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.

Joachim Darjes: Elements of metaphysics 1 - Connecting substances

While in previous post I discussed Darjesian notions of entity and substance, when regarded in abstraction from other entities and substances, now we shall see what happens when entities and substances are connected to one another. Darjes notes that these connections fall into three general classes, depending on whether the connection exists between mere entities, between both mere entities and substances or between substances. In case of the first kind of connection the things connected are regarded as just being impenetrable to one another. Example of a such a connection would be placing many entities into a same space so as to form a figure.

If in addition to or in place of mere entities substances are added to the connection, Darjes notes, we get forces to the equation. Such connections involve either the existence of substances or then their states – for instance, existence of certain substances might be connected with a state of one substance. In other words, such substantial connections concern various interactions between substances, for example, when one substance acts upon a passive substance or when one substance removes obstacles stopping some substance from acting.

All connections involving mere entities are extrinsic in the sense that it doesn't affect entities if we e.g. arrange them to form a figure. Thus, these connections are completely contingent. One might think that the case might be different with substantial connections, but Darjes notes that this is not so – substances can exist independently of one another, so there is no necessity that e.g. a substance affects another substance. Because no connections between entities or substances is necessary, Darjes says, these connections must ultimately be dependent on some necessary entity.

A particular type of connection Darjes mentions is the relationship between cause and what is caused. Like always, Darjes makes interesting divisions rarely seen in previous Wolffian philosophy. Thus,he notes when discussing cause or caused, one can firstly regard cause and caused as mere subjects – that is, as a material cause and caused – secondly as containing a reason for the possibility of something or having a reason of possibility in some other entity – this is what Darjes calls active/passive causating reason – and finally, as containing or having in something else a reason for actuality – active or passive causality. Like many other Wolffians before him, Darjes goes into great lengths in describing various causal notions, such as principal cause and instrumental cause or mediate and immediate cause, and we need not follow him in such a detail.

Just like almost all Wolffians thus far, Darjes defines the notion of space through the spatial relations an entity could have. Indeed, spatial relations are based on certain connections between entities, in which one entity cannot take the place of the other entities. Such space is then no true entity, but merely an abstraction out of real entities and their relationships. While spatial relationships are completely external to the entities or substances, if one adds activities to the equation, the connection becomes at least more internal. Darjes speaks of presence, by which he means the factor of one substance affecting another – the more a substance affects another, the more present it is to that other substance. Darjesian presence is then a much stronger relationship than mere spatial closeness – if one unites entities by bringing them close to one another, the union is merely external, while a union involving substances being present to one another is internal.

Before moving to more particular parts of metaphysics, Darjes finally considers the notions of infinity and finity, which he defines simply through the notion of perfection – finite entity is such that something can be more perfect than it, while an infinite entity is as perfect as is possible. The finity of an entity does not mean it couldn't be also perfect in some measure. It just isn't completely perfect and all perfection it has must belong also to the infinite entity. It is then immediately clear that all passivity, incompleteness and possibility of non-existence are signs of finity.

Already at this place in metaphysics, Darjes introduces Wolffian aposteriori and apriori proofs of God's existence, although he is, of course, not yet speaking of God. He notes, firstly, that since all finite entities are contingent, they must ultimately depend on a necessary infinite entity. Hence, if finite entities exist, an infinite entity surely must exist also (aposteriori proof). Since it is clearly possible that a finite entity would exist, an infinite entity must also be a possibility. Because infinite entity can be only impossible or necessary, it must then exist necessarily (apriori proof). We see here a similar dual role played by the two proofs as in Wolff's theology, the difference being that Darjes has to assume only the possibility of something finite.

Infinite and finite form then the major division of entities. Infinite entity is essentially unique, so no further division of that species is possible. Finite entities, on the other hand, can divide into further subspecies, depending on whether they are simple or complex.

Baumgarten: Metaphysics – Kinds of substances

The primary classification of things in metaphysical treatises has long been that of substances and accidences and Baumgarten's Metaphysics makes no exception. Substances are things that can exist without being attached to something else, while accidences have to exist in something else, namely, in substances. Furthermore, Baumgarten adds, accidences are not just something externally connected to a substance, but a substance must contain some reason why such accidences exist within it. In other words, substance is a force that in a sense causes its accidences – if completely, they are its essentials and attributes, if partially, they are its modes.

