BR TO “NOTES ON POINCARE, RMM Nov. ’05, 7 DEC. 1905”
BRACERS 53271. ALS. La Chaux-de-Fonds Bib., Suisse. Russell–Couturat 1: #192
Edited by A.-F. Schmid

Notes on Poincaré, RMM Nov. ’05

p. 818. The principle of induction as here stated has the defect of being false. $\aleph_0$ is a whole number ; but it has the property that a class of $\aleph_0$ numbers contains a part similar to itself, while does not belong to 1, and does not belong to n + 1 if it does not belong to n. If mathematics depends on the principle here stated, its theorems may occasionally be true by accident, but we can have no reason for believing them.

p. 819. Definitions by postulates are always to be transformed into nominal definitions before they can be admitted. This is not always possible ; when it is not, the definition by postulates is wrong substantially as well as formally. This point is relevant in answering P’s criticisms as to existence of induction.

p. 821 (and again p. 830). The accusation of circularity in Definitions of numbers rests on supposing that when a proposition contains two constituents, it cannot be enunciated without presupposition that we know what two is, and so on. Take e. g. the Df. of 1, which is as follows : « 1 is the class of all classes u which are such that the proposition « “x is a u” is equivalent, for all values of x, to “x is identical with c” » is not false for all values of c. » Here « x is a u » may be supposed involve one in the indefinite article ; but this phrase is defined as a whole, as follows : « x is a u » means « “φ(x) is true, and u^a has to φ the relation of defined class to defining property” is not false for all values of φ ». It is doubtless true that the various entities occurring in such definitions are each one, or rather that the class of which one of them is the only constituent is a unit class. But this fact is not presupposed in the definition.

p. 823. The circle as regard 0 and Ʌ is easily avoided. Put

Ʌ = ??{~ (x ⸧ x)} Df      0 = ι‘Ʌ Df

It is unwise to put Ʌ = ℩‘??{~ (Ǝx) . x ε u} Df

p. 825. « La pasigraphie ne nous préserve pas de l’erreur. Pourquoi ? Est-ce parce que les règles de la logique sont trompeuses ? Évidemment non. » It should be « Évidemment oui ». It is plain to me that the hitherto current logical assumptions are erroneous.

p. 833. The definition of finite numbers [M. Poincaré keeps on forgetting that there are infinite numbers] must not be effected by postulates, but nominally. I. e. « x is a finite number » is to mean « 0 ε s : n ε s . ⸧_n . n + 1 ε s : ⸧_s . x ε s ». This defines a class of x’s satisfying the definition ; the class is not null, for we have 0 ε s : n ε s . ⸧_n . n + 1 ε s : ⸧_s . 0 ε s,^b whence 0 is a member of the class. Hence 1, 2, 3, … are members of it. This is the demonstration of existence.

p. 835. That every number can be obtained by additions of 1 starting from 0, is a proposition equivalent to mathematical induction ; this appears by examining what we mean when we say it can be obtained by additions of 1. We plainly mean, in this case, a finite number of additions of 1, and thus we have to bring in the definition of finite number by induction in order to explain what we mean. If an infinite number of additions were permitted, the statement would be false.

Further note on p. 825.

The reason things go wrong is that the indemonstrables from which we start are wrong. These are vouched for by intuition (as M. Poincaré rights urges), not by pasigraphy. Thus intuition is the offender, not pasigraphy. I have constructed a different set of indemonstrables, from which, so far as I can discover, pasigraphy will not elicit any contradiction.

Notes

^a[consists of all] dans cette formule, devant n ε s, Russell a d’abord écrit et barré ~ Ǝ.