The significance of Our Knowledge of the External World is that it proposed a new method for philosophizing. Although the method suggested was not strictly novel (it had been previously used by such mathematical logicians as George Boole, Giuseppe Peano, and Gottlob Frege), it was modified by Bertrand Russell and transferred from mathematical to philosophical subject matter. Russell called it the “logical-analytic” method or the method of “logical atomism.” Since the publication of these lectures, the method has been taken over and radically modified and broadened by a large school of philosophers who call themselves “analysts” and who constitute an important group in the modern philosophical world.

The analytic method, as Russell formulated it, is less ambitious than that of classical philosophy in that it does not claim to determine the nature of reality or the universe as a whole. What it does is both less speculative and less sweeping, but more scientific. It uses the techniques of modern logic and modern mathematics and employs such concepts as class, relation, and order for the purpose of clarifying and solving some of the perennial problems of philosophy that have not yet yielded to satisfactory solution. There are many such problems, but Russell here considers those concerned with the nature of the external world, with how the world of physics is related to the world of ordinary sense experience, and with what is meant by space, time, continuity, infinity, and causation. The book consists of illustrations of the application of the logical-analytic method to these problems for the purpose of showing its fruitfulness. Russell insists that his results are to be taken as tentative and incomplete, but he believes that if any modification in his method is found necessary, this will be discovered by the use of the very method that he is advocating.

The Faults of Traditional Logic

The philosophies of two typical representatives of the classical school—F. H. Bradley and Henri Bergson—are first examined in order to show the errors that these men made. Bradley found the world of everyday life to be full of contradictions, and he concluded that it must be a world of appearance only, not of reality. His error lay in his attempt to determine the character of the world by pure reasoning rather than by going to experience and examining what he found. Bergson believed that reality is characterized fundamentally by growth and change; he then concluded that logic, mathematics, and physics are too static to represent such a world and that a special method called “intuition” must be employed. His mistake was twofold: First, he supposed that because life is marked by change and evolution, the universe as a whole must be so described; he failed to realize that philosophy is general and does not draw conclusions on the basis of any of the special sciences. Second, his emphasis on life suggests that he believed philosophy to be concerned with problems of human destiny. However, such is not the case; philosophy is concerned with knowledge for its own sake, not with making people happier.

One reason why the logical-analytic method was not more widely employed earlier, according to Russell, is that it was new and only gradually replaced some of the earlier and erroneous conceptions of logic. Traditional (syllogistic) logic had been quite generally abandoned as inadequate. The inductive logic of Francis Bacon and John Stuart Mill had been shown to be unsatisfactory because it cannot really show why people believe in such uniformities as that the sun will rise every morning. Belief in universal causation cannot be a priori because it is very complicated when formulated precisely; nor can it be a postulate, for then it would be incapable of justifying any inference; nor can it be proved inductively without assuming the very principle that one is trying to prove. Russell claims that Georg Wilhelm Friedrich Hegel made the mistake of confusing logic with metaphysics, and that only mathematical logic provides the tool by which one may hope to solve philosophical problems.

The Logical-Analytical Method

Mathematical logic is both a branch of mathematics and a logic that is specially applicable to mathematics. Its main feature is its formal character, its independence of all specific subject matter. Looked at formally, it can be said to be concerned with propositions. The main types of propositions are atomic propositions, such as “Socrates is a man”; molecular propositions, which are atomic propositions unless connected by and, if, or unless; and general propositions, such as “All men are mortal.” In addition, there must be some knowledge of general truths that is not derived from empirical evidence. For example, if we are to know that all men are mortal, we must also know that the men we have examined are all the men there are. This we can never derive from experience because empirical evidence gives us only particular truths. Thus, there are certain general truths that, if they are known, must be either self-evident or inferred from other general truths.

It is obvious that one of the oldest problems of philosophy is the problem concerning knowledge of the external world. In applying the logical-analytical method to this problem, Russell finds that what we have to begin with (something that is always vague, complex, and inexact) is knowledge of two kinds: First, our acquaintance with the particular objects of everyday life—furniture, houses, people, and so on; and second, our knowledge of the laws of logic. These may be called...

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Sense-Data and Physics

Another philosophical problem that yields to Rusell when his logical-analytic method is applied to it is that of the relation between the world of physics and the world of sense-data. According to physics, the world consists of material bodies, located in space and time and having a high degree of rigidity and permanence. Although the bodies themselves may change, they are made up of particles that are themselves unchanging and indestructible. On the other hand, the world of sense-data for any one of us comes and goes; even such permanent things as mountains become data for us only when we see them; they are not known with certainty to exist at other moments. The problem of “getting these worlds together” involves the attempt to construct things, a single space and a single time out of the fleeting data of experience.

