples. He also emphasises that the idea of examples is not to teach that e.g. this particular figure is a square, but to make the student learn how to recognise squares.

In addition to examples, Wolff underlines that especially in the case of arithmetics, a well planned presentation of the mathematical symbols is important in making the student follow what they are taught. Indeed, he insists, the very symbols themselves make us quickly understand what concepts are being spoken of (e.g. 3 + 3 + 3 + 3 = 12 tells a seasoned reader instantly that this is a question of combining a certain number of threes and that the = indicates the result of this combination).

Furthermore, Wolff adds, it helps us to comprehend intricate theorems, if we investigate what they mean in case of concrete examples. In the case of solutions, this means especially making calculations with specific numbers or drawing real figures. Such a repetitive practice of solutions ascertains that we have a capacity to use them in real life.

The second grade of knowledge, Wolff defines, comes about in being convinced of something. In mathematics, this requires going through proofs or demonstrations, and because demonstrations cannot be followed without understanding what is proven, the second grade of knowledge presupposes the first one. Demonstrations, Wolff says, consist of chains of reasoning or syllogisms, good grasp of which presupposes capacity in making judgements and concepts, but also in lower faculties of sense, imagination and memory. Thus, he suggests as a mediating step for students not yet able to follow demonstrations using mechanical proofs, in which e.g. a geometric figure is drawn and the theorem is ascertained through measuring devices and other instruments. Wolff immediately adds that such mechanical proofs tell us only that a theorem works in this particular case, but does not give us universal assurance.

The proper demonstrations, Wolff notes, can also be aided by sensuous means. In geometry this can be done by an image showing what kind of points, lines and figures are being discussed about. In case of arithmetic, a similar effect can be reached by using actual numbers, instead of letters, as long as the numbers are selected in such a manner that they themselves do not have properties that might simplify the proofs too much. Furthermore, the proofs themselves can be set out in such a manner that the student grasps easily what deductive moves are being made and how what is assumed is used in the proof. Finally, Wolff points out that in case of problems, it is helpful to turn their solutions into theorems, when following the proofs of these solutions.

The third and highest grade of knowledge, according to Wolff, is one in which we can use our knowledge to discover new, still unknown matters. He suggests that the way proofs are set up in mathematics is also helpful for learning how new truths are discovered – we just assume that the theorem is not known beforehand or turn these theorems into problems. Since all proofs are based on definitions and previously known propositions, the more one knows these, the more truths one is able to find out. Furthermore, Wolff adds, the earlier mentioned mechanical proofs can help us to discern unknown truths in individual cases, although we then have to learn to prove them universally.

The reason why Wolff goes through these three grades of knowledge is that often the student of mathematics is not learning a mathematical discipline just for the sake of the information, but also for generally improving their own understanding. Indeed, Wolff suggests that mathematics is especially suited for this task and that all students should therefore start by learning mathematics. If this is the motivation of the student, the first grade of knowledge will not be enough, since it at most trains our attention and faculty of conceiving, and indeed, a student just learning mathematical truths will forget them eventually, if they do not use them daily, like engineers do.

The true worth of mathematical studies, then, lies in second and third grades of knowledge or skills of understanding and making one’s own demonstrations. Wolff insists that these skills should not be then left unused, but applied also in other disciplines. He is eager to point out that his own philosophical works provide ample opportunities for this, since they are presented in the form of demonstrations. Indeed, Wolff emphasises, the philosophical method is precisely the same as the mathematical method.

In the rest of the book, Wolff goes painstakingly through the various parts of mathematics and suggests what parts of his mathematical work students with different ambitions and purposes should especially concentrate on. He points out that all students should have at least some grasp on arithmetics and geometry, since the rest of mathematics is essentially based on them. Furthermore, he instructs a student especially interested in discovering new truths to learn algebra, since it is a convenient tool for finding solutions to problems. In case of more applied fields of mathematics, he especially emphasises the importance of astronomy, since many important practical topics hinge on being able to calculate the apparent movements of stars.

Christian August Crusius: Draft of necessary truths of reason, in so far as they are set opposite to contingent ones - Measuring quantities

A common topic in ontologies of Crusius’ time, not that usual in modern ontologies, is quantities - back then, general philosophers were keen to explain what mathematics is all about, while nowadays this question is more and more left for special branch called philosophy of mathematics. Crusius follows the tradition and starts by defining quantity as such a property of a thing, by which something is posited more than once.

