Scientific Method in Philosophy is a collection of lectures delivered by Bertrand Russell that seeks to demonstrate the nature, capacity, and limitations of the logical-analytic method in philosophy. The work represents Russell’s attempt to articulate a philosophical approach that could yield objective scientific knowledge, in contrast to the temperament-dependent systems of past philosophers. The book is presented as a series of lectures, beginning with an examination of current tendencies in philosophy and proceeding through detailed analyses of logic, our knowledge of the external world, the nature of matter, the problem of infinity, and the notion of cause.
The author explicitly states that the work is tentative and incomplete, claiming no finality except on the theory of the mathematical infinite developed by Georg Cantor. Russell acknowledges contributions from Dr. Whitehead, particularly regarding the definition of points, the treatment of instants and “things,” and the conception of physics as a construction rather than an inference, as detailed in the fourth volume of Principia Mathematica. He also acknowledges unpublished logical discoveries by Ludwig Wittgenstein. The work draws on the mathematical achievements of Cantor, who removed the difficulty of the mathematical infinite, and Gottlob Frege, whose writings provide the first complete example of the logical-analytic method in philosophy.
The book is situated in the early twentieth century, a period when philosophy was dominated by the classical tradition descending from Kant and Hegel, and by evolutionism derived from Darwin through Herbert Spencer, William James, and Henri Bergson. Russell positions his own approach—which he calls “logical atomism”—as a third alternative, representing an advance comparable to Galileo’s in physics.
Context
The work emerges from a philosophical landscape in which three principal tendencies competed for dominance. The classical tradition, descending from Greek faith in reason and geometry’s a priori method, had accepted Aristotelian logic uncritically. Russell illustrates this tradition through Mr. Bradley’s Appearance and Reality, which condemns relations as self-contradictory, arguing that relations require new connecting relations endlessly. The classical tradition’s logic is constructive through negation, whereas true logic, Russell argues, is analytic, showing possibilities rather than legislating what is. The doctrine of the universe as “organic unity” and the doctrine that all reality is “mental” or “spiritual” are both fallacious, in Russell’s view.
The second tendency, evolutionism, had become the prevailing creed, dominating politics and literature. Darwin’s Origin of Species swept away natural kinds, and evolutionism assumed progress from amoeba to man—a hasty generalisation in Russell’s assessment. Bergson’s version rejects both mechanism and radical finalism, since both make time useless; his finalism depends on life as a continuous stream where all divisions are artificial. Russell criticises evolutionism on two grounds: its truth does not follow from scientific facts, and its interests are practical rather than theoretical. Philosophy, he insists, must renounce mundane desires and aim at understanding general aspects of the world.
The work also responds to the common opposition between instinct and reason that had shaped modern thought. In the eighteenth century, reason was favoured, but under Rousseau and the romantic movement, instinct was preferred—first by rebels against artificial government and thought, then by those feeling that science menaced creeds tied to a spiritual outlook. Bergson, under the name “intuition,” had raised instinct to sole arbiter of metaphysical truth. Russell argues this opposition is mainly illusory: instinct, intuition, or insight first leads to beliefs which reason confirms or confutes; confirmation consists of agreement with other equally instinctive beliefs. Reason is a harmonising, controlling force, not a creative one.
The book is written against the background of developments in mathematics and physics that had transformed the intellectual landscape. The discovery of incommensurables by the Pythagoreans, the development of Cartesian geometry, the rise of the electromagnetic theory of matter, and the principle of relativity all inform Russell’s analysis. The problem of infinity, which had troubled philosophers from Zeno to Bergson, had been definitively solved by Cantor, whose work first appeared in 1882. Frege’s logical analysis of number, published in his Die Grundlagen der Arithmetik (1884), provided the foundation for Russell’s approach to logic and mathematics.
Structure of the Book
The book is organised as a series of lectures that proceed from general considerations about the state of philosophy to specific technical analyses of logic, knowledge, matter, infinity, and causation. The opening lecture, “Current Tendencies,” surveys the three principal types of philosophy—the classical tradition, evolutionism, and logical atomism—and argues for the superiority of the logical-analytic method. Russell discusses the opposition between instinct and reason, criticises Bergson’s advocacy of intuition, and argues that philosophy must pursue ethical neutrality, seeking facts rather than attempting to demonstrate that the world has desirable ethical characteristics.
The second lecture argues that all genuinely philosophical problems reduce to problems of logic. Russell distinguishes knowledge of logical forms from knowledge of existing things, explains the role of inference, and criticises traditional logic for assuming that all propositions have the subject-predicate form. He introduces the classification of relations into symmetrical, non-symmetrical, and asymmetrical, and into transitive, non-transitive, and intransitive, showing that asymmetrical relations are involved in all series and that denying them condemns the world of sense as illusory. The lecture also introduces the distinction between atomic and molecular propositions, and between atomic and general propositions, arguing that some general knowledge is independent of empirical evidence.
The subsequent lectures turn to the problem of our knowledge of the external world. Russell begins with the data of common knowledge, distinguishing between hard data (particular facts of sense and general truths of logic) and soft data (psychologically derivative but logically primitive beliefs). He poses the problem of whether the existence of anything other than our own hard data can be inferred from those data, and distinguishes two problems: whether objects of sense exist unperceived, and whether other objects inferable from sense can exist. The discussion leads to the conclusion that the common-sense belief in persistent things arises from combining a belief that something persists with the fact that sensations change.
The fourth lecture addresses the apparent difference between matter as physics describes it and things as they appear in sensation. Russell argues that the only possible justification for physics must exhibit matter as a logical construction from sense-data. He traces the history of the belief in indestructible “things” from the common-sense belief in fairly permanent and rigid bodies through the atomic theory to the electromagnetic theory. He proposes that a “thing” may be defined as a certain series of appearances, connected by continuity and causal laws, and invokes Occam’s razor: entities are not to be multiplied without necessity.
The fifth lecture continues the analysis of matter and motion, discussing the mathematical definition of continuity and addressing Zeno’s paradoxes. Russell argues against the notion of consecutive points or instants, showing that the hypothesis of consecutive points and instants leads to unacceptable conclusions. He presents the mathematical definition of continuity, which admits points and instants while denying infinitesimals, and discusses three answers to the objection that we see fast motion directly: the physiological, psychological, and logical answers.
The sixth lecture considers the problem of infinity historically, beginning with Kant’s first antinomy. Russell criticises Kant’s argument that the world must have a beginning in time because an infinite series can never be completed by successive synthesis, showing that Kant only proves at most that an infinite series cannot be completed in a finite time. The lecture then traces the problem of infinity from Pythagoras through Zeno to Cantor, discussing the discovery of incommensurables and its fatal consequences for the Pythagorean philosophy that number is the constitutive essence of all things.
The seventh lecture presents the positive theory of infinity. Russell distinguishes philosophy from mathematics, criticises Sigwart’s definition of number, and defines two properties of infinite numbers: reflexiveness (not increased by adding 1) and non-inductiveness. He quotes Leibniz and Galileo on the paradoxes of infinity, and credits Frege with the logical definition of numbers and Cantor with the theory of infinite numbers.