Now, substances with modes are variable or they have states, which can change into other states. Like all things in Baumgarten's system, these changes also require grounding in some forces. Changes effected by forces are then activities of substances having these forces. Such activities might be connected with changes in the active substance itself, but they might also link to changes in other things: these other things then have a passion. In latter case, the forces might act alone to produce a certain effect and then we speak of real actions and passions, or then the passive substance also has some activity at the same time as it has passions, and then we speak of ideal actions and passions. The division of real and ideal actions and passions is of importance in relation to Baumgarten's thoughts about causality.

Because all substances have forces, all of them have also activities – if nothing else, then at least activities towards themselves. Furthermore, activity does not define just the essence of substances, but also their mutual presence – substances are present to one another, Baumgarten says, when they happen to interact with one another.

Baumgarten divides substances, in quite a Wolffian manner, into complex and simple substances (Baumgarten does admit that we can also have complexes of accidences, but these are of secondary importance in comparison with complexes of substances). Not so Wolffian is Baumgarten's endorsement of Leibnizian term ”monad” as the name of the ultimate simple substances. Rounding up the division of substances is the division of simple substances into finite and infinite substances, in which infinite substance has all the positive properties in highest grade and thus exists necessarily and immutably – this is something we will return to in Baumgarten's theology – while finite substances change their states and have restrictions.

This concludes Baumgarten's account of the substances or primary entities of the world, and like with Wolff, we can already discern the outlines of the three concrete metaphysical disciples. But before moving away from ontology, we still have to discuss Baumgarten's account of basic relations of entities.

Simplicity itself

As familiar as was his account of complex entities, as familiar is also Wolff's description of simple entities, which in many cases simply have characteristics opposite to characteristics of complex substances. Previously I characterised Wolffian simple things as units of forces, which is quite correct still in light of Latin ontology, but one must not assume that complex entities could not then be described in terms of forces. Instead, the notion of force is something common to both simple and complex entities.

To understand what Wolff means by a force, one must begin with the notion of modes that we know to be characteristics that can be changed without changing the essential identity of a thing. Now, consistent collections of modes define a certain state. Such states, if they happen to be instantiated, belong to some thing, which can then be called the subject of these states, which are then adjunct to the subject. Note that the notion of subject, just like the notion of essence, is context dependent: in geometry we might take certain figure as stable, while in physics this figure could also be mutable.

In some cases, the change of states can be explained through the subject of change – then the change can be called an action of the subject, while in the opposite case it could be called passion. Thus, while if I voluntarily jump from a plane, the subsequent fall is my action, if on the other hand I am thrown from a plane, the fall is my passion. A subject undergoing an action can be called an agent, while a subject undergoing a passion can be called patient.

Furthermore, corresponding to action and passion, a thing has corresponding possibilities for action and passion or active potentiality and passive potentiality, the former of which Wolff also calls faculty. Without these potentialities actions and passions could not occur, but as mere possibilities they still require something in order to be activated.

In case of actions, this activating element is finally called force. What a force is or how it will be generated should not yet be apparent from this nominal definition. Still, it is quite clear from the definition that it makes no sense to speak of a force if there is no action that it activates, unless there is some opposing force resisting this activation.

This is as far as conceptual analysis takes us. From empirical considerations Wolff concludes that we could describe force as consisting of conatus. Conatus is a peculiar notion, common to many early modern thinkers, such as Spinoza, meaning a sort of life force of a thing that aimed at preventing the destruction of the thing. In physical contexts, conatus was often identified with impetus, the habit of bodies to remain in the same state of movement – this tendency was thought to be due to some internal yearning of bodies.

One obvious aim of this talk of conatus or impetus is to introduce the possibility to quantify forces – forces can be connected to the actions they trigger, and we can thus present forces as vectors. Because of their quantitative nature, forces can be combined (basic principle for this possibility is easily seen in a so-called parallelogram of force). Thus, we can regard forces of composite entities as combinations of forces of simple entities.

