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.

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.