An evaluation of Russell’s attack on Hegel

L.M. Geerdink

December 9, 2009

1 Introduction

Around 1900 B.A.W. Russell started to renounce the idealism that was dominantat that time in England. His strong interest in mathematics and its foundationsconvinced him that the monism that idealism postulated was false. He believedthat mathematics could not be understood without accepting a pluralism of enti-ties.1

The so called English Idealists, e.g. F.H. Bradley and J.M.E. McTaggart, werevery influential at that time, and Russell could not avoid engaging them in order tofurther his own logicist program, i.e. the program of proving that all mathematicscould be deduced from formal logic.

Part of this attack on English Idealism was an attack on all philosophical pro-grams that presupposed term-logic. Russell believed that using term-logic com-mitted one to a metaphysics of substances and attributes, which would directlylead to either a monadistic philosophy or a monistic philosophy. This attack onmonadistic and monistic philosophies is most strongly carried out in his book thePrinciples of Mathematics (Russell, 1937).

Monadistic philosophies transform all statements of relation into a multitude ofsubject-predicate sentences. We will not be concerned with Monadistic philosophyhere, but according to Russell these doctrines where for instance expounded byG.W. Leibniz and R.H. Lotze.2 Monistic philosophies, on the other hand, tryto reduce a relation between two or more objects to a statement about a singlesubject that encompasses all relata. It is this subject of which something is thenpredicated. Examples of monist philosophers are Spinoza, Bradley and G.F.W.Hegel.

Here we will only examine Russell’s attack on Hegel’s philosophy. This attackon Hegel was made explicit in Russell’s article Logic as the essence of philosophy

1Russell recounts this development in his essay Logical Atomism (Russell, 1959)2Lotze was a German logician and philosopher, who lived from 1817 till 1881.

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(Russell, 1914) and seems to have been very influential.3 We will first take acloser look at the exact critique that Russell gives on Hegel’s system. Then wewill take a quick detour, and try to determine why it is so important for Russellto refute all monistic philosophies. We will then try to evaluate this attack onHegel. In order to see if this attack on Hegel is justified we will study parts ofHegel’s Phanomenolgie des Geistes (Hegel, 2003) and Logik (Hegel, 1812). We willconclude this investigation in aporia. It seems that we can only show that Hegel’smetaphysics and the modern logic of relations and foundations of mathematics areinconsistent with each other.

2 Russell attack on Hegel’s use of term-logic

Russell believed that Hegel’s metaphysical system was ultimately the result ofHegel’s use of term-logic. According to Russell ‘[. . . ] Hegel’s doctrine, that philo-sophical propositions must be of the form, “the Absolute is such-and-such,” de-pends upon the traditional belief in the universality of the subject-predicate form’(Russell, 1914, p. 48). In order to evaluate Russell’s attack, we will first takea quick look at what term-logic is. We will then see why Russell believes thatterm-logic leads to either Monadism or Monism. Lastly we will see why Russellbelieves that Monism is inconsistent, and thus false.

Term-logic, or traditional logic, states that each proposition is made up of twoterms, the subject and the predicate. The sentence is true when the predicateholds for the subject. For example, the proposition “all men are mortal” is madeup of two terms: the subject all men and the universal predicate mortality. Thisproposition is true when it holds for all men that they are mortal.

Russell believes that Hegel’s term-logic leads to his metaphysical Monism.‘Now the traditional logic holds that every proposition ascribes a predicate toa subject, and from this it easily follows that there can be only one subject, theAbsolute, for if there were two, the proposition that there were two would notascribe a predicate to either’ (Russell, 1914, p. 48).4

3The exact influence that Russell has had on the analytical tradition goes beyond the scopeof this essay and will have to wait for another occasion. What we do know is that the analyticaltradition dislikes Hegel’s philosophy and does not really study it. I believe that this is in a largepart a direct consequence of Russell’s attack. Other philosophers that might have been a sourceof this dislike of Hegel in the analytic tradition are G.E. Moore, A.J. Ayer and K.R. Popper. Itis interesting to note that F.L.G. Frege, who is considered the father of modern logic, does noteven mention Hegel in any of his works. This is interesting because Frege also needs a pluralisticuniverse for his logicist program, just like Russell. That Russell does attack monistic philosophiesand Frege does not might be an indicator of the differences in philosophical climate in Germanyand England during these developments.

