Chapter 17: Retreat from Pythagoras ,n.4 The aesthetic pleasure to be derived from an elegant piece of mathematical reasoning remains. But here, too, there were disappointments. The solution of the contradictions mentioned in an earlier chapter seemed to be only possible by adopting theories which might be true but were not beautiful. I felt about the contradictions much as an earnest Catholic must feel about wicked Popes. And the splendid certainty which I had always hoped to find in mathematics was lost in a bewildering maze. All this would have made me sad but for the fact that the ascetic mood had begun to fade. It had had so strong a hold upon me that Dante’s Vita Nuova appeared to me psychologically quite natural, and its strange symbolism appealed to me as emotionally satisfying. But this mood began to pass, and was finally dispelled by the First World War. Source: My Philosophical Development, 1959, by Bertrand Russell More info.: https://russell-j.com/beginner/BR_MPD_17-040.HTM
この頃の私の数学に対する態度は、「数学の研究」と名付けられた論文に表われている。これは1907年に「The New Quarterly」誌 に発表され、また、(私の)『 Philosophical Essays(哲学論文集)』(1910年刊)に再録された。この論文集からの(以下の)いくつかの引用が、当時私の感じていたことを例証している。
============================= 数学は、正しく見れば、真理を持つだけでなく、最高の美を持っている。その美は、彫刻の美のように、冷たく簡素な(禁欲的な)美であり、我々(人間)の本性の弱い部分に訴えることなく、絵画や音楽のような立派な装飾を持たず、しかるに(yet)この上なく純粋で ただ最高の芸術のみが示すことができるような厳しい完全性を持ちうるのである。喜びの真の精神、 精神の高揚、人間以上のものであるという感じは、- これこそ最高の卓越性の証拠であり- 詩の中においてと同様に確実に(as surely as)、数学の中に見出すことができる。数学における最上のものは、単に課題として学ぶ価値があるだけでなく、日常的な思考の一部として吸収され(同化され)、繰り返し繰り返しそれを心に思うことによって、常に新たな励ましをもたらしてくれる。 実生活(real life )は大多数の人にとって長期にわたる次善のもの( a long second-best)であり、理想と可能との間における絶えざる妥協(の産物)である。しかし、純粋理性の世界は妥協を知らず,実際上の制限を知らず、 全て偉大な作品が生まれいづる(ところの)完全なものへの熱情的な憧憬を見事な建造物に具体化(具現化)する創造的活動を妨げる何ものをも知らない(のである)。人間の情念から遠く離れ、自然の事実からさえも遠く離れ、 幾世代もの人々が、一つの秩序ある宇宙を造り出してきた(きている)。 そこには純粋な思考がその自然の住処(すみか:s in its natural home)を見出すことができ、我々(人間)の持つ高貴な衝動の少なくとも一つが、現実世界の悲惨な流浪(he dreary exile of the actual world.)から逃れることができるのである。
非人間的なるものについての熟考(沈思黙考)、我々の精神によって作り出されたものではない材料をとり扱うことができるという発見、とりわけ、美が(人間の)内的世界に属するように外的世界にも属していると理解する(悟る)ことは、外的な力がほぼ全能(the all-but omnipotence of alien forces)であるのを認めることからあまりにもしばしば生ずる、あのおそるべき無力感、虚弱感、敵意ある諸力の中に追放されているという意識を克服するための主な手段である。運命の支配に対して -運命の支配というのは、(上述の)諸力を単に文学的に擬人化(personification)したものにすぎない- おごそかな美を展示して我々と和解させることは、悲劇の任務とするところである。しかし、数学は人間的なるものからさらに一層我々を引き離し、現実世界のみならずあらゆる可能世界が従わなければならない(ところの)絶対的必然性の領域に我々を連れてゆく。そして まさにこの所に、数学はその住みかを建てる、あるいはむしろ、永遠の昔から立っている住みかを見出す のであり、そこでは我々の理想は完全に満たされ、我々の最上の希望は阻止されないのである。
Chapter 17: Retreat from Pythagoras ,n.3 My attitude to mathematics at this time was expressed in an article called ‘The Study of Mathematics’, which was printed in The New Quarterly in 1907, and reprinted in Philosophical Essays (1910). Some quotations from this essay illustrate what I then felt: Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world. The contemplation of what is non-human, the discovery that our minds are capable of dealing with material not created by them, above all, the realization that beauty belongs to the outer world as to the inner, are the chief means of overcoming the terrible sense of impotence, of weakness, of exile amid hostile powers, which is too apt to result from acknowledging the all-but omnipotence of alien forces. To reconcile us, by the exhibition of its awful beauty, to the reign of Fate — which is merely the literary personification of these forces — is the task of tragedy. But mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world must conform; and even here it builds a habitation, or rather finds a habitation eternally standing, where our ideals are fully satisfied and our best hopes are not thwarted. Too often it is said that there is no absolute truth but only opinion and private judgment; that each