Crusius notes that at least complex concrete things naturally have a quantity - they consist of many things. Furthermore, even simple concrete things have quantifiable features - they have forces, and even though they are indivisible, they still are spatial and thus have some magnitude. Then again, some abstractions are not quantifiable, Crusius says: there are no levels of existence, but all existing things exist as much as others. Crusius also notes in passing the possibility of infinite quantities, but at once declares that we finite beings cannot really know anything about them.

Quantities come in different types, Crusius continues, for instance, quantity of a force differs from a quantity of an extension. The difference between these types becomes important, when we start to measure the quantities. Measuring, Crusius says, involves determining a relation of a quantity to some known quantity. As such, this kind of comparison is possible only between quantities of the same type (there’s no sense in measuring weight with a ruler). Still, Crusius admits, quantities of different type can be compared indirectly. Firstly, we can compare them through relations of quantities - for instance, we can say that punishments should be proportional to the crimes punished. Secondly, the comparison can be done through causal links, for example, the resistance of a body can be compared with the striving of a soul, because one has the effect of hindering the other.

To determine a quantity perfectly, Crusius says, we must represent its parts distinctly. This requires expressing the quantity as a number of distinctly thought units. These units might be naturally distinct - for instance, when we count things distinguished by natural limits, like cows - or arbitrarily chosen, for example, when we compare length of a thing to a measuring stick. Since a given quantity might not be expressible as a number of arbitrarily chosen units, Crusius also introduces fractions (no mention of irrational numbers, though).

An extreme case of natural units, for Crusius, is naturally provided by simple substances. Crusius admits that measuring complex substances by their simple parts is impossible, since we do not perceive these ultimate constituents. Still, he continues, understanding the nature of these simple parts can help us in picking suitable units for measurement: for instance, when we note that movement should be ideally measured by checking how many simple substances move through smallest measures of space, we can surmise that movement could be measured by checking how many things move through a certain space.

Crusius spends the majority of the rest of the chapter discussing a hotly debated topic of the time, namely, the so-called question of living forces. The point of the debate, at least as conceived by Crusius, is how to measure the quantity of an action, such as movement. Crusius’ take is that while abstractly taken this quantity can be expressed as a multiple of the strength of the action (in case of movement, mass of the moving object) and its velocity, we must also account for the resistance encountered by the action and thus use the square of velocity to determine the action.

Christian August Crusius: Draft of necessary truths of reason, in so far as they are set opposite to contingent ones - One and the same

From the very start of metaphysics, in the book with that name by Aristotle, the concept of one was regarded as an important topic. Thus, it is no wonder that Crusius would consider it. Indeed, he thinks that we have many different notions of one, distinguished by what concepts they are opposed to. First of these is the concept of one thing in contrast to several things. This concept, he insists, cannot really be defined, but can only be exemplified. In essence, a thing, no matter what it is, is always one or a unit, because we can think what it would be like, if it were multiplied into several similar things.

Another concept of one, Crusius continues is that of something unified in contrast to what is disunified. By unification Crusius means a relation where things are so intrinsically related under certain conditions that when one is assumed to exist, the other must be assumed to exist also. Unification can come in many forms, the primary ones of which are unification merely in our thoughts and unification in real existence. Crusius notes that we cannot really know all the subdivisions of unification and mentions only a few examples, such as metaphysical unification of one thing subsisting in another (e.g. property in a substance), existential unification of two perfect things connected so as to become inseparable, such as a hand and a torso, and moral unification where two persons are united by having common goals. In any case, Crusius emphasises, all cases of real existential unification are ultimately based on causal interactions. Thus, he insists that Leibnizian pre-established harmony would be no real unification of body and soul.

A third notion of one is connected with the notion of identity. Crusius defines the concept of identity as the opposite of difference, where two things are different if in one is something that is not in the other - Crusius notes in passing that this notion of “not” or denial is again something simple, which we cannot really define. Identity as the denial of difference can then be just similarity, where things share something, but also identity in a strict sense, where one thing - here is the connection to one - is represented through two concepts, of which one is found to contain nothing that wouldn’t be contained in the other.

Crusius considers the question, when we can know that the objects of two concepts are identical. The criterion he suggests is that one should be able to replace what is thought in one concept with what is thought in the other without any consequence. He also emphasises that mere same essence is no true criterion of identity, since we could have substances that are just numerically different, that is, that would agree in their absolute properties, but would be e.g. in different spaces at the same time.

Although Crusius' definition of identity might seem rather rigid, he does admit that identity can fluctuate according to the viewpoint chosen. For instance, when we are considering whether things at different points of time are the same thing, we might get different results depending on what we focus on: corpse is in a sense different from a living body – they have different essence - but in another sense they can be identical, because they share the same matter. Then again, if an essence of a thing consists of a certain relations of parts, the thing can remain identical, despite its parts being replaced by different, but similar parts.