4As was stated in the introduction Russell actually does not believe that term-logic necessarily

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Let’s take a closer look at why Russell believes that term-logic leads to Monism.The problem starts when term-logic is confronted with a relational proposition,i.e. a proposition that asserts a relation between two or more entities. Let’s takethe proposition “a is greater than b” as our example. The problem for term-logichere is that it seems impossible to divide this proposition into two, i.e. a subjectand a predicate. The most natural division seems to be a division into three, thetwo terms a and b and the universal relation x is greater then y. Now term-logiccould try to analyze this sentence as “The universal predicate being greater then bholds for the subject a”. This is the Monadistic solution.5 The other way of tryingto deal with it is taking a and b to be a composite subject ab and predicate theuniversal diversity in magnitude with itself 6 of this subject. This is the Monisticsolution. If we try to solve the tension in the second, monistic, way, we will beforced to conclude that there is only one subject, the Absolute. This is so, becausewhenever we seem to have two entities and ask what the relation between thesetwo entities is, term-logic will force us to assert that these two entities are reallya composite entity.

Now Russell believes the Monistic metaphysics to be self-contradictory. Sinceit presupposes term-logic, it presupposes that there are at least two things, thesubject and the predicate. ‘For if the Absolute has predicates, then there arepredicates; but the proposition “there are predicates” is not one which the presenttheory can admit’ (Russell, 1937, 448). The Monist will indeed not admit the truthof the proposition “there are predicates” and thus is driven to the view that allpropositions are contradictory. ‘And hence we find monists driven to the view thatthe only true whole, the Absolute, has no parts at all, and that no propositions inregard to it or anything else are quite true—a view which, in the mere statement,unavoidably contradicts itself’ (Russell, 1937, p. 226).

3 Hegel’s Absolute and the pluralism of mathe-

matics

It seems that the logical conflict between Russell and Hegel can be reduced to theopposition between monism and pluralism. Hegel’s term-logic leads to monism,Russell’s modern logic absolutely needs pluralism. Russell thus attacks Hegelbecause pluralism is a necessary condition for mathematics. Any philosophy that

leads to a monistic metaphysics. It could also lead to a Monadistic metaphysics where there isa plurality of subjects and where each subject contains the whole universe.

5This analysis will lead to all entities being like Leibnizian Monads, because all subjects willultimately contain all their relations with other entities as part of their essence.

6This is Russell’s own example. It is clear that it is difficult to find an acceptable predicatethat holds for the complex subject here.

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denies pluralism must be false, since it cannot understand mathematics.The reason why Russell absolutely needs pluralism is that the logicist program

of deducing mathematics from logic seems to need the existence of a relation ofthe form “y is between x and z” (Russell, 1937, p. 217). This relation is necessaryfor the logicist program because it is only able to build up the ordering of thenatural numbers by making use of this relation.7 This relation between x, y andz must ultimately be analyzed as a relation between two different asymmetricalrelations, a relation between x and y (the relation of being smaller than) and arelation between y and z (the relation of being greater than). It is important tosee that these relations are asymmetrical. If x is smaller than y, then it cannotalso be the case that y is smaller then x. Saying that the subject xy has diversityin magnitude with itself simply does not preserve this asymmetry.

In order to have this asymmetry we absolutely need a plurality of entities.Since these relations are asymmetrical, x 6= y, i.e. if this relation holds betweenanything, then there are at the very least two different objects. Since the existenceof these asymmetrical relations is a necessary condition of the logicist program itis committed to a pluralistic metaphysics. Without pluralism it will be impossibleto deduce the natural numbers from logic.

4 Hegel’s term-logic

We have seen above that Russell attacks Hegel’s metaphysics on Hegel’s im-plicit use of term-logic. Russell believes this metaphysics to be ultimately self-contradictory, because it seems committed to the idea that the only true thing isthe Absolute, and that every proposition about this Absolute contains a contra-diction and is thus false.

In order to evaluate Russell’s argument we need to see if Hegel is indeed com-mitted to the doctrine that Russell ascribes to him. We will first examine if Hegelindeed uses term-logic and if this use of term-logic indeed leads him to accept onlyone subject, the Absolute. We will then see if Russell is correct in stating thatfor Hegel all propositions ultimately contain a self-contradiction and are thereforefalse.

It seems fair to ascribe a great role to term-logic in Hegel’s system. Hegel neverexplicitly states this, but the most telling sign of this is the fact that Hegel alwaysopposes two terms in his philosophical sentences. The dialectical movement nevertakes place between more than two terms. That this seems the only logical choiceshows the implicit influence that term logic has on our thinking.8

7Russell shows in the Principles of Mathematics that all other methods of trying to build uporder can be reduced to this relation of betweenness.

