Russell en wiskunde

Russell’s logicism (the view that mathematics, or a part of mathematics can be reduced to logic) was an important part of his philosophy. Both in his Platonist and in his Logical Atomist phase, logic was an important part of Russell’s view on a priori truths. And, his mathematical logic was the logic of his philosophical program. So, Russell’s logic was the logic of Russell’s philosophical system. And Russell’s philosophy of mathematics and logic was also meant as a refutation of Kant’s idea that mathematical judgments cannot be understood as logical propositions. Furthermore, Russell held that his philosophy of mathematics shows that the forms of the Anschauung are irrelevant for mathematics. And he also held that it follows from his philosophy that space and time are objectively real and totally independent of any subject whatsoever. In the first sub-section of this section I shall discuss Russell’s logicism. And in the second sub-section his arguments against Kant shall be discussed. In the third sub-section I shall show a bit of the formal details of Russell’s philosophy of mathematics.

What were Russell’s arguments against Kant’s philosophy of mathematics?[1]

Russell writes that Kant thought that all propositions of mathematics are synthetic.[2] Kant deduces from this that mathematics can therefore be proved by loc alone. So, mathematical reasoning is based on something different. The axioms of mathematics are synthetic a priori truths, and the arguments based on them within mathematics are not purely logical. Russell also adds that the propositions of mathematics are about space, the eternal form of the Anschauung, and time, the internal form of the Anschauung. Geometry has space as its source, and arithmetic has time as its source. And any object must have space and time as its form. And since mathematics is about pure space and pure time, and since all objects must have space and time as their form, mathematics is necessarily applicable to all experience.

Russell stresses that the point here is that a priori intuitions provide us with the way of reasoning and inference. For the pure intuitions enables the subject to reason in a way which cannot be got from logic alone. Geometry, for example, needs figures for its proofs. Figures are of the essence in geometrical proofs. Russell held that space and time are subjective. He also concludes that the antinomies are used by Kant to show that if space and time are outside of experience, they are self-contradictory.

In order to refute Kant’s position, Russell must ask two questions: is mathematical reasoning the same as logical reasoning? And Are space and time contradictory?

Kant was of the opinion that mathematical propositions are synthetic a priori, and formal logical reasoning is analytical and a priori. Russell holds that since mathematical reasoning in the eighteenth century was defective, Kant thought that pure intuition was necessary for mathematical reasoning. But, Russell claims, since we now have the correct logic, geometry and arithmetic can be deduced from logic.

Kant might still say that a priori Anschauung shows that only three-dimensional Euclidean space is the space in which the actual objects of experience appear and are. And Euclidean three-dimensional space is the definition of an existent, or it is ‘an entity having some relation to existents’. But, says Russell’s, this makes no difference for the philosophy of mathematics, for mathematics is neutral about the question whether its objects exist or not.

With the conclusion Russell says that he has, on the bases of the arguments just given, shown that Kant is wrong about mathematics. Intuition is irrelevant for mathematics.

He then discusses the antinomies. Russell only discusses the first two antinomies. He takes them to be about infinity and continuity. Russell holds that these are not spatio-temporal.

The first antinomy is about whether the world has a beginning in space and time. So, Russell concludes, it is not about space and time, but about what is in them. Russell discusses both the thesis and the anti-thesis. He quotes Kant the following translation:

For assume that the world has no beginning in time, and then an eternity has passed away (abgelaufen) before every given point of time, and consequently an infinite series of conditions in the world has happened. But the infinity of a series consists in this, that it can never be completed by successive synthesis. Consequently an infinite past series of things in the world (Weltreihe) is impossible, and a beginning of the world is a necessary condition of its existence, which was first to be proved.[3]

According to Russell this argument makes use of a covert appeal to causality and the first cause. And Russell argued that Kant does not understand the class-concept any. Kant holds that when an event is preceded by another event it is such that the preceding event is definable by extension, which cannot be done in the case of an extension having an infinite number. Completion by successive enumeration cannot be done in this fashion. But, Russell, says, such a class is definable as ‘the class of terms having a specified relation to a specified term’. And if, in such a case, the class is infinite, it does not provide us with a contradiction. For classes can be defined intensionally.

Kant saw these previous events, in Russell’s view, as causes. He adds that if a cause is prior to its effects, and if the antinomy is about events, instead of moments, it is probably valid. But, he later argues, in chapter VII that causes and effects are logically on the same level. And therefore the thesis of the first antinomy is invalid.

