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 - Simple and complex

Concepts of simple and complex substances were of great interest to Wolffians, being one of the primary divisions of substances, and Crusius seems eager to show where Wolffians wen’t wrong with them. He firstly notes that just like the concepts of part and whole, on which the two former concepts are based on, can actually mean very different things. Starting with the parts, these can mean, Crusius says, any group of things we can represent as forming also a one thing, which then is the respective whole. Furthermore, these parts can be actual or such that they can be separated elsewhere than in our thoughts, but they can also be mere thought parts, which can be distinguished in our thinking, but not really separated.

Simple is then for Crusius something that has no parts - in some sense, while complex is something that has parts - again, in some sense. Since the notion of parts was already twofold, this same duality continues with the notions of simple and complex: something may be simple or complex just based on mere thoughts, but also based on something outside our thought.

Even in case of actual simplicity, Crusius notes, there are various levels of simplicity. The epitome of simplicity, he thinks, is God, who is not just a simple substance - that is, something, which cannot be separated into further substances - but also has a simple essence in the sense that no property could be removed from his essence. This is not always the case, Crusius says, because substance can be simple, like a human soul, without having a simple essence. Even a complex substance, like air, Crusius notes, is simpler than, say, a human body, because the former has only integral parts - parts that all have the same essence - but no physical parts - parts that have a different essence from one another.

Crusius also notes that it is a different thing, if something is simple as such or has nothing separable in it, than if something is simple on the condition that the current world exists, Crusius notes that we cannot really distinguish between the two cases and neither can any finite being, but God might be able to do it.

Every force is in some subject, Crusius insists, because no subjectless forces could be thought of. On this basis Crusius argues that in case of complex substances, their force must be determined by forces of their parts. Crusius then concludes that if a complex substance wouldn’t ultimately consist of simple substances, the constituent forces would have no immediate subject where to subsist, which he thinks is absurd. Despite the seeming complexity of the argument, it appears to just assume what it sets out to prove: that the existence of a complex thing must be based on the existence of simple things.

Crusius is especially keen to distance his notion of a simple substance from a mathematical understanding of simplicity. Mathematics, he says, considers only abstract magnitudes, not other determinations of things. In other words, he rephrases, mathematics is only about the concept of space and its possible divisions. Thus, it was natural for mathematicians to assume the existence of points, which should have even no parts that could be thought of as being outside one another. Yet, Crusius states, no true simple substance is simple in the mathematical sense, but is spatial - they just cannot be physically divided further.

Crusius goes thus straight against the Wolffian notion of elements, which are more like non-spatial forces. If we would accept such non-spatial substances, how could we account for spatial matter being generated from them, especially as any concrete matter would require an infinite amount of them? Furthermore, he continues, we couldn’t even say how such pointlike substances could touch one another, as there are always further points between any two points.

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.

Andreas Rüdiger: True and false sense - Sensational mathematics

I have often wondered where Kant got the idea of dividing judgments into analytic and synthetic, analytic referring to judgments where the content of the predicate was included in the content of the subject and synthetic referring then obviously to judgements where this inclusion did not hold. It's not any difficulty in the definitions I am speaking of, but of the nomenclature that would have in Kant's days reminded the reader of two different methods of reasoning, analysis and synthesis.

Originally analysis and synthesis were used by Greek geometers as referring to processes that mirrored one another. In analysis, one assumed that the required conclusion – proposition to be proved or a figure to be constructed – was already known or in existence. One then had to go through the conditions of this conclusion in order to find self-evident principles on which the conclusion could be based.

If analysis moved from conclusions to premisses, the synthesis moved the other way. One began from some principles already assumed or demonstrated to be true and from methods that one already knew how to use, and from these principles and methods set out to prove some new theorem or to draw a new sort of figure.

A certain step in the evolution of the mathematical methods into Kantian judgement types is symbolised by Rüdiger's notion of analysis and synthesis. Just like in the tradition, Rüdiger uses analysis for a method moving from consequences to principles behind them. Yet, he also calls such method judicial and separates it from synthetic method, which he describes as invention. That is, analysis does not produce any new information, just like in Kant's analytic judgement predicate does not reveal anything that wouldn't already be in the subject. Instead, analysis merely determines whether a given proposition is clearly true or at least probable.

Rüdiger's account of synthesis or invention of new and informative truths includes even more aberrations from the traditional account. For Rüdiger, synthesis might involve also mere probable conclusions that are based on the correspondence of various sensations – for instance, by seeing that a certain effect follows always from certain conditions, we may conclude that a new occurence of similar conditions would probably lead to similar effects. Here we see Rüdiger's empiricist leanings, but he does not restrict synthesis to mere empirical generalisations – in addition he also accepts necessary demonstrations.

Rüdiger divides demonstrations into three classes, according to three components required for thinking. One type is based on the verbal form of thinking and grammar: for instance, we deduce from the statement that Jane hit Mary the related statement that Mary was hit by Jane. The second type contains various forms of reasoning, such as traditional syllogisms, but the common element Rüdiger suggests is that all of them are based on the relations of ideas – we might name these forms logical.

By far the most interesting is the third type of reasoning, the mathematical. Leibniz and Wolff had thought that mathematics was based on inevitable axioms and even empiricists like Hume grouped mathematics with logical reasoning. Rüdiger, on the other hand, clearly separates logic and mathematics. Logical reasoning is based on the relations of ideas, while mathematics is based on the sensuous element of thinking.

Rüdiger's position shares some interesting similarities with Kant's ideas on mathematics. Both Kant and Rüdiger are convinced that mathematics are not mere logic, but synthetic or inventive. True, Kant speaks of mathematics as based on intuitions, while Rüdiger speaks of sensations, but this might not be as great a difference as it first seems. Rüdigerian concept of sensation is clearly more extensive than Kant's and would probably include also what Kant called pure intuitions. Indeed, Rüdiger also separates mathematical reasoning from mere empirical generalisations – mathematical truths are not mere probabilities.

The reason behind Rüdiger's desire to separate mathematics from logic is also of interest. Once again Spinoza is the devil that one wants to excommunicate. Spinoza's Ethics is supposedly philosophy in a mathematical form, but Rüdiger notes that this is intrinsically impossible. Mathematics can rely on certain sensations, when it constructs its definitions and divisions – it can tell that triangle is a meaningful concept, because it can draw triangles. Philosophy, on other hand, does not have a similar possibility for infallibly finding sensations for its concepts – a very Kantian thought.

So much for Rüdiger this time. Next time we are back with Wolffian philosophy.