of us is conditioned, in his view of the world, by his own peculiarities, his own taste and bias; that there is no external kingdom of truth to which, by patience, by discipline, we may at last obtain admittance, but only truth for me, for you, for every separate person. By this habit of mind one of the chief ends of human effort is denied, and the supreme virtue of candour, of fearless acknowledgment of what is, disappears from our moral vision. In a world so full of evil and suffering, retirement into the cloister of contemplation, to the enjoyment of delights which, however noble, must always be for the few only, cannot but appear as a somewhat selfish refusal to share the burden imposed upon others by accidents in which justice plays no part. Have any of us the right, we ask, to withdraw from present evils, to leave our fellowmen unaided, while we live a life which, though arduous and austere, is yet plainly good in its own nature? All this, though I still remember the pleasure of believing it, has come to seem to me largely nonsense, partly for technical reasons and partly from a change in my general outlook upon the world. Mathematics has ceased to seem to me non-human in its subject-matter. I have come to believe, though very reluctantly, that it consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal. I think that the timelessness of mathematics has none of the sublimity that it once seemed to me to have, but consists merely in the fact that the pure mathematician is not talking about time. I cannot any longer find any mystical satisfaction in the contemplation of mathematical truth. Source: My Philosophical Development, 1959, by Bertrand Russell More info.: https://russell-j.com/beginner//BR_MPD_17-030.HTM
Chapter 17: Retreat from Pythagoras ,n.2 My interest in the applications of mathematics was gradually replaced by an interest in the principles upon which mathematics is based. This change came about through a wish to refute mathematical scepticism. A great deal of the argumentation that I had been told to accept was obviously fallacious, and I read whatever books I could find that seemed to offer a firmer foundation for mathematical beliefs. This kind of research led me gradually further and further from applied mathematics into more and more abstract regions, and finally into mathematical logic. I came to think of mathematics, not primarily as a tool for understanding and manipulating the sensible world, but as an abstract edifice subsisting in a Platonic heaven and only reaching the world of sense in an impure and degraded form. My general outlook, in the early years of this century, was profoundly ascetic. I disliked the real world and sought refuge in a timeless world, without change or decay or the will-o’-the-wisp of progress. Although this outlook was very serious and sincere, I sometimes expressed it in a frivolous manner. My brother-in-law, Logan Pearsall Smith, had a set of questions that he used to ask people. One of them was, ‘What do you particularly like?’ I replied, ‘Mathematics and the sea, and theology and heraldry, the two former because they are inhuman, the two latter because they are absurd’. This answer, however, took the form that it did from a desire to win the approval of the questioner. Source: My Philosophical Development, 1959, by Bertrand Russell More info.: https://russell-j.com/beginner//BR_MPD_17-020.HTM
