Knowledge and the Philosophy of Number

What Numbers Are and How They Are Known

Keith Hossack

• Publisher: Bloomsbury Academic
• Year: 2020
• ISBN:
  9781350102927
• Edition: 1
• Topic: Mathematics
• Subtopic: History & Philosophy of Mathematics
• Index: Mathematics

If numbers were objects, how could there be human knowledge of number? Numbers are not physical objects: must we conclude that we have a mysterious power of perceiving the abstract realm? Or should we instead conclude that numbers are fictions? This book argues that numbers are not objects: they are magnitude properties. Properties are not fictions and we certainly have scientific knowledge of them. Much is already known about magnitude properties such as inertial mass and electric charge, and much continues to be discovered. The book says the same is true of numbers. In the theory of magnitudes, the categorial distinction between quantity and individual is of central importance, for magnitudes are properties of quantities, not properties of individuals. Quantity entails divisibility, so the logic of quantity needs mereology, the a priori logic of part and whole. The three species of quantity are pluralities, continua and series, and the book presents three variants of mereology, one for each species of quantity. Given Euclid's axioms of equality, it is possible without the use of set theory to deduce the axioms of the natural, real and ordinal numbers from the respective mereologies of pluralities, continua and series. Knowledge and the Philosophy of Number carries out these deductions, arriving at a metaphysics of number that makes room for our a priori knowledge of mathematical reality.

1

Properties

The thesis of this book is that numbers are magnitudes. A system of magnitudes is by definition a plurality of properties that are determinates of a determinable and that have a characteristic algebra. The different types of number are different systems of magnitudes. Magnitudes are properties, and the present chapter makes the case for property realism – the metaphysical theory that properties are entities that really exist.

Section 1.1 recalls Aristotle’s doctrine that in a judgement there is a part of reality that is the subject, and another part of reality which is predicated. Section 1.2 lists the principal theories of predication, including property realism and various alternatives. Section 1.3 discusses Davidson’s theory that there are no predicables and that no entities at all need be mentioned in the semantics of predicates. His theory is criticized on the grounds that it entangles us in the semantic paradoxes. Section 1.4 makes the case for property realism and criticizes two versions of nominalism on the grounds that they cannot explain the difference between a natural class and a merely miscellaneous collection. Section 1.5 argues that there are many different types of properties. Section 1.6 introduces the type of properties that are magnitudes. Section 1.7 says that extensive magnitudes have a common algebraic structure and are needed in the theory of ratio and proportion, which is indispensable in the physical sciences. Section 1.8 says that many kinds of extensive magnitudes are known to exist and that numbers are extensive magnitudes too.

1.1 Predicables

If numbers are a part of reality, and numbers are properties, then properties are a part of reality. But does it even make sense to speak of parts of reality? Parmenides thought not – he taught that reality is One:

[The One] is now, all at once, one and continuous. … Nor is it divisible, since it is all alike; nor is there any more or less of it in one place which might prevent it from holding together, but all is full of what is. (Frag. B 8.5–6, 8.22–24. In Cornford 1939: 35–9)

But the doctrine that there is only one thing is absurd. Reality is a many, namely the totality of what Aristotle calls ‘things themselves’(1941a: 1a20).

Aristotle sees classification as a central task of science. Each special science taxonomizes the things in the part of reality that fall within its special province; Aristotle thought that metaphysics, the most general of all sciences, must taxonomize at the very highest level of generality. He first divides all the ‘things that are’ into two categories: the predicables ‘such as man or horse’ and the impredicables, ‘such as the individual man or horse’ (1941a: 1b4).

Definition. An entity is called a predicable if it can be predicated of a subject.

Aristotle’s doctrine is that reality divides into two highest kinds: the category of things that can be predicated and the category of things that cannot be predicated.