Parallelogram of forces: when forces F1 and F2 are the only forces affecting a thing the resulting movement is described by their sum

The mathematics of forces is one step in Wolff's project of quantifying philosophy. A final step is taken with the notion of grade, which Wolff defines as a characteristic of qualities that can be used to distinguish different (spatial or temporal) instances of same quality (thus, two apples might have a different tinge of green). Now, Wolff notes that it is possible to create at least a fictitious quantification for the grades (just think of a temperature scale – if a temperature of air rises two grades, this does not happen because of adding two individual grades of warmth to air). Because qualities were originally the only impediment of the quantification program, Wolff thinks he has solved the problem suggested by his critics.

The final piece in separating complex and simple entities is the notion of substance. Here Wolff begins by distinguishing between what is mutable (that which can be changed without it losing its essential identity) and what is only perdurable (that which can exist for a time without losing its essential identity). Now, Wolff's interest lies in perdurable things: cows, shadows, colours, you name it. Some of these perdurable entities are not mutable, some of them are. According to Wolff, this distinction among perdurables captures the traditional distinction between accidences and substances. This might need some explanation. Consider a traditional example of an accidence, such as certain shade of colour. It can definitely exist for a while, say, on some surface, but when you try to change it, it will change into a different shade. Then again, a substance, like a cow, will not be destroyed, if you paint it black – thus, it is not just perdurable, but also mutable.

Wolff's definition clearly is not meant as a strict division, but more as a hierarchy of substantiality – that is, we can speak of what is more accidental or substantial. Thus, we can change e.g. shape of a certain blob of colour, so that it will still remain a blob of this colour. Then again certain modifications of cow, such as tearing it apart, will undoubtedly destroy it. In addition, Wolff suggests we may define as proper substances those perdurables that will endure through any humanly conceivable change – these are essentially the simple substances. Then again, complex substances are in comparison accidental, because all their essential characteristics, such as figure and magnitude, are mere accidents. Thus, they can be only secondary substances.

Wolff ends his account of simple substances with a consideration of infinities. The characterisation of an infinite substance contains no surprises – infinite substance is incomparable with finite substances, but we can say that it has some analogical or eminent characteristics (eminence appears to be just a roundabout way to say that we really do not understand what it is). Then again, Wolff also makes some interesting remarks on mathematical infinities and infinitesimals. To put short, he admits that no mathematical infinities or infinitesimals actually exist, but also suggests that such fictions are useful in e.g. differential calculus.

So much for simple substances, now it is only relations we have to speak of.

Christian Wolff: Reasonable thoughts on God, world, and human soul, furthermore, of all things in general - Units of force

I have always found the concept of infinity, as used by many historical philosophers, to be rather confusing. This is probably due to my own background as a student in mathematics, where by infinity one means a number so great that all the regular numbers are way little in comparison. Because the infinities of which these philosophers speak – particularly God – are supposedly something beyond numbers, I have tried to avoid the ambiguous term. Indeed, one of my articles was once rejected, because the reviewer had something against me speaking of perfections, instead of infinities. Yet, I still feel that the two concepts are rather close. Infinite substance, for instance, is something ”way awesome”, superior in all relevant senses compared to a mere finite substance – shouldn't we then say that it is perfect in comparison with the finite substances?

Wolff describes infinity of a thing as a lack of all bounds (Schranke), and while he never explicitly defines what he means by these bounds, he in several occasions appears to relate them with how a thing is determined and classified. The identification of boundaries and determination resembles Hegel's statement that ”all determination is negation”; Hegel says that he borrowed the statement from Spinoza, and it would be interesting to know how widely it had circulated.

The identification of boundaries/negations and determinations may be difficult to understand. With Wolff, we should remember the notion of essence: a possible structure, which can be actualised in concrete things. Essence determines then at least some characteristics of an actual thing, but other characteristics might be determined by its relations to other things. These relations, then, are what bound the essence and divide it into different species: if the essence in question would be a hole in the wall, the windows and the doors would be differentiated by their differing relations to persons using them. The word ”boundary” is here used somewhat metaphorically: boundaries of a figure are also its relations to the things surrounding it. Essence and all the relations determine then an individual thing completely, because by knowing the essence of a thing and its relations to other things we know everything there is to know about the thing.

