Principia Mathematica, Vol 2

by Bertrand Russell, Alfred North Whitehead

An Unabridged, Digitally Enlarged Printing Of Volume II Of III With Additional Errata To Volume Part III - CARDINAL ARITHMETIC - Definition And Logical Properties Of Cardinal Numbers - Addition, Multiplication And Exponentiation - Finite And Infinite - Part IV - RELATION ARITHMETIC - Ordinal Similarity And Relation-Numbers - Addition Of Relations, And The Product Of Two Relations - The Principle Of First Differences, And The Multiplication And Exponentiation Of Relations - Arithmetic And Relation-Numbers - Part V -SERIES - General Theory Of Series - On Sections, Segments, Stretches, And Derivatives - On Convergence, And The Limits Of Functions

Genres: Mathematics, Philosophy, Science, Nonfiction

806 pages, Paperback

First published January 1, 1912

About the author

Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, was a Welsh philosopher, historian, logician, mathematician, advocate for social reform, pacifist, and prominent rationalist. Although he was usually regarded as English, as he spent the majority of his life in England, he was born in Wales, where he also died.

He was awarded the Nobel Prize in Literature in 1950 "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought."

Reviews

[Laura · Rating 4 out of 5]

Volume 2 demonstrates that fundamental concepts of number theory and arithmetic can be derived purely from logic and set theory without assuming any arithmetic axioms. It shows how both finite and infinite numbers emerge from the logical system built in Volume 1, vindicating the logicist thesis that mathematics is reducible to logic.

CONTENTS OF VOLUME II

PREFATORY STATEMENT OF SYMBOLIC CONVENTIONS ix

PART III. CARDINAL ARITHMETIC.

Summary of Part III 3

Section A. Definition and Logical Properties of Cardinal Numbers 4

*100. Definition and elementary properties of cardinal numbers 13

*101. On 0 and 1 and 2 19

*102. On cardinal numbers of assigned types 24

*103. Homogeneous cardinals 36

*104. Ascending cardinals 42

*105. Descending cardinals 52

*106. Cardinals of relational types 60

Section B. Addition, Multiplication and Exponentiation 66

*110. The arithmetical sum of two classes and of two cardinals 75

*111. Double similarity 88

*112. The arithmetical sum of a class of classes 97

*113. On the arithmetical product of two classes or of two cardinals 105

*114. The arithmetical product of a class of classes 124

*115. Multiplicative classes and arithmetical classes 135

*116. Exponentiation 143

*117. Greater and less 171

General note on cardinal correlators 185

Section C. Finite and Infinite 187

*118. Arithmetical substitution and uniform formal numbers 193

*119. Subtraction 201

*120. Inductive cardinals 207

*121. Intervals 233

*122. Progressions 253

*123. \(\aleph_{0}\) 268

*124. Reflexive classes and cardinals 278

*125. The axiom of infinity 289

*126. On typically indefinite inductive cardinals 293

PART IV. RELATION-ARITHMETIC.

Summary of Part IV 301

Section A. Ordinal Similarity and Relation-Numbers 303

*150. Internal transformation of a relation 306
*151. Ordinal similarity 319
*152. Definition and elementary properties of relation-numbers 330
*153. The relation-numbers \(0_{r}\), \(2_{r}\) and \(1_{s}\) 334
*154. Relation-numbers of assigned types 339
*155. Homogeneous relation-numbers 344

Section B. Addition of Relations, and the product of two relations 347

*160. The sum of two relations 351
*161. Addition of a term to a relation 357
*162. The sum of the relations of a field 362
*163. Relations of mutually exclusive relations 369
*164. Double likeness 376
*165. Relations of relations of couples 386
*166. The product of two relations 396

Section C. The Principle of First Differences, and the multiplication
and exponentiation of relations 403

*170. On the relation of first differences among the sub-classes of a
given class 411
*171. The principle of first differences (continued) 423
*172. The product of the relations of a field 428
*173. The product of the relations of a field (continued) 443
*174. The associative law of relational multiplication 447
*176. Exponentiation 458
*177. Propositions connecting \(P_{\text{df}}\) with products and powers 471

Section D. Arithmetic of Relation-Numbers 473

*180. The sum of two relation-numbers 477
*181. On the addition of unity to a relation-number 482
*182. On separated relations 487
*183. The sum of the relation-numbers of a field 496
*184. The product of two relation-numbers 501
*185. The product of the relation-numbers of a field 505
*186. Powers of relation-numbers 507

PART V. SERIES.

