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.
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