Introduction to Mathematical Philosophy

by Bertrand Russell

Logical propositions are such as can be known a priori, without study of the actual world. We only know from a study of empirical facts that Socrates is a man, but we know the correctness of the syllogism in its abstract form (i.e. when it is stated in terms of variables) without needing any appeal to experience. This is a characteristic, not of logical propositions in themselves, but of the way in which we know them. It has, however, a bearing upon the question what their nature may be, since there are some kinds of propositions which it would be very difficult to suppose we could know without experience.

It is clear that the definition of "logic" or "mathematics" must be sought by trying to give a new definition of the old notion of "analytic" propositions. Although we can no longer be satisfied to define logical propositions as those that follow from the law of contradiction, we can and must still admit that they are a wholly different class of propositions from those that we come to know empirically. They all have the characteristic which, a moment ago, we agreed to call "tautology." This, combined with the fact that they can be expressed wholly in terms of variables and logical constants (a logical constant being something which remains constant in a proposition even when all its constituents are changed)—will give the definition of logic or pure mathematics. For the moment, I do not know how to define "tautology."[44] It would be easy to offer a definition which might seem satisfactory for a while; but I know of none that I feel to be satisfactory, in spite of feeling thoroughly familiar with the characteristic of which a definition is wanted. At this point, therefore, for the moment, we reach the frontier of knowledge on our backward journey into the logical foundations of mathematics.

[44]The importance of "tautology" for a definition of mathematics was pointed out to me by my former pupil Ludwig Wittgenstein, who was working on the problem. I do not know whether he has solved it, or even whether he is alive or dead.

We have now come to an end of our somewhat summary introduction to mathematical philosophy. It is impossible to convey adequately the ideas that are concerned in this subject so long as we abstain from the use of logical symbols. Since ordinary language has no words that naturally express exactly what we wish to express, it is necessary, so long as we adhere to ordinary language, to strain words into unusual meanings; and the reader is sure, after a time if not at first, to lapse into attaching the usual meanings to words, thus arriving at wrong notions as to what is intended to be said. Moreover, ordinary grammar and syntax is extraordinarily misleading. This is the case, e.g., as regards numbers; "ten men" is grammatically the same form as "white men," so that 10 might be thought to be an adjective qualifying "men." It is the case, again, wherever propositional functions are involved, and in particular as regards existence and descriptions. Because language is misleading, as well as because it is diffuse and inexact when applied to logic (for which it was never intended), logical symbolism is absolutely necessary to any exact or thorough treatment of our subject. Those readers, therefore, who wish to acquire a mastery of the principles of mathematics, will, it is to be hoped, not shrink from the labour of mastering the symbols—a labour which is, in fact, much less than might be thought. As the above hasty survey must have made evident, there are innumerable unsolved problems in the subject, and much work needs to be done. If any student is led into a serious study of mathematical logic by this little book, it will have served the chief purpose for which it has been written.