The eighth lecture applies the method of philosophic analysis to the notion of cause, with applications to the free-will problem. Russell inquires what is meant by a causal law, the evidence that causal laws have held, the evidence they will continue to hold, how scientific causality differs from common sense, and the new light on free will. He defines a causal law as a general proposition allowing inference of one thing or event from another, and discusses the evidence for causal laws, noting that observed uniformities are not free from exceptions.
Main Arguments
The central argument of the work is that the logical-analytic method, whose first complete example is in Frege’s writings, can yield objective scientific knowledge in philosophy. Russell argues that past philosophical systems serve as hypotheses and aids to imagination but fail to make philosophy a science independent of the philosopher’s temperament. The central problem is the relation between crude sense-data and the space, time, and matter of mathematical physics.
Russell argues that all genuinely philosophical problems reduce to problems of logic. Traditional logic failed by assuming only one simple proposition form: subject-predicate. This error sourced many metaphysical mistakes, including the belief that all facts consist in a thing having a quality, which rendered philosophers unable to account for science and daily life. The belief that all facts are subject-predicate led philosophers to convict the world of sense of unreality for a super-sensible “real” world. Russell shows that relational propositions have a different form, and that asymmetrical relations—involved in all series—cannot be reduced to common or different properties. Admitting relations removes the logical grounds for calling the world of sense illusory.
Russell argues against the classical tradition’s logic, which is constructive through negation, and in favour of a true logic that is analytic, showing possibilities rather than legislating what is. He criticises the doctrine of the universe as “organic unity” and the doctrine that all reality is “mental” or “spiritual” as fallacious. He also criticises evolutionism, arguing that its truth does not follow from scientific facts and that its interests are practical rather than theoretical. Philosophy must renounce mundane desires and aim at understanding general aspects of the world.
Against Bergson’s advocacy of intuition, Russell argues that the opposition between instinct and reason is mainly illusory. Instinct, intuition, or insight first leads to beliefs which reason confirms or confutes; confirmation consists of agreement with other equally instinctive beliefs. Reason is a harmonising, controlling force, not a creative one. Instinct is liable to error; it is least liable in practical matters aiding survival, but even then reserve or flattery can mislead. Reason tests beliefs by mutual compatibility and examines possible sources of error. Russell argues that intuition is an aspect of instinct, admirable in customary surroundings but incompetent when surroundings change. Philosophy, aiming at theoretical understanding, demands liberation from instinct; intellect proves superior to intuition here.
Russell argues that philosophy cannot satisfy the hope of demonstrating that the world has desirable ethical characteristics. The difference between a good and bad world is in particular characteristics, not sufficiently abstract for philosophy. Ethical interests must remain in the background. A similar change occurred in other sciences: astronomy was studied for astrology; physics in Plato’s Timaeus is full of ethical notions; modern physicists are not concerned with ethical attributes. In philosophy, ethical neutrality has been seldom sought; the belief that good and evil afford a key to understanding must be driven forth. Those who forget good and evil and seek facts are more likely to achieve good.
On the problem of the external world, Russell argues that the only possible justification for physics must exhibit matter as a logical construction from sense-data, unless there were some wholly a priori principle for inferring unknown entities from known ones. The common-sense belief in fairly permanent and rigid bodies is a piece of audacious metaphysical theorising, since objects are not continually present to sensation. A “thing” may be defined as a certain series of appearances, connected by continuity and causal laws. This extrusion of permanent things exemplifies Occam’s razor: entities are not to be multiplied without necessity.
Russell argues against the notion of consecutive points or instants in motion, showing that this hypothesis leads to Zeno’s paradoxes. The mathematical definition of continuity admits points and instants while denying infinitesimals. He argues that infinite numbers must be admitted, citing fractions less than 1, where no two are consecutive. The difficulties of infinity are sham difficulties arising from etymology and confusion with the “true” infinite; genuine difficulties are resolved by Cantor’s theory.
On causation, Russell argues that a causal law is a general proposition allowing inference of one thing or event from another. The principle “same cause, same effect” is too narrow. A causal law must state temporal relations and should state how much earlier or later. The evidence for causal laws consists in discovered approximate uniformities, but observed uniformities are not free from exceptions; when one fails, a wider uniformity can be found.
Key Concepts
Logical atomism is Russell’s term for the philosophical approach he advocates, representing an advance like Galileo’s in physics. It holds that philosophy should analyse complex facts into their simplest components—atomic facts—which are known empirically through sense-perception. Atomic facts have no facts as constituents; an atomic proposition expresses an atomic fact, and atomic facts determine their assertion or denial. Pure logic is independent of atomic facts.
Atomic and molecular propositions are distinguished in Russell’s logic. Atomic propositions express atomic facts and are known empirically without inference. Molecular propositions contain conjunctions such as if, or, and; they connect two propositions, not depending on their truth but on the second being inferable from the first. All inference depends on such propositions. General propositions include “all men are mortal” (positive) and “some men are philosophers” (negative). General truths cannot be inferred from particular truths alone; they require general premisses, so some general knowledge is independent of empirical evidence.
Asymmetrical relations are relations that never hold both ways, such as husband, father, before, greater. They are involved in all series, and denying them condemns the world of sense as illusory. Relations are classified as symmetrical (holds between A and B and B and A), non-symmetrical (not symmetrical), and asymmetrical (never holds both ways). A second classification divides relations into transitive, non-transitive, and intransitive. Symmetrical relations might be reduced to common or different properties, but asymmetrical relations cannot—mere difference of magnitude fails to explain “greater than.”
Hard and soft data are distinguished in Russell’s analysis of common knowledge. Hard data resist critical reflection and include particular facts of sense, general truths of logic, some recent memory facts, some introspective facts, spatial and temporal relations, and facts of comparison. Soft data become doubtful on reflection and include beliefs in persisting sensible objects, other people’s minds, and testimony. The world for reconstruction is fragmentary, slightly more extensive than Descartes’s world.
Occam’s razor is the principle that entities are not to be multiplied without necessity. Russell invokes it to argue that the assumption of a constant entity, the wall-paper, which “has” changing colours is gratuitous metaphysics; we may define the wall-paper as the series of its aspects. More generally, a “thing” is a certain series of aspects; to say an aspect is of a thing means it is one of those which, taken serially, are the thing.
Reflexiveness and non-inductiveness are two properties of infinite numbers. Reflexiveness means not increased by adding 1, illustrated by matching natural numbers with numbers starting from 1, and with even numbers. Non-inductiveness is explained via hereditary properties and mathematical induction. An inductive property belongs to 0 and is hereditary; inductive numbers are those possessing all inductive properties. Infinite numbers are non-inductive, lacking an immediate predecessor. It is known all reflexive numbers are non-inductive, but not vice versa.