Chapter 17: Retreat from Pythagoras ,n.1 My philosophical development, since the early years of the present century, may be broadly described as a gradual retreat from Pythagoras. The Pythagoreans had a peculiar form of mysticism which was bound up with mathematics. This form of mysticism greatly affected Plato and had, I think, more influence upon him than is generally acknowledged. I had, for a time, a very similar outlook and found in the nature of mathematical logic, as I then supposed its nature to be, something profoundly satisfying in some important emotional respects. As a boy, my interest in mathematics was more simple and ordinary: it had more affinity with Thales than with Pythagoras. I was delighted when I found things in the real world obeying mathematical laws. I liked the lever and the pulley and the fact that falling bodies describe parabolas. Although I could not play billiards, I liked the mathematical theory of how billiard balls behave. On one occasion, when I had a new tutor, I spun a penny and he said, ‘Why does the penny spin?’ I replied, ‘Because I make a couple with my fingers’. He was surprised and remarked, ‘What do you know about couples?’ I replied airily, ‘Oh, I know all about couples’. When, on one occasion, I had to mark the tennis court myself, I used the theorem of Pythagoras to make sure that the lines were at right angles with each other. An uncle of mine took me to call on Tyndall, the eminent physicist. While they were talking to each other, I had to find my own amusement. I got hold of two walking-sticks, each with a crook. I balanced them on one finger, inclining them in opposite directions so that they crossed each other at a certain point. Tyndall looked round and asked what I was doing. I replied that I was thinking of a practical way of determining the centre of gravity, because the centre of gravity of each stick must be vertically below my finger and therefore at the point where the sticks crossed each other. Presumably in consequence of this remark, Tyndall gave me one of his books, The Forms of Water, I hoped, at that time, that all science could become mathematical, including psychology. The parallelogram of forces shows that a body acted on by two forces simultaneously will pursue a middle course, inclining more towards the stronger force. I hoped that there might be a similar ‘parallelogram of motives’– a foolish idea, since a man who comes to a fork in the road and is equally attracted to both roads, does not go across the fields between them. Science had not then arrived at the ‘all-or-nothing principle’ of which the importance was only discovered during the present century. I thought, when I was young, that two divergent attractions would lead to a Whig compromise, whereas it has appeared since that very often one of them prevails completely. This has justified Dr Johnson in the opinion that the Devil, not the Almighty, was the first Whig. Source: My Philosophical Development, 1959, by Bertrand Russell More info.: https://russell-j.com/beginner//BR_MPD_17-010.HTM
生(なま)の事実から科学への移行において、我々(人間)は演繹的論理の推論形式に加えて種々の推論形式を必要とする。 伝統的には帰納法がこの目的に役立つだろうと考えられて来たが、それは誤まりであった。というのは、真である前提から出発する帰納的推論の結論は、真であるよりも偽である場合のほうが多いからである。感覚(で得られるもの)から科学への移行に必要な(要求される)推論の原理は分析によって獲得されなければならない。必要とされる分析(the analysis involved)は、誰もが実際上疑うことのないような種類の推論の分析である(訳注:どういった推論なら誰もが疑わない推論と認めるだろうか、ということについての分析)。たとえば、ある瞬間に猫が炉の前の敷物(hearth-rug)の上にいるのを目撃し、別の瞬間にその猫が戸口(部屋あるいは家の出入り口)にいるのを見たならば、たとえわれわれが猫が動くのを見なかったとしても、猫は炉と戸口との中間の諸点を通った(はずだ)という推論である。科学的推論を分析するという仕事が適切に遂行されるならば、そういう推論の具体的事例は、(a)誰も本気では疑わないようなものであり、(b)感覚的事実を基にして、感覚的事実を超えたものを我々が信じることができるためには、不可欠な推論である,ということが明らかになるであろう(it will appear that)。 そういった研究の結果(成果)は、哲学よりはむしろ科学であるとみなされるべきものである。即ち、そういう結果を受けいれるための理由は、科学的研究に適用される通常の理由であり、何らかの形而上学的理論から得られるかけ離れた理由ではない(のである)。より具体的に言うと、あまりにもしばしば,また,あまりにも無益に、向う見ずな哲学者達が主張してきたような、確実性の主張(訳注:私の主張は絶対に正しい、といったような主張)はまったく存在していない(のである)。
