The Principles of Mathematics, by Bertrand Russell. + nemokamas atvežimas! | knygos.lt

Aprašymas

Excerpt:

PREFACE.
The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles, is undertaken in Parts II.—VII. of this Volume, and will be established by strict symbolic reasoning in Volume ii. -The demonstration of this thesis has, if I am not mistaken, all the certainty and precision of which mathematical demonstrations are capable. As the thesis is very recent among mathematicians, and is almost universally denied by philosophers, I have undertaken, in this volume, to defend its various parts, as occasion arose, against such adverse theories as appeared most widely held or most difficult to disprove. I have also endeavoured to present, in language as untechnical as possible, the more important stages in the deductions by which the thesis is established.

The other object of this work, which occupies Part I., is the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task, and I cannot flatter myself that I have done more than indicate a vast field of inquiry, and give a sample of the methods by which the inquiry may be conducted. The discussion of indefinables—which forms the chief part of philosophical logic^is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple. Where, as in the present case, the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them ; there is a process analogous to that which resulted in the discovery of Neptune, with the difference that the final stage—the search with a mental telescope for the entity which has been inferred—is often the most difficult part of the undertaking. In the case of classes, I must confess, I have failed to perceive any concept fulfilling the conditions

I requisite for the notion of class. And the contradiction discussed in Chapter X proves that something is amiss, but what this is I have hitherto failed to discover.

The second volume, in which I have had the great good fortune to secure the collaboration of Mr A. N. Whitehead, will be addressed exclusively to mathematicians; it will contain chains of deductions, from the premisses of symbolic logic through Arithmetic, finite and infinite, to Geometry, in an order similar to that adopted in the present volume; it will also contain various original developments, in which the method of Professor Peano, as supplemented by the Logic of Relations, has shown itself a powerful instrument of mathematical investigation. 1 The present volume, which may be regarded either as a commentary

j upon, or as an introduction to, the second volume, is addressed in equal measure to the philosopher and to the mathematician; but some parts will be more interesting to the one, others to the other. I should advise mathematicians, unless they are specially interested in Symbolic Logic, to begin with Part IV., and only refer to earlier parts as occasion arises. The following portions are more specially philosophical: Part I. (omitting Chapter II); Part II., Chapters XI, XV, XVI, XVII; Part III.; Part IV., §207, Chapters XXVI, XXVII, XXXI; Part V., Chapters XLI, XLII, XLIII; Part VI., Chapters I, II, III ; Part VII., Chapters LIII, LIV, LV, LVII, LVIII; and the two Appendices, which belong to Part I., and should be read in connection with it.