The logical definition of number was discovered by Frege. Frege asks what kind of object number can be ascribed to, noting that physical things can be regarded as one or many. He concludes number is non-sensible and objective, applying to all of mathematics and logic. Numbers are properties of general terms, not physical things or mental occurrences. Two general terms applying to the same class have the same number of instances. The number of terms in a given class means the class of all classes that are similar to the given class. This definition yields the usual arithmetical properties, applies to finite and infinite numbers, and handles 0 and 1 without difficulty.
The doctrine that classes are merely symbolic is introduced through the contradictions that led Russell to this view. Nothing said significantly about things can be said significantly about classes. Statements nominally about classes can be reduced to statements about what follows from the hypothesis of anything’s having the defining property of the class. If classes are merely symbolic, then numbers are not actual entities; propositions in which numbers occur have only a certain logical form. Words like or, not, if, there is, identity, greater, plus, nothing, everything, function are formal, requiring a context for meaning. “Logical constants” are not entities.
Causal laws are defined as general propositions allowing inference of one thing or event from another. The inferred thing is a particular, but since we cannot be acquainted with a particular except by its being given, the inferred particular must be described, not named. What is constant in a causal law is the relation between what is given and what is inferred. A causal law must state temporal relations and should state how much earlier or later. The general scheme: whenever things occur in certain relations to each other, then a thing having a fixed relation to these things will occur at a date fixed relatively to their dates.
Themes
A recurring theme is the contrast between the scientific and the mystical attitudes in philosophy. Russell contrasts scientific (Ionian) and mystical (Sicilian) attitudes, noting that Pythagoras combined both—his proposition on right-angled triangles and his belief that it is wicked to eat beans. Plato used logic to defeat common sense, leaving the field clear for mysticism. Russell maintains that insight, untested and unsupported, is an insufficient guarantee of truth, though much important truth is first suggested by it. The philosophy inspired by the scientific spirit must deal with dry, abstract matters, not practical problems; it offers triumphs like Newton’s and Darwin’s, a sense of power and reliable progress.
The theme of ethical neutrality in philosophy runs through the work. Russell argues that philosophy cannot satisfy the hope of demonstrating that the world has desirable ethical characteristics. The difference between a good and bad world is in particular characteristics, not sufficiently abstract for philosophy. Love and hate are ethical opposites but closely analogous attitudes to philosophy; their difference belongs to psychology. Ethical interests must remain in the background. The belief that good and evil afford a key to understanding must be driven forth. Those who forget good and evil and seek facts are more likely to achieve good.
The extension of factual knowledge has had two effects, as in the Renaissance: distrust of wide systems—theories come and go, each a temporary makeshift, the ideal of an all-embracing synthesis recedes, leading to despair and slovenly intellect—and new powers—man’s physical control over natural forces increases, bringing optimism about practice, forgetting fixed limits like death, and no philosophy is tolerated that sets bounds to man’s wishes. Russell accepts the welcoming of new fact and suspicion of dogmatism as an advance, but both practical pretensions and theoretical despair go too far. Most greatness is called forth by thwarting hopes through natural obstacles; by pretending omnipotence, man becomes trivial.
The theme of the relation between instinct and reason is developed through the discussion of Bergson’s advocacy of intuition. Russell argues that the opposition between instinct and reason is mainly illusory: instinct, intuition, or insight first leads to beliefs which reason confirms or confutes. Conflict arises with single beliefs held instinctively with such determination that no inconsistency leads to abandonment. Instinct is liable to error; it is least liable in practical matters aiding survival. Those recommending intuition should return to running wild, dyeing themselves with woad, and living on hips and haws. Philosophy is a highly refined pursuit demanding liberation from instinct; intellect proves superior to intuition here.
The theme of the reconstruction of common-sense beliefs through logical analysis is central. The common-sense belief in persistent things arises from combining a belief that something persists with the fact that sensations change. The belief in indestructible “things” took the form of atomism, motivated not by empirical success but by an instinctive belief in something permanent beneath changes. If the a priori belief in permanence had not existed, the same laws could have been formulated without it. The task is to reconstruct the conception of matter without a priori beliefs.
The theme of the limitations of philosophy is also present. Russell surveys hopes: philosophy cannot satisfy the hope of demonstrating the world has desirable ethical characteristics. The philosophy inspired by the scientific spirit must deal with dry, abstract matters, not practical problems. Ultimate metaphysical truth can be discovered by combining science’s hopefulness, patience, and open-mindedness with Greek feeling for beauty in abstract logic.
The Author’s Conclusions
Russell concludes that the logical-analytic method in philosophy, whose first complete example is in Frege’s writings, yields objective scientific knowledge. Past philosophical systems serve as hypotheses and aids to imagination but fail to make philosophy a science independent of the philosopher’s temperament. The central problem is the relation between crude sense-data and the space, time, and matter of mathematical physics.
Russell concludes that all genuinely philosophical problems reduce to problems of logic. Traditional logic failed by assuming only one simple proposition form: subject-predicate. This error sourced many metaphysical mistakes, including the belief that all facts consist in a thing having a quality. Admitting relations removes the logical grounds for calling the world of sense illusory. Modern logic enlarges abstract imagination, unlike classical logic which put thought in fetters; it introduces the same advance as Galileo introduced into physics.
On the external world, Russell concludes that the only possible justification for physics must exhibit matter as a logical construction from sense-data. A “thing” may be defined as a certain series of appearances, connected by continuity and causal laws. This extrusion of permanent things exemplifies Occam’s razor: entities are not to be multiplied without necessity. The belief in indestructible “things” was motivated not by empirical success but by an instinctive belief in something permanent beneath changes.
On infinity, Russell concludes that infinite numbers must be admitted. The difficulties of infinity are sham difficulties arising from etymology and confusion with the “true” infinite. Cantor has shown there are infinite numbers of different sizes. The logical definition of numbers was discovered by Frege, while the theory of infinite numbers was discovered by Cantor.
On causation, Russell concludes that a causal law is a general proposition allowing inference of one thing or event from another. The principle “same cause, same effect” is too narrow. Observed uniformities are not free from exceptions, but when one fails, a wider uniformity can be found.
The work concludes with the application of the method of philosophic analysis to the notion of cause, with applications to the free-will problem. The nature of philosophic analysis is to start from common knowledge (data), reduce them to simple, precise propositions arranged in deductive chains, and estimate the degree of certainty of beliefs. The lecture inquires what is meant by a causal law, the evidence that causal laws have held, the evidence they will continue to hold, how scientific causality differs from common sense, and the new light on free will.
Significance
The work is significant for its articulation of logical atomism as a philosophical approach that could yield objective scientific knowledge. Russell positions this approach as an advance comparable to Galileo’s in physics, representing a third alternative to the classical tradition descending from Kant and Hegel and to evolutionism derived from Darwin. The work demonstrates the application of the logical-analytic method to central philosophical problems, including the nature of matter, the problem of infinity, and the notion of causation.
