Can mathematics be reduced to logic1 without having to appeal to principles peculiar2 to mathematics? There is a whole school, abounding3 in ardor4 and full of faith, striving to prove it. They have their own special language, which is without words, using only signs. This language is understood only by the initiates5, so that commoners are disposed to bow to the trenchant6 affirmations of the adepts7. It is perhaps not unprofitable to examine these affirmations somewhat closely, to see if they justify8 the peremptory9 tone with which they are presented.
But to make clear the nature of the question it is necessary to enter upon certain historical details and in particular to recall the character of the works of Cantor.
Since long ago the notion of infinity11 had been introduced into mathematics; but this infinite was what philosophers call a becoming. The mathematical infinite was only a quantity capable of increasing beyond all limit: it was a variable quantity of which it could not be said that it had passed all limits, but only that it could pass them.
Cantor has undertaken to introduce into mathematics an actual infinite, that is to say a quantity which not only is capable of passing all limits, but which is regarded as having already passed them. He has set himself questions like these: Are there more points in space than whole numbers? Are there more points in space than points in a plane? etc.
And then the number of whole numbers, that of the points of space, etc., constitutes what he calls a transfinite cardinal12 number, that is to say a cardinal number greater than all the ordinary cardinal numbers. And he has occupied himself in comparing these transfinite cardinal numbers. In arranging in a proper order the elements of an aggregate13 containing an infinity of them, he has also imagined what he calls transfinite ordinal numbers upon which I shall not dwell.
Many mathematicians15 followed his lead and set a series of questions of the sort. They so familiarized themselves with transfinite numbers that they have come to make the theory of finite numbers depend upon that of Cantor’s cardinal numbers. In their eyes, to teach arithmetic in a way truly logical, one should begin by establishing the general properties of transfinite cardinal numbers, then distinguish among them a very small class, that of the ordinary whole numbers. Thanks to this détour, one might succeed in proving all the propositions relative to this little class (that is to say all our arithmetic and our algebra) without using any principle foreign to logic. This method is evidently contrary to all sane16 psychology17; it is certainly not in this way that the human mind proceeded in constructing mathematics; so its authors do not dream, I think, of introducing it into secondary teaching. But is it at least logic, or, better, is it correct? It may be doubted.
The geometers who have employed it are however very numerous. They have accumulated formulas and they have thought to free themselves from what was not pure logic by writing memoirs19 where the formulas no longer alternate with explanatory discourse20 as in the books of ordinary mathematics, but where this discourse has completely disappeared.
Unfortunately they have reached contradictory21 results, what are called the cantorian antinomies, to which we shall have occasion to return. These contradictions have not discouraged them and they have tried to modify their rules so as to make those disappear which had already shown themselves, without being sure, for all that, that new ones would not manifest themselves.
It is time to administer justice on these exaggerations. I do not hope to convince them; for they have lived too long in this atmosphere. Besides, when one of their demonstrations23 has been refuted, we are sure to see it resurrected with insignificant24 alterations25, and some of them have already risen several times from their ashes. Such long ago was the Lern?an hydra26 with its famous heads which always grew again. Hercules got through, since his hydra had only nine heads, or eleven; but here there are too many, some in England, some in Germany, in Italy, in France, and he would have to give up the struggle. So I appeal only to men of good judgment27 unprejudiced.
1
In these latter years numerous works have been published on pure mathematics and the philosophy of mathematics, trying to separate and isolate28 the logical elements of mathematical reasoning. These works have been analyzed29 and expounded30 very clearly by M. Couturat in a book entitled: The Principles of Mathematics.
For M. Couturat, the new works, and in particular those of Russell and Peano, have finally settled the controversy31, so long pending32 between Leibnitz and Kant. They have shown that there are no synthetic33 judgments34 a priori (Kant’s phrase to designate judgments which can neither be demonstrated analytically35, nor reduced to identities, nor established experimentally), they have shown that mathematics is entirely37 reducible to logic and that intuition here plays no r?le.
This is what M. Couturat has set forth38 in the work just cited; this he says still more explicitly39 in his Kant jubilee40 discourse, so that I heard my neighbor whisper: “I well see this is the centenary of Kant’s death.”
