Maarten Maartensz

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CHAPTER IX

THE WORLD OF UNIVERSALS

Note 1: At the end of the last chapter I noted:

It seems a pretty wild metaphysical assumption to introduce yet a third world, 'a world which is neither mental nor physical', and I much like to avoid it, not only because of the principle of parsimony, but because it is difficult to make sense of.

I also noted that

So there is, in a way, a realm between experience and reality, namely the supposed meanings of our experiences (which, to remind the reader, may be contradictory or fantastical statements or pictures), which arises because every experience is taken to be an experience of something, and we are quite capable of saying or imagining what things should be like or should not be like so that our experiences could be true.

But it seems to me this realm of meanings, that may be alternatively described as whatever people may think of, whether real or fantastical, or possible or impossible, very much depends on the human mind, and consists of whatever hypothetical or conceivable things that are represented by our experiences, and which are, apart from their possible real existence, and apart from the experiences, signs or symbols we use to represent them, items in our memories, which indeed may well be called thoughts or ideas.

And in chapter 5, note 16 I remarked on universals:

I agree with Russell there are universals, and it is well to indicate my reasons: it seems to me that there are clearly recurring identical things, such as two patches of the same colour or two instances of the same shape or two occurrences of a straight line. Those who deny universals often do so by insisting that there must be some particular difference between different instances of what seems to be the same, but I do not see any reason to admit that, and I also insist that the only way I can make sense of supposedly real things like rays of light, that move along straight (or geodesic) lines, is to suppose that this and that straight (or geodesic) line are both identical in being the shortest distance between the points they connect. And if this is admitted, one has admitted universals.

So I agree with Russell that there are universals, and I agree with Plato that there are ideas or meanings, but disagree with Russell that universals form a third world that is neither mental nor real, and I disagree with Plato that the ideas or meanings human beings may think of, if they do not represent anything real, represent anything other than a supposed thing, that we may and do suppose simply because we assume our terms and experiences to be meaningful even if false, and indeed are capable of thinking of - imaginary, fantastic, hypothetical - things, apart from whatever real existence these things may have.

It seems well to put these differences clearly at the beginning of this chapter, since this and the next chapter are concerned with these topics. Back.

Note 2: It is not very pleasant to have to disagree on matters of historical detail so often, but here it seems fair to remark that, while Plato did give what seems to be the first theory of universals in Western philosophy that was written down, he also got it in large part from his teacher Socrates - as Plato himself almost certainly would have insisted - who did not write about philosophy but only talked about it. Back.

Note 3: Let's note a number of things:

  • It is natural to consider the assumption of universals in part as an answer to a problem about what general terms refer to i.e. terms which refer or may refer to several things: a general term refers to several things because each of these things contains an identical instance of the same thing, and this last thing, that may occur at the same time and in the same sense in several places and things is a universal.
  • One should, however, be clearly aware that this involves at least two distinct hypotheses that should be kept separate, viz. a hypothesis about language (that it contains general terms, which do or may refer to several distinct things), and a hypothesis about what language may be about (that it contains a kind of thing that may recur identically at several places or in several things).
  • Also, it is important to try to keep separate the problem of universals (which, as stated, concerns an explanation of how general terms refer and what they refer to) from other problems, such as what terms like idea, form or essence might refer to.

Back.

Note 4: Here yet another complication is introduced somewhat surreptitiously, namely the problems whether and in what sense there are any particular things and whether what Russell calls 'the world of sense' consists of particulars.

Let's first state what I understand by a particular. Particulars are defined in similar terms as universals, namely as things that occur at one place and one time only, if they occur at all. In other words, whereas particulars are unique things, universals are non-unique things (with the complication that what starts out as a universal, such as "dodo" or "dinosaur", may end up as satisfying the definition of particular, if we consider the last dodo or the last dinosaur).

This does not yet answer the question whether there are any particulars, although it may seem to do. Let us carefully note that particulars occur "at one place and one time only" in the sense that if at any given time they occur at one place, at that time they do not occur at another place. This does not exclude that at a later time the same particular may not occur at another place, while this introduces another complication, namely that what is by definition a particular may seem to behave much like a universal if one disregards the references to time.

To say that "the world of sense" consists of particulars is puzzling, at the very least, for it seems to contain - at least on our definitions - both particulars, like Socrates, and universals, like whiteness. It is also true that when one attributes any one world of sense to a person, a place and a time, as when saying "Russell perceived a table in his room in Cambridge on January 1, 1910 at 12.00 hours GMT", what one thus attributes to a person seem to be a quite particular experience - but then such a quite particular experience may represent and refer to (as does e.g. "table" in the example) universal things.

Also, if one disregards the specific attribution of a time and a place and a person to statements of experience, it seems obvious that statements of experience generally involve at least one universal term (as Russell will point out below).

