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 - Finite and infinite

A key pair of ontological concepts in traditional metaphysics has been that of infinity and finity, thus, it is no wonder Crusius investigates them also. Simply put, finite is that which has limits, while infinite is that which has no limits. These definitions leave it still unclear what having limits actually means. Crusius explains that limit means an end to that where thing is thought, that is, something where the essence of the thing cannot continue to a higher grade, cannot spread into larger space or cannot have a longer duration. This definition implies, Crusius says, that a limited or finite thing can be multiplied in grade, space or duration, in other words, something greater than it can be thought.

Infinite in some aspects, on the other hand, is such that greater of it cannot be thought. Crusius continues that there are then three types of infinity, as there are three aspects involved with every existing substance. Firstly, every substance has an essence, which is ultimately based on its fundamental forces: here, infinity means that a substance is capable of all possible actions. Secondly, in addition to essence, substances exist in space and time, both of which have their own types of infinity: immeasurability, where a substance occupies all possible spaces, and eternity, where the duration of a substance has no beginning or end.

Crusius also notes that none of the three types of infinity should be confused with what could be called infinity of progression, which is no true infinity, but a mere series of ever greater things, which still always remains finite. For instance, a thing generated at some point of time could continue existing without any end and still its duration would always have been just finite. Crusius suggests that such an infinity of progression is the only way we humans can think also the infinity of the past: we set out a past moment, then a still further past moment etc.

Interestingly, while Hoffmann rejected the so-called ontological - or as he called it, Cartesian - proof of God’s existence, Crusius appears to accept it. He starts with the notion of a substance with an infinite essence, that is, with capacity to do anything - such a substance has then an infinite grade of perfection. Now, Crusius continues, if a substance should be capable of everything, it should be capable of ensuring its own existence, or existence belongs to the perfection of the infinite essence. This means, he concludes, that a substance with infinite essence must necessarily exist everywhere and at every time.

Crusius argued earlier that necessary things can only be simple - otherwise they could be broken - and this must then apply also to an infinite substance. Its simplicity then implies that an infinite substance cannot be reduced. In fact, he points out, there can be no quantitative relation between the infinite substance and finite substances. In fact, nothing could be added to a finite substance to make it infinite, and finite and infinite substances differ by their essence.

Crusius makes the remark that one might think as an infinite force a determined capacity for doing a certain type of action in as great a magnitude as possible. Force of an infinite substance is not of this sort, he clarifies, but a general capacity to do anything whatsoever, even what any of these determined infinite forces could do. Indeed, an infinite substance should have only one force, which it can then apply in different manners. Of course, Crusius admits, the infinite substance cannot do anything impossible, but this is more of a clarification than any real limitation. The infinite substance does not then need any instruments for its actions, but if it so chooses, it can use them. In fact, since no effect of the actions of the infinite substance could be the highest possible, it can freely choose the magnitude of its effects.

Infinite substance should be able to do anything that just is possible. Crusius argues that creation of all finite simple substances is one of the things the infinite substance has done. True, he admits, it is inconceivable to us mere humans how an infinite substance has done this, but as contingent they must have been created by something, and since a finite mind cannot apprehend an infinite substance, it is understandable that we cannot fathom everything it could do. Crusius notes that there is also nothing contradictory in finite substances creating finite simple substances, although it is again inconceivable how they could have done it. Still, he assures the reader, we should not assume any finite substance to have such a power, because this assumption would undermine our ability to investigate natural causality, which is based on the premiss that finite substances can only bring about something by combining existing substances or by dividing existing combinations.

Crusius also considers the question, whether there could exist at the same time an infinite amount of things. His first point is that we certainly can always think of a number greater than any given number, thus, we shouldn’t be able to think any infinite number (of course, nowadays mathematicians do think of infinite numbers or cardinalities, but since they also form a never ending series of infinities, these would not actually be infinite in the sense Crusius means; still, this is a distinction that we can ignore when speaking of what Crusius had in mind).

Now, although we cannot think of an infinite number, this might not imply anything for the possibility of an infinite amount of real things. Here the crucial question is, Crusius suggests, whether this infinite amount is meant to be added up from actually different, perfect things. If it is, Crusius insists, we should be able to divide this amount into two groups. Since neither subgroup is the greatest, they are both of a finite amount, but then an infinity would be made up of two finities, which contradicts the idea that an infinity cannot be quantitatively compared with something finite.

Then again, Crusius notes, the previous argument works only if it is really distinct things that are added up and divided into groups. Thus, God might be able to think at once an infinite amount of possible things, since these possible things are not really distinct. We finite beings cannot comprehend how God can do it, but this does not restrict God’s capacities.

A far simpler question, Crusius thinks, is that of an infinite series of causes and effects, because Crusius smells a contradiction in that notion. In a series of causes and effects, he argues, all terms are either generated or not. If not, the series has a first cause and is therefore finite. Then again, if they all are generated, then the individual members have all not existed at some point and therefore the whole series has not existed at some point and has thus a beginning. Key part of this argument is clearly the move from all parts of a whole to the whole itself. Crusius notes that this move does not work in all cases - if parts of a whole weigh 1 kg, then the whole will definitely not weigh 1 kg. He unconvincingly tries to argue that usually and in this particular case this move is guaranteed by a principle of non-contradiction, because whole just is parts taken together.

Whatever the validity of the argument, Crusius believes he has shown that a series of causes and effects cannot be really infinite. He does admit it can have an infinity of progression, that is, it could have more members. These members could also be added to the beginning of the series, that is, we could think that the series began from an earlier point than it does, but this just means that it is completely arbitrary where such a series begins.

