1
the logicians have attempted to answer the preceding considerations. for that, a transformation of logistic was necessary, and russell in particular has modified on certain points his original views. without entering into the details of the debate, i should like to return to the two questions to my mind most important: have the rules of logistic demonstrated their fruitfulness and infallibility? is it true they afford means of proving the principle of complete induction without any appeal to intuition?
2
the infallibility of logistic
on the question of fertility, it seems m. couturat has na?ve illusions. logistic, according to him, lends invention ‘stilts and wings,’ and on the next page: ”ten years ago, peano published the first edition of his formulaire.” how is that, ten years of wings and not to have flown!
i have the highest esteem for peano, who has done very pretty things (for instance his ‘space-filling curve,’ a phrase now discarded); but after all he has not gone further nor higher nor quicker than the majority of wingless mathematicians, and would have done just as well with his legs.
on the contrary i see in logistic only shackles for the inventor. it is no aid to conciseness — far from it, and if twenty-seven equations were necessary to establish that 1 is a number, how many would be needed to prove a real theorem? if we distinguish, with whitehead, the individual x, the class of which the only member is x and which shall be called ιx, then the class of which the only member is the class of which the only member is x and which shall be called μx, do you think these distinctions, useful as they may be, go far to quicken our pace?
logistic forces us to say all that is ordinarily left to be understood; it makes us advance step by step; this is perhaps surer but not quicker.
it is not wings you logisticians give us, but leading-strings. and then we have the right to require that these leading-strings prevent our falling. this will be their only excuse. when a bond does not bear much interest, it should at least be an investment for a father of a family.
should your rules be followed blindly? yes, else only intuition could enable us to distinguish among them; but then they must be infallible; for only in an infallible authority can one have a blind confidence. this, therefore, is for you a necessity. infallible you shall be, or not at all.
you have no right to say to us: “it is true we make mistakes, but so do you.” for us to blunder is a misfortune, a very great misfortune; for you it is death.
nor may you ask: does the infallibility of arithmetic prevent errors in addition? the rules of calculation are infallible, and yet we see those blunder who do not apply these rules; but in checking their calculation it is at once seen where they went wrong. here it is not at all the case; the logicians have applied their rules, and they have fallen into contradiction; and so true is this, that they are preparing to change these rules and to “sacrifice the notion of class.” why change them if they were infallible?
“we are not obliged,” you say, “to solve hic et nunc all possible problems.” oh, we do not ask so much of you. if, in face of a problem, you would give no solution, we should have nothing to say; but on the contrary you give us two of them and those contradictory, and consequently at least one false; this it is which is failure.
russell seeks to reconcile these contradictions, which can only be done, according to him, “by restricting or even sacrificing the notion of class.” and m. couturat, discovering the success of his attempt, adds: “if the logicians succeed where others have failed, m. poincaré will remember this phrase, and give the honor of the solution to logistic.”
but no! logistic exists, it has its code which has already had four editions; or rather this code is logistic itself. is mr. russell preparing to show that one at least of the two contradictory reasonings has transgressed the code? not at all; he is preparing to change these laws and to abrogate a certain number of them. if he succeeds, i shall give the honor of it to russell’s intuition and not to the peanian logistic which he will have destroyed.
3
the liberty of contradiction
i made two principal objections to the definition of whole number adopted in logistic. what says m. couturat to the first of these objections?
what does the word exist mean in mathematics? it means, i said, to be free from contradiction. this m. couturat contests. “logical existence,” says he, “is quite another thing from the absence of contradiction. it consists in the fact that a class is not empty.” to say: a‘s exist, is, by definition, to affirm that the class a is not null.
and doubtless to affirm that the class a is not null, is, by definition, to affirm that a‘s exist. but one of the two affirmations is as denuded of meaning as the other, if they do not both signify, either that one may see or touch a‘s which is the meaning physicists or naturalists give them, or that one may conceive an a without being drawn into contradictions, which is the meaning given them by logicians and mathematicians.
