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«PETER SIMONS Appeared in Johannes Marek and Maria Reicher, eds., Experience and Analysis. Proceedings of the 2004 Wittgenstein Symposium. Vienna: ...»

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Against Set Theory

PETER SIMONS

Appeared in Johannes Marek and Maria Reicher, eds., Experience and Analysis.

Proceedings of the 2004 Wittgenstein Symposium. Vienna: öbv&hpt, 2005, 143–152.

Since the rise of modern science in the seventeenth century, philosophy has suffered from

a need for legitimation. If it is science that provides us with our advanced knowledge of

the world, what role is there left for philosophy? Several answers have been given. One answer is that of Wittgenstein. Philosophy is not in the knowledge business at all. In a world in which no one misused language, there would be no need for the restorative activity called ‘philosophy’. Philosophy, like medicine, is a response to imperfection.

Among the positive answers are three. One assimilates philosophy to natural science and denies there is any sharp distinction. We find this among empiricists such as Mill, Quine and Armstrong. Another attempts to discern some special subject matter and/or method for philosophy, inaccessible to and prior to science. Kant’s critical philosophy, Husserl’s phenomenology and some strains of linguistic-analytic philosophy are like this. Finally there are those who would assimilate philosophy to the formal sciences of mathematics and logic. This was most popular among seventeenth century rationalists who attempted to do philosophy more geometrico but has its modern echoes in philosophy in the phenomenon we may call ‘math envy’. Applying set theory promises the philosopher partial relief from math envy, because it can be used to convince sceptics that philosophy too can be hard science. This paper is about why philosophers should stand up on their own and overcome their besottedness with sets.

Set theory in mathematics Set theory was created single-handedly by Georg Cantor as recently as 130 years ago.

Prompted in part by suggestive ideas of Bolzano and adopting Bolzano’s term ‘Menge’, it served as a vehicle for exploring the transfinite. In the 1870s and 1880s Cantor established the use of one-one correspondence as indicating the size or cardinality of a collection of objects, and it was these collections that he termed ‘sets’. Here is Cantor’s

famous 1895 characterization of what a set is:

Unter einer “Menge” verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder unseres Denkens (welche die “Elemente” von M genannt werden) zu einem Ganzen.1 Among the main early achievements of Cantor’s set theory were the demonstration that there are more real numbers than natural, rational or algebraic numbers, the demonstration that continua of any dimension are equinumerous with continua of one dimension, and the use of diagonalization to generate an unending sequence of infinite cardinalities. From here he developed the concepts of order type and ordinal number. Set theory soon ran into problems of paradox however, first Burali-Forti’s paradox of the greatest ordinal, possibly discovered by Cantor as early as 1895, Cantor’s own paradox of the greatest cardinal, discovered by him in 1899, and the Russell-Zermelo paradox of

1901. Nor was Cantor’s theory or its subsequent development by Zermelo and others able to answer the continuum problem, namely whether the cardinality of the continuum is or is not the next greater cardinality than the cardinality of the natural numbers, and the results of Gödel and Cohen showed that the set theory developed to date was incapable of giving an answer either way.

Although Cantor himself was not much bothered by the paradoxes, others such as Frege and Russell took them seriously as threatening their attempts to provide a logical foundation for mathematics. Modern mathematical logic in all its complexity emerged from the efforts to salvage as much as possible from the wreckage, and this logic was soon in employment by Russell and others as a vehicle for tackling philosophical as well as logical and mathematical problems. The prestige quite rightfully earned by mathematical logic among philosophers and logicians from Frege to Turing, and its dutiful application to philosophy by Russell, Whitehead, Carnap, Reichenbach, Quine 1. “By a ‘set’ we understand any collection into a whole M of definite and well distinguished objects m of our intuition or our thought, which are called the ‘elements’ of M.” and others, help to explain why set theory has acquired an almost unassailable status among philosophers as “hard” theory, to be respected and used but not questioned.

Set theory found application for example in the twentieth-century developments of measure theory (including probability theory) and point-set topology, and by the 1960s it was commonplace in textbooks and articles to present other branches of mathematics using set theory as a general framework. Even as mathematics, set theory, despite this common use of some of its terminology as a lingua franca, is a not especially natural, fruitful or useful part of the subject. Traditional number theory, geometry, analysis, function theory, and all mathematics up until Cantor were developed in happy ignorance of set theory. Contrast this historical fact with this takeover bid by a leading modern set

theorist:

All branches of mathematics are developed, consciously or unconsciously, in set theory or in some part of it.2 (Levy 1979, 1f.; my italics) In his 1911 An Introduction to Mathematics, the mathematical counterpart in its classic clarity and accessibility to Russell’s Problems of Philosophy, Whitehead was easily able to give a balanced and attractive introduction to the subject without once mentioning sets.





Modern logicians of the highest calibre, such as Church and Curry, present their work set-free. Most mathematicians are profoundly uninterested in foundational issues. Among those that are interested, a majority choose to present their views in terms of some form of set theory (which of the several set theories available they use varies). They prefer the simplicity and convenience of sets to the notational and conceptual complexities of type theory, though as Frege, Russell, Lesniewski, Church and others recognized, philosophically, types are in some ways more natural than sets. Some mathematicians and philosophers have claimed that category theory makes a better and more mathematically suitable or powerful foundation than set theory.