The work is also significant for its engagement with the mathematical achievements of Frege and Cantor. Russell credits Frege with the logical definition of numbers and Cantor with the theory of infinite numbers, and he draws on their work to resolve philosophical problems that had troubled philosophers from Zeno to Bergson. The work acknowledges unpublished logical discoveries by Ludwig Wittgenstein, indicating the collaborative and evolving nature of the logical-analytic approach.
The work’s treatment of the problem of the external world, arguing that matter must be exhibited as a logical construction from sense-data, represents a significant contribution to epistemology and the philosophy of science. Russell’s invocation of Occam’s razor to argue against the assumption of permanent “things” beneath changing appearances offers a distinctive resolution to the problem of the relation between sense-data and the objects of physics.
The work’s discussion of the notion of cause, with its analysis of causal laws and their evidence, contributes to the ongoing philosophical debate about causation and free will. Russell’s definition of a causal law as a general proposition allowing inference of one thing or event from another, and his insistence that the principle “same cause, same effect” is too narrow, offer a precise framework for understanding scientific causality.
The work is significant for its insistence on ethical neutrality in philosophy. Russell argues that the belief that good and evil afford a key to understanding the world must be driven forth, and that those who forget good and evil and seek facts are more likely to achieve good. This position aligns philosophy with the sciences, which achieved success through ethical neutrality, and represents a distinctive contribution to the understanding of philosophy’s proper method and aims.
Detailed Section-by-Section Summary
Preface and Lecture I: The Nature of the Logical-Analytic Method
Russell opens by stating that the lectures aim to demonstrate the nature, capacity, and limitations of the logical-analytic method in philosophy, a method whose first complete example is found in Frege’s writings. This method, he argues, is capable of yielding objective scientific knowledge, unlike past philosophical systems which serve merely as hypotheses and aids to imagination but fail to make philosophy a science independent of the philosopher’s temperament. The central problem is the relation between crude sense-data and the space, time, and matter of mathematical physics. Russell credits Dr. Whitehead with key contributions, including the definition of points, the treatment of instants and “things,” and the conception of physics as a construction rather than an inference, as detailed in the fourth volume of Principia Mathematica. He also acknowledges that Georg Cantor removed the difficulty of the mathematical infinite, and that the work is tentative and incomplete, claiming no finality except on Cantor’s theory.
Current Tendencies in Philosophy
Russell observes that philosophy has made greater claims and achieved fewer results than other learning, from Thales to Anaximander. He identifies three principal types of philosophy: the classical tradition (from Kant and Hegel), evolutionism (from Darwin, via Spencer, James, and Bergson), and “logical atomism,” which he advocates, representing an advance like Galileo’s in physics. The classical tradition descends from Greek faith in reason and geometry’s a priori method, but it accepted Aristotelian logic uncritically. Russell illustrates this with Mr. Bradley’s Appearance and Reality, which condemns relations as self-contradictory, arguing that relations require new connecting relations endlessly. He finds such arguments produce bewilderment, not conviction. The classical tradition’s logic is constructive through negation, whereas true logic is analytic, showing possibilities rather than legislating what is. The doctrines of the universe as an “organic unity” and of all reality as “mental” are fallacious, being the child of Greek belief in reason and mediaeval belief in the universe’s tidiness.
Evolutionism and Its Critics
Evolutionism, the prevailing creed dominating politics and literature, was swept in by Darwin’s Origin of Species, which destroyed the notion of fixed natural kinds. It assumes progress from amoeba to man, but Russell calls this a hasty generalisation. Bergson rejects both mechanism and radical finalism, since both make time useless; his finalism depends on life as a continuous stream where all divisions are artificial. Russell criticises evolutionism on two grounds: its truth does not follow from scientific facts, and its interests are practical, not theoretical. Philosophy, he insists, must renounce mundane desires and aim at understanding the general aspects of the world. He contrasts the scientific (Ionian) and mystical (Sicilian) attitudes, noting that Pythagoras combined both—his proposition on right-angled triangles and his belief that it is wicked to eat beans. Plato used logic to defeat common sense, leaving the field clear for mysticism.
Instinct, Intuition, and Reason
Russell states he knows nothing of the mystic’s world’s reality and does not deny it, but maintains that insight, untested and unsupported, is an insufficient guarantee of truth, though much important truth is first suggested by it. He discusses the common opposition between instinct and reason: in the eighteenth century, reason was favoured, but under Rousseau and the romantic movement, instinct was preferred, first by rebels against artificial government and thought, then by those feeling science menaced creeds. Bergson, under the name “intuition,” has raised instinct to sole arbiter of metaphysical truth. Russell argues this opposition is mainly illusory: instinct, intuition, or insight first leads to beliefs which reason confirms or confutes; confirmation consists of agreement with other equally instinctive beliefs. Reason is a harmonising, controlling force, not a creative one. Instinct is liable to error; it is least liable in practical matters aiding survival, like discerning friendship and hostility, but even then reserve or flattery can mislead. Reason tests beliefs by mutual compatibility and examines possible sources of error.
Bergson’s Intuition and Its Failings
Applying this to Bergson’s advocacy of “intuition” versus “intellect,” Russell notes Bergson says there are two ways of knowing: moving round the object (relative) or entering into it (absolute); the second is “intellectual sympathy,” illustrated by self-knowledge of our personality flowing through time. Bergson’s philosophy reports intuition’s knowledge and condemns science and common sense. Russell counters that we only know of the struggle for survival through intellect; if intellect misleads, that history is untrue. If evolution occurred as Darwin believed, all faculties developed under practical utility. Intuition is seen at its best where directly useful, like judging characters; the savage deceived by false friendship may die, while mathematical incompetence is not fatal. Both intuition and intellect developed because useful when giving truth. Intellect in civilised man has been developed beyond individual utility; intuition diminishes as civilisation increases, being greater in children, the uneducated, and dogs. Those recommending intuition should return to running wild, dyeing themselves with woad, and living on hips and haws.
The Claim of Intuition’s Infallibility
Russell examines intuition’s claimed infallibility. Self-knowledge, Bergson’s best instance, is proverbially rare; men are unconscious of their meannesses, vanities, and envies. Intuition has convincingness lacking to intellect, but if fallible, its subjective certainty becomes a demerit. Love is another example: people think they see into another’s soul, yet deception is constantly practised, and experience proves the supposed insight illusory. Bergson says intellect deals only with resemblance to the past, while intuition apprehends uniqueness and novelty. Russell agrees there is uniqueness at every moment, not fully expressible by intellectual concepts, but direct acquaintance is given fully in sensation, not requiring a special intuition faculty. Intuition is an aspect of instinct, admirable in customary surroundings but incompetent when surroundings change, as with a hen with ducklings who is helpless when they take to water. Philosophy, aiming at theoretical understanding, is not practically important to animals, savages, or most civilised men; intuition is best in older activities like self-preservation and love. Philosophy is a highly refined pursuit demanding liberation from instinct; intellect proves superior to intuition here.
