If previously have been defined all these notions but one, then this last will be by definition the thing which verifies these postulates.
Thus certain indemonstrable a.s.sumptions of mathematics would be only disguised definitions. This point of view is often legitimate; and I have myself admitted it in regard for instance to Euclid"s postulate.
The other a.s.sumptions of geometry do not suffice to completely define distance; the distance then will be, by definition, among all the magnitudes which satisfy these other a.s.sumptions, that which is such as to make Euclid"s postulate true.
Well the logicians suppose true for the principle of complete induction what I admit for Euclid"s postulate; they want to see in it only a disguised definition.
But to give them this right, two conditions must be fulfilled. Stuart Mill says every definition implies an a.s.sumption, that by which the existence of the defined object is affirmed. According to that, it would no longer be the a.s.sumption which might be a disguised definition, it would on the contrary be the definition which would be a disguised a.s.sumption. Stuart Mill meant the word existence in a material and empirical sense; he meant to say that in defining the circle we affirm there are round things in nature.
Under this form, his opinion is inadmissible. Mathematics is independent of the existence of material objects; in mathematics the word exist can have only one meaning, it means free from contradiction. Thus rectified, Stuart Mill"s thought becomes exact; in defining a thing, we affirm that the definition implies no contradiction.
If therefore we have a system of postulates, and if we can demonstrate that these postulates imply no contradiction, we shall have the right to consider them as representing the definition of one of the notions entering therein. If we can not demonstrate that, it must be admitted without proof, and that then will be an a.s.sumption; so that, seeking the definition under the postulate, we should find the a.s.sumption under the definition.
Usually, to show that a definition implies no contradiction, we proceed by _example_, we try to make an example of a thing satisfying the definition. Take the case of a definition by postulates; we wish to define a notion _A_, and we say that, by definition, an _A_ is anything for which certain postulates are true. If we can prove directly that all these postulates are true of a certain object _B_, the definition will be justified; the object _B_ will be an _example_ of an _A_. We shall be certain that the postulates are not contradictory, since there are cases where they are all true at the same time.
But such a direct demonstration by example is not always possible.
To establish that the postulates imply no contradiction, it is then necessary to consider all the propositions deducible from these postulates considered as premises, and to show that, among these propositions, no two are contradictory. If these propositions are finite in number, a direct verification is possible. This case is infrequent and uninteresting. If these propositions are infinite in number, this direct verification can no longer be made; recourse must be had to procedures where in general it is necessary to invoke just this principle of complete induction which is precisely the thing to be proved.
This is an explanation of one of the conditions the logicians should satisfy, _and further on we shall see they have not done it_.
V
There is a second. When we give a definition, it is to use it.
We therefore shall find in the sequel of the exposition the word defined; have we the right to affirm, of the thing represented by this word, the postulate which has served for definition? Yes, evidently, if the word has retained its meaning, if we do not attribute to it implicitly a different meaning. Now this is what sometimes happens and it is usually difficult to perceive it; it is needful to see how this word comes into our discourse, and if the gate by which it has entered does not imply in reality a definition other than that stated.
This difficulty presents itself in all the applications of mathematics.
The mathematical notion has been given a definition very refined and very rigorous; and for the pure mathematician all doubt has disappeared; but if one wishes to apply it to the physical sciences for instance, it is no longer a question of this pure notion, but of a concrete object which is often only a rough image of it. To say that this object satisfies, at least approximately, the definition, is to state a new truth, which experience alone can put beyond doubt, and which no longer has the character of a conventional postulate.
But without going beyond pure mathematics, we also meet the same difficulty.
You give a subtile definition of numbers; then, once this definition given, you think no more of it; because, in reality, it is not it which has taught you what number is; you long ago knew that, and when the word number further on is found under your pen, you give it the same sense as the first comer. To know what is this meaning and whether it is the same in this phrase or that, it is needful to see how you have been led to speak of number and to introduce this word into these two phrases. I shall not for the moment dilate upon this point, because we shall have occasion to return to it.
Thus consider a word of which we have given explicitly a definition _A_; afterwards in the discourse we make a use of it which implicitly supposes another definition _B_. It is possible that these two definitions designate the same thing. But that this is so is a new truth which must either be demonstrated or admitted as an independent a.s.sumption.
_We shall see farther on that the logicians have not fulfilled the second condition any better than the first._
VI
The definitions of number are very numerous and very different; I forego the enumeration even of the names of their authors. We should not be astonished that there are so many. If one among them was satisfactory, no new one would be given. If each new philosopher occupying himself with this question has thought he must invent another one, this was because he was not satisfied with those of his predecessors, and he was not satisfied with them because he thought he saw a pet.i.tio principii.
I have always felt, in reading the writings devoted to this problem, a profound feeling of discomfort; I was always expecting to run against a pet.i.tio principii, and when I did not immediately perceive it, I feared I had overlooked it.
This is because it is impossible to give a definition without using a sentence, and difficult to make a sentence without using a number word, or at least the word several, or at least a word in the plural. And then the declivity is slippery and at each instant there is risk of a fall into pet.i.tio principii.
I shall devote my attention in what follows only to those of these definitions where the pet.i.tio principii is most ably concealed.
VII
PASIGRAPHY
The symbolic language created by Peano plays a very grand role 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 t.i.tle 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, ent.i.tled: _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 const.i.tutes 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 rea.s.sure 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 pet.i.tio 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 cla.s.s. And what is the null cla.s.s? 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.