BR TO “NOTES ON POINCARE'S SECOND ARTICLE”, 7 DEC. 1905
BRACERS 53271. ALS. La Chaux-de-Fonds Bib., Suisse. Russell–Couturat 1: #193
Edited by A.-F. Schmid

Notes on Poincaré’s 2nd article.

N° XXVI. Couturat’s remark quoted here is incautious. It does not hold of any theorem in the proof of which $\aleph_0$ occurs. P’s remarks in XXVII are absurd : instance of use of 3 dimensions in proving the uniqueness of the harmonic conjugate in projective geometry [quadrilateral construction].

N° XXVIII. P’s remarks on geometrical existence-theorems are sound. The answer to all this is that there is not the slightest reason for wishing to avoid mathematical induction, since its use does not constitute an appeal to intuition.

N° XXIX. All this talk about the N^th syllogism is nonsense. We don’t count syllogisms, or prove that if the passage to the n^th syllogism does not take us to a contradiction, the passage to the (N + 1)^th does not.

N° XXXI. « Nous avons beaucoup à y apprendre ». Evidently.

N° XXVI (end). Whatever is known to be true in the theory of aggregates is proved without appeal to intuition. The appeal to intuition is only useful for helping lazy people to believe what is doubtful. As for the contradictions, I have solved them.

It is a pity Poincaré has not read Whitehead on Cardinals, Amer. J. XXIV [the Dfs. of 1, 2, 3 given there are faulty ; they wrongly omit existence.]

Sketch of theory of finite and infinite

Nc‘u = ??{(ƎR) . R ε 1 → 1 . D‘R = u . $\breve {D}$‘R = v} Df

α + 1 = ??{(Ǝu, x) . Nc‘u = α . x ~ ε u . v = u ∪ ι‘x} Df

Ʌ = ??(x ≠ x) Df

0 = ι‘Ʌ Df

1 = ??{(Ǝc) : x ε u . ≡_x . x = c} Df

⊢ . 1 = 0 + 1 [Easily proved]

Ncfin = ??{0 ε s : m ε s . ⸧_m . m + 1 ε s : ⸧_s n ε s} Df

⊢ . 0 ε Ncfin

Dem.

⊢ :. 0 ε s : m ε s . ⸧_m . m + 1 ε s : ⸧_s . 0 ε s ⸫ ⸧ ⊢ . Prop

⊢ . Ǝ!Ncfin ⊢ . 1 ε Ncfin [⊢ . 1 = 0 + 1 . ⸧ ⊢ . Prop ]

In this way, any given number which is finite can be proved finite. The non-contradictoriness of the Df of Ncfin follows from the fact that 0, 1, … are members of Ncfin.

All this is easily translated into the language of the new substitutional theory. If people object to 0, we can put 1 for 0 in the Df of Ncfin ; then 1, 2,.. ε Ncfin and things proceed as before.