For the moment we only register these problems. Back.

Note 5: All of this is true in so far as Plato is concerned. However, though these Platonic constructions may be imposed on the concept of a universal, they do not need to be imposed on it, and it is well to try to avoid whatever is peculiar to Plato's philosophy while discussing the problem of universals. Back.

Note 6: In view of the previous note, I agree. Back.

Note 7: As remarked in note 4, it is, at least, somewhat problematic to describe the things given in sensation as particulars, both for the reason given in note 4, and for the reason that it seems we are normally sensing universals: we see red tomatoes, green grass, tall people sitting on green grass eating red tomatoes and so on, and each of the terms in such statements are universals even if the specific experiences the statements describe themselves are also, when specifically attributed to a person, a place and a time, particulars in the sense that they cannot identically recur (simply because even if the very same thing seems to happen at a later time, at least the time differs).

It is well to keep apart what Russell seems to confuse: I may here and now have a particular experience of whiteness, but the whiteness given in this particular experience is a universal, in that some previous or later time, possibly at a different place, I may see the same whiteness again, in another particular experience, while also the whiteness I see, say in this patch of snow, is just the same as the whiteness I see in another patch, just as this patch of snow and that patch of snow both have the property of being snow. Back.

Note 8: In this paragraph Russell classifies kinds of words that occur in English and other Indo-European languages by saying whether they stand for universals or particulars.

This is OK as far as it goes (which is not very far), but my problem with it is that of notes 7 and 4: While I am willing to make the assumption that there are, in reality, unique things, that occur at any time on one place only, it seems to me that these do not occur in experience, even if each experience is unique.

In other words, whenever I see you, who are a unique person, I have a unique experience of a unique person - which nevertheless is given to me as one of the kind of my experiences that may be fairly described as "seeing you".

So it seems to me that the hypothesis that there are, in reality, particulars is a sensible one, and also that each of our experiences is particular in the sense that it happens to one person at one place and one time, but I do not believe that any of the things that make up my experiences is a particular: as far as I can see they are all universals, though some of these universals of my experience, such as my seeing you, may represent a real particular, such as you. Back.

Note 9: "Strictly speaking", to use an expression Russell liked to use when weighing gnats on the balance of reason, this is false, for there are English sentences, even of one or two letters, that are made up of terms that denote particulars. An example is the sentence "I." or "Me" in reply to "Who did it?". And such questions can equally well be answered by perfectly grammatical English sentences which consist of only a proper name.

However, I agree that less simple English sentences will refer to at least one universal. Back.

Note 10: In view of the previous two notes this is at least misleading. I prefer to say that all our experiences are in terms of universals even if they represent real particulars which, then, never occur in our experiences, even though each experience as such is a real particular. Back.

Note 11: If it were true, perhaps this might be a good argument in favour of the existence of philosophers. But it is not true, for any speaker of a normal language is aware of general words, that refer to possibly very many individuals of some kind, and any speaker of a normal language who has not been brainwashed by a course of philosophy will explain general words in terms of some characteristic that each individual instance shares - in other words, in terms of universals, though he may not use that name. Back.

Note 12: This, like the earlier part of the sentence - "We do not naturally dwell upon those words in a sentence which do not stand for particulars" - draws attention to at least two important facts about language: it seems based upon a hypothesis of real particulars qualified by real universals, and human beings normally think and imagine in terms of this hypothesis, and organise their language around it.

Of course, there is a simple underlying reason for this: the particulars people presume are generally at one and the same time instances or bearers of many universal qualities and relations, and hence the reference to a particular involves generally much more implied and supposed information than the reference to a universal. Back.

Note 13: As remarked in note 11, this is not so, and it seems far more accurate to describe all human experiences as being in terms of kinds of things (that may be unique or not) rather than in terms of particular and unique things.

Perhaps this also is the place to give a simple clarification of kinds, to relate them to what may be called world-structures, and to make a remark on the distinction between universals and particulars. In the rest of this note I presuppose some elementary knowledge of logic and set-theory.

A kind of things X, as I use the term, is a class of things for which there are three properties, A, B and C, such that (i) every thing in X that is both A and C is also B, but (ii) there are things in X that are A and not C and not B. In set-theoretical terms, accordingly, where I presume that every property corresponds to a set of things having that property, the powerset of X (which is the set of all possible subsets of X) contains three definable subsets such that the first two are not related by inclusion while the intersection of the first and the third are included in the second.

The properties A, B and C are whatever is represented by any formula with at least one free variable, and subsets are 'definable' in the language one uses if there is a formula in the language with at least one free variable that is co-extensive with the subset.

The condition C clearly amounts in effect to the removal of the logical counter-examples to the generalization that all things that are A are B, and thus, again in set-theoretical terms C = (X - (A and not B)).