INDEX

Aggregates, [12]
Alephs, [83], [92], [97], [125]
Aliorelatives, [32]
All, [158] ff.
Analysis, [4]
Ancestors, [25], [33]
Argument of a function, [47], [108]
Arithmetising of mathematics, [4]
Associative law, [58], [94]
Axioms, [1]
Between, [38] ff., [58]
Bolzano, [138] n.
Boots and socks, [126]
Boundary, [70], [98], [99]
Cantor, Georg, [77], [79], [85] n., [86], [89],
[95], [102], [136]
Classes, [12], [137], [181] ff.;
reflexive, [80], [127], [138];
similar, [15], [16]
Clifford, W. K., [76]
Collections, infinite, [13]
Commutative law, [58], [94]
Conjunction, [147]
Consecutiveness, [37], [38], [81]
Constants, [202]
Construction, method of, [73]
Continuity, [86], [97] ff.;
Cantorian, [102] ff.;
Dedekindian, [101] ff.;
in philosophy, [105];
of functions, [106] ff.
Contradictions, [135] ff.
Convergence, [115]
Converse, [16], [32], [49]
Correlators, [54]
Counterparts, objective, [61]
Counting, [14], [16]
Dedekind, [69], [99], [138] n.
Deduction, [144] ff.
Definition, [3];
extensional and intensional, [12]
Derivatives, [100]
Descriptions, [139], [144]
Descriptions, [167]
Dimensions, [29]
Disjunction, [147]
Distributive law, [58], [94]
Diversity, [87]
Domain, [16], [32], [49]
Equivalence, [183]
Euclid, [67]
Existence, [164], [171], [177]
Exponentiation, [94], [120]
Extension of a relation, [60]
Fictions, logical, [14] n., [45], [137]
Field of a relation, [32], [53]
Finite, [27]
Flux, [105]
Form, [198]
Fractions, [37], [64]
Frege, [7], [10], [25] n., [77], [95], [146] n.
Functions, [46];
descriptive, [46], [180];
intensional and extensional, [186];
predicative, [189];
propositional, [46], [144];
propositional, [155];
Gap, Dedekindian, [70] ff., [99]
Generalisation, [156]
Geometry, [29], [59], [67], [74], [100], [145];
analytical, [4], [86]
Greater and less, [65], [90]
Hegel, [107]
Hereditary properties, [21]
Implication, [146], [153];
formal, [163]
Incommensurables, [4], [66]
Incompatibility, [147] ff., [200]
Incomplete symbols, [182]
Indiscernibles, [192]
Individuals, [132], [141], [173]
Induction, mathematical, [20] ff., [87], [93],
[185]
Inductive properties, [21]
Inference, [148]
Infinite, [28]; of rationals, [65];
Cantorian, [65];
of cardinals, [77] ff.;
and series and ordinals, [89] ff.
Infinity, axiom of, [66] n., [77], [131] ff.,
[202]
Instances, [156]
Integers, positive and negative, [64]
Intervals, [115]
Intuition, [145]
Irrationals, [66], [72]
Kant, [145]
Leibniz, [80], [107], [192]
Lewis, C. I., [153], [154]
Likeness, [52]
Limit, [29], [69] ff., [97] ff.;
of functions, [106] ff.
Limiting points, [99]
Logic, [159], [65], [194] ff.;
mathematical, [v], [201], [206]
Logicising of mathematics, [7]
Maps, [52], [60] ff., [80]
Mathematics, [194] ff.
Maximum, [70], [98]
Median class, [104]
Meinong, [169]
Method, [vi]
Minimum, [70], [98]
Modality, [165]
Multiplication, [118] ff.
Multiplicative axiom, [92], [117] ff.
Names, [173], [182]
Necessity, [165]
Neighbourhood, [109]
Nicod, [148], [149], [151]
Null-class, [23], [132]
Number, cardinal, [10] ff., [56], [77] ff., [95];
complex, [74] ff.;
finite, [20] ff.;
inductive, [27], [78], [131];
infinite, [77] ff.;
irrational, [66], [72];
maximum? [135];
multipliable, [130];
natural, [2] ff., [22];
non-inductive, [88], [127];
real, [66], [72], [84];
reflexive, [80], [127];
relation, [56], [94];
serial, [57]
Occam, [184]
Occurrences, primary and secondary,
[179]
Ontological proof, [203]
Order 29ff.; cyclic, [40]
Oscillation, ultimate, [111]
Parmenides, [138]
Particulars, [140] ff., [173]
Peano, [5] ff., [23], [24], [78], [81], [131], [163]
Peirce, [32] n.
Permutations, [50]
Philosophy, mathematical, [v], [1]
Plato, [138]
Plurality, [10]
Poincaré, [27]
Points, [59]
Posterity, [22] ff., [32]; proper, [36]
Postulates, [71], [73]
Precedent, [98]
Premisses of arithmetic, [5]
Primitive ideas and propositions, [5], [202]
Progressions, [8], [81] ff.
Propositions, [155]; analytic, [204];
elementary, [161]
Pythagoras, [4], [67]
Quantity, [97], [195]
Ratios, [64], [71], [84], [133]
Reducibility, axiom of, [191]
Referent, [48]
Relation numbers, [56] ff.
Relations, asymmetrical [31], [42];
connected, [32];
many-one, [15];
one-many, [15], [45];
one-one, [15], [47], [79];
reflexive, [16];
serial, [34];
similar, [52];
squares of, [32];
symmetrical, [16], [44];
transitive, [16], [32]
Relatum, [48]
Representatives, [120]
Rigour, [144]
Royce, [80]
Section, Dedekindian, [69] ff.;
ultimate, [111]
Segments, [72], [98]
Selections, [117]
Sequent, [98]
Series, [29] ff.; closed, [103];
compact, [66], [93], [100];
condensed in itself, [102];
Dedekindian, [71], [73], [101];
generation of, [41];
infinite, [89];
perfect, [102], [103];
well-ordered, [92], [123]
Sheffer, [148]
Similarity, of classes, [15] ff.;
of relations, [83];
of relations, [52]
Some, [158] ff.
Space, [61], [86], [140]
Structure, [60] ff.
Sub-classes, [84] ff.
Subjects, [142]
Subtraction, [87]
Successor of a number, [23], [35]
Syllogism, [197]
Tautology, [203], [205]
The, [167], [172] ff.
Time, [61], [86], [140]
Truth-function, [147]
Truth-value, [146]
Types, logical, [53], [135] ff., [185], [188]
Unreality, [168]
Value of a function, [47], [108]
Variables, [10], [161], [199]
Veblen, [58]
Verbs, [141]
Weierstrass, [97], [107]
Wells, H. G., [114]
Whitehead, [64], [76], [107], [119]
Wittgenstein, [205] n.
Zermelo, [123], [129]
Zero, [65]

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TRANSCRIBER'S NOTES

This ebook was produced using scanned images and OCR text generously provided by the University of Toronto through the Internet Archive.

Minor typographical corrections and presentational changes have been made without comment.