The symbolic language created by Peano plays a very grand rôle in these new researches. It is capable of rendering some service, but I think M. Couturat attaches to it an exaggerated importance which must astonish Peano himself.
The essential element of this language is certain algebraic signs which represent the different conjunctions: if, and, or, therefore. That these signs may be convenient is possible; but that they are destined to revolutionize all philosophy is a different matter. It is difficult to admit that the word _if_ acquires, when written C, a virtue it had not when written if. This invention of Peano was first called _pasigraphy_, that is to say the art of writing a treatise on mathematics without using a single word of ordinary language. This name defined its range very exactly. Later, it was raised to a more eminent dignity by conferring on it the title of _logistic_. This word is, it appears, employed at the Military Academy, to designate the art of the quartermaster of cavalry, the art of marching and cantoning troops; but here no confusion need be feared, and it is at once seen that this new name implies the design of revolutionizing logic.
We may see the new method at work in a mathematical memoir by Burali-Forti, entitled: _Una Questione sui numeri transfiniti_, inserted in Volume XI of the _Rendiconti del circolo matematico di Palermo_.
I begin by saying this memoir is very interesting, and my taking it here as example is precisely because it is the most important of all those written in the new language. Besides, the uninitiated may read it, thanks to an Italian interlinear translation.
What constitutes the importance of this memoir is that it has given the first example of those antinomies met in the study of transfinite numbers and making since some years the despair of mathematicians. The aim, says Burali-Forti, of this note is to show there may be two transfinite numbers (ordinals), _a_ and _b_, such that _a_ is neither equal to, greater than, nor less than _b_.
To reassure the reader, to comprehend the considerations which follow, he has no need of knowing what a transfinite ordinal number is.
Now, Cantor had precisely proved that between two transfinite numbers as between two finite, there can be no other relation than equality or inequality in one sense or the other. But it is not of the substance of this memoir that I wish to speak here; that would carry me much too far from my subject; I only wish to consider the form, and just to ask if this form makes it gain much in rigor and whether it thus compensates for the efforts it imposes upon the writer and the reader.
First we see Burali-Forti define the number 1 as follows:
1 = [iota]T'{Ko[(n_](u, h)[epsilon](u[epsilon]Un)},
a definition eminently fitted to give an idea of the number 1 to persons who had never heard speak of it.
I understand Peanian too ill to dare risk a critique, but still I fear this definition contains a petitio principii, considering that I see the figure 1 in the first member and Un in letters in the second.
However that may be, Burali-Forti starts from this definition and, after a short calculation, reaches the equation:
(27) 1[epsilon]No,
which tells us that One is a number.
And since we are on these definitions of the first numbers, we recall that M. Couturat has also defined 0 and 1.
What is zero? It is the number of elements of the null class. And what is the null class? It is that containing no element.
To define zero by null, and null by no, is really to abuse the wealth of language; so M. Couturat has introduced an improvement in his definition, by writing:
0 = [iota][Lambda]:[phi]x = [Lambda]·[inverted c]·[Lambda] = (x[epsilon][phi]x),
which means: zero is the number of things satisfying a condition never satisfied.
But as never means _in no case_ I do not see that the progress is great.
I hasten to add that the definition M. Couturat gives of the number 1 is more satisfactory.
One, says he in substance, is the number of elements in a class 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.