Infinite thing is then something that is not bounded, that is, it is determined only by its own essence and not by any relations to other things. In other words, an infinite thing cannot be distinguished from other infinite things. Instead, it is completely inclassifiable – it cannot be put into a same class with finite things, because their nature or essence is too dissimilar. Thus, all we can do to describe an infinite thing is to use meaningless superlatives – it is beyond anything we can imagine, or indeed, just ”way awesome”.

How does the division of infinite and finite things then relate to the earlied division of simple and complex things? Wolff notes that infinite things cannot really change, and by change he means specifically a change of the bounding relations: an essence of a thing cannot be changed, but at most one can replace a thing having one essence with another thing having a different essence. Infinite thing is then all that it is ”at once” and not by going through successive stages. Then again, a complex thing might change e.g. its spatial characteristics, which for Wolff are essentially relations to other things. Thus, an infinite thing, as atemporal, must be simple.

Complex and infinite things are then two classes with no common members, but are there any finite simple things? Well, all the complex things, says Wolff, must be founded on some simple things, the combination of which has generated the complex thing. Now, a combination of simple things is undoubtedly a relation of them, and furthermore, a relation which might change. Thus, the simple things that are the final constituents of complex things must be capable of change and therefore finite.

Boundaries of complex things can be spatial or relate to the number of things it consists of, but what about boundaries of a simple thing, which is not spatial and does not consist of other objects? Remember that by boundary Wolff refers to a non-essential classification caused by the relations of a thing to other things. He apparently seems to think that such a classification must at least be analogical with the relations of magnitudes. A good example of this sort of scale would be one consisting of temperatures: temperatures do not consist of smaller temperatures, although they can be related like one number relates to another. Wolff calls quantities of such scale grades: this concept was used later by Kant and Hegel.

Wolff shares with Kant and Hegel also the idea of relating grades to forces (Kraft). Indeed, beyond numeric and spatial magnitudes, it is rather difficult to imagine any quantities, but those which measure the effects of a thing. For instance, temperature can be quantified, because a certain grade of temperature has a clear effect on the size of certain substances. Thus, Hegel later suggested that all grade-scales are not just similarly structured as scales of numeric and spatial magnitudes, but also essentially connected to such.

Wolff's simple, but finite things are thus indivisible units of forces. In this Wolff appears to move beyond Leibnizian monadology, where the ontological units were characterised by perception, and towards the identification of activity as the most essential characteristic of true existence, which is a common theme in German idealists.

By a force, furthermore, Wolff does not mean a mere capacity, the activation of which is completely contingent. Instead, force is active and causes some effects, unless it is countered by a contrary force. Furthermore, the force of a finite thing is bounded or has a definite grade. In other words, the activity of a simple, finite thing is somehow limited. This limitation is not essential, and the simple, finite thing could well change it, which proves the possibility of applying temporal terms to these things.

Wolff does not stop here, but suggests that simple things are constantly striving towards changing their boundaries. Wolff's only justification for this statement appears to be the principle of contradiction: a thing cannot counteract its own actions. The justification appears once again

somewhat loose: although the thing itself cannot nullify its own force, other things might well affect the thing, that is, if the opposing force is strong enough, the simple thing becomes to a standstill or even starts to become weaker. Wolff also apparently thinks that static states of standstill are merely transitory phenomena, which cannot hinder the almost constant change of the strength of the forces.

We could thus picture a finite simple substance through a graph where every moment of time is connected with some grade in the scale measuring the quantity of the force. The graph goes up, when the force achieves its goals, and when it goes down, it is hindered by other forces. In the shadowy distance above, there is the infinity, unreachable by mere finite things.

Wolff does not say as much, but it appears reasonable to suppose that it is this infinity towards which the finite substances probably strive. The notion of infinity thus produces an objective criteria for making value judgements in the realm of finite substances: the more the finite things resemble the infinite thing, the better they are. We shall see next time what sort of value scale of things Wolff suggests.