Can we subscribe41 to this conclusive42 condemnation43? I think not, and I shall try to show why.
2
What strikes us first in the new mathematics is its purely44 formal character: “We think,” says Hilbert, “three sorts of things, which we shall call points, straights and planes. We convene45 that a straight shall be determined46 by two points, and that in place of saying this straight is determined by these two points, we may say it passes through these two points, or that these two points are situated47 on this straight.” What these things are, not only we do not know, but we should not seek to know. We have no need to, and one who never had seen either point or straight or plane could geometrize as well as we. That the phrase to pass through, or the phrase to be situated upon may arouse in us no image, the first is simply a synonym48 of to be determined and the second of to determine.
Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous49 to know what it means. The geometer might be replaced by the logic piano imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary50 Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician14 know what he does.
I do not make this formal character of his geometry a reproach to Hilbert. This is the way he should go, given the problem he set himself. He wished to reduce to a minimum the number of the fundamental assumptions of geometry and completely enumerate51 them; now, in reasonings where our mind remains52 active, in those where intuition still plays a part, in living reasonings, so to speak, it is difficult not to introduce an assumption or a postulate53 which passes unperceived. It is therefore only after having carried back all the geometric reasonings to a form purely mechanical that he could be sure of having accomplished54 his design and finished his work.
What Hilbert did for geometry, others have tried to do for arithmetic and analysis. Even if they had entirely succeeded, would the Kantians be finally condemned55 to silence? Perhaps not, for in reducing mathematical thought to an empty form, it is certainly mutilated.
Even admitting it were established that all the theorems could be deduced by procedures purely analytic36, by simple logical combinations of a finite number of assumptions, and that these assumptions are only conventions; the philosopher would still have the right to investigate the origins of these conventions, to see why they have been judged preferable to the contrary conventions.
And then the logical correctness of the reasonings leading from the assumptions to the theorems is not the only thing which should occupy us. The rules of perfect logic, are they the whole of mathematics? As well say the whole art of playing chess reduces to the rules of the moves of the pieces. Among all the constructs which can be built up of the materials furnished by logic, choice must be made; the true geometer makes this choice judiciously56 because he is guided by a sure instinct, or by some vague consciousness of I know not what more profound and more hidden geometry, which alone gives value to the edifice57 constructed.
To seek the origin of this instinct, to study the laws of this deep geometry, felt, not stated, would also be a fine employment for the philosophers who do not want logic to be all. But it is not at this point of view I wish to put myself, it is not thus I wish to consider the question. The instinct mentioned is necessary for the inventor, but it would seem at first we might do without it in studying the science once created. Well, what I wish to investigate is if it be true that, the principles of logic once admitted, one can, I do not say discover, but demonstrate, all the mathematical verities58 without making a new appeal to intuition.
3
I once said no to this question:12 should our reply be modified by the recent works? My saying no was because “the principle of complete induction59” seemed to me at once necessary to the mathematician and irreducible to logic. The statement of this principle is: “If a property be true of the number 1, and if we establish that it is true of n + 1 provided it be of n, it will be true of all the whole numbers.” Therein I see the mathematical reasoning par10 excellence60. I did not mean to say, as has been supposed, that all mathematical reasonings can be reduced to an application of this principle. Examining these reasonings closely, we there should see applied61 many other analogous62 principles, presenting the same essential characteristics. In this category of principles, that of complete induction is only the simplest of all and this is why I have chosen it as type.
The current name, principle of complete induction, is not justified63. This mode of reasoning is none the less a true mathematical induction which differs from ordinary induction only by its certitude.
12 See Science and Hypothesis, chapter I.
4
Definitions and Assumptions
The existence of such principles is a difficulty for the uncompromising logicians; how do they pretend to get out of it? The principle of complete induction, they say, is not an assumption properly so called or a synthetic judgment a priori; it is just simply the definition of whole number. It is therefore a simple convention. To discuss this way of looking at it, we must examine a little closely the relations between definitions and assumptions.
Let us go back first to an article by M. Couturat on mathematical definitions which appeared in l’Enseignement mathématique, a magazine published by Gauthier-Villars and by Georg at Geneva. We shall see there a distinction between the direct definition and the definition by postulates64.