Chapter 16: Non-Demonstrative Inference , n.25 In the transition from crude fact to science, we need forms of inference additional to those of deductive logic. Traditionally, it was supposed that induction would serve this purpose, but this was an error, since it can be shown that the conclusions of inductive inferences from true premisses are more often false than true. The principles of inference required for the transition from sense to science are to be attained by analysis. The analysis involved is that of the kinds of inference which nobody, in fact, questions: as, for example, that if, at one moment, you see your cat on the hearth-rug and, at another, you see it in a doorway, it has passed over intermediate positions although you did not see it doing so. If the work of analysing scientific inference has been properly performed, it will appear that concrete instances of such inference are (a) such as no one honestly doubts, and (b) such as are essential if, on the basis of sensible facts, we are to believe things which go beyond this basis. The outcome of such work is to be regarded rather as science than as philosophy. That is to say, the reasons for accepting it are the ordinary reasons applied in scientific work, not remote reasons derived from some metaphysical theory. More especially, there is no such claim to certainty as has, too often and too uselessly, been made by rash philosophers. Source: My Philosophical Development, 1959, by Bertrand Russell More info.: https://russell-j.com/beginner//BR_MPD_16-250.HTM
Chapter 16: Non-Demonstrative Inference , n.24 The method of Cartesian doubt, which appealed to me when I was young and may still serve as a tool in the work of logical dissection, no longer seems to me to have fundamental validity. Universal scepticism cannot be refuted, but also cannot be accepted. I have come to accept the facts of sense and the broad truth of science as things which the philosopher should take as data, since, though their truth is not quite certain, it has a higher degree of probability than anything likely to be achieved in philosophical speculation. Source: My Philosophical Development, 1959, by Bertrand Russell More info.: https://russell-j.com/beginner//BR_MPD_16-240.HTM
Chapter 16: Non-Demonstrative Inference , n.23 I have been much criticized for applying the methods of mathematical logic to the interpretation of physics, but, in this matter, I am wholly unrepentant. It was Whitehead who first showed me what was possible in this field. Mathematical physics works with a space composed of points, a time composed of instants and a matter composed of punctual particles. No modem mathematical physicist supposes that there are such things in nature. But it is possible, given a higgledy-piggledy collection of things destitute of the smooth properties that mathematicians like, to make structures composed of these things and having the properties which are convenient to the mathematician. It is because this is possible that mathematical physics is more than an idle amusement. And it is mathematical logic which shows how such structures are to be made. For this reason, mathematical logic is an essential tool in constructing the bridge between sense and science of which I spoke above. Source: My Philosophical Development, 1959, by Bertrand Russell More info.: https://russell-j.com/beginner//BR_MPD_16-230.HTM