8This can be contrasted with Russell, who allows a proposition to contain a multitude of

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In the preface to his Phanomenologie des Geistes, Hegel discusses the relationbetween subject and predicate in thinking. He first makes a distinction betweenphilosophical thinking and mathematical thinking9, and tries to show that it isonly in philosophical thinking that we can see the necessity of the movement ofthought and thus have real understanding.

Mathematical thinking is defective,10 because:

In solchem unwirklichen Elemente gibt es denn auch nur unwirklichesWahres, d.h. fixierte, tote Satze; bei jedem derselben kann aufgehortwerden; der folgende fangt fur sich von neuem an, ohne daß der erstesich selbst zum andern fortbewegte und ohne daß auf diese Weise einnotwendiger Zusammenhang durch die Natur der Sache selbst entstunde.(Hegel, 2003, p. 40)

What is wrong with the relation between subject and predicate as understoodby mathematical thinking is, according to Hegel, that the predicate does not ex-press the essence of the subject. By attributing the predicate to the independentsubject we do not understand why it is precisely this predicate that belongs to thesubject, or why it is necessary that this predicate is predicated of the subject inthe first place.

In the Phanomeonolgie des Geistes, Hegel gives the example of trying to un-derstand what a right-angled triangle is. A right-angled triangle is a triangle thathas one of its interior angles measuring 90o. The proof that a right-angled trianglecan be constructed from a given straight line by ruler and compass can be foundin Euclid’s Elements. The problem with this proof is that: ‘die Bewegung desmathematischen Beweises gehort nicht dem an, was Gegenstand ist, sondern istein der Sache außerliches Tun’ (Hegel, 2003, p. 38). The proof does not showthe necessity of each step taken in the proof. It does not follow from trying tounderstand the essence of the right-angled triangle. That any of the proof is anecessary moment in the construction of the triangle is only seen at the end of theproof, when the construction of the triangle is completed:

Was das Erkennen betrifft, so wird vors erste die Notwendigkeit derKonstruktion nicht eingesehen. Sie geht nicht aus dem Begriffe desTheorems hervor, sondern wird geboten, und man hat dieser Vorschrift,

terms.9Since I take Russell’s philosophy to be a clear example of what Hegel calls mathematical

thinking, we will not discuss Hegel’s critique on formalistic thinking here.10Please be aware that Hegel does not believe mathematical knowledge to be false. Of course

Hegel believes that 3 + 6 = 9 is true, but it is true in an uninteresting way. Mathematicalknowledge is defective knowledge, since it does not concern the Absolute knowing itself. Math-ematics concerns the knowing of something that is outside, and therefore it is not philosophicalknowledge and thus falls outside the concept of Science (Wissenschaft).

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gerade diese Linien, deren unendliche andere gezogen werden konnten,zu ziehen, blindlings zu gehorchen, ohne etwas weiter zu wissen, alsden guten Glauben zu haben, daß dies zu Fuhrung des Beweiseszweckmaßig sein werde. (Hegel, 2003, p. 39)

Philosophical thinking, on the other hand, understands that we must under-stand the predicate to express the essence of the subject it is predicated of. Thus,we can now see that the sentence does not express that the predicate holds for thesubject, but that the subject and the predicate is essentially one and the same.

Formell kann das Gesagte so ausgedruckt werden, daß die Natur desUrteils oder Satzes uberhaupt, die den Unterschied des Subjekts undPradikats in sich schließt, durch den spekulativen Satz zerstort wird,und der identische Satz, zu dem der erstere wird, den Gegenstoß zujenem Verhaltnisse enthalt. (Hegel, 2003, p. 54)

This asserting of the identity of subject and predicate starts the dialecticalmovement. The identity that we try to assert between subject and predicatealways contains a contradiction, it tries to say about two that they are one. Thiscontradiction is resolved by understanding that there are not two subjects, butthat the difference between the two subjects should be negated by seeing that thesetwo things are really different aspects of a single whole. After seeing this, we canunderstand that both the subject and the predicate and their contradictory naturewere only necessary moments of this encompassing whole. We now appreciate thenecessity with which the latter has come forth out of the former.

We will look at one of these movements in detail, to see how it works. Inhis Logik, Hegel famously shows that ‘Das Seyn, das unbestimmte Unmittelbareist in der That Nichts, und nicht mehr noch weniger als Nichts’ (Hegel, 1812, p.22). This contradiction that Seyn is Nichts is resolved, by seeing that both arenecessary moments of an encompassing whole, das Werden.

