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.

Same as usual

I have until now been silent about the structure Wolff gives to his ontology. We have actually passed already through two sections. First of them showed us the principles governing whatever there can be, while the second then introduced the actual topic of the book, namely, the possibilities or essences. The third section, beginning now, deals then with general affects of the essences – by affects Wolff means all characteristics of a thing, whether they be caused by the internal structure of the thing or by its contacts with other things.

The first type of affect Wolff considers is identity. For the basic definition Wolff uses the so-called principles of the identity of indiscernibles and the indiscernibility of identicals. That is, whenever we can substitute name of one thing with name of an apparently different thing, whatever is predicated of it, the assumedly separate things are actually identical or one thing; and whenever two names refer to same thing, we can substitute one for the other in every context.

Both sides of the principle can be doubted. The identity principle appears at first sight to say that whenever two things have exactly same qualities, they can be identified. The possibility of two exactly similar particles at different points of space seems then a difficulty. We would essentially have to fall back to Leibnizian conclusion that no such exactly similar entities exist. Yet, we can offer a weaker reading of the principle, which manages to circumvent the problem, that is, we can suppose that the predicates in question include relational predicates. Then we can simply point out that of the two particles, one of them, call it A, satisfies the predicate of being identical with A, while the other particle fails to satisfy this predicate – the only fault then being that the whole question becomes rather trivial.

The indiscernibility principle seems even more suspect. We just need to think of a statement like ”Everyone admires Spiderman” and compare it with a statement ”Everyone admires Peter Parker”. Clearly people can admire Spiderman without even knowing that he is Peter Parker. Such problems led the early analytical philosophers to clearly distinguish between extensional and intensional uses of concepts. Words like ”admire” or ”believe” are dependent on the intensions or meanings of concepts – when we admire someone, we actually admire the person as described by our notion of her. Thus, it is more about the case of identity of intensions, for instance, a person thought to be Spiderman is not identical with the same person when he is thought as Peter Parker. In cases where we can instead of intensions speak merely of extensions or the actual things, no matter how they are described, the indiscernibility principle works well.

The problems with confusing intensions and extensions raise the interesting point that identity and indiscerniblity principles are rather poor criteria for recognizing identities – we cannot really go and test with every predicate whether each one of them either fits both names or not. In fact, the whole idea of testing is rather misleading. Before the identity of morning and evening stars was discovered, we would have said that while morning star appears in the morning, evening star never does, making it obvious that the two cannot be identical. It was only after the identity was determined that we could see that certain apparently true predications of morning and evening stars were actually false.

Identities should then be determined through some other, more robust criteria. Problem is whether these criteria are tools to determine independently true identities or whether they actually constitute what is identical. That is, different criteria give different results for certain identities. For instance, one could define the identity of human being from identity of the materials out of which the human body consists, while another person could define it through memories. Now, it could be possible that human body is constantly changing its atoms and that an old person had not a single atom common with a child who had lived earlier, although the old person well remember having lived as the child. Then again, while a blow on the head won't change the atoms of the body in a significant manner, it might purge one's mind of many memories. Thus, there are cases where one criterion will point out an identity, while another doesn't.

In such cases, it might seem natural to ask which one of the identity criteria is correct – and even if neither of them would be correct in every case, we often just assume that there is one completely right criterion of identity. Yet, it also makes sense to question the meaningfulness of such problems – could it be that there are many viable criteria, none of which would be the only truth? Then we could accept one criterion in some cases where it fits quite well and another in other cases: different criteria would be answers to different questions. This would not mean a complete freedom in choosing what to take as identical. Indeed, while it would be in a sense free to specify what one means, when one is looking for identities, this task would usually have an answer clearly independent of us – the concepts would determine only the questions asked, not their answers. Furthermore, even if the notion of one true identification criterion was rejected, this wouldn't cancel the possibility that some criteria might be more natural than others.

Getting back to Wolff, it is difficult to decide which side of the fight he would take. Mainly, he just appears to take his definition of identity granted, which might suggest that he would believe identity to be an independent ontological relation that would hold no matter what our criteria of identity are. Then again, Wolff's main interest appears to lie in finding a definition of identity that works in mathematics. This suggests a certain level of relativity – two mathematical expressions may well be identical, even if what these expressions physically say isn't (say, if the two expressions refer to quantities of different things).

Whatever the case, Wolff clearly admits that identity is a relation not just between (possible or actual) things, because he at once talks of an identity between determinations of different things (for instance, when two different berries have the same shade of red). This identity of determinations clearly differs from the identity of individuals – redness of one berry can occupy different space from redness of another berry.

Now, Wolff continues, if those characteristics of two things are identical that can be used to discern them in themselves (that is, not through relations it has to other things), then the things are similar. Later on, Wolff also explicates that similarity can be defined through identity of essence. The problem lies in deciding what can be taken as the characteristics required in the first definition. Clearly any quantitative characteristics won't do, because we cannot e.g. differentiate a one inch square from a one mile square, unless we can see that one is bigger than the other. Otherwise, the Wolffian requirements of similarity appear to be quite subjective. That is, in different circumstances, different characteristics can serve as marks of similarity, or what is taken as essence depends on what we think as essential. Again, Wolff emphasizes similarity as used in mathematics, for instance, in case of seeing two similar figures we look at their shape (not their size, but also not the material from which they are made).

So much for identities and similarities, next time we shall see what Wolff has to say about universals.