1
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 classic logic, and he has put forth on the subject views which are original and at times well warranted.
first, russell subordinates the logic of classes to that of propositions, while the logic of aristotle was above all the logic of classes and took as its point of departure the relation of subject to predicate. the classic 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 classic 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 associative 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 accumulate the most surprising results, some of which even may be true.
2
we see how much richer the new logic is than the classic 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 classes’ 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 entitled treatise on logic? they have not changed nature; they have only changed place.
3
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 assumptions 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.
4
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.
5
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 assumptions, 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 assumption is the principle of complete induction.
m. couturat considers these assumptions as disguised definitions; they constitute the definition by postulates of zero, of successor, and of integer.
but we have seen that for a definition by postulates to be acceptable we must be able to prove that it implies no contradiction.
is this the case here? not at all.
the demonstration can not be made by example. we can not take a part of the integers, for instance the first three, and prove they satisfy the definition.
if i take the series 0, 1, 2, i see it fulfils the assumptions 1, 2, 4 and 5; but to satisfy assumption 3 it still is necessary that 3 be an integer, and consequently that the series 0, 1, 2, 3, fulfil the assumptions; we might prove that it satisfies assumptions 1, 2, 4, 5, but assumption 3 requires besides that 4 be an integer and that the series 0, 1, 2, 3, 4 fulfil the assumptions, and so on.
it is therefore impossible to demonstrate the assumptions for certain integers without proving them for all; we must give up proof by example.
it is necessary then to take all the consequences of our assumptions and see if they contain no contradiction.
if these consequences were finite in number, this would be easy; but they are infinite in number; they are the whole of mathematics, or at least all arithmetic.
what then is to be done? perhaps strictly we could repeat the reasoning of number iii.
but as we have said, this reasoning is complete induction, and it is precisely the principle of complete induction whose justification would be the point in question.
6
the logic of hilbert
i come now to the capital work of hilbert which he communicated to the congress of mathematicians at heidelberg, and of which a french translation by m. pierre boutroux appeared in l’enseignement mathématique, while an english translation due to halsted appeared in the monist.13 in this work, which contains profound thoughts, the author’s aim is analogous to that of russell, but on many points he diverges from his predecessor.
“but,” he says (monist, p. 340), “on attentive consideration we become aware that in the usual exposition of the laws of logic certain fundamental concepts of arithmetic are already employed; for example, the concept of the aggregate, in part also the concept of number.
“we fall thus into a vicious circle and therefore to avoid paradoxes a partly simultaneous development of the laws of logic and arithmetic is requisite.”
we have seen above that what hilbert says of the principles of logic in the usual exposition applies likewise to the logic of russell. so for russell logic is prior to arithmetic; for hilbert they are ‘simultaneous.’ we shall find further on other differences still greater, but we shall point them out as we come to them. i prefer to follow step by step the development of hilbert’s thought, quoting textually the most important passages.
“let us take as the basis of our consideration first of all a thought-thing 1 (one)” (p. 341). notice that in so doing we in no wise imply the notion of number, because it is understood that 1 is here only a symbol and that we do not at all seek to know its meaning. “the taking of this thing together with itself respectively two, three or more times. . . . ” ah! this time it is no longer the same; if we introduce the words ‘two,’ ‘three,’ and above all ‘more,’ ‘several,’ we introduce the notion of number; and then the definition of finite whole number which we shall presently find, will come too late. our author was too circumspect not to perceive this begging of the question. so at the end of his work he tries to proceed to a truly patching-up process.
hilbert then introduces two simple objects 1 and =, and considers all the combinations of these two objects, all the combinations of their combinations, etc. it goes without saying that we must forget the ordinary meaning of these two signs and not attribute any to them.
afterwards he separates these combinations into two classes, the class of the existent and the class of the non-existent, and till further orders this separation is entirely arbitrary. every affirmative statement tells us that a certain combination belongs to the class of the existent; every negative statement tells us that a certain combination belongs to the class of the non-existent.
13 ‘the foundations of logic and arithmetic,’ monist, xv., 338-352.
7
note now a difference of the highest importance. for russell any object whatsoever, which he designates by x, is an object absolutely undetermined and about which he supposes nothing; for hilbert it is one of the combinations formed with the symbols 1 and =; he could not conceive of the introduction of anything other than combinations of objects already defined. moreover hilbert formulates his thought in the neatest way, and i think i must reproduce in extenso his statement (p. 348):
“in the assumptions the arbitraries (as equivalent for the concept ‘every’ and ‘all’ in the customary logic) represent only those thought-things and their combinations with one another, which at this stage are laid down as fundamental or are to be newly defined. therefore in the deduction of inferences from the assumptions, the arbitraries, which occur in the assumptions, can be replaced only by such thought-things and their combinations.