The contradiction is derived by trying to determine what the Absolute is.Now we know from the Phanomonology, that Being is the Absolute.11 But, sinceBeing is the Absolute, we assert a relation, namely identity, between two thing,the Absolute stands in the relation of identity to Being. Since they are two,we have Being and something that is different from it, the Absolute. Since theAbsolute is different from Being, the Absolute is not-Being. Thus, Being is not-Being (Hegel, 1812, p. 36).12 To avoid the contradiction, thinking can try to think

11See the preface of the Logik why thinking is justified in taking this as its starting point.12I have reconstructed Hegel’s proof like this, because here it is most clearly seen how the

movement comes forth out of the difference between subject and predicate. Earlier in the LogikHegel tries to show that since pure Being does not give any determination it is the same asnon-Being, i.e. Being is non-Being. Here it is the contradiction between Being and non-Being

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of the Absolute as being not-Being. Thus, the Absolute is Being, namely Beingnot-Being. Again we have asserted that Being is not-Being.

As was said above, this contradiction is resolved by seeing that both are nec-essary moments of an encompassing whole, not-Being becoming Being and Beingbecoming not-Being. Thus the Absolute is not-Being becoming Being and Beingbecoming not-Being. This determining of the Absolute will however form a newcontradiction, the infiniteness of the Absolute will contradict the restless changingof the not-Being becoming Being, and so the dialectical movement continues fromwithin and shows the necessity of its movement.

Hegel’s metaphysics is build up by this dialectical movement of thought. Itneeds the contradictions in order to move from the necessary moments of the wholetowards the whole, i.e. the Absolute. This dialectical movement needs term-logicin order to function, it needs to oppose two terms, which it consequently sublates13

into a whole. The moment we accept relations between terms, what Hegel callsmathematical thinking, we will have ‘nur unwirkliches Wahres, d.h. fixierte, toteSatze’.

5 Does Russell actually refute Hegel?

As we have seen in section 2, Russell believes Hegel’s system to be ultimatelyself-contradictory. Any predicate we ascribe to the Absolute will result into a falseproposition.

But, as we have also seen, it is precisely this contradictory nature of term-logicthat is necessary for Hegel’s metaphysics. Absolute knowing is the movement ofthe Absolute thinking itself. It sees the sublating of the contradictions betweensubject and predicate into a subject of a higher order as necessary moments of themovement that is the pure concept of Absolute Knowledge. From within Hegel’ssystem it is not problematic to believe that all propositions contain a contradiction,it is necessary. This is Hegel’s famous law of contradiction.

On the other hand, assuming term-logic in order to accept Hegel’s metaphysicswill make it impossible to understand modern logic and modern developments inmathematics. Modern logic and modern set theory14 were build on the foundationsof the logicist program and presupposes its pluralism of entities. As we haveseen, without pluralism we cannot build up the order of the natural numbers,

that starts the movement, instead of the contradiction between the Absolute and Being. Boththese movements can be found in the Logik and I believe that they are essentially the same.

13I use the term “sublating” as a translation of the Hegelian “aufhebung”. The aufhebung isthe negation of the difference between subject and predicate into a whole that encompasses themboth.

14This can be seen by the heavy use modern set theory makes of predicate logic.

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and therefore we cannot understand the modern understanding of the foundationsof mathematics if we accept the universality of subject-predicate logic. Subject-predicate logic cannot understand the asymmetrical relations that the foundationsof mathematics absolutely need. Having to deny these modern developments seemsa very high price to pay.

Thus it seems impossible to find neutral ground to decide between these twostandpoints. From the standpoint of Hegel’s metaphysics, the relational thinking ofmathematics is defective. I(t does not show its own necessity. From the standpointof modern logic, Hegel’s system is flawed since it denies asymmetrical relations.

What is clear is that we have to choose between the relational logic of thelogicist program and Hegel’s dialectical movement of the Absolute via traditionallogic. It seems that we cannot accept both, unless we find a way to sublate theircontradiction.15

References

Hegel, G. (1812). Wissenschaft der Logik. Nuernberg: Johann Leonhard Schrag.

Hegel, G. (2003). Phaenomenology des Geistes. Stuttgart: Philipp Reclam jun.

Russell, B. (1914). Logic as the essence of philosophy. In Our knowledge of theexternal world. London: Allen and Unwin.

Russell, B. (1937). The principles of mathematics. New York and London: W.W.Norton and Company, 2nd edition.

Russell, B. (1959). Logical atomism. In Marsh, R., editor, Logic and Knowledge.London: Allen and Unwin.

15I am grateful to Peter Sperber for commenting on an earlier draft of this paper.

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