Crusius chooses at this point to give a list of simplest concepts. This list is a development of a similar one from Hoffmann, and we could consider it to be a precursor of Kant’s list of categories. Crusius' list contains the following concepts:

• Subsistence, that is, the relation between a property and its subject
• Relation of one thing being spatially within or outside of another thing
• Succession
• Causality
• Relation of one thing being figuratively outside of another thing, in the sense of not being its part, property or determination
• Oneness in opposition to plurality
• Relation of things being unified
• Thing’s being somewhere in space.

Crusius also clarifies further the relation of these simple concepts to the seemingly simple concepts of sensation, like colours. Crusius’ idea is that while the above mentioned simple concepts are the ultimate result of analysing more complex concepts for any understanding, it is we humans who are incapable of analysing sensations just because they are caused by unknown activities affecting us in a confused manner, so that we cannot distinguish these causes from one another. Crusius also distinguishes the simple concepts from indeterminate, symbolic concepts, which we cannot really think, but which we can only represent by saying what it is not and what its relations to other things are.

Quantities and qualities

A continuing element in Wolff's ontological studies is his habit of bringing in mathematical examples to substantiate the correctness of his analysis. Indeed, Wolff often ends an investigation of some ontological concept by noting that his conclusions concur with the way how the concept has been used in mathematics. For instance, the analysis of similarity works, because it can be applied to similarity of geometric figures.

All this happens not just for the sake of Wolff's love of mathematics, but it is a part of a larger plan, meant to show that the method of mathematics is useful even in philosophical questions. This argument might have been Wolff's answer to criticism of Rüdigerand Hoffman that philosophy as a study of causal relations exceeds the capacities of mathematics as a study of quantities. We shall see later how Wolff conceived the argument go through, when we look at Wolff's discussion of forces.

For now, it is enough to note how Wolff introduces the very notion of quantity. We have to begin with the idea of unity – idea that things with certain features form an inseparable whole. There is no criterion to say when a thing or entity is such an unity, Wolff says, because all things just are unities, or being equals oneness. Here Wolff is following a tradition beginning from Aristotelian Metaphysics and inscribed in the medieval notion of unity as a transcendental – a property of all things.

What is important in this unity of a things is that we can then collect several of such unities or form a multiplicity. In such multiplicities, we can then abstract from the differences of the entities and concentrate on their common features – we can pick out cows on a field and forget the differences in their colouring. Then this multiplicity forms also a unity or is a whole, of which the original unities were parts. Thus, we can get examples of all the different integers. With integers out of the way, Wolff can then define fractions, and in general, all rational numbers through the notion of ratios of integers and then irrational numbers and generally all numbers geometrically, through the notion of ratios of straight lines – every number has to unit a ratio that a straight line has to another straight line.

Numbers have then, for Wolff, a special connection to quantities, which Wolff defines in a rather peculiar manner as that by which one can discern similar things. Wolff is here thinking about the mathematical notion of similarity, according to which e.g. two figures can be similar, even if their sizes are different. Now, noting what shape a figure has requires only a look on this figure itself. Then again, determining what size it is requires relating the figure to something else, for instance, to say that it is twice the size of that figure. Quantities are then in some sense relational features,because by choosing some quantity of the same type as the unit, we can give a precise numeric expression to that quantity. Quantities can thus be also defined as indeterminate numbers or numbers as determinate quantities.

Wolff also uses the idea of quantity to define notions like equality and inequality (respectively, sameness and difference of quantities), greater and less, addition and multiplication. Furthermore, he uses the opportunity to argue for certain basic truths of mathematics, such as the transitivity of equality (that is, the fact that if A equals be B and B equals C, then A equals C). But what is important for now is the definition of the apparent limit of the mathematical cognition, that is, qualities.

I suggested that Wolff takes quantities as relational, but this is only partially true. Certainly the precise numerical expression of quantity is determined by a relation to some given unity. Then again, Wolff is quite sure that a thing has intrinsically the quantity it does have, and only this determination of the quantity requires relating. Then again, we can define another type of intrinsic features, which do not require such relating, but which can be recognised immediately. It is this second type of intrinsic features that defines the class of qualities. At least essential features and attributes of things are qualities, while modes are either qualities or quantities.

At first sight qualities cannot then be expressed numerically, but as we shall see, Wolff attempts to prove otherwise. We shall not consider this topic for a few posts, and indeed, next time I shall look at what Wolff has to say about truth and perfection.

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.