Summary of Part V. 513

Section A. General Theory of Series 516

*200. Relations contained in diversity 518
*201. Transitive relations 525
*202. Connected relations 533
*204. Elementary properties of series 547
*205. Maximum and minimum points 559
*206. Sequent points 577
*207. Limits 594
*208. The correlation of series 605

Section B. On Sections, Segments, Stretches, and Derivatives 612

*210. On series of classes generated by the relation of inclusion 615

*211. On sections and segments 624

*212. The series of segments 651

*213. Sectional relations 668

*214. Dedekindian relations 684

*215. Stretches 691

*216. Derivatives 700

*217. On segments of sums and converses 710

Section C. On Convergence, and the Limits of Functions 715

*230. On convergents 720

*231. Limiting sections and ultimate oscillation of a function 727

*232. On the oscillation of a function as the argument approaches a
given limit 737

*233. On the limits of functions 745

*234. Continuity of functions 753

I made the formatting and post-processing this book for DistributedProofreaders and Project Gutenberg will publish it.

[William Bies · Rating 4 out of 5]

At last, we prove that 1+1=2! The reason we do not get around to a seemly simple proposition such as this until well into vol. II, of course, is that before doing so one has to have defined one’s terms. Whitehead and Russell want to work with absolute rigor and in extreme generality. Thus, it takes the better part of vol. I to build one’s way up to a satisfactory and purely logical definition of 0,1,2. Here in vol. II, we define the concept of number itself and the arithmetical operations that can be performed on them. The basic approach draws on Frege’s concept of equinumerosity. For Whitehead and Russell, though, it is not quite so easy since, first, we want to embrace the entire hierarchy of infinite quantities in Cantor’s transfinitum and second, we want to take especial care to preclude any possible contradictions by means of a theory of types.

All this complication leaves this reviewer wondering whether there is a useful simplified concept of number so that we can employ ordinary arithmetic without worrying about all the complexities of type theory. The answer is yes: this is what Whitehead and Russell call formal numbers [see pp. 292-298].

At this juncture, one may well raise the overall question as to whether Whitehead and Russell do anything interesting in vol. II: their relation arithmetic reduces to Cantor’s ordinal arithmetic for well-ordered relations but works in principle for any relations, yet Whitehead-Russell do not demonstrate that the generalization leads to anything new and interesting, e.g., for non-serial relations. For instance, it is not even clear that anything very deep has ever been done with Cantor’s ordinal arithmetic itself. If so, then what cause have we of generalizing it? In a similar vein, it is far from evident that all the complications of type lead to anything of interest. Whitehead and Russell’s propositions are usually trivial, which means one might as well derive them on the fly as needed rather than build up a huge stock of such cases. Thus they purvey a wealth of new concepts but hardly any results, none of which is deep.

Later sections in Part V get into interesting concepts such as limits, convergence and continuity but note: treated here in full generality, we won’t get to the reals until Part VI in vol. III. To get a handle on what the authors are up to here, let us remark that what general topologists now refer to as a net seems to be just a transfinite series in Whitehead-Russell’s terminology? Now, what the authors go on to do may be of interest as it should show equivalence of Cauchy and Dedekind’s methods of defining the reals [p. 651ff]. Maybe all the material about segments becomes interesting for series involving transfinite ordinals! Also, we know that derivatives in the set-theoretic sense are interesting because of Cantor’s work [p. 700ff]. So maybe all the formalism here is not altogether useless. Unfortunately, it is not spelled out with any examples of situations where, as functional analysts are aware, sequences are insufficient and one needs to avail oneself of the full machinery of nets.

Now the authors discuss convergence and limits of functions without topology! [p. 715ff.] They seem here really to be doing functional analysis in nuce along the lines of Kelly’s textbook on general topology. Their serial relations amount to what we call nets. But everything here is too abstract as again not illustrated by means of any substantial examples. It remains doubtful whether anything is gained by working in such generality, for topological language avails itself of spatial intuition and hence will be easier to comprehend than Whitehead-Russell’s relation language, in which the basic concept is an order relation. For instance, the conventional designations of lim sup and lim inf are easier to understand than Whitehead-Russell’s ‘ultimate oscillation’ though they seem to be essentially the same notion, though with respect to an arbitrary order and not just to the real number line [p. 727]. It will be observed that practically nothing in the basic theory of continuous functions requires the use of numbers themselves (hence Whitehead-Russell recognize that they are covertly doing functional analysis here, cf. p. 755).

A handful of questions for reflection:

1) Whitehead-Russell’s way of defining things is reminiscent of functors in homological algebra (which itself was criticized as empty formalism when first introduced but has since shown its itself productive of good ideas). If someone were to follow up on the Principia Mathematica in earnest, would it play out into something worthwhile as homological algebra seems to have, despite its naysayers?