“also we must duly remember, that through the super-addition and making fundamental of a new thought-thing the preceding assumptions undergo an enlargement of their validity, and where necessary, are to be subjected to a change in conformity with the sense.”
the contrast with russell’s view-point is complete. for this philosopher we may substitute for x not only objects already known, but anything.
russell is faithful to his point of view, which is that of comprehension. he starts from the general idea of being, and enriches it more and more while restricting it, by adding new qualities. hilbert on the contrary recognizes as possible beings only combinations of objects already known; so that (looking at only one side of his thought) we might say he takes the view-point of extension.
8
let us continue with the exposition of hilbert’s ideas. he introduces two assumptions which he states in his symbolic language but which signify, in the language of the uninitiated, that every quality is equal to itself and that every operation performed upon two identical quantities gives identical results.
so stated, they are evident, but thus to present them would be to misrepresent hilbert’s thought. for him mathematics has to combine only pure symbols, and a true mathematician should reason upon them without preconceptions as to their meaning. so his assumptions are not for him what they are for the common people.
he considers them as representing the definition by postulates of the symbol (=) heretofore void of all signification. but to justify this definition we must show that these two assumptions lead to no contradiction. for this hilbert used the reasoning of our number iii, without appearing to perceive that he is using complete induction.
9
the end of hilbert’s memoir is altogether enigmatic and i shall not lay stress upon it. contradictions accumulate; we feel that the author is dimly conscious of the petitio principii he has committed, and that he seeks vainly to patch up the holes in his argument.
what does this mean? at the point of proving that the definition of the whole number by the assumption of complete induction implies no contradiction, hilbert withdraws as russell and couturat withdrew, because the difficulty is too great.
10
geometry
geometry, says m. couturat, is a vast body of doctrine wherein the principle of complete induction does not enter. that is true in a certain measure; we can not say it is entirely absent, but it enters very slightly. if we refer to the rational geometry of dr. halsted (new york, john wiley and sons, 1904) built up in accordance with the principles of hilbert, we see the principle of induction enter for the first time on page 114 (unless i have made an oversight, which is quite possible).14
so geometry, which only a few years ago seemed the domain where the reign of intuition was uncontested, is to-day the realm where the logicians seem to triumph. nothing could better measure the importance of the geometric works of hilbert and the profound impress they have left on our conceptions.
but be not deceived. what is after all the fundamental theorem of geometry? it is that the assumptions of geometry imply no contradiction, and this we can not prove without the principle of induction.
how does hilbert demonstrate this essential point? by leaning upon analysis and through it upon arithmetic and through it upon the principle of induction.
and if ever one invents another demonstration, it will still be necessary to lean upon this principle, since the possible consequences of the assumptions, of which it is necessary to show that they are not contradictory, are infinite in number.
14 second ed., 1907, p. 86; french ed., 1911, p. 97. g. b. h.
11
conclusion
our conclusion straightway is that the principle of induction can not be regarded as the disguised definition of the entire world.
here are three truths: (1) the principle of complete induction; (2) euclid’s postulate; (3) the physical law according to which phosphorus melts at 44° (cited by m. le roy).
these are said to be three disguised definitions: the first, that of the whole number; the second, that of the straight line; the third, that of phosphorus.
i grant it for the second; i do not admit it for the other two. i must explain the reason for this apparent inconsistency.
first, we have seen that a definition is acceptable only on condition that it implies no contradiction. we have shown likewise that for the first definition this demonstration is impossible; on the other hand, we have just recalled that for the second hilbert has given a complete proof.
as to the third, evidently it implies no contradiction. does this mean that the definition guarantees, as it should, the existence of the object defined? we are here no longer in the mathematical sciences, but in the physical, and the word existence has no longer the same meaning. it no longer signifies absence of contradiction; it means objective existence.
you already see a first reason for the distinction i made between the three cases; there is a second. in the applications we have to make of these three concepts, do they present themselves to us as defined by these three postulates?
the possible applications of the principle of induction are innumerable; take, for example, one of those we have expounded above, and where it is sought to prove that an aggregate of assumptions can lead to no contradiction. for this we consider one of the series of syllogisms we may go on with in starting from these assumptions as premises. when we have finished the nth syllogism, we see we can make still another and this is the n + 1th. thus the number n serves to count a series of successive operations; it is a number obtainable by successive additions. this therefore is a number from which we may go back to unity by successive subtractions. evidently we could not do this if we had n = n ? 1, since then by subtraction we should always obtain again the same number. so the way we have been led to consider this number n implies a definition of the finite whole number and this definition is the following: a finite whole number is that which can be obtained by successive additions; it is such that n is not equal to n ? 1.