for m. couturat, “it is not non-contradiction that proves existence, but it is existence that proves non-contradiction.” to establish the existence of a class, it is necessary therefore to establish, by an example, that there is an individual belonging to this class: “but, it will be said, how is the existence of this individual proved? must not this existence be established, in order that the existence of the class of which it is a part may be deduced? well, no; however paradoxical may appear the assertion, we never demonstrate the existence of an individual. individuals, just because they are individuals, are always considered as existent. . . . we never have to express that an individual exists, absolutely speaking, but only that it exists in a class.” m. couturat finds his own assertion paradoxical, and he will certainly not be the only one. yet it must have a meaning. it doubtless means that the existence of an individual, alone in the world, and of which nothing is affirmed, can not involve contradiction; in so far as it is all alone it evidently will not embarrass any one. well, so let it be; we shall admit the existence of the individual, ‘absolutely speaking,’ but nothing more. it remains to prove the existence of the individual ‘in a class,’ and for that it will always be necessary to prove that the affirmation, “such an individual belongs to such a class,” is neither contradictory in itself, nor to the other postulates adopted.
“it is then,” continues m. couturat, “arbitrary and misleading to maintain that a definition is valid only if we first prove it is not contradictory.” one could not claim in prouder and more energetic terms the liberty of contradiction. “in any case, the onus probandi rests upon those who believe that these principles are contradictory.” postulates are presumed to be compatible until the contrary is proved, just as the accused person is presumed innocent. needless to add that i do not assent to this claim. but, you say, the demonstration you require of us is impossible, and you can not ask us to jump over the moon. pardon me; that is impossible for you, but not for us, who admit the principle of induction as a synthetic judgment a priori. and that would be necessary for you, as for us.
to demonstrate that a system of postulates implies no contradiction, it is necessary to apply the principle of complete induction; this mode of reasoning not only has nothing ‘bizarre’ about it, but it is the only correct one. it is not ‘unlikely’ that it has ever been employed; and it is not hard to find ‘examples and precedents’ of it. i have cited two such instances borrowed from hilbert’s article. he is not the only one to have used it, and those who have not done so have been wrong. what i have blamed hilbert for is not his having recourse to it (a born mathematician such as he could not fail to see a demonstration was necessary and this the only one possible), but his having recourse without recognizing the reasoning by recurrence.
4
the second objection
i pointed out a second error of logistic in hilbert’s article. to-day hilbert is excommunicated and m. couturat no longer regards him as of the logistic cult; so he asks if i have found the same fault among the orthodox. no, i have not seen it in the pages i have read; i know not whether i should find it in the three hundred pages they have written which i have no desire to read.
only, they must commit it the day they wish to make any application of mathematics. this science has not as sole object the eternal contemplation of its own navel; it has to do with nature and some day it will touch it. then it will be necessary to shake off purely verbal definitions and to stop paying oneself with words.
to go back to the example of hilbert: always the point at issue is reasoning by recurrence and the question of knowing whether a system of postulates is not contradictory. m. couturat will doubtless say that then this does not touch him, but it perhaps will interest those who do not claim, as he does, the liberty of contradiction.
we wish to establish, as above, that we shall never encounter contradiction after any number of deductions whatever, provided this number be finite. for that, it is necessary to apply the principle of induction. should we here understand by finite number every number to which by definition the principle of induction applies? evidently not, else we should be led to most embarrassing consequences. to have the right to lay down a system of postulates, we must be sure they are not contradictory. this is a truth admitted by most scientists; i should have written by all before reading m. couturat’s last article. but what does this signify? does it mean that we must be sure of not meeting contradiction after a finite number of propositions, the finite number being by definition that which has all properties of recurrent nature, so that if one of these properties fails — if, for instance, we come upon a contradiction — we shall agree to say that the number in question is not finite? in other words, do we mean that we must be sure not to meet contradictions, on condition of agreeing to stop just when we are about to encounter one? to state such a proposition is enough to condemn it.
so, hilbert’s reasoning not only assumes the principle of induction, but it supposes that this principle is given us not as a simple definition, but as a synthetic judgment a priori.
to sum up:
a demonstration is necessary.
the only demonstration possible is the proof by recurrence.
this is legitimate only if we admit the principle of induction and if we regard it not as a definition but as a synthetic judgment.