“The definition by postulates,” says M. Couturat, “applies not to a single notion, but to a system of notions; it consists in enumerating65 the fundamental relations which unite them and which enable us to demonstrate all their other properties; these relations are postulates.”
If previously66 have been defined all these notions but one, then this last will be by definition the thing which verifies these postulates. Thus certain indemonstrable assumptions of mathematics would be only disguised definitions. This point of view is often legitimate67; and I have myself admitted it in regard for instance to Euclid’s postulate.
The other assumptions of geometry do not suffice to completely define distance; the distance then will be, by definition, among all the magnitudes which satisfy these other assumptions, 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 assumption, that by which the existence of the defined object is affirmed. According to that, it would no longer be the assumption which might be a disguised definition, it would on the contrary be the definition which would be a disguised assumption. 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 rectified68, 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 assumption; so that, seeking the definition under the postulate, we should find the assumption 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 demonstration22 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 premises69, 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 invoke70 just this principle of complete induction which is precisely71 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.
5
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 implicitly72 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 dilate74 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 assumption.
We shall see farther on that the logicians have not fulfilled the second condition any better than the first.
6
The definitions of number are very numerous and very different; I forego the enumeration75 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 predecessors76, and he was not satisfied with them because he thought he saw a petitio principii.
I have always felt, in reading the writings devoted77 to this problem, a profound feeling of discomfort78; I was always expecting to run against a petitio 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 plural79. And then the declivity80 is slippery and at each instant there is risk of a fall into petitio principii.
I shall devote my attention in what follows only to those of these definitions where the petitio principii is most ably concealed81.
7
Pasigraphy
The symbolic82 language created by Peano plays a very grand r?le in these new researches. It is capable of rendering83 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 destined84 to revolutionize all philosophy is a different matter. It is difficult to admit that the word if acquires, when written C, a virtue85 it had not when written if. This invention of Peano was first called pasigraphy, that is to say the art of writing a treatise86 on mathematics without using a single word of ordinary language. This name defined its range very exactly. Later, it was raised to a more eminent87 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 cavalry88, 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 memoir18 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 reassure89 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 rigor73 and whether it thus compensates90 for the efforts it imposes upon the writer and the reader.
First we see Burali-Forti define the number 1 as follows:
a definition eminently91 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:
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:
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.
8
But to return to the memoir of Burali-Forti; I have said his conclusions are in direct opposition92 to those of Cantor. Now, one day M. Hadamard came to see me and the talk fell upon this antinomy.
“Burali-Forti’s reasoning,” I said, “does it not seem to you irreproachable93?” “No, and on the contrary I find nothing to object to in that of Cantor. Besides, Burali-Forti had no right to speak of the aggregate of all the ordinal numbers.”
“Pardon, he had the right, since he could always put
I should like to know who was to prevent him, and can it be said a thing does not exist, when we have called it Ω?”
It was in vain, I could not convince him (which besides would have been sad, since he was right). Was it merely because I do not speak the Peanian with enough eloquence94? Perhaps; but between ourselves I do not think so.
Thus, despite all this pasigraphic apparatus95, the question was not solved. What does that prove? In so far as it is a question only of proving one a number, pasigraphy suffices, but if a difficulty presents itself, if there is an antinomy to solve, pasigraphy becomes impotent.