Permanent things can be constructed if we can find some way of connecting what we commonly call “appearances of the same thing.” Although some sort of continuity among appearances is necessary, it is not sufficient to define a thing. What is needed in addition is that the appearances of a single thing obey certain physical laws that the appearances of different things do not.

In much the same way, the dimensionless points (no one has ever seen a perfect point) of physics must be constructed out of the surfaces and volumes of our sensory experience. Using the fact that spaces can be observed to enclose other spaces (like Chinese boxes), Russell constructs “enclosure-series,” which can be called “point-producers” because points can be defined by means of them. Russell does not, however, define a point as the lower limit of an enclosure-series, for there may be no such limit. Instead he defines it as the series itself; thus, a point is a logical entity constructed out of the immediate data of experience.

Time and Continuity

It can now be seen how Russell derives the concept of durationless instants of time. Events of our experience are not instantaneous; they occupy a finite time. Furthermore, different events may overlap in time because one event may begin before the other, but there may be a common time during which they both occur. If we restrict ourselves to the time that is common to more and more overlapping events, we get durations that are shorter and shorter. Then we can define an instant of time as the class of all events that are simultaneous with one another. To state accurately when an event happens, we need only to specify the class of events that defines the instant of its happening.

Russell recalls that since the days of Zeno, philosophers have wondered about the problem of permanence and change, and especially about the apparent paradox of motion. Some motions, such as that of the second hand on a watch, seem to be continuous because we can actually see the movement; other motions, such as that of the hour hand, seem to be discontinuous because we can observe only a broken series of positions. Russell’s solution to this apparent contradiction in the nature of motion lies in the mathematical theory of continuity.

Continuity is regarded as, first, a property of a series of elements, of terms arranged in an order, such as numbers arranged in order of magnitude. Second, a continuous series must in every case be “compact,” which means...

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Zeno’s Paradoxes

After a discussion of German philosopher Immanuel Kant’s conception of infinity, Russell turns to an analysis of Zeno of Elea’s famous paradoxes. Zeno had four arguments designed to show the impossibility of motion—all based on his theory of the infinite divisibility of space and time. His first argument was designed to prove that a runner cannot get to the end of a race course, for if space is infinitely divisible, he will have to cover an infinite number of points in a finite time, and this is impossible. According to Russell, Zeno’s error came from thinking that the points must be covered “one by one.” This would, apparently, require an infinitely long time, for the series has infinitely many elements. However, the...

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Infinite and Finite Numbers

According to Russell’s analysis, infinite numbers differ from finite numbers in two ways: First, infinite numbers are reflexive, while finite numbers are not, and second, finite numbers are inductive, while infinite numbers are not. A number is reflexive when it is not increased by adding 1 to it. Therefore, the number of members in the class 0, 1, 2, 3 . . . n is the same as the number in the class 1, 2, 3, 4 . . . n plus 1. Even more surprising, the number of even numbers is equal to the total number of numbers (even and odd). It is therefore possible in the case of infinite numbers for a part of a class to equal the whole class.

Infinite numbers are also noninductive. Inductive...

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Causation and Free Will

Russell’s final application of the logical-analytic method is to the problem of causation and free will. Causation is a complicated relation that must be described by a carefully worded formula: “Whenever things occur in certain relations to each other (among which their time-relations must be included), then a thing having a fixed relation to these things will occur at a date fixed relatively to their dates.” The evidence for causation in the past is the observation of repeated uniformities, together with knowledge of the fact that where there appears to be an exception to such uniformities, we can always find wider ones that will include both the successes and failures. The only guarantee we could have that such a causal...

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Additional Reading

Dejnozka, Jan. Bertrand Russell on Modality and Logical Relevance. Avebury Series on Philosophy. Aldershot: Ashgate, 1999. This work presents a criticism and interpretation of modality and logical relevance in the work of Bertrand Russell. Includes index.

Grayling, A. C. Russell. Past Masters series. Oxford: Oxford University Press, 1996. This work, part of a series on great thinkers, covers the life and accomplishments of Russell. Includes an index.

Irvine, A. D., ed. Bertrand Russell: Critical Assessments. London: Routledge, 1999. This book critically...

(The entire section is 390 words.)