5
the cantor antinomies
now to examine russell’s new memoir. this memoir was written with the view to conquer the difficulties raised by those cantor antinomies to which frequent allusion has already been made. cantor thought he could construct a science of the infinite; others went on in the way he opened, but they soon ran foul of strange contradictions. these antinomies are already numerous, but the most celebrated are:
1. the burali-forti antinomy;
2. the zermelo-k?nig antinomy;
3. the richard antinomy.
cantor proved that the ordinal numbers (the question is of transfinite ordinal numbers, a new notion introduced by him) can be ranged in a linear series; that is to say that of two unequal ordinals one is always less than the other. burali-forti proves the contrary; and in fact he says in substance that if one could range all the ordinals in a linear series, this series would define an ordinal greater than all the others; we could afterwards adjoin 1 and would obtain again an ordinal which would be still greater, and this is contradictory.
we shall return later to the zermelo-k?nig antinomy which is of a slightly different nature. the richard antinomy15 is as follows: consider all the decimal numbers definable by a finite number of words; these decimal numbers form an aggregate e, and it is easy to see that this aggregate is countable, that is to say we can number the different decimal numbers of this assemblage from 1 to infinity. suppose the numbering effected, and define a number n as follows: if the nth decimal of the nth number of the assemblage e is
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
the nth decimal of n shall be:
1, 2, 3, 4, 5, 6, 7, 8, 1, 1
as we see, n is not equal to the nth number of e, and as n is arbitrary, n does not appertain to e and yet n should belong to this assemblage since we have defined it with a finite number of words.
we shall later see that m. richard has himself given with much sagacity the explanation of his paradox and that this extends, mutatis mutandis, to the other like paradoxes. again, russell cites another quite amusing paradox: what is the least whole number which can not be defined by a phrase composed of less than a hundred english words?
this number exists; and in fact the numbers capable of being defined by a like phrase are evidently finite in number since the words of the english language are not infinite in number. therefore among them will be one less than all the others. and, on the other hand, this number does not exist, because its definition implies contradiction. this number, in fact, is defined by the phrase in italics which is composed of less than a hundred english words; and by definition this number should not be capable of definition by a like phrase.
15 revue générale des sciences, june 30, 1905.
6
zigzag theory and no-class theory
what is mr. russell’s attitude in presence of these contradictions? after having analyzed those of which we have just spoken, and cited still others, after having given them a form recalling epimenides, he does not hesitate to conclude: “a propositional function of one variable does not always determine a class.” a propositional function (that is to say a definition) does not always determine a class. a ‘propositional function’ or ‘norm’ may be ‘non-predicative.’ and this does not mean that these non-predicative propositions determine an empty class, a null class; this does not mean that there is no value of x satisfying the definition and capable of being one of the elements of the class. the elements exist, but they have no right to unite in a syndicate to form a class.
but this is only the beginning and it is needful to know how to recognize whether a definition is or is not predicative. to solve this problem russell hesitates between three theories which he calls
a. the zigzag theory;
b. the theory of limitation of size;
c. the no-class theory.
according to the zigzag theory “definitions (propositional functions) determine a class when they are very simple and cease to do so only when they are complicated and obscure.” who, now, is to decide whether a definition may be regarded as simple enough to be acceptable? to this question there is no answer, if it be not the loyal avowal of a complete inability: “the rules which enable us to recognize whether these definitions are predicative would be extremely complicated and can not commend themselves by any plausible reason. this is a fault which might be remedied by greater ingenuity or by using distinctions not yet pointed out. but hitherto in seeking these rules, i have not been able to find any other directing principle than the absence of contradiction.”
this theory therefore remains very obscure; in this night a single light — the word zigzag. what russell calls the ‘zigzaginess’ is doubtless the particular characteristic which distinguishes the argument of epimenides.
according to the theory of limitation of size, a class would cease to have the right to exist if it were too extended. perhaps it might be infinite, but it should not be too much so. but we always meet again the same difficulty; at what precise moment does it begin to be too much so? of course this difficulty is not solved and russell passes on to the third theory.