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logic
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| n.逻辑(学)；逻辑性 | |
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peculiar
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| adj.古怪的，异常的；特殊的，特有的 | |
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abounding
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| adj.丰富的，大量的v.大量存在，充满，富于( abound的现在分词 ) | |
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ardor
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| n.热情，狂热 | |
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initiates
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| v.开始( initiate的第三人称单数 )；传授；发起；接纳新成员 | |
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trenchant
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| adj.尖刻的，清晰的 | |
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adepts
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| n.专家，能手( adept的名词复数 ) | |
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justify
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| vt.证明…正当(或有理)，为…辩护 | |
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peremptory
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| adj.紧急的，专横的，断然的 | |
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par
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infinity
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| n.无限，无穷，大量 | |
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cardinal
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aggregate
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mathematician
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mathematicians
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| 数学家( mathematician的名词复数 ) | |
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sane
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psychology
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memoir
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| n.[pl.]回忆录，自传；记事录 | |
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memoirs
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| n.回忆录；回忆录传( mem,自oir的名词复数) | |
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discourse
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contradictory
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demonstration
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demonstrations
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| 证明( demonstration的名词复数 )； 表明； 表达； 游行示威 | |
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insignificant
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alterations
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hydra
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judgment
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isolate
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analyzed
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expounded
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controversy
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pending
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| prep.直到，等待…期间；adj.待定的；迫近的 | |
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synthetic
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judgments
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analytically
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analytic
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entirely
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forth
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explicitly
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jubilee
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subscribe
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| vi.(to)订阅，订购；同意；vt.捐助，赞助 | |
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conclusive
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condemnation
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purely
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convene
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determined
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situated
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synonym
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advantageous
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legendary
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| adj.传奇（中）的，闻名遐迩的；n.传奇（文学） | |
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enumerate
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| v.列举，计算，枚举，数 | |
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remains
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| n.剩余物，残留物；遗体，遗迹 | |
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postulate
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| n.假定，基本条件；vt.要求，假定 | |
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accomplished
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| adj.有才艺的；有造诣的；达到了的 | |
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condemned
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| adj. 被责难的, 被宣告有罪的 动词condemn的过去式和过去分词 | |
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judiciously
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| adv.明断地，明智而审慎地 | |
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edifice
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| n.宏伟的建筑物(如宫殿，教室) | |
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verities
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| n.真实( verity的名词复数 )；事实；真理；真实的陈述 | |
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induction
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| n.感应，感应现象 | |
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excellence
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| n.优秀，杰出，(pl.)优点，美德 | |
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applied
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| adj.应用的；v.应用，适用 | |
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analogous
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| adj.相似的；类似的 | |
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justified
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| a.正当的，有理的 | |
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postulates
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| v.假定，假设( postulate的第三人称单数 ) | |
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enumerating
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| v.列举，枚举，数( enumerate的现在分词 ) | |
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previously
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| adv.以前，先前（地） | |
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legitimate
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| adj.合法的，合理的，合乎逻辑的；v.使合法 | |
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rectified
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| [医]矫正的，调整的 | |
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premises
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| n.建筑物，房屋 | |
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invoke
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| v.求助于(神、法律)；恳求，乞求 | |
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precisely
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| adv.恰好，正好，精确地，细致地 | |
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implicitly
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| adv. 含蓄地, 暗中地, 毫不保留地 | |
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rigor
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| n.严酷，严格，严厉 | |
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dilate
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| vt.使膨胀，使扩大 | |
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enumeration
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| n.计数，列举；细目；详表；点查 | |
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predecessors
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| n.前任( predecessor的名词复数 )；前辈；（被取代的）原有事物；前身 | |
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devoted
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| adj.忠诚的，忠实的，热心的，献身于...的 | |
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discomfort
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| n.不舒服，不安，难过，困难，不方便 | |
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plural
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| n.复数；复数形式；adj.复数的 | |
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declivity
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| n.下坡，倾斜面 | |
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concealed
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| a.隐藏的，隐蔽的 | |
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symbolic
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| adj.象征性的，符号的，象征主义的 | |
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rendering
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| n.表现，描写 | |
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destined
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| adj.命中注定的；(for)以…为目的地的 | |
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virtue
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| n.德行，美德；贞操；优点；功效，效力 | |
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treatise
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| n.专著；(专题)论文 | |
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eminent
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| adj.显赫的，杰出的，有名的，优良的 | |
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cavalry
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| n.骑兵；轻装甲部队 | |
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reassure
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| v.使放心，使消除疑虑 | |
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compensates
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| 补偿，报酬( compensate的第三人称单数 )； 给（某人）赔偿（或赔款） | |
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eminently
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| adv.突出地；显著地；不寻常地 | |
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opposition
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| n.反对，敌对 | |
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irreproachable
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| adj.不可指责的，无过失的 | |
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eloquence
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| n.雄辩；口才，修辞 | |
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apparatus
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| n.装置，器械；器具，设备 | |
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