1.2 Different Accounts of Predication

Property realism agrees with Aristotle that the category of predicables is needed in the account of what it is for something to fall under or satisfy a predicate. For example, according to property realism, the predicable wisdom, ‘wisdom itself’ as Plato would say, is a being of a fundamentally different kind from the individual beings such as Socrates who are instances of wisdom. There are three versions of property realism: (i) the extremely abundant theory says that for every class, there is a property which ‘unites the class’ – every member of the class instantiates the property (Lewis 1983); (ii) the moderately abundant theory says that for every possible predicate there exists a corresponding property:

Every meaningful predicate stands for a property or relation, and it is sufficient for the actual existence of a property or relation that there could be a predicate with appropriate satisfaction conditions. (Hale 2013: 133)

(iii) the sparse theory says that properties are in much shorter supply, for it is only the natural classes that have members united by a common property, so only those predicates express a property whose extension is a natural class. All realist theories agree that if ‘wise’ predicates the property wisdom, then y satisfies ‘x is wise’ if y instantiates wisdom. Sparse property realism is the theory I shall be advocating. Alternative theories of predication may be classified as follows.¹

1. Davidson’s theory is a nominalist theory that denies the existence of properties. According to Davidson, the theory of satisfaction does not need to mention any entities whatsoever: the semantics of a predicate is given simply by stating the satisfaction condition of the predicate. When we have given the satisfaction condition for each predicate, we have given the whole semantics of predication for the language, without needing to mention any non-linguistic entities.

2. The extension theory is another nominalist theory. It says that there is indeed a specific part of reality associated with each predicate, but this is just its extension, in other words the plurality (alternatively, the set) of things that fall under the predicate. On the extension theory, y satisfies ‘x is wise’ if y is an element of the set of wise persons: there is no need to invoke a separate category of predicables. To deal with counterfactual situations, the extension theory may invoke the predicate’s intension, the supposed function that maps each possible world to the extension the predicate would have were that world actual.

3. The paradigms theory is a nominalist theory that says the specific part of reality associated with a predicate is its collection of paradigms, which can be just a few archetypal instances of the predicate, not its whole extension. On this account, y satisfies ‘x is wise’ if y relevantly resembles paradigm wise people, such as Socrates.

4. On Frege’s theory, the entities that can be predicated are categorially different from other things. Frege calls them Concepts: they are not properties, but functions from objects to the truth values – the True and the False. On Frege’s theory, y satisfies ‘x is wise’ if the value of the Concept presented by ‘x is wise’ is the True for the argument y.

1.3 Criticism of Davidson

According to Davidson, there is no need to postulate predicable entities, for a semantic theory that conforms to Tarski’s ‘Convention T’ can be ‘enough for an interpreter to go on’ (1984: xiv). ‘Convention T’ is the requirement that a semantic theory for a language ℒ must entail for each sentence s a theorem of the following form:

s is true in ℒ if and only if p.

where p is the sentence of the metalanguage that translates s. If the semantic theory of ℒ is given in ℒ itself, Convention T reduces to the disquotational requirement that the theory prove every instance of the following schema:

‘s’ is true in ℒ if and only if s.

Such a semantic theory should be finitely axiomatizable if it is to serve its interpretative purpose. A theory might typically include axioms of three types: axioms of the first type say for each referring term in ℒ what its referent is, axioms of the second type say for each predicate what its satisfaction condition is and axioms of the third type are compositional. Such a semantic theory for English, if given in English, could include the following axioms:

(1) ‘Snow’ refers in English to snow.

(2) ∀y (y satisfies ‘x is white’ in English ↔ y is white.)

(3) A subject–predicate sentence is true in English if and only if the referent of the subject term satisfies the predicate.

These three axioms allow the theory to prove the following familiar theorem:

‘Snow is white’ is true in English if and only if snow is white.

Axioms of the first type look outside language: axiom (1) is ontologically committed to the existence of snow. But axioms of the second type do not commit us to the existence of any entity outside language. According to Davidson, the satisfaction relation is fully characterized by the collection of all disquotational axioms like (2) – one axiom for each predicate of the language: ‘The recursive definition of satisfaction must run through every primitive predicate in turn’ (1984: 47). Beyond the totality of these axioms there is nothing more that can be said, or that needs to be said, about what satisfaction is.

But the collection of axioms like (2) cannot be the whole explanation of what it is for a thing to satisfy a predicate. According to the disquotational theory, for each one-place predicate A(x) of English, the satisfaction relation must meet the following condition:

(4) ∀y (y satisfies ‘A(x)’ in ℒ ↔ A(y))

For example, Socrates satisfies ‘x is wise’ in English if and only if he is wise. But this account of satisfaction cannot be correct, for ‘x does not satisfy x’ is itself a one-place predicate of English, so substituting it for ‘A(x)’ in (4) yields:

∀y (y satisfies ‘x does not satisfy x’ ↔ y does ...