Ethical Neutrality in Philosophy
Russell surveys hopes: philosophy cannot satisfy the hope of demonstrating the world has desirable ethical characteristics. The difference between a good and bad world is in particular characteristics, not sufficiently abstract for philosophy. Love and hate are ethical opposites but closely analogous attitudes to philosophy; their difference belongs to psychology. Ethical interests must remain in the background. A similar change occurred in other sciences: astronomy was studied for astrology; physics in Plato’s Timaeus is full of ethical notions; modern physicists are not concerned with ethical attributes. Psychology’s scientific attitude is recent; ethical neutrality has been essential to success. In philosophy, ethical neutrality has been seldom sought; the belief that good and evil afford a key to understanding must be driven forth. Those who forget good and evil and seek facts are more likely to achieve good.
The Effects of Extended Factual Knowledge
The extension of factual knowledge has had two effects, as in the Renaissance: distrust of wide systems—theories come and go, each a temporary makeshift, the ideal of an all-embracing synthesis recedes, leading to despair and slovenly intellect—and new powers—man’s physical control over natural forces increases, bringing optimism about practice, forgetting fixed limits like death, and no philosophy is tolerated that sets bounds to man’s wishes. Russell accepts the welcoming of new fact and suspicion of dogmatism as an advance, but both practical pretensions and theoretical despair go too far. Most greatness is called forth by thwarting hopes through natural obstacles; by pretending omnipotence, man becomes trivial. Ultimate metaphysical truth can be discovered by combining science’s hopefulness, patience, and open-mindedness with Greek feeling for beauty in abstract logic. The philosophy inspired by the scientific spirit must deal with dry, abstract matters, not practical problems; it offers triumphs like Newton’s and Darwin’s, a sense of power and reliable progress.
Logic and the Forms of Propositions
Lecture II begins with the assertion that all genuinely philosophical problems reduce to problems of logic. Logic in the Middle Ages meant scholastic technical terms and syllogistic rules; Aristotle spoke, and humbler men repeated. This tradition is still set in examinations as a “propaedeutic,” training in solemn humbug. The first extension was induction by Bacon and Galileo—Bacon theoretically and largely mistakenly, Galileo in actual use. Induction, though important as a method, does not remain when its work is done; in a perfected science, everything ought to be deductive. Knowledge of logical forms is distinct from knowledge of existing things; the form of “Socrates drank the hemlock” is abstract, not an existing thing. One can understand a sentence’s constituents without its form, or its form without constituents. Understanding requires both, conveying information by relating known objects according to a known form. Philosophical logic extracts this implicit knowledge. In inference, form alone is essential; subject-matter only secures premiss truth. All inferences, fully stated, are instances of purely general, purely formal implications.
The Failure of Subject-Predicate Logic
Traditional logic failed by assuming only one simple proposition form: subject-predicate. This is not universal; relational propositions like “this is bigger than that” have a different form. This error sourced many metaphysical mistakes. The belief that all facts consist in a thing having a quality rendered philosophers unable to account for science and daily life, often because they sought to convict the world of sense of unreality for a super-sensible “real” world. This conviction arises from certain moods, the source of mysticism and metaphysics; reason then seeks logical justification. This “malicious” logic, borrowing from Mr Santayana, is seen in mystics like Plato, Spinoza, and Hegel. Such logic is not candid, and reading the book of Nature with hostility prevents understanding. Traditional logic, holding all propositions subject-predicate, cannot admit relations. Refutation comes from “asymmetrical” relations. Relations are classified as symmetrical (holds between A and B and B and A), non-symmetrical (not symmetrical), and asymmetrical (never holds both ways, e.g., husband, father, before, greater). A second classification divides relations into transitive (A to B, B to C, implies A to C), non-transitive, and intransitive (never, e.g., father). Symmetrical relations might be reduced to common or different properties, but asymmetrical relations cannot—mere difference of magnitude fails to explain “greater than.” Asymmetrical relations are involved in all series; denying them condemns these as appearance. Admitting relations removes logical grounds for calling the world of sense illusory.
Atomic and Molecular Propositions
Relations can have more than two terms, as in jealousy among three or a wish involving four people. A “fact” means a thing having a quality or things having a relation, never simple. Atomic facts have no facts as constituents. An assertion expresses a fact; a proposition is a form of words either true or false. Atomic propositions express atomic facts, which determine their assertion or denial. Atomic facts are known empirically, without inference, via sense-perception. Pure logic is independent of atomic facts. “Molecular” propositions contain conjunctions (if, or, and). For example, “If it rains, I shall bring my umbrella” connects two propositions, not depending on their truth, but on the second being inferable from the first. All inference depends on such propositions.
General Propositions and the Limits of Empiricism
General propositions include “all men are mortal” (positive) and “some men are philosophers” (negative). General truths cannot be inferred from particular truths alone; they require general premisses. Thus, some general knowledge is independent of empirical evidence, refuting older empiricists. Such knowledge is found in logic, e.g., “If anything has a certain property…” This is self-evident and purely formal. Logic has two parts: first, investigating what propositions are and their forms; second, supremely general propositions asserting truths of certain forms, merging into pure mathematics. The first part is more difficult and philosophically important. The problem of judgment requires admitting complicated forms. If all judgments were true, belief could be apprehension of a fact. But error (e.g., believing Charles I died in his bed) makes this impossible; a false belief has no objective fact. Thus, belief needs a different logical form than a two-term relation. Modern logic enlarges abstract imagination, unlike classical logic which put thought in fetters; it introduces the same advance as Galileo introduced into physics.
The Problem of the External World
The lecture then turns to “Our Knowledge of the External World.” Doubt about the world of sense arises from Indian mysticism, Parmenides, Plato, Berkeley, and modern physics. The mystic cannot be refuted if merely reporting revelation, but when denying reality, he is questioned. Berkeley’s attack, reinforced by physiology, suggests sense-objects depend on physiological conditions. The discrepancy between physics and sense will be shown more apparent than real. The investigation starts from “data”—matters of common knowledge (furniture, history, physical science). Philosophy cannot use a superfine knowledge to condemn all common knowledge; universal scepticism is logically irrefutable but practically barren.