Because all series of causes and effects are thus finite, Crusius says, the essence of an infinite substance cannot consist of such a series of changes. Even more, he insists, the infinite substance cannot go through any series of changes, because it would undermine its eternal perfection. Crusius might be arguing here against the idea that God could be persuaded by a series of reasons to do something. In any case, he notes that an infinite substance should be immediately everything it can be.

The lack of changes in the infinite substance means according to Crusius, firstly, that all the actions of the infinite substance must be fundamental, free actions. Secondly, the infinite substance cannot be affected by a finite substance, at least not directly. Crusius does admit that finite substance could hinder actions of the infinite substance by not fulfilling certain conditions the infinite substance has placed for its own action. Furthermore, finite substance could resist finite effects generated by the infinite substance.

Although Crusius speaks against the idea of an infinite series of causes and effects, his attitude toward an infinite duration is quite the opposite. Indeed, he is committed to the idea that the first, uncaused cause has existed an infinite amount of time or eternally. One might argue that Crusius’ commitment should fall to the very same argument he himself used against the infinite series of causes and effects, creating then a dilemma reminiscent of Kant’s third antinomy, where we cannot accept either that there is an uncaused cause nor that there isn’t. Crusius’ solution is once again to differentiate between actual and merely possible. A series of causes and effects involves an actual succession of things, while an infinite duration consists only of possible succession of things, whereas nothing really changes during the existence of an eternal substance.

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

It has taken Crusius this long to finally get to the definitions of two concepts mentioned in the very title of his book: necessity and contingency. Like definitions of modalities tend to do, Crusius’ are ultimately circular: necessary is what cannot be otherwise, while contingent is what could be otherwise. Yet, he at once gives a more substantial explanation of the terms, linked to causal terms: necessary is such that no cause could make it otherwise, while contingent is such that requires a cause making it so, without which it would be otherwise or not at all.

Crusius also suggests a criterion for recognising what is necessary and contingent: if we cannot think contradictory of something, it must be necessary, while if we can, it must be contingent. Crusius underlines that this criterion is not foolproof and definitely does not reveal the essence of necessity and contingency. Indeed, he points out, even a fatalist could think that a world might be otherwise, even if his worldview would mean that the world is necessarily what it is.

Crusius notes that necessity and contingency can concern both the essence and the existence of a thing. In case of essence, he clarifies, the question is whether a thing cannot have different properties, because changing them would make the existence of the thing impossible - this means that the essence is necessary.

Necessity and contingency of the existence of a thing, then, refer to the respective impossibility and possibility of the non-being of the thing. Crusius notes that necessary existence of a thing can be either independent necessity, where the thing exists continuously and is necessary in all circumstances, or consecutive necessity, where the thing exists necessarily in certain circumstances, when it is generated. Respectively, contingent existence of a thing can be either dependent contingency, where thing doesn’t always exist, but must have an origin, and consecutive contingency, where thing is generated in such a manner that it might have been otherwise or even not at all.

Crusius’s concepts of independent and consecutive necessity resemble the concepts of absolute and hypothetical necessity in Wolffian tradition. Yet, Crusius sees a difference. He defines absolute necessity as something, which as such cannot be otherwise. The most obvious type of absolute necessity, Crusius notes, is such where contradictory opposite of something contradicts the very principle of non-contradiction - something which many Wolffians also recognised - but it is not the only type, since similar forms of necessity should arise also from the other two basic principles.

The hypothetical necessity Crusius defines as such where something cannot be otherwise in certain circumstances that are based on a series of conditions, ultimately caused by something that at the moment of its occurrence could have happened otherwise. This concept of hypothetical necessity is obviously a form of consecutive necessity, but with a distinct characteristic that it is based on some free action. On the other hand, if no such free action is to be found behind consecutive necessity, then it will be, Crusius insists, just absolute necessity. This distinction is especially geared against the notion that God would have had to create the best possible world, without any free choice in the matter, which would make the existence of this world absolutely necessary.

Crusius thinks that only such things exist with absolute necessity, where their non-existence would contradict their essence - it is not yet made explicit, but he is clearly implying that only God fits this requirement. All other things are contingent or they could be thought to not exist, so they must have been generated by something else, that is, God. With something of a sleight of a hand, Crusius notes that since these other things cannot then be necessary in the absolute sense of the world - they could fail to exist - but they still have consecutive necessity, being necessitated by the creative act of God, this act of creation must have been free.

Crusius still has some loose ends to tie. Firstly, he notes that complex substances are always generated from their constituents, so that all absolutely necessary substances must be perfectly simple. Secondly, he defines moral versions of all the modal notions he has introduced: this is clearly something that influenced Kant in defining ethical categories of modality. Thus, Crusius begins from the notion of moral existence - what is a goal of a free person, or in effect, what should be - goes on to define moral impossibility - that something shouldn’t be - moral possibility - of which cannot be said that it should or that it shouldn’t be - and finally moral necessity - that something should be done, because of a presupposed goal. These notions and their somewhat strange definitions, where moral impossibility is not contrasted with moral necessity, but with moral existence, parallels closely Kant’s later division of the ethical categories of modality.

Crusius finally discusses the notion of coercion (Zwang), where the necessity of some action is caused by something outside the thing acting. He is especially interested to show that lack of coercion by an external thing still does not mean something would not be necessary - even an uncoerced necessity would be real necessity.