Now, having power-sets one may introduce world-structures or valued power-sets as follows, using a function f: The world-structure of X induced by f, in short vp(X,f) = {Y: Y is a subset of X and there is a real number y such that f(Y)=y and all subsets Z of Y are such that f(Z) <= f(Y)}.

The underlying intuition is that X and its subsets are there as a matter of logic, while the function f, that assigns real numbers to sets, implicitly defines a possible reality while preserving the relations of inclusion (in that whenever Z inc Y, f(Z) <= f(Y)).

Simple choices for the values of f are {0,1} (where the simplest choice that all subsets of X have f(X)=1 corresponds to excluding no possibility and corresponds to all that's logically there excluding none), or real numbers from the interval (0,1) inclusive, which yields the basis for probabilities.

Accordingly, what we are trying to know (generally by making guesses, deriving conclusions, and comparing these conclusions with information from the supposed word-structure) are world-structures, and specifically the values assigned by f - which may intuitively be understood as the real values of the logical possibilities that Mother Nature or Father Divinity used to set up the reality.

Next, as I draw the distinction between universals and particulars, the only real difference I used is that particulars are unique and universals are not unique. I have made no assumptions about what universals or particulars are apart from this, and I also have not assumed universals must correspond to one class of grammatical terms and particulars to another. Indeed, as a matter of logic my fundamental assumptions may be put thus:

  • everything is either a unique original or a copy (of an original)
  • there are both unique originals and copies in the real world

In the selection of Russell he remarks that 'the study of philosophy' draws attention to the existence of universals. I remarked on this in Note 11, and add here that it is likewise for physics, according to which elementary particles, at least, are, properly speaking, in terms of my definition, universals: there is no difference apart from location between this and that elementary particle. This remark is of some conceptual importance, because it illustrates that universals are not necessarily abstruse mental concepts or associated with adjectives, verbs or prepositions.

Finally, on my theory of universals particulars can be seen as arising from combinations of universals that happen to be unique for some reason, and all things are first and foremost kinds. And on my theory every world that contains at least two things, whatever they may be, contains universals, but no world needs to contain particulars (if it is sufficiently large to contain some copies of anything in it). Back.

Note 14: This is true, albeit with qualifications (such as the mediaeval logicians, who had extensive theories about what they called syncategorematic terms, i.e. such terms as "and" and "in"). Back.

Note 15: Though Russell seems to be speaking about Spinoza, I think he is really speaking about Leibniz, in whose texts the doctrine Russell discusses is much more explicit than in Spinoza's texts.

There is another though related reason why the existence of relations has been denied: it is difficult to make sense of relations. Where are the referents of "between" or "gives to" in the real world? Can one point to them, as one can point to particular persons? Can they exist by themselves at all?

These and other similar questions are not easy to answer, and it is true that any non-trivial use of natural language seems to presuppose answers to them, yet very few users of natural language seem aware of such questions or capable of answering them in a clear way, while it is also an interesting fact - that concerns human cognitive capacities - that only in the 19th Century a few people (notably De Morgan and Peirce) became aware of the need of a logic of relations and of a sensible way of giving such a logic.

This is an interesting fact in that all mathematics and all reasoning involves relations. Back.

Note 16: Indeed this is the kind of doctrine Berkeley and Hume had, but I don't believe it is a "largely true account of our actual mental processes".

My principal reasons are (1) that I flatly deny that our images of particular things are particulars and (2) that the only way I can account for the apparent identity in colour between two patches of white(ness) is to assume that they have something in common.

In addition, if it is true that each of our experiences is unique in as much as it is a unique event befalling a unique person, it is also true that each of our experiences consists of universals (or kinds) that represent real particulars or real universals. Back.

Note 17: This is a rather famous - within the context of the problem of universals - argument of Russell. It may well be asked whether it does not commit the fallacy of begging the question (for if there are no universals, resemblance cannot be a universal), but there is a related argument that is correct. It is this: If there are two things with the same quality or relation, there are universals (by definition), and there are in fact many pairs of things with the same quality or relation.

Thus you and I are both human beings, and in the same sense; we are both a child of our parents, again in the same sense, even if our parents are different and we are different; two straight lines are straight in the same sense of both being the shortest distance between two points, even if the points and thereby the lines differ, and so on and on.

And one fundamental problem with trying to base universals on particulars and a particular relation of resemblance (like some of the so-called nominalists, who deny the real existence of universals) is that it seems to necessarily involve an infinite regress: if this and that thing resemble each other in respect X, this again must be a particular resemblance, while a third thing that resembles the second thing in what seems to be respect X, on the present nominalistic theory only can resemble it in a respect similar to but not identical with X'. This may have some plausiblity when one reflects that this green and that green may be both greenish without being both exactly the same shade of green, but it ceases to be plausible when one also reflects that the resemblances between three greenish patches also at best can resemble each other without being the same, for now we have three greenish patches, all particular, and two particular relations of resemblance (in respect of being greenish), that also at most resemble each other, but each of which is as particular as the patches of green they relate. So, while the rejection of universal may seem to be based on simplicity, it is soon forced to duplicate entities that need not be duplicated if one assumes universals. Back.