Russell rejects the reasoning given in the antithesis. The argument goes as follows: Kant has no proof that space and time are themselves infinite. And no such proof exists. In fact such a proof cannot be given, since it depends on the axiom that there is a moment before every other moment, and a point in space beyond any given space. But, and this is crucial, the question whether the world is bounded by empty space cannot be solved in this way. In order to do that one needs a philosophy of causality. This will be made clear later on.

The second antinomy is about the question whether physical reality consists of elementary parts that cannot be divided further, or whether they can be divided indefinitely.

The thesis of the second antinomy is that every complex substance in the world is made up from simple parts. The antithesis is that no complex thing in the world is made from simple parts. In other words, every substance can always be divided, and every part of a substance can always be divided infinitely.

Russell argues that the thesis, as provided by Kant, is about things in space and time, and not about space and time. But, he says it is clear that the thesis can be expanded to include every collection, including space and time. The reason for this expansion is that this thesis is concerned with whole and part. It does not have a special relation to space and time, since it can also hold for numbers, such as the numbers between 1 and 2. Russell says that the proof of the thesis is valid, but terms and concepts ought to be substituted for substances. Russell adds that the notion that the relations between substances are accidental ought to be replaced by the notion that relations imply terms. Furthermore, Russell also points out that complexity implies relations.

Russell disagreed with Kant that the notion that space does not consists of simple parts, but of spaces is self-evident. But, the argument of the thesis is applicable to both space and other collections. It also demonstrates the existence of simple points, and these simple points are the elements out of which space is composed.

Kant argues that however much space is divided it will always only yield spaces, and not points. According to Russell this rests on the presupposition that if something is composed of points, it is composed of a finite number of points. But if one allows infinity of points, one can both hold that no division of space will yield a point, and still also claim that space is composed of points. And every finite space consists of simple parts, but not of a finite number of simple parts, it consists of an infinity of simple parts. And this holds for many collections. It is not specifically about space. It also applies to arithmetic. Russell therefore concludes that the thesis of the second antinomy should be a postulate of logic, and the antithesis should be rejected.

So, Russell concludes that the first and the second antinomy are not specifically about space and time; they are about continuous series (such as the real numbers). And he also claimed that, with the arguments contra Kant, he has also shown that absolute space exists.

In part VII of the Principles, Russell points out that if one accepts that space is composed of points, then one has to accept the possibility of matter. And this possibility must be discussed in the context of the question ‘what is matter?’ Furthermore, Russell adds that this question is about how matter occurs in rational Dynamics. The actual existence of matter is therefore a separate issue. [4]

The existence of space does not imply in and of itself that there are entities in space. Therefore it is the case that these entities must be grounded in something other than space itself, seen in isolation. Russell wants to ground these entities in the evidence of the senses.

Russell accepts the existence of the following terms: instants, points, terms which occupy instants but not points, terms which occupy points and instants.[5] Occupying a point or an instant is a fundamental relation, and therefore cannot be explained entirely by analysis. It is asymmetrical, indefinable, intransitive, and simple. And bits of matter are in space and time, as terms which occupy points and instants, but do not form the whole of the class of entities which occupy points and instants.

All unities are propositions or propositional concepts. But the unity of a proposition, which is in Russell’s view indefinable, is lost by analysis. And no enumeration of constituents can restore this unity. Russell concludes from this that nothing that exists is a unity, so if things are unities, things do not exist.[6] So, matter cannot be understood in terms substances.

Russell is not satisfied with the idea that matter consists of substances. He rejects the tradition of seeing things as substances with accidents. Instead, Russell accepts only the unity of absolute simples, and of wholes consisting of parts with relations. This complexity is conceptual. Russell means by this that is can be analyzed by the means of logic, but it is not dependant on the mind, and is therefore real. This leads Russell to the question about the difference between matter and secondary qualities. He concludes that matter and secondary qualities both belong to the class of things.[7]

Matter exists in space and time, and is fundamentally connected with them. A piece of matter cannot be at two places at the same moment. Nor can two bits of matter be at the same place at the same time. But, a piece of matter can occupy two moments at the same place, or two places at different moments. So, a division of time does not mean that matter can be divided in the same way, but a division of space does mean that matter can be divided in the same way. So, if something (matter) has extension it can always be divided. This property of spatial divisibility is what distinguishes matter from everything else. One and the same colour may occur at different places at the same time, but two colours cannot be at the same place at the same time. And pairs of qualities, such as hardness and colour, may occur at the same place at the same time. Matter and colour are indirectly related. The relation between matter and colour is only they occur at the same place.

