One, says he in substance, is the number of elements in a cla.s.s in which any two elements are identical.

It is more satisfactory, I have said, in this sense that to define 1, he does not use the word one; in compensation, he uses the word two. But I fear, if asked what is two, M. Couturat would have to use the word one.

VIII

But to return to the memoir of Burali-Forti; I have said his conclusions are in direct opposition to those of Cantor. Now, one day M. Hadamard came to see me and the talk fell upon this antinomy.

"Burali-Forti's reasoning," I said, "does it not seem to you irreproachable?" "No, and on the contrary I find nothing to object to in that of Cantor. Besides, Burali-Forti had no right to speak of the aggregate of _all_ the ordinal numbers."

"Pardon, he had the right, since he could always put

[Omega] = T'(No,[epsilon]>).

I should like to know who was to prevent him, and can it be said a thing does not exist, when we have called it [Omega]?"

It was in vain, I could not convince him (which besides would have been sad, since he was right). Was it merely because I do not speak the Peanian with enough eloquence? Perhaps; but between ourselves I do not think so.

Thus, despite all this pasigraphic apparatus, the question was not solved. What does that prove? In so far as it is a question only of proving one a number, pasigraphy suffices, but if a difficulty presents itself, if there is an antinomy to solve, pasigraphy becomes impotent.

CHAPTER IV

THE NEW LOGICS

I

_The Russell Logic_

To justify its pretensions, logic had to change. We have seen new logics arise of which the most interesting is that of Russell. It seems he has nothing new to write about formal logic, as if Aristotle there had touched bottom. But the domain Russell attributes to logic is infinitely more extended than that of the cla.s.sic logic, and he has put forth on the subject views which are original and at times well warranted.

First, Russell subordinates the logic of cla.s.ses to that of propositions, while the logic of Aristotle was above all the logic of cla.s.ses and took as its point of departure the relation of subject to predicate. The cla.s.sic syllogism, "Socrates is a man," etc., gives place to the hypothetical syllogism: "If _A_ is true, _B_ is true; now if _B_ is true, _C_ is true," etc. And this is, I think, a most happy idea, because the cla.s.sic syllogism is easy to carry back to the hypothetical syllogism, while the inverse transformation is not without difficulty.

And then this is not all. Russell's logic of propositions is the study of the laws of combination of the conjunctions _if_, _and_, _or_, and the negation _not_.

In adding here two other conjunctions, _and_ and _or_, Russell opens to logic a new field. The symbols _and_, _or_ follow the same laws as the two signs and +, that is to say the commutative a.s.sociative and distributive laws. Thus _and_ represents logical multiplication, while _or_ represents logical addition. This also is very interesting.

Russell reaches the conclusion that any false proposition implies all other propositions true or false. M. Couturat says this conclusion will at first seem paradoxical. It is sufficient however to have corrected a bad thesis in mathematics to recognize how right Russell is. The candidate often is at great pains to get the first false equation; but that once obtained, it is only sport then for him to acc.u.mulate the most surprising results, some of which even may be true.

II

We see how much richer the new logic is than the cla.s.sic logic; the symbols are multiplied and allow of varied combinations _which are no longer limited in number_. Has one the right to give this extension to the meaning of the word _logic_? It would be useless to examine this question and to seek with Russell a mere quarrel about words. Grant him what he demands; but be not astonished if certain verities declared irreducible to logic in the old sense of the word find themselves now reducible to logic in the new sense--something very different.

A great number of new notions have been introduced, and these are not simply combinations of the old. Russell knows this, and not only at the beginning of the first chapter, 'The Logic of Propositions,' but at the beginning of the second and third, 'The Logic of Cla.s.ses' and 'The Logic of Relations,' he introduces new words that he declares indefinable.

And this is not all; he likewise introduces principles he declares indemonstrable. But these indemonstrable principles are appeals to intuition, synthetic judgments _a priori_. We regard them as intuitive when we meet them more or less explicitly enunciated in mathematical treatises; have they changed character because the meaning of the word logic has been enlarged and we now find them in a book ent.i.tled _Treatise on Logic_? _They have not changed nature; they have only changed place._

III

Could these principles be considered as disguised definitions? It would then be necessary to have some way of proving that they imply no contradiction. It would be necessary to establish that, however far one followed the series of deductions, he would never be exposed to contradicting himself.

We might attempt to reason as follows: We can verify that the operations of the new logic applied to premises exempt from contradiction can only give consequences equally exempt from contradiction. If therefore after _n_ operations we have not met contradiction, we shall not encounter it after _n_ + 1. Thus it is impossible that there should be a moment when contradiction _begins_, which shows we shall never meet it. Have we the right to reason in this way? No, for this would be to make use of complete induction; and _remember, we do not yet know the principle of complete induction_.

We therefore have not the right to regard these a.s.sumptions as disguised definitions and only one resource remains for us, to admit a new act of intuition for each of them. Moreover I believe this is indeed the thought of Russell and M. Couturat.

Thus each of the nine indefinable notions and of the twenty indemonstrable propositions (I believe if it were I that did the counting, I should have found some more) which are the foundation of the new logic, logic in the broad sense, presupposes a new and independent act of our intuition and (why not say it?) a veritable synthetic judgment _a priori_. On this point all seem agreed, but what Russell claims, and _what seems to me doubtful, is that after these appeals to intuition, that will be the end of it; we need make no others and can build all mathematics without the intervention of any new element_.

IV

M. Couturat often repeats that this new logic is altogether independent of the idea of number. I shall not amuse myself by counting how many numeral adjectives his exposition contains, both cardinal and ordinal, or indefinite adjectives such as several. We may cite, however, some examples:

"The logical product of _two_ or _more_ propositions is....";

"All propositions are capable only of _two_ values, true and false";

"The relative product of _two_ relations is a relation";

"A relation exists between two terms," etc., etc.

Sometimes this inconvenience would not be unavoidable, but sometimes also it is essential. A relation is incomprehensible without two terms; it is impossible to have the intuition of the relation, without having at the same time that of its two terms, and without noticing they are two, because, if the relation is to be conceivable, it is necessary that there be two and only two.

V

_Arithmetic_

I reach what M. Couturat calls the _ordinal theory_ which is the foundation of arithmetic properly so called. M. Couturat begins by stating Peano's five a.s.sumptions, which are independent, as has been proved by Peano and Padoa.

1. Zero is an integer.

2. Zero is not the successor of any integer.

3. The successor of an integer is an integer. To this it would be proper to add,

Every integer has a successor.

4. Two integers are equal if their successors are.

The fifth a.s.sumption is the principle of complete induction.

M. Couturat considers these a.s.sumptions as disguised definitions; they const.i.tute the definition by postulates of zero, of successor, and of integer.