科学的知識のいくつかの体系の分析から始めよう。科学的知識は全て、何らかの算法(the methods of some calculus 計算手段)によって容易に操作できることを目的とした,人工的に作られた存在(entities 実体、存在)を用いる。 その科学が進歩していればしているほど、それは真実である。 経験科学の中では、そのことは物理学において最も完全に真実(事実)である。 物理学のような進歩した科学においては、哲学者(訳注:特に、科学哲学者)にとって、科学を一つの演繹体系として示すという予備的な作業がある。その演繹体系は、ある一定の原理から出発し、残りはその原理から論理的に導出され、また,その演繹体系は、当該科学が扱う全てのものが少なくとも理論的に定義可能であるところの現実的あるいは仮説的な存在から出発する。この(予備的な)仕事が十分になしとげられたならば、分析の後に残留物(残滓/カス)として残るそれらの原理や存在を、当該科学全体に対する抵当(人質)として取っておくとができ、哲学者はもはや、その科学を構成している他の複雑な知識に関心を持つ必要がなくなる。 しかし、いかなる経験科学も単に一貫性のある(整合性のある)お伽話であろうとするものではない。 科学は、現実世界に適用可能でありかつ現実世界への関係のゆえに(正しいと)信ぜられるところの(諸)陳述から構成されることを目指している。たとえば、科学の最も抽象的な部分でさえ、たとえば一般相対性論のようなものでさえ、観察された事実ゆえに受け入れられている。従って(thus その結果として)、哲学者は、観察された事実と科学的抽象との関係を探究することを強いられる。この探究は長期間の骨の折れる任務である。この任務の困難の一つは、我々の出発点である常識が既に、粗末かつ原始的な種類(の理論)ではあるけれども、理論によって影響されているということである。我々が自ら観察していると思っていること(もの)は、 我々が実際に観察している以上のもの(何らかのものがつけ加わってしまっているもの)であり、その(付け加えられた)何ものかは、常識の形而上学と科学とによって加えられたものである。私は常識の(に含まれる)形而上学及び科学を全てしりぞけなければならないと言おうとしているのではない。ただ、それらも我々が吟味しなければならないものの一つであると言っているだけである。それは二つの極、すなわち一方では定式化された科学と、 他方では純粋な(混じりけのない)観察との、いずれにも属しないものである。
Chapter 16: Non-Demonstrative Inference , n.22 Let us begin with the analysis of some body of scientific knowledge. All scientific knowledge uses artificially manufactured entities of which the purpose is to be easily manipulated by the methods of some calculus. The more advanced the science, the more true this is. Among empirical sciences, it is most completely true in physics. In an advanced science, such as physics, there is, for the philosopher, a preliminary labour of exhibiting the science as a deductive system starting with certain principles from which the rest follows logically and with certain real or supposed entities in terms of which everything dealt with by the science in question can, at least theoretically, be defined. If this labour has been adequately performed, the principles and entities, which remain as the residue after analysis, can be taken as hostages for the whole science in question, and the philosopher need no longer concern himself with the rest of the complicated knowledge which constitutes that science. But no empirical science is intended merely as a coherent fairy-tale. It is intended to consist of statements having application to the real world and believed because of their relation to that world. Even the most abstract parts of science, such, for instance, as the general theory of relativity, are accepted because of observed facts. The philosopher is thus compelled to investigate the relation between observed facts and scientific abstractions. This is a long and arduous task. One of the reasons for its difficulty is that common sense, which is our starting-point, is already infected with theory, though of a crude and primitive kind. What we think that we observe is more than what we in fact observe, the ‘more’ being added by common-sense metaphysics and science. I am not suggesting that we should wholly reject the metaphysics and science of common sense, but only that it is part of what we have to examine. It does not belong to either of the two poles of formulated science, on the one hand, or unmixed observation, on the other. Source: My Philosophical Development, 1959, by Bertrand Russell More info.: https://russell-j.com/beginner//BR_MPD_16-220.HTM
『数学原理(プリンキピア・マテマティカ)』に取り組んでいた時以来ずっと、当初はほとんど意識していなかったが次第に私の思考において明確になってきたあるひとつの方法を私は持っていた。その方法とは、感覚の世界と科学の世界との間に橋を架けようとする試みである。私は両者(感覚の世界と科学の世界)を、その大体の輪郭において(おおざっぱに言って)、疑う余地がないものとして受けいれる(受け入れている)。(そうして)アルプスの(一つの)山を貫くトンネルを掘ろうとする時のように、作業(研究)は両端から進められ、最後に中間で出会うことによってその労働が報われる(crowned by 王冠を得る)という希望をもって進まなければならない。
Chapter 16: Non-Demonstrative Inference , n.21 Ever since I was engaged on Principia Mathematica, I have had a certain method of which at first I was scarcely conscious, but which has gradually become more explicit in my thinking. The method consists in an attempt to build a bridge between the world of sense and the world of science. I accept both as, in broad outline, not to be questioned. As in making a tunnel through an Alpine mountain, work must proceed from both ends in the hope that at last the labour will be crowned by a meeting in the middle. Source: My Philosophical Development, 1959, by Bertrand Russell More info.: https://russell-j.com/beginner//