in the no-classes theory it is forbidden to speak the word ‘class’ and this word must be replaced by various periphrases. what a change for logistic which talks only of classes and classes of classes! it becomes necessary to remake the whole of logistic. imagine how a page of logistic would look upon suppressing all the propositions where it is a question of class. there would only be some scattered survivors in the midst of a blank page. apparent rari nantes in gurgite vasto.
be that as it may, we see how russell hesitates and the modifications to which he submits the fundamental principles he has hitherto adopted. criteria are needed to decide whether a definition is too complex or too extended, and these criteria can only be justified by an appeal to intuition.
it is toward the no-classes theory that russell finally inclines. be that as it may, logistic is to be remade and it is not clear how much of it can be saved. needless to add that cantorism and logistic are alone under consideration; real mathematics, that which is good for something, may continue to develop in accordance with its own principles without bothering about the storms which rage outside it, and go on step by step with its usual conquests which are final and which it never has to abandon.
7
the true solution
what choice ought we to make among these different theories? it seems to me that the solution is contained in a letter of m. richard of which i have spoken above, to be found in the revue générale des sciences of june 30, 1905. after having set forth the antinomy we have called richard’s antinomy, he gives its explanation. recall what has already been said of this antinomy. e is the aggregate of all the numbers definable by a finite number of words, without introducing the notion of the aggregate e itself. else the definition of e would contain a vicious circle; we must not define e by the aggregate e itself.
now we have defined n with a finite number of words, it is true, but with the aid of the notion of the aggregate e. and this is why n is not part of e. in the example selected by m. richard, the conclusion presents itself with complete evidence and the evidence will appear still stronger on consulting the text of the letter itself. but the same explanation holds good for the other antinomies, as is easily verified. thus the definitions which should be regarded as not predicative are those which contain a vicious circle. and the preceding examples sufficiently show what i mean by that. is it this which russell calls the ‘zigzaginess’? i put the question without answering it.
8
the demonstrations of the principle of induction
let us now examine the pretended demonstrations of the principle of induction and in particular those of whitehead and of burali-forti.
we shall speak of whitehead’s first, and take advantage of certain new terms happily introduced by russell in his recent memoir. call recurrent class every class containing zero, and containing n + 1 if it contains n. call inductive number every number which is a part of all the recurrent classes. upon what condition will this latter definition, which plays an essential r?le in whitehead’s proof, be ‘predicative’ and consequently acceptable?
in accordance with what has been said, it is necessary to understand by all the recurrent classes, all those in whose definition the notion of inductive number does not enter. else we fall again upon the vicious circle which has engendered the antinomies.
now whitehead has not taken this precaution. whitehead’s reasoning is therefore fallacious; it is the same which led to the antinomies. it was illegitimate when it gave false results; it remains illegitimate when by chance it leads to a true result.
a definition containing a vicious circle defines nothing. it is of no use to say, we are sure, whatever meaning we may give to our definition, zero at least belongs to the class of inductive numbers; it is not a question of knowing whether this class is void, but whether it can be rigorously deliminated. a ‘non-predicative’ class is not an empty class, it is a class whose boundary is undetermined. needless to add that this particular objection leaves in force the general objections applicable to all the demonstrations.
9
burali-forti has given another demonstration.16 but he is obliged to assume two postulates: first, there always exists at least one infinite class. the second is thus expressed:
the first postulate is not more evident than the principle to be proved. the second not only is not evident, but it is false, as whitehead has shown; as moreover any recruit would see at the first glance, if the axiom had been stated in intelligible language, since it means that the number of combinations which can be formed with several objects is less than the number of these objects.
16 in his article ‘le classi finite,’ atti di torino, vol. xxxii.
