The Existence of the World

An Introduction to Ontology

Reinhardt Grossmann

• Publisher: Routledge
• Year: 2019
• Print ISBN: 9780367194055
• eBook ISBN: 9780429514180
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

Originally published in 1992. The history of Western philosophy can be seen as a battle between those that insist that the "physical universe" exists and those would claim that there is a much larger "world" which contains atemporal and nonspatial things as well. The central part of this book, and the battle, concerns the existence of universals. Starting with the mediaeval definition of the issue found in Porphry and Boethius, the author then considers modern and contemporary versions of the battle. He concludes that what is at stake between naturalists and ontologists is the existence and nature of a number of important categories, like structures, relations, sets, numbers and so on.

CHAPTER III

The Structure of the World: The Categories

Categories

Individual things are temporal. They constitute the physical universe. Their properties, as we have seen, are not temporal. Nor are they spatial. The discovery of this monumental fact is the discovery of the world. The world, unlike the universe, consists of individual things and of all other kinds of entity. It consists, therefore, of individual things and of properties. But the realization that there are atemporal things, that there is a world in addition to the physical universe, immediately raises the question of whether or not there are other atemporal things. Does the world consist of a great many kinds of abstract entity? The answer to this question is affirmative: as it turns out, the world is a rather complicated web of individual things and of kinds of abstract thing. There are many more abstract kinds of entity than the first ontologists dreamed of. I shall call these kinds of thing ‘categories’. The world consists of individuals and of a number of categories of abstract things.

I believe that there are altogether seven categories, namely, individuals, properties, relations, structures, sets, quantifiers, and facts. Of course, there may be more or there may be less. How does one decide? There is no ‘decision procedure’ that allows us to decide once and for all, by some mechanical method, how many categories there are. Nor are categorial (ontological) inquiries of a more sublime and indubitable nature than ordinary ones, as many philosophers used to think. All we can do is to argue piecemeal that things of a certain kind do not belong to a given category, because things of this kind have properties different from the properties of the things of the category. For example, an individual apple is not a property because properties are exemplified by things, while apples are not. We have argued in great detail, to give another example, that the property of being an apple is not an individual thing because it does not exist in time and space. Thus apples and the property of being an apple must belong to different categories. To sort out the categories is a painstaking chore. As a matter of fact, until little more than a hundred years ago, few categories were known to philosophers. In the main, ontology revolved around the two categories of individual thing and property of individual thing. These two are of course, roughly speaking, the categories of the Greek (Aristotelian and Platonic) tradition.

In the following sections, we shall discuss the seven categories I just mentioned. Each one of these categories deserves much more space than we have here available, so that only the barest hint of the complexity of the ontological issues surrounding their existence can be given. I shall try to select for each category a particularly interesting or important problem and discuss it in some detail, leaving out many other questions raised by the possible existence of the category.

Structures

Our prime example of an individual thing was a billiard ball. But this choice is not quite correct. It was really a billiard ball without spatial parts and at a moment. The billiard ball as a whole with all of its spatial parts and temporal phases is not a simple individual thing, but is a complicated structure consisting of individual things. It consists of these temporal phases, and during every phase, it has numerous spatial parts. Are there also true individuals, that is, simple individual things? Are there things which are temporal and/or spatial and at the same time simple? I think so.

A square drawn on the black board is a spatially simple thing. It is not temporally simple, since it lasts for some time. For example, it exists before I sneezed and it continues to exist afterwards. But let us leave time out of the picture for a moment. I said that the square is spatially simple. But how could this be? Can we not distinguish its left side from its right side, drawing in our imagination a straight line through its middle? Of course, we can. But an imaginary line is not a real line, just as an imaginary giant is not a real one. The square, as it is drawn, does not have a left side and a right side. After you have actually drawn a line through the middle, it has these two sides. As it is, it has no sides. If you actually draw, in addition, a horizontal line, the square will have four parts, namely, four smaller squares. But before any lines are drawn, it has not parts and is therefore (spatially) simple.

Are there also temporally simple individuals? Are there individuals which have no duration? A flash of light is often cited as a temporally simple individual. I think that the most obvious example is rather a mental act. You are on your hands and knees on the carpet under your desk, looking for the pencil which you just dropped. Suddenly you see the pencil behind one of the legs of the desk. This act of seeing the pencil has no duration; as soon as it occurs, it is gone. You cannot stop ‘half-way through this act’. It appears and then it is gone. What holds for it, it seems, holds for all mental acts (as distinguished, of course, from mental processes). The thought that you have forgotten to lock your car occurs and as soon as it occurs, it is gone. It has no duration. The sudden wish that you were in Paris, sitting in a sidewalk cafe, has no duration. And so on.

If these considerations are correct, then there are simple individual things. But the ordinary perceptual objects around us, like the two white billiard balls of our paradigm, are complex individuals. As a consequence, they belong to the category of structure: they are spatio-temporal structures. This distinction between simple and complex individual things calls for some comments. When we started our inquiry, we divided the things there are into things that are temporal and into things that are not. We called the former individuals. But we now see that this is not the only way in which we can divide entities up into broad categories. Among the individuals there are, some are simple and some are complex. And if we wish to stress this feature of things, we may distinguish between simple individuals, on the one hand, and temporal structures, on the other. We must note that there are other kinds of structure as well; not all structures are temporal. For example, the structure formed by the natural numbers 1, 2, 3, etc., in this order, is not temporal. As a matter of fact, there are many kinds of structure which are not temporal. Let me stress: whether or not we separate complex individuals from simple ones and assign the former to the category of structure depends entirely on our interest and purpose.