Motion and time explain other properties of matter. Russell holds that every bit of matter must persist temporally. And it is at rest or in motion. So, if it moves, then its positions at different moments in time are continuous series which occur is space.[8] This means that matter does not bring with it a logical difference in the sense that it is the subject or predicate of a substance, or that it is a substance. The logical difference between matter and everything else is the relation between matter and space and time. And space consists of points, and time consists of instants.
Russell adds that we now have sufficient insight to see a part of rational Dynamics in terms of logicism. Space can be seen as an n-dimensional series and time as a one-dimensional series. Furthermore, the only relevant function of material points as far as Dynamics is concerned, is that they establish correlations between all moments in time and some spatial points. So, a point of matter brings about a correlation between all moments in time, and some spatial points. This means that it is possible to replace a material point by a many-one relation which has a one-dimensional series as its domain. The converse domain then consists of a three-dimensional series. In order to have a kinematical, material universe one has to say that the class of these relations is restricted by the condition that the intersection of any pair of these relations is the empty-class. Since the one-dimensional and the three-dimensional series are continuous, and given that the many-one relation is a continuous function, Russell has now given all of the conditions which one needs for a kinematic system of particles. And, since he sees the notion of class and the notion of class-membership as logical constants, he has given this system in terms of logical constants only.[9]

In the next section the Principles Russell discusses motion. He says that motion of any entity consists of the occurrence of that entity at a continuous series of places. And this occurs at a continuous series of moments. So, change is the difference in truth or falsehood of a proposition about an entity and a moment in time, and another proposition about the same entity at a different moment in time.[10] And the only difference between these propositions is that T occurs in the first one, and T’ in the second one. If a series of these propositions constitute a continuous series, and such a series is correlated with a continuous series of movements, then, change is continuous. Russell adds that existence at some moments, but not during all moments is what change amounts to in this view. So, motion is the occupation of different places during different moments of time, and motion is continuous.

In the next section Russell discusses causality. More specifically, it is about the question if causality plays a role in Dynamics. Causality could only occur in Dynamics as force. One might just take a descriptive view on force in Dynamics. But, Russell argues, that would make inferences from an occurrence at time T, and another occurrence at time T’ impossible. For such inferences contain relations of implication between events which occur at different moments in time. And in Russell’s view such relations can only be causal. But, in Dynamics only the ‘whole configuration of the material universe’ as a datum, seems to be the only form of causality which can be accepted. In order to see if causality in Dynamics is of this form, or if there is such a thing as causality between particulars within this context, Russell examines the notion of causality.

Russell starts this task with the question about the logical nature of causal proposition. Succession in time is not a direct relation between events. But, temporal succession can only be relations between moments, and not between events. So, one needs another notion of causality. And no causal relation involves reference to constant, particular parts of time. Causal relations themselves are eternal. So, if one analyses the causal relation between A and B, one says that A and B need not exist, but if A should exist at a moment in time, then B exists at later moment in time.

Since ‘A causes B’ has this form, neither A nor B needs to exist. For causes are implications. There can be series of causes which are parallel to other series of causes. And perhaps only one of them is actually realized. But, when either A or B exists, the other exists as well. And if one of them exists, then the other one also exits. And it exists either later or earlier. So, there is a connection between causality and time.

So, whatever exists in time has at least one causal relation. But, existence is not a necessary feature of causality, since non-existing items might have a causal relation. But, except space and time, every possible existent is a term which has at least one causal relation with at least one other term. Numbers are not included in this definition.

The most important question that Russell needs to discuss in the last part of the chapter on causality is, according to himself, the question if there are causal relations between individual events, or if there are only causal relations between different states of the universe, such that the first state is before the next one.

In order to answer this question, Russell states that causal relations between events do not hold between events and two times, but between events and three moments in time. Furthermore, in order to state any causal relation correctly, one needs to state the entire state of the universe at two of the three moments which are related to the events between which the causal relation holds. In the words of Russell:

Causality, generally, is the principle in virtue of which, from a sufficient number of events at a sufficient number of moments, one or more events at one or more new moments can be inferred. [11]

So, let e1 events occur at t1, and e2 events at t2, and let e3 events occur at t3, …, and let en events occur at tn. One can now conclude that en+1th event will occur at tn+1. In other words, the implication: if e1 events occur at t1, and e2 events at t2, and let e3 events occur at t3, …, and en events occur at tn, then en+1th event will occur at tn+1.

