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 a.s.sumptions 1, 2, 4 and 5; but to satisfy a.s.sumption 3 it still is necessary that 3 be an integer, and consequently that the series 0, 1, 2, 3, fulfil the a.s.sumptions; we might prove that it satisfies a.s.sumptions 1, 2, 4, 5, but a.s.sumption 3 requires besides that 4 be an integer and that the series 0, 1, 2, 3, 4 fulfil the a.s.sumptions, and so on.
It is therefore impossible to demonstrate the a.s.sumptions 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 a.s.sumptions 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.
VI
_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 mathematique_, while an English translation due to Halsted appeared in _The Monist_.[13] In this work, which contains profound thoughts, the author"s aim is a.n.a.logous to that of Russell, but on many points he diverges from his predecessor.
[13] "The Foundations of Logic and Arithmetic," _Monist_, XV., 338-352.
"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 pa.s.sages.
"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 circ.u.mspect 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 cla.s.ses, the cla.s.s of the existent and the cla.s.s 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 cla.s.s of the existent; every negative statement tells us that a certain combination belongs to the cla.s.s of the non-existent.
VII
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 a.s.sumptions 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 a.s.sumptions, the arbitraries, which occur in the a.s.sumptions, 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 a.s.sumptions 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 subst.i.tute 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.
VIII
Let us continue with the exposition of Hilbert"s ideas. He introduces two a.s.sumptions 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 quant.i.ties 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 a.s.sumptions 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 a.s.sumptions lead to no contradiction. For this Hilbert used the reasoning of our number III, without appearing to perceive that he is using complete induction.
IX
The end of Hilbert"s memoir is altogether enigmatic and I shall not lay stress upon it. Contradictions acc.u.mulate; we feel that the author is dimly conscious of the _pet.i.tio 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 a.s.sumption of complete induction implies no contradiction, Hilbert withdraws as Russell and Couturat withdrew, because the difficulty is too great.
X
_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]
[14] Second ed., 1907, p. 86; French ed., 1911, p. 97. G. B. H.
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 a.s.sumptions 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 a.n.a.lysis 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 a.s.sumptions, of which it is necessary to show that they are not contradictory, are infinite in number.
XI
_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.