2) Is the Whitehead-Russell approach to foundations fruitful for the rest of mathematics or just a rabbit hole they like to go down? Along the same lines, is anything Whitehead-Russell do in Principia Mathematica deep? This reviewer is not so sure; for depth, one would want to see something pleasing and unexpected come out, not just an array of over-refined technical distinctions that are never pursued into any interesting consequences.

3) The whole problem of extensive versus intensive definition of properties comes down to this: is their concept of mathematics = everything extrinsically definable adequate? For what purpose, that is? As a bare and formal language game one could so restrict oneself, but what about for physics, viz., for comprehending the world? We know that natural language is a powerful instrument, maybe we need another kind of logic to describe the real world? In other words, might we have proper inclusions: mathematics < physics < life-world, where physics would rest upon a more expressive language than Whitehead-Russell type mathematics that is nevertheless simpler than natural language, but self-contained and useful for the analysis of natural processes (not involving intervention of human beings)?

To summarize our evaluation after reading vol. II, let us ask whether a closer study of the Principia Mathematica would facilitate comprehension of any major branch of mathematics other than set theory and logic? Probably not, a good textbook on real analysis such as Folland or papa Rudin should be sufficient for most purposes. The present approach of Whitehead and Russell may be said to be hampered by 1) an excessive accumulation of special notations; 2) outdated terminology different from what has since become standard; 3) lack of perspective: too many propositions are labeled as important; and 4) a focus almost exclusively on treating things in far more generality than is needed for any substantial applications. Hence, one gets bogged down in minutiae that have no bearing on any interesting examples.

Thus, unless one wants to become a logician, it is doubtful that devoting more than a cursory attention to the Principia Mathematica would repay the effort. It may be well that somebody tried to work out the theory of types in minute detail as far as Whitehead and Russell did but it seems that any major advance in logic would be more likely to stem from an original idea (namely, a new basic concept beyond that of set or class membership, such as may be found in abundance in medieval supposition theory) or philosophical criticism of the construction of existing concepts (as in the age-old problem of universals) than from attempting to take the framework of the present Principia Mathematica itself any further. For there is little evidence that any deep ideas in a conventional field of mathematics, say in functional analysis or algebraic topology, have been stimulated by Whitehead and Russell’s work as we find it here.

Therefore, this reviewer’s judgment: spend as much time as one can stand on the Principia Mathematica for the sake of culture but don’t get mired in the minor details which will probably never lead to anything of substantial interest, then move on to more arresting ideas taken from somewhere else in the vast domain of the rest of mathematics.

[A YOGAM · Rating 5 out of 5]

„Principia Mathematica“ ist nicht einfach nur ein Buch; es ist ein intellektuelles Monument, dessen schiere Ambition noch heute Ehrfurcht gebietet. Alfred North Whitehead und Bertrand Russell unternahmen hier den Angst einflößend komplexen Versuch, die gesamte Mathematik aus einer Handvoll rein logischer Axiome abzuleiten. Über Tausende von Seiten erstreckt sich ein Geflecht aus Symbolen und Beweisen von legendärer Dichte, das den Gipfel des logizistischen Traums darstellt und das Fundament für die analytische Philosophie und die moderne Logik legte.
Der wahre, unsterbliche Stellenwert dieses Werkes bemisst sich jedoch ironischerweise an seinem vermeintlich größten Kritiker. Dass sich ein Logiker vom Kaliber eines Kurt Gödel berufen fühlte, seine revolutionären Unvollständigkeitssätze ausgerechnet in direkter Auseinandersetzung mit dem System der „Principia Mathematica“ zu entwickeln, ist die ultimative Verbeugung. Gödel widerlegte nicht irgendein System; er wählte den am weitesten entwickelten, rigorosesten und einflussreichsten Versuch der Geschichte. Gödels Arbeit dokumentiert daher weniger das Scheitern der „Principia Mathematica“ als vielmehr deren unangefochtene historische Dominanz als der Maßstab, an dem sich selbst die größten Geister messen lassen mussten.

[Mitchell A. Solsky · Rating 5 out of 5]

I spent the better part of 3 months in a pure fixation over the revised vol 1 & 2. For my money it’s the perfect antidote to Gödel - though I have great admiration for Gödel.

[Richard Marney · Rating 5 out of 5]

Two down, one to go.

See comments from Volume 1. A one week break, then on to volume III.

Having time and quiet time at that to explore this series is thanks to being locked-down on and off for 16 months!

A proper review will come when the full set is completed and the project work for which the reading undertaken published.