In order to explain Russell’s logicism, we must first get a better understanding of the theory of propositions that is closely connected with his logicism. Although sentences and language point to propositions, propositions are fundamentally non-linguistic. So, propositions consist of individuals, and universals. Relations are among those universals. In fact, Russell’s rejection of both Aristotelian, Leibnizian, and Hegelian philosophy is closely connected to his notion of relations.

Propositions of logic, generalized propositions are not complexes consisting of all the propositions that fall under it, such that these propositions are parts of the generalized proposition. Generalized propositions have variables, and these variables are connected to the propositions that fall under the variable in a way that resembles the logical function of any. [12]

Russell held that, since logicism holds, the idea that mathematical judgments are not based on logical principles alone is false. This means that the forms of the Anschauung are not necessarily involved in mathematical judgments. But Russell went further. The idea that space and time are empirically real but transcendentally ideal was rejected by him. In Russell’s view space and time have being independent of the subject.

The logic of the Principia Mathematica was seen, by Russell, as the a priori part of every judgment in so far as that judgment was mathematical. And this kind of a prioricity needed to be connected with the notion of causality.

3.2. As we have seen, Russell was a logicist. The kind of logicism that Russell held was of the following kind: He said that a mathematical proposition contains only variables and logical constants. In the chapter Definition of Pure Mathematics pure mathematics is defined as follows:

Pure mathematics is the class of all propositions of the form ‘p implies q’, where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants.[13]

And mathematical propositions are characterized by the fact that they contain implications. To quote Russell again:

We assert always in mathematics that if a certain assertion p is true of any entity x, or of any set of entities x, y,z, … , then some other assertion q is true of those entities; but we do not assert either p or q  separately of our entities. We assert a relation between the assertion p and q, which I shall call formal implication.[14]

So, according to Russell, mathematical assertions consist of variables, and logical constants, and express implications. What does this mean for Russell’s theory of proof?

In the Principia Mathematica Russell and Whitehead provide us with a theory of proof. They say, as Russell also wrote in Introduction to Mathematical Philosophy, that a proof consists of a relation between a premise and a conclusion. And the relation between the premises and the conclusion of a proof is a relation of implication. So, a series of statements is a proof, if and only if, the premise(s) imply the conclusion.

Russell and Whitehead say that p implies q means that the truth of p implies the truth of q. And this means that:

p ⊃ q = def.  ~p v q. (This is number * 1.01 in PM.)

Russell and Whitehead explain implications as a combination of negation and disjunction. So, if ~p v q is true, and if p is true, then: q is true. Therefore, the truth of p implies the truth of q. In the system of the PM deductions are implications of the premises and definitions of the system.

The propositional calculus of the PM consists of five primitive propositions. These are, together with anything implied by a true premise is true Primitive proposition (Pp), these premises[15]are:

(1)   |- :  p v p . ⊃ . p                                 Pp

(2)    |- :  q . ⊃ . p v q                                Pp

(3)   |- :  p v q . ⊃ .  q v p                          Pp

(4)   |- :  p v (q v r) . ⊃ .  q v (p v r)                Pp

(5)   |- :.  q ⊃  r  . ⊃ : p v q .  ⊃  . p v r  Pp

Together with the definition of implication one can now understand an example of what a Russellian proof looks like:

*2.03. |- :  p ⊃ ~q . ⊃ .  q ⊃ ~p

            |- :  ~p v  ~q . ⊃ .  ~q v ~p      [Perm.: ~p/ p,  ~q/q, and (1). (*1.01)]

              |- :  p ⊃ ~q . ⊃ .  q ⊃ ~p

The proof works by saying that ~p v  ~q, and ~q v ~p is asserted. And, next, one replaces the propositions with disjunctions by ones with implications (as is allowed according to definition 1.01).

In part I section B, Russell and Whitehead provides us with the premises of predicate logic.[16] These are:

*9.1. |- :     Φx . ⊃ .  (∃z)  Φz             Pp

*9.11. |- : Φx v Φy . ⊃ .  (∃z)  Φz      Pp

And if one also takes the following definitions into account, we can discuss an example of another deduction. The definitions are:

*2.1. |-  : ~p v  p

*9.01. |- : ~{ (x) . Φx} . = . (∃x) .  ~ Φx

*9.05. |- : (∃x)  . Φx . v . p : = . (∃x)  Φx v p

The proof I would like to show is:

*9.2. |- : (x) . Φx . ⊃ . Φy

Proof.