10
zermelo’s assumption
a famous demonstration by zermelo rests upon the following assumption: in any aggregate (or the same in each aggregate of an assemblage of aggregates) we can always choose at random an element (even if this assemblage of aggregates should contain an infinity of aggregates). this assumption had been applied a thousand times without being stated, but, once stated, it aroused doubts. some mathematicians, for instance m. borel, resolutely reject it; others admire it. let us see what, according to his last article, russell thinks of it. he does not speak out, but his reflections are very suggestive.
and first a picturesque example: suppose we have as many pairs of shoes as there are whole numbers, and so that we can number the pairs from one to infinity, how many shoes shall we have? will the number of shoes be equal to the number of pairs? yes, if in each pair the right shoe is distinguishable from the left; it will in fact suffice to give the number 2n ? 1 to the right shoe of the nth pair, and the number 2n to the left shoe of the nth pair. no, if the right shoe is just like the left, because a similar operation would become impossible — unless we admit zermelo’s assumption, since then we could choose at random in each pair the shoe to be regarded as the right.
11
conclusions
a demonstration truly founded upon the principles of analytic logic will be composed of a series of propositions. some, serving as premises, will be identities or definitions; the others will be deduced from the premises step by step. but though the bond between each proposition and the following is immediately evident, it will not at first sight appear how we get from the first to the last, which we may be tempted to regard as a new truth. but if we replace successively the different expressions therein by their definition and if this operation be carried as far as possible, there will finally remain only identities, so that all will reduce to an immense tautology. logic therefore remains sterile unless made fruitful by intuition.
this i wrote long ago; logistic professes the contrary and thinks it has proved it by actually proving new truths. by what mechanism? why in applying to their reasonings the procedure just described — namely, replacing the terms defined by their definitions — do we not see them dissolve into identities like ordinary reasonings? it is because this procedure is not applicable to them. and why? because their definitions are not predicative and present this sort of hidden vicious circle which i have pointed out above; non-predicative definitions can not be substituted for the terms defined. under these conditions logistic is not sterile, it engenders antinomies.
it is the belief in the existence of the actual infinite which has given birth to those non-predicative definitions. let me explain. in these definitions the word ‘all’ figures, as is seen in the examples cited above. the word ‘all’ has a very precise meaning when it is a question of a finite number of objects; to have another one, when the objects are infinite in number, would require there being an actual (given complete) infinity. otherwise all these objects could not be conceived as postulated anteriorly to their definition, and then if the definition of a notion n depends upon all the objects a, it may be infected with a vicious circle, if among the objects a are some indefinable without the intervention of the notion n itself.
the rules of formal logic express simply the properties of all possible classifications. but for them to be applicable it is necessary that these classifications be immutable and that we have no need to modify them in the course of the reasoning. if we have to classify only a finite number of objects, it is easy to keep our classifications without change. if the objects are indefinite in number, that is to say if one is constantly exposed to seeing new and unforeseen objects arise, it may happen that the appearance of a new object may require the classification to be modified, and thus it is we are exposed to antinomies. there is no actual (given complete) infinity. the cantorians have forgotten this, and they have fallen into contradiction. it is true that cantorism has been of service, but this was when applied to a real problem whose terms were precisely defined, and then we could advance without fear.
logistic also forgot it, like the cantorians, and encountered the same difficulties. but the question is to know whether they went this way by accident or whether it was a necessity for them. for me, the question is not doubtful; belief in an actual infinity is essential in the russell logic. it is just this which distinguishes it from the hilbert logic. hilbert takes the view-point of extension, precisely in order to avoid the cantorian antinomies. russell takes the view-point of comprehension. consequently for him the genus is anterior to the species, and the summum genus is anterior to all. that would not be inconvenient if the summum genus was finite; but if it is infinite, it is necessary to postulate the infinite, that is to say to regard the infinite as actual (given complete). and we have not only infinite classes; when we pass from the genus to the species in restricting the concept by new conditions, these conditions are still infinite in number. because they express generally that the envisaged object presents such or such a relation with all the objects of an infinite class.
but that is ancient history. russell has perceived the peril and takes counsel. he is about to change everything, and, what is easily understood, he is preparing not only to introduce new principles which shall allow of operations formerly forbidden, but he is preparing to forbid operations he formerly thought legitimate. not content to adore what he burned, he is about to burn what he adored, which is more serious. he does not add a new wing to the building, he saps its foundation.
the old logistic is dead, so much so that already the zigzag theory and the no-classes theory are disputing over the succession. to judge of the new, we shall await its coming.