Hard and Soft Data
The passage discusses the analysis of common knowledge, distinguishing between different grades of certainty. Data can only be criticised by other data, not by an outside standard. The most certain knowledge is that from our own personal sensible acquaintance (“evidence of the senses”). Knowledge from testimony, like history and geography, has varying certainty: doubts about Napoleon’s existence are a joke, but Agamemnon’s historicity is debatable. In science, the law of gravitation has certainty like Napoleon’s existence, while latest speculations on matter have slight probability. These varying degrees of certainty are themselves part of our data. Analysis reveals some knowledge is derivative (believed because of something else) and some primitive (believed on its own account). Senses give primitive knowledge. Psychologists show much that seems given in sense is actually inferred, especially space-perceptions: we infer “real” size and shape from apparent size and shape. When hearing speech, we supply missing parts by unconscious inference; in a foreign language, we seem deaf. Derivative beliefs arise psychologically without logical inference, e.g., judging a man is angry from a frown. Such knowledge is derivative psychologically but logically primitive. If logically primitive beliefs cannot be deduced from psychologically primitive ones, confidence diminishes. Belief that tables and chairs persist when unperceived is logically primitive but psychologically derivative; it needs justification. This leads to a vague distinction between “hard” data (resisting critical reflection) and “soft” data (becoming doubtful). Hardest data are particular facts of sense and general truths of logic. Verbal doubt is possible, but real doubt would be pathological. Psychologically derivative but logically primitive beliefs are mostly soft data. Hard data include facts of sense, laws of logic, some recent memory facts, some introspective facts, spatial/temporal relations (e.g., swift motion in specious present), and facts of comparison (likeness of colours). Excluded are beliefs in persisting sensible objects and other people’s minds, and testimony. The world for reconstruction is fragmentary, slightly more extensive than Descartes’s world.
The Question of Inference from Sense-Data
The problem is: Can the existence of anything other than our own hard data be inferred from those data? “External” must not mean “spatially external,” since visible coloured surfaces are spatially external. The question of “independent of ourselves” is ambiguous due to “independent” and “self.” Logical dependence means one thing is part of another; causal dependence means one occurs without the other. The question reduces to whether objects of sense exist when not perceived. Two distinct problems arise: (1) do objects of sense exist unperceived? (2) can other objects, inferable from sense, exist? The latter is the “thing in itself” in philosophy and matter in physics. Sensation (mental awareness) must be distinguished from sensible object (the patch of colour, hardness, sound). The felt passivity of sensation only suggests the sensation has an outside cause, not the sensible object. The common-sense belief in persistent things arises from combining a belief that something persists with the fact sensations change. A table viewed from different places presents different appearances. Stated in terms of sensible objects alone, we perceive a series of changing visible objects correlated with muscular sensations. Other changes come from shutting one eye, blue spectacles, microscopes, atmosphere, physiological changes. The blue spectacles case: the blue glass is known by touch, not sight. To account for blue appearances, we might assume the spectacles persist when untouched, or that ceased objects still have effects. The latter hypothesis is possible, not a priori impossible.
A New Start for Reconstruction
Generally, in so far as physics or common sense is verifiable, it must be interpretable in terms of actual sense-data alone, since verification consists in occurrence of an expected sense-datum (e.g., eclipse). But at this level of doubt, testimony cannot be accepted; belief in other minds needs justification. The felt passivity of sensation is irrelevant. The naive belief that things persist is dispelled by changing appearances, but stating this assumes common-sense world. The bare outcome is that gradual changes in sense-data correlate with others. The assumption of persistence may be replaced by persistence of effects. Testimony depends on other minds. The thing-in-itself theory confuses sensation with its object; a patch of colour is not mental. The theory fails to realise the radical reconstruction needed. There are no “illusions of sense”; objects of sense in dreams are real, but called unreal due to unusual connections with other objects. As we walk round a table, aspects change; but “same place” presupposes solved difficulties. The passage ends by proposing a new start: construct a possible (not necessary) hypothesis to explain facts, then pare away what is superfluous.
The Gulf Between Physics and Perception
The fourth lecture opens by confronting a fundamental tension: the world as physics describes it—a world of particles, forces, and mathematical laws—seems utterly unlike the world of immediate sensation, with its colours, sounds, and textures. Men of science often dismiss these immediate data as “merely subjective” while simultaneously asserting the objective truth of the physics they have inferred from those very data. Russell argues that this attitude requires justification. The only possible justification, he contends, must show that matter is a logical construction from sense-data, unless one posits a wholly a priori principle for inferring unknown entities from known ones. Physicists, he notes, are largely unconscious of this gulf between the two worlds, while psychologists are acutely aware of it but lack the mathematical tools to bridge it.
The Failure of the Common-Sense World
The world of physics originated from the common-sense belief in permanent, rigid bodies like tables, chairs, and stones. Russell points out that this belief is itself a piece of audacious metaphysical theorising, since objects are not continually present to our senses. This problem, acute since Berkeley’s time, is generally ignored. Furthermore, the supposed permanence and rigidity of matter are contradicted by experience: breath, smoke, and clouds are visible but not touchable, and ice melts into water, requiring the hypothesis that water is the same thing in a new form. The early physicists’ pursuit of indestructible atoms was a response to this instinctive belief in something permanent beneath change, not a conclusion drawn from empirical success. Russell argues that if this a priori belief in permanence did not exist, the same physical laws could be formulated without it. When ice melts, we only know that one appearance is replaced by another; there is no logical reason, except prejudice, to regard them as manifestations of the same underlying substance.
Reconstructing the Notion of a “Thing”
Russell’s task is to reconstruct the conception of matter without such a priori beliefs. He proposes that a “thing” may be defined as a certain series of appearances, connected by continuity and causal laws. For example, a fading wall-paper is not a constant entity that “has” different colours at different times; rather, it is the series of its colour-aspects. To say an aspect is of a thing means it is one of those which, taken serially, constitute the thing. This extrusion of permanent things is an application of “Occam’s razor”: entities are not to be multiplied without necessity. Although we find it easier to imagine a wall-paper with changing colours than to think of a series of colours, Russell insists that what is easy in thought is not what is most free from unwarrantable assumptions.
The Difficulty of Collecting Sense-Data into Series
There are significant difficulties in collecting sense-data into series. Common sense regards a human body as one thing, while physics sees it as a continually changing collection of particles. The problem is to determine by what principles we select certain data and call them appearances of the same thing. Resemblance is insufficient, as two different things can be exactly alike. Continuity is also insufficient: it is not a necessary condition, as unobserved intermediate states are hypothetical, and it is not sufficient, as one can pass by continuous gradations from any one drop of the sea to any other. While physics successfully assumes continuity, it can only be a necessary condition for two appearances belonging to the same thing, not a sufficient one. What is further wanted is the fulfilment of causal laws. Physics has found it empirically possible to collect sense-data into series that obey its laws, and if this grouping is unambiguous, there must be only one way of doing so.
The Illusion of Consecutive Points and Instants
The argument then turns against the notion of consecutive points or instants in motion. If a speck moved from one point to a next point at the next instant, Zeno’s paradoxes would become unavoidable. A simple paradox illustrates this: a moving speck cannot be at the same point at two consecutive instants, nor travel further than from one point to the next. It would have to pass from one point to the next, implying a single, definite velocity for all motions. Since this conclusion is false, the hypothesis of consecutive points and instants must be rejected. The difficulty lies in imagining infinitesimal distances, as infinite divisibility yields only finite distances; there is no “end” to the process. The phrase “a distance smaller than any finite distance” is a logical blunder, falsely suggesting a single distance smaller than all finite ones. In continuous motion, the body occupies a position at each instant, but between any two positions there are infinitely many nearer ones. It is not at rest, since rest means being in the same position throughout a finite period.