We can speak of a category of structure because there are atemporal structures. Otherwise, we would merely have simple and complex individuals. What are the essential characteristics of structures? How do structures differ from other kinds of thing?

It is clear from what I have said that structures are complex things. But not all complex things are structures. For example, sets and facts are also complex, but they are not structures in our technical sense of the term. Structures differ from sets in that the elements of a set are not related to each other while the parts of a structure always are. For example, to the structure consisting of the natural numbers in the order of size, arranged from 1, 2, 3, etc., to 10, there corresponds the set of these natural numbers arranged in no particular order. Structures also differ from facts, although both kinds are complex. The most important difference is that facts stand in certain relations to each other, while structures do not. Two facts may be conjoined, for example, by the and of conjunction, but there is no conjunction for structures.

Most importantly, structures differ from other kinds of complex thing by the fact that they can be isomorphic to each other. (But we can extend this primary notion of isomorphism also to sets and states of affairs.) Two structures S₁ and S₂ are said to be isomorphic if and only if the following three conditions are fulfilled:

(i)For every non-relational part of S₁ there is precisely one nonrelational part of S₂, and conversely;

(ii)for every relation of S₁ there is precisely one relation of S₂, and conversely;

(iii)the parts of S₁ which correspond to S₂ stand in the relations of S, to each other which correspond to the relations of S₂, and conversely.

An illustration may make clear how simple the notion of isomorphism really is. Imagine the two series of even and of odd natural numbers arranged according to size:

2 4 6 8 10 …

1 3 5 7 9 …

These two series are isomorphic to each other. The first condition is fulfilled: to every even number there corresponds precisely one odd number, and conversely. Secondly, to the relation of being the next larger even number there corresponds the relation of being the next larger odd number. And thirdly, whenever two even numbers stand in their relation to each other, the corresponding odd numbers stand in their relation to each other. For example, four is coordinated with three, and six is coordinated with five; and since six is the next larger even number after four, five is the next larger odd number after three.

The notion of isomorphism explicates what we mean, speaking precisely, by saying that two complex things have the same structure: sameness of structure amounts to an isomorphism between the respective complexes. Most importantly, when there exists an isomorphism between two structures, then there exists a unique kind of ‘similarity’ between them. Our two white billiard balls are similar to each other in that they share the same color; they are both white. And they are also similar in virtue of the fact that they share the same shape. Similarity, as most often understood, is sameness in regard to properties or relations. Roughly speaking, the more properties and relations two things share, the more similar they are to each other. But we now see that there also exists a similarity between certain things which is of an entirely different sort. Two structures can be similar to each other, not by sharing common properties or relations, but by being isomorphic to each other. We have discovered a new and most fascinating kind of similarity.

This kind of similarity, firmly anchored in the nature of structures, is responsible for the fact that structures can be used to represent other structures. Language, for example, is a structure which, if it is to succeed, must have some sort of isomorphism to the world. This has suggested to many recent and contemporary philosophers that one can ‘read off’ the structure of the world from the structure of language. I believe that there is a kernel of truth in this approach to ontology. But the difficult problem is to discover not just a rough-and-ready similarity between language and world, but a more detailed delineation of the structure of the world. On the other hand, the necessary isomorphism between language and world has also been used to philosophize by means of an ‘ideal language’. What these philosophers aspired to is the construction of a schema of a language which reflects the categorial structure of the world better than our ordinary spoken languages do. Because of these connections between language and ontology one speaks nowadays of a ‘Linguistic Turn’ in recent philosophy.

Structures are all around us. Most sciences study structures of one sort or another. The (real) numbers form a complicated web held together by the familiar relations of sum, product, etc. Arithmetic, we may say, is nothing but the ‘science’ that deals with this web. Of course, molecules are structures studied by chemistry, and atoms are structures studied by physics. As soon as one thinks about it, structures appear everywhere in the sciences, and in arithmetic and geometry as well. There even exists a general theory of structures, namely, algebra. It must be emphasized, however, that from our point of view set theory and ontology do not primarily deal with certain sorts of structures. Sets, as I said earlier, are not structures, since their members are not connected by any relations. Nevertheless, a certain minimal sort of ‘similarity’ obtains even among sets: two sets may be said to be similar to each other if and only if they have the same number of members. Expressed more technically, they may be said to be similar to each other if and only if their members can be coordinated one-to-one. Ontology, as we shall see, deals primarily with facts. The world is its topic, and the world is a fact.

Relations

Structures are things in relations. Witho...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• I The Discovery of the World: Timeless Being
• II The Battle Over the World: Universals
• III The Structure of the World: The Categories
• IV The Substratum of the World: Existence
• V The Enigma of the World: Negation
• Bibliography
• Index