|- . *2.1 . ⊃ |- .  ~Φy v  Φy

|- . *9.1 . ⊃ |- .  ~Φy v  Φy . ⊃  . (∃x)  ~Φx v   Φy

|- . (1) . (2) . *1.11 . ⊃ |- . (∃x)  .  ~Φx v  Φy

[(3).(*9.05)]                  |- : (∃x)  .  ~Φx . v  . Φy

[(4).(*9.01.*1.01)]        |- : (x) . Φx . ⊃ . Φy

The next important issue for the purpose of this article is Russell’s notion of what numbers are. According to Russell numbers are properties of classes. And classes have the same number when there is a one-one correspondence between the members of those classes. One might wonder if this definition is circular, since it appears to use the notion ‘1’. But this is not the case. For one-one correspondence is defined as: a relation is one-one, if x and x’ are related to y and y’ and x and x’ are identical, and y and y’ are identical. [17] So a number is a class of similar classes. [18]

After having provided us with a definition of number, Russell explains addition and multiplication. Arithmetical addition is a special case of logical addition. And logical addition is the same as disjunction. So, the sum of p and q is ‘p or q’. And if u and v are classes, then their sum is ‘u or v’. Addition may also be defined as a specific case of a logical product.[19]

But Russell’s notion of arithmetic can be discussed in a bit more detail if we take his discussion of Peano Arithmetic into account. Peano’s theory of arithmetic works with three fundamental notions and five fundamental propositions. The fundamental notions are: 0, finite integer, and successor of. The fundamental propositions are:

1.      0 is a number.

2.      If a is a number, the successor of a is a number.

3.      If two numbers have the same successor, these two numbers are identical.

4.      0 is not the successor of any number.

5.      If s be a class to which o belongs, and also the successor of every number belonging to s, then every number belongs to s.[20]

For various reasons which fall outside the scope of this article, Russell rejected Peano Arithmetic and replaced it by the following theory[21]:

There is a class of entities that belongs to aleph-0, such that:

1.      0 is the class of classes whose only member is the null-class.

2.      A number is the class of all classes similar to anyone of themselves.

3.      1 is the class of all classes which are not null, and are such that, if x belongs to the class, the class without x is the null-class or, if x and y belong to the class, then they are identical.

4.      Having shown that if two classes are similar, and a class of one term is added to each, the sums are the same. We define that, given any number, n+1 is the number that is the result of adding a unit to a class of n terms.

5.      Finite numbers are those numbers belonging to every class s to which 0 belongs, and to which, if n belongs, n + 1 belongs.

As the observant reader will have noticed, all of Peano’s propositions are satisfied by Russell’s theory.

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[1] The following explanation is based on Russell (1903), p. 457-461, Chapter III: Kant’s theory of space.

[2] Kant writes about judgemtns, but Russell atributes the notion of synthetic propositions to Kant.

[3] Quotation as given in Russell (1903), p. 459.

[4] See Russell (103), p. 465, Part VII, Matter and Motion, chapter III Matter.

[5] This list is taken directly from Russell (1903), p. 465.

[6] Russell (1903), p 466-467.

[7] Russell (1903), p. 467. Russell adds that the only  classes as far as matter is concerned  are: things, predicates, and relations.

[8] Russell (1903), p. 468.

[9] Russell (1903), p. 468.

[10] This formulation is a direct paraphrase of the corresponding remark in Russell (1903), p. 469.

[11] Russell (1903), p. 478.

[12] See Hylton (1990).

[13] Russell, The Principles of Mathematics (London, 1903), Chapter I, p. 3.

[14] Russell, The Principles of Mathematics (London, 1903), Chapter I, p. 5.

 

[15] Russell, and Whitehead, Principia Mathematica (London, new York, 1910), p; 13.

[16] To which of course the Theory of Types is applicable.

[17] PoM, p. 113.

[18] PoM, p. 117.

[19]PoM, p. 117.: ‘ As the class of terms belonging to every class in which both u and v are contained.’

[20] Formalized, and with the axioms about multiplication Peano Arithmetic becomes: Let ‘S’ be the successor-function.

(x) (y) (Sx = Sy à x = y)

(x) ~( Sx = 0)

(x) ( x + 0 =x)

(x) (y) ((x + Sy) = (S(x + y))

(x) ( x . 0 = x)

(x) (y) (( x . Sy) = (x . y +x))

(Φ0 & (n) (Φn à S(Φn)) à (n) (Φn))))

[21] Observe that Russell’s theory is in accordance with logicism: all constants are only logical constants.