The Mathematical Definition of Continuity
The mathematical definition of continuity is precise: a particle at point P at time t is continuous if, for any small portion of its path containing P, there exist instants earlier and later such that throughout the interval the particle lies between the boundaries of that portion, however small it is made. This admits points and instants while denying infinitesimals. Philosophers like Bergson object that we see fast motion directly, not merely infer it from successive positions. The mathematical theory seems to say nothing happens except being in different places at different times, yet the second-hand’s motion is perceptible while the hour-hand’s is not. Three answers are given. First, the physiological answer: sensations die away gradually, so a swift motion is seen as a finite extent of positions with diminishing vividness. Second, the psychological answer: the world of physics is inferred from sensation, so we must construct logical constructions from sense-data having the mathematical properties of particles, points, and instants. Even in immediate sense-data, we must distinguish instantaneous states forming a compact series, as two sense-data can be really different when we cannot perceive the difference. Third, the logical answer: the Bergsonian view that motion is indivisible rests on the doctrine that analysis falsifies, a view contrary to plain truth. Change requires a succession of states, and analysis must end with terms that are not changes, driving us to instants without duration.
Four Questions on Continuity
Russell then lists four questions: Are mathematically continuous series logically possible? Are sense-data composed of mutually external units? Does assuming points and instants make the account fictitious? Is there empirical evidence for continuity of sense? For the second, immediate experience cannot prove or disprove mutual externality. For the third, mathematics formally adopts absolute space and time, but this is a convenient fiction; we should abstain from assuming or denying points and instants, adopting a relational theory. Things may be composed of point-instant-particles, but these are not data; we can construct points and instants as logical functions of finite elements. For the fourth, there is no sufficient empirical reason to believe the world of sense is continuous; the hypothesis is consistent and simpler, but unverifiable, since imperceptible differences could hide finite jumps. If instants are defined as groups of simultaneous events, compactness requires an infinite number of events in any finite period, which may be possible but faces difficulties.
Kant’s First Antinomy
The sixth lecture turns to the problem of infinity considered historically, beginning with Kant’s first antinomy. Kant’s thesis states the world has a beginning in time and limits in space; the antithesis states it is infinite. Kant professes to prove both, but modern logic shows neither can be proved. To rescue the world of sense, it suffices to destroy the proof of finitude. Kant’s argument is that if the world had no beginning, an infinite series of states has passed, but an infinite series can never be completed by successive synthesis. Russell criticises this: infinity is primarily a property of classes, not defined by “completion,” and the word “synthesis” introduces a reference to mind. Kant only proves, at most, that an infinite series cannot be completed in a finite time. His blunder arose from reversing the series in imagination—going backwards from the present, the mental series has no end, while the physical series has an end but no beginning.
Kant’s Second Antinomy and the Problem of Incommensurables
The passage then analyzes Kant’s second antinomy, which links continuity to infinity. The thesis claims every complex substance consists of simple parts; the antithesis claims no complex thing consists of simple parts. Russell finds a fallacy in the antithesis’s proof, specifically in Kant’s statement “Space does not consist of simple parts, but of spaces.” Russell argues geometry can regard space as made of points, which are simple, so this possibility vitiates Kant’s argument. Russell conjectures Kant was influenced by two fallacious grounds: spatial order cannot be reduced to points, and infinite divisibility means points can never be reached by successive division. Russell counters that an infinite set of points can be given all at once by its defining concept. The problem of infinity first arose with Pythagoras, who applied number to geometry, believing space is composed of indivisible points and time of indivisible instants. This led to conflict with their discovery of incommensurables, as the diagonal of a square is incommensurable with its side. This was fatal to Pythagoras’s philosophy that number is the constitutive essence of all things, and the secret was kept, with Hippasos of Metapontion said to be shipwrecked for disclosing it. The ancients reconstructed geometry without numerical measurement, as seen in Euclid, while the moderns reasserted numerical measurement by extending arithmetic to include “irrational” numbers. The existence of incommensurables proves every finite length contains an infinite number of points, a property characterizing infinite collections. Philosophers from Zeno to Bergson based metaphysics on the supposed impossibility of infinite collections, with no material addition until Bolzano’s Paradoxien des Unendlichen and the definitive solution due to Georg Cantor.
Parmenides and Zeno
The passage then discusses Parmenides, Zeno’s master, who expounded his views in a poem divided into “the way of truth” and “the way of opinion.” Reality, according to Parmenides, is uncreated, indestructible, unchanging, and indivisible. His fundamental principle is that “it is the same thing that can be thought and that can be.” Parmenides introduced the conception of a reality behind the passing illusions of sense, influencing Plato, Spinoza, and Hegel. Zeno’s relation to Parmenides is explained by Plato, where Zeno says his writings were meant to protect Parmenides’s arguments by retorting upon the partisans of the many that their hypothesis appears in a more ridiculous light. Zeno’s four arguments against motion were intended to exhibit contradictions from supposing change exists. Russell notes we only know these arguments through Aristotle, who stated them to refute them. Zeno’s arguments seem to be ad hominem, assuming premisses granted by opponents. Russell concludes Zeno’s polemic is directed against the view that space and time consist of points and instants, and as against the view that a finite stretch consists of a finite number of points and instants, his arguments are perfectly valid. In a fragment, Zeno argues if things are a many, they must be finite in number, but also infinite, since there will always be other things between them. The weak point is the assumption of the impossibility of definite infinite numbers.
The Four Arguments Against Motion
The first of Zeno’s arguments is the race-course: you cannot traverse an infinite number of points in a finite time. Russell analyzes the words “one by one,” noting that if all points are concerned, you do not touch them one by one, as no two points are next each other. The argument’s apparent force lies in the mistaken supposition that there cannot be anything beyond the whole of an infinite series, which is false since 1 is beyond the infinite series 1/2, 3/4, 7/8, 15/16. The second argument concerns Achilles and the tortoise: Achilles will never overtake the tortoise because he must first reach the place from which the tortoise started. Russell notes this shows that if Achilles overtakes the tortoise, it must be after an infinite number of instants, but an infinite number of instants do not make an infinitely long time. The third argument, the arrow, states the arrow in flight is at rest, since at any given moment it occupies a space equal to itself. This argument assumes a finite part of time consists of a finite series of successive instants. The solution lies in the theory of continuous series: there is no next position or next moment. The fourth argument, the stadium, involves three rows of bodies, one at rest while two move with equal velocity in opposite directions. The argument attempts to prove half of a given time is equal to double that time. Russell clarifies that if time consists of consecutive instants and motion of consecutive points, the fastest motion is at a consecutive point each instant. B is opposite a fresh A every instant, so the number of A’s passed equals the number of instants. But B passes twice as many C’s, yet cannot pass more than one per instant, leading to Zeno’s conclusion.