that granted, what do we do? we show that if there has been no contradiction up to the nth syllogism, no more will there be up to the n + 1th, and we conclude there never will be. you say: i have the right to draw this conclusion, since the whole numbers are by definition those for which a like reasoning is legitimate. but that implies another definition of the whole number, which is as follows: a whole number is that on which we may reason by recurrence. in the particular case it is that of which we may say that, if the absence of contradiction up to the time of a syllogism of which the number is an integer carries with it the absence of contradiction up to the time of the syllogism whose number is the following integer, we need fear no contradiction for any of the syllogisms whose number is an integer.
the two definitions are not identical; they are doubtless equivalent, but only in virtue of a synthetic judgment a priori; we can not pass from one to the other by a purely logical procedure. consequently we have no right to adopt the second, after having introduced the whole number by a way that presupposes the first.
on the other hand, what happens with regard to the straight line? i have already explained this so often that i hesitate to repeat it again, and shall confine myself to a brief recapitulation of my thought. we have not, as in the preceding case, two equivalent definitions logically irreducible one to the other. we have only one expressible in words. will it be said there is another which we feel without being able to word it, since we have the intuition of the straight line or since we represent to ourselves the straight line? first of all, we can not represent it to ourselves in geometric space, but only in representative space, and then we can represent to ourselves just as well the objects which possess the other properties of the straight line, save that of satisfying euclid’s postulate. these objects are ‘the non-euclidean straights,’ which from a certain point of view are not entities void of sense, but circles (true circles of true space) orthogonal to a certain sphere. if, among these objects equally capable of representation, it is the first (the euclidean straights) which we call straights, and not the latter (the non-euclidean straights), this is properly by definition.
and arriving finally at the third example, the definition of phosphorus, we see the true definition would be: phosphorus is the bit of matter i see in yonder flask.
12
and since i am on this subject, still another word. of the phosphorus example i said: “this proposition is a real verifiable physical law, because it means that all bodies having all the other properties of phosphorus, save its point of fusion, melt like it at 44°.” and it was answered: “no, this law is not verifiable, because if it were shown that two bodies resembling phosphorus melt one at 44° and the other at 50°, it might always be said that doubtless, besides the point of fusion, there is some other unknown property by which they differ.”
that was not quite what i meant to say. i should have written, “all bodies possessing such and such properties finite in number (to wit, the properties of phosphorus stated in the books on chemistry, the fusion-point excepted) melt at 44°.”
and the better to make evident the difference between the case of the straight and that of phosphorus, one more remark. the straight has in nature many images more or less imperfect, of which the chief are the light rays and the rotation axis of the solid. suppose we find the ray of light does not satisfy euclid’s postulate (for example by showing that a star has a negative parallax), what shall we do? shall we conclude that the straight being by definition the trajectory of light does not satisfy the postulate, or, on the other hand, that the straight by definition satisfying the postulate, the ray of light is not straight?
assuredly we are free to adopt the one or the other definition and consequently the one or the other conclusion; but to adopt the first would be stupid, because the ray of light probably satisfies only imperfectly not merely euclid’s postulate, but the other properties of the straight line, so that if it deviates from the euclidean straight, it deviates no less from the rotation axis of solids which is another imperfect image of the straight line; while finally it is doubtless subject to change, so that such a line which yesterday was straight will cease to be straight to-morrow if some physical circumstance has changed.
suppose now we find that phosphorus does not melt at 44°, but at 43.9°. shall we conclude that phosphorus being by definition that which melts at 44°, this body that we did call phosphorus is not true phosphorus, or, on the other hand, that phosphorous melts at 43.9°? here again we are free to adopt the one or the other definition and consequently the one or the other conclusion; but to adopt the first would be stupid because we can not be changing the name of a substance every time we determine a new decimal of its fusion-point.
13
to sum up, russell and hilbert have each made a vigorous effort; they have each written a work full of original views, profound and often well warranted. these two works give us much to think about and we have much to learn from them. among their results, some, many even, are solid and destined to live.
but to say that they have finally settled the debate between kant and leibnitz and ruined the kantian theory of mathematics is evidently incorrect. i do not know whether they really believed they had done it, but if they believed so, they deceived themselves.