Note 18: Indeed, though it is fair to remark that in all probability by "the rationalists" Russell meant Leibniz in particular (who indeed denied the existence of relations, with rather amazing metaphysical effects). Back.

Note 19: This cannot be all correct, though it is correct procedure. The reason is that some universals must be merely mental, namely whenever we have a false idea of a universal. Back.

Note 20: Russell's argument is more directed against idealism than it is directed in favour of universals. It seems more sensible to argue like this: Terms representing relations, if true, represent something, and the something they represent, if it is an item of the independently existing reality, has itself independent existence (though relations cannot exist without the things they relate). Back.

Note 21: These are the kind of difficulties mentioned in note 15, and the kind of reasons Russell had for believing in what he called "the world of universals".

But it seems to me Russell is mistaken, and the answer to his question 'Where and when does this relation "north of" exist?' is simply: in such pairs as Edinburgh and London, Stockholm and Rome etc.

What makes this problematic is that relations involve several things to be true of and simply cannot exist without things to be true of, for which reason a relation like 'north of' will indeed never be found hovering about all by itself, without any pair of things such that the one is north of the other. This may be problematic (and indeed it is, as pointed out in note 15) but surely all this is less problematic than introducing a kind of entity that is 'Nowhere and nowhen' and forms a special world all by itself. Back.

Note 22: For the reasons given in the previous note, this seems nonsense to me, and the introduction of such a kind of entities, as are "neither in space nor in time, neither material nor mental" should give anyone pause and raise doubts, even a philosopher. Back.

Note 23: This is simply not so. The reason that moved people to hold universals are mental has been usually the same as to hold that things are mental: since we can think of them they must be mental in some way, whereas they found it difficult or impossible to believe they existed really. (This is the same line of argument that leads to the conclusion that colours and tastes are mental.) Back.

Note 24: Yes, I agree the same ambiguity is involved, but that is the ambiguity I discussed in the previous note. Back.

Note 25: I don't believe so, apart from the sense discussed in notes 8, 7 and 4. That is, I believe that whereas my experiences and your experiences are different in being, respectively, mine and yours, it is quite possible, if difficult to prove, that they are otherwise the same (at least to some extent). Indeed, to the extent that you and I understand each other it seems we must be thinking the same thoughts, in some appropriate sense of 'the same', just as we may have the same appreciations of the same food, the same joke, or the same music. Back.

Note 26: As I argued several times, for me thoughts are universals and may be thoughts of particulars or universals.And it seems very difficult to account for our thoughts about universals if our thoughts never are universals, especially when one considers such thoughts of non-existing universals as "3000 square circles" or "the qualities of a living Dodo" or "the chess-parties of griphons and mermaids", or - if the reader thinks these are too problematic - "the thoughts I will have tomorrow, if I am conscious" (which certainly will be more than one, and certainly do not exist yet). Back.

Note 27: Apart from other difficulties with this position I mentioned already (parsimony, a mistaken analysis of universals) this involves the difficulty that universals are part and parcel of our every thought and feeling: we see colours and shapes, hear tones and tunes, feel pains and pleasures, and so on, and each of these are first and foremost certain kinds of things and universals in our sense, even though they do occur at a particular time in a particular place. (Besides, on my definition of universal - what exists in the same way at different places on the same time - universals do involve time.) Back.

Note 28: In short, it is Russell's version of Plato's heaven. It seems to me pretty incredible in either version, but both thinkers have a better argument for such a heaven than is given in the present chapter: the problem what the terms and propositions of mathematics represent.

For there may not be triangles, straight lines, circles, real numbers of what have you in reality (although it seems there are) but nevertheless mathematically inclined people may, it seems, formulate propositions about such things, that may be true or false. If true, what are these propositions true of, especially when we take notice of the fact that such truths are not local and temporal, but always and everywhere?

Here I merely raise the problem and indicate one of Russell's and Plato's respectable mathematical reasons to believe in a Platonic world: it is the world of such things as make mathematics true.(But an answer along the lines of Note 13 may be sketched: the mathematical truths are those which are true in any world-structure because they correspond to the relations of inclusion that exist in any world-structure if anything exists in it at all. Whatever non-mathematical truths hold in a world-structure over and above the mathematical ones depends on the function f that may be seen as specifying for each subset to which it applies how many of that set there really are.) Back.

Note 29: This is mainly romantic exaggeration, like much else in Russell's paragraph. That's not necessarily a fault, since a dose of romanticism tends to make things more pleasant, as long as one is clear that what one gets is not the sober truth. Back.