The Grounds for Zeno’s Arguments
Russell notes that Zeno’s arguments have grounds for almost all theories of space, time, and infinity. All his arguments are valid on the assumption that finite spaces and times consist of a finite number of points and instants. Escape is possible by maintaining the number of points and instants is infinite, by denying they exist at all, or by denying the reality of space and time. Zeno himself, as a Parmenides supporter, drew the last deduction. Bergson denied that space and time consist of points and instants. Russell argues infinite numbers must be admitted, citing fractions less than 1, where no two are consecutive. He distinguishes sham difficulties of infinity from genuine ones, criticising Kant’s first antinomy for treating the past as harder to be infinite than the future. The difficulty derives from the idea of counting; an unending series can form a whole, with new terms beyond it. Infinite numbers are not increased by adding one or doubling, which contradicts mental habits, not logic.
The Positive Theory of Infinity
The next lecture, “The Positive Theory of Infinity,” begins by contrasting philosophy and mathematics: both are general and a priori, but mathematics is synthetic, moving from simple to complex, while philosophy is analytic, moving from complex to abstract. The philosophical question “What is a number?” is pre-eminent. Russell criticises Sigwart’s definition of number as “a plurality thought as held together and closed,” noting it fails to reach the necessary abstraction. He argues counting is not the logical source of number, as it presupposes knowledge of numbers. Two propositions are involved in counting: the number of numbers from 1 to n is n, which is false for infinite numbers, and the number of numbers used as names equals the number of objects, which is true. Russell defines two properties of infinite numbers: reflexiveness, not increased by adding 1, and non-inductiveness. Reflexiveness is illustrated by matching natural numbers with numbers starting from 1, and with even numbers. He quotes Leibniz, who saw a contradiction in the whole not being greater than its part, and a long passage from Galileo’s Dialogues on Motion, where Salviati, Sagredo, and Simplicius discuss square numbers, concluding that equality, majority, and minority have no place in infinites. Russell notes Galileo’s solution is not right; Cantor has shown there are infinite numbers of different sizes. Non-inductiveness is explained via hereditary properties and mathematical induction. An inductive property belongs to 0 and is hereditary. Inductive numbers are those possessing all inductive properties. Infinite numbers are non-inductive, lacking an immediate predecessor. The property of being increased by 1 is inductive, but reflexive numbers are not increased by 1. It is known all reflexive numbers are non-inductive, but not vice versa. The logical definition of numbers was discovered by Gottlob Frege of Jena, in his Die Grundlagen der Arithmetik (1884), while the theory of infinite numbers was discovered by Georg Cantor (1882-3).
Frege’s Logical Analysis of Number
Frege’s work begins with the modern demand for logical strictness in mathematics, which forces a critical examination of what number actually is. He demonstrates the inadequacy of both Kant’s theory of synthetic a priori judgments and Mill’s empirical theory. The central question becomes: to what kind of object can number be ascribed? Physical things cannot be the proper subjects, since they can be regarded as one or many without contradiction—a tree’s foliage is one, but its thousand leaves are many; a pair of boots is one, but two boots are two. The number ascribed must be unambiguous, so physical things are ruled out.
Frege also rejects the view that number is psychological and subjective. He insists that number is as little an object of psychology as the North Sea is. He carefully distinguishes the objective from the palpable, spatial, and actual—the earth’s axis is objective but not actual. Number, he concludes, is non-sensible and objective, applying to the whole of mathematics and logic. Frege’s great merit is accepting both denials—that mathematical objects are neither subjective nor physical—and thereby recognising a third realm: the world of logic.
No number, not even 1, applies to physical things, but only to general terms or descriptions like “man” or “satellite of the earth.” The unity required for number is the unity of the general term itself. Zero is a property of the general term “satellite of Venus.” Thus, numbers are properties of general terms, not of physical things or mental occurrences. We may take the subject of a number statement to be the class of objects to which the general term applies. Two general terms applying to the same class have the same number of instances. For infinite classes, enumeration is impossible, so description by a general characteristic is the only possible method.
Frege then asks when two collections have the same number. A one-one relation—such as husband to wife in Christian countries—between all terms of two collections means they have the same number. This leads to defining the number of a given collection as the class of all collections similar to it. The formal definition is: “The number of terms in a given class” means “the class of all classes that are similar to the given class.” This definition yields the usual arithmetical properties, applies to both finite and infinite numbers, and handles 0 and 1 without difficulty.
The definition may seem odd, but this is common when an unanalysed idea is first resolved. Definitions are somewhat arbitrary; the real desideratum is that numbers satisfy the formulae of arithmetic. The simplest set fulfilling this requirement is the one introduced.
The Doctrine That Classes Are Not Entities
The text then addresses the doctrine that there are no such objects as classes. Russell was led to this view by contradictions. Nothing said significantly about things can be said significantly about classes. For example, “Adam is fond of apples” cannot be meaningfully transformed into “Mankind is fond of apples” in the same sense. Classes cannot have the same kind of reality as things. The text cites the Chinese philosopher Hui Tzu (third or fourth century B.C.), who maintained “a bay horse and a dun cow are three” because taken separately they are two, and taken together they are one. This is only a quibble if collections of things are not themselves things.
Statements nominally about classes can be reduced to statements about what follows from the hypothesis of anything’s having the defining property of the class. If classes are merely symbolic, then numbers are not actual entities; propositions in which numbers occur have only a certain logical form. Words like or, not, if, there is, identity, greater, plus, nothing, everything, and function are formal, requiring a context for meaning. “Logical constants” are not entities. This has important bearing on logic and philosophy, showing how they differ from the special sciences. The text notes it is making use of unpublished work by Ludwig Wittgenstein.
Lecture VIII: On the Notion of Cause
The excerpt then begins Lecture VIII, “On the Notion of Cause, with Applications to the Free-Will Problem.” It states the nature of philosophic analysis: starting from common knowledge (data), we reduce them to simple, precise propositions arranged in deductive chains. These initial propositions are premisses, distinct from data. Analysis also serves to estimate the degree of certainty of beliefs, as some premisses may be doubtful.
The lecture will apply this method to “cause,” inquiring: I. what is meant by a causal law; II. the evidence that causal laws have held; III. the evidence they will continue to hold; IV. how scientific causality differs from common sense; V. the new light on free will.
A “causal law” is a general proposition allowing inference of one thing or event from another—thunder from lightning, a footprint from a human being. The inferred thing is a particular, not an abstract object. Since we cannot be acquainted with a particular except by its being given, the inferred particular must be described, not named. What is constant in a causal law is the relation between what is given and what is inferred, not the objects themselves. The principle “same cause, same effect” is too narrow.
A causal law must state temporal relations (succession or coexistence) and should state how much earlier or later. The general scheme is: “Whenever things occur in certain relations to each other… then a thing having a fixed relation to these things will occur at a date fixed relatively to their dates.” The inferred object may be later, earlier, or simultaneous.
On evidence for causal laws: we discover approximate uniformities, such as lightning followed by thunder. Experienced past uniformity leads to expectation, a psychological account of “animal belief” in causation, as observed in horses and dogs. This repeats Hume. Observed uniformities are not free from exceptions, but when one fails, a wider uniformity can be found.