Even within mathematics some uses of set theory are questionable. It is usual to

2. So Euclid, Archimedes, Ptolemy, Al Khwarizmi, Newton, Leibniz, Euler, Gauß, Cauchy and many others were all doing unconscious set theory.

interpret numbers of various kinds as sets, but as Benacerraf (1965) showed, this leads to the pseudo-question which sets the numbers should be. Numbers are old and very useful, whereas sets are new and problematic. To explicate numbers as sets is to explain the clear by the obscure. The obscurity turns on the fact that set theory has no natural interpretation. Despite Lewis’s (1991) heroic attempt to tame set theory via mereology, he was left with the mystery of the singleton: what distinguishes a from {a}? Singletons enable set theory to do what upset Goodman, to build many things out of the same materials, so one thing can occur more than once in a set. For example in {{a,b},{a,c}} a occurs twice. In Lewis’s regimentation this set comes out as {{{a} ∪ {b}} ∪ {{a} ∪ {c}}}. Without singletons it would just collapse into {a,b,c}. Both singletons and the empty set stuck like a fishbone in the craw of the twentieth century logician most vehemently opposed to set theory, Stanislaw Lesniewski.

False idols in philosophy

Leaving the status of set theory within mathematics now aside, let us consider its use, that is to say, its misuse, within and around philosophy. Philosophers and others have had the wool pulled over their eyes by authorities in set theory, who try to present sets as something wholly natural and uncontroversial. The technique, usually applied on or about page 1 of a textbook of set theory, is to claim that we are already familiar with sets under some other names or guises, and then trade on this supposed familiarity to sell us a bill of fare which is ontologically far from neutral and far from benign. Here are some

authorities:3

Consider a collection of concrete objects, for instance of the apples, oranges etc. in a fruit shop. We may call it a set of fruit, the individual apples etc. being the members (or elements) of the set. Conceiving the collection as a new single concept is an elementary intellectual act.4 (Fraenkel 1953, 4)

3. I am not being ironical: these are from some of the best books on set theory.

4. So one can buy a set from a fruiterer or supermarket.

a pack of wolves, a bunch of grapes, or a flock of pigeons are all examples of sets of things.5 (Halmos 1960, 1) In our examples, sets consisted of concrete and familiar objects, but once we have sets, we can form sets of sets, e.g. the set of all football teams.6 (Van Dalen, Doets and de Swart 1978, 1) Intuitively speaking, a set is a definite collection, a plurality of objects of any kind, which is itself apprehended as a single object. For example, think of a lot of sheep grazing in a field. They are a collection of sheep, a plurality of individual objects.

However, we may (and often do) think of them—it—as a single object: a herd of sheep.7 (Machover 1996, 10) What has gone wrong in each of these cases is the attribution of inappropriate properties such as causal powers and location to

Abstract

entities.8 The problem is one of misappropriation. There are indeed concrete collections of sheep, grapes, apples, wolves, pigeons etc., into herds, bunches, piles, packs, flocks etc., but none of them are sets. They are concrete collectives with their own sorts of membership and persistence conditions, which vary widely and which need their own ontological treatment independently of mathematics. Typically they have causal powers and locations and non-extensional identity conditions. A somewhat different case are those collections of things which can be presented as sheer pluralities, for example by listing them, like Russell and Whitehead, or the several people who happen to be in a certain room now. Such pluralities, or manifolds as I once called them (Simons 1980), are what Russell called ‘classes as many’

5. So one can be chased, attacked and even eaten by a set, oneself eat a set and absorb vitamins from it, press a set and make wine out of it, and have to clean up the droppings a set leaves on a statue.

6. So Juventus, Barcelona and Manchester United are very expensive sets.

7. So sets may safely graze on the field behind my house, and grow in the springtime by the addition of new members: lambs.

8. That it need not be so is magnificently demonstrated by Jech 1997, who unapologetically launches page 1 with axioms and then proceeds to deduce theorems: pure mathematics at its purest.

and Cantor called ‘Vielheiten’. The hegemony of set theory among philosophers concerned about collective entities has still to be broken. Among linguists, plurals have however gained a fair foothold, and the logician George Boolos persuasively championed pluralities as an ontologically economical way to interpret monadic second-order logic,9 so the picture is not wholly bleak.

One effect of set theory in ontology has thus been to cripple the development of an adequate ontology of collective entities. This however is far from the worst of its effects. In general the employment of set theory, usually hand in hand with modeltheoretic semantics, has been to persuade many philosophers that the rich panoply of entities the world throws at us can be reduced to individuals and sets of various sorts, for example sets as properties, sets of ordered tuples as relations, sets of possible worlds as propositions, and so on and so forth. It is hard to know where to start in revealing the scope of the damage caused to ontology by the thoughtless or supposedly scientifically economic reduction of various entities to sets. The most open-eyed proponent of this approach has been Quine, who to his credit has never shirked from confronting the issues, has never tried to sell sets under a false bill of fare, and is prepared to accept the absurdities as they arise, for example taking physical objects as sets of quadruples of real numbers, and thus as all composed of set-theoretic complications of the empty set.

(Quine 1976) You, too, are a set, according to Quine, albeit a very complex one.

One area of ontology which has been beholden to set theory and has suffered from this is the philosophy of space and time, in particular the ontology of continua. Since the time of Bolzano, and especially since Cantor, it has been commonplace to regard continua as sets of points, in plain contradistinction to the mereological conception of geometrical entities assumed by Euclid. This has the following strange consequence, noticed and approved by Bolzano. If two bodies are continua, then they can only touch if one of them is topologically open and the other is topologically closed at the place or places of contact. Attempts by the more ontologically fastidious such as Brentano, Whitehead and Menger to account for continua otherwise than as treating them as sets of

9. See the papers on plurals in Boolos 1998.

points have met with scant approval among mathematicians, and only marginally more among philosophers.

The wider damage done to ontology through the hegemony of predicate logic and its model-theoretic interpretations is discussed in Barry Smith’s contribution to this volume and I shall not go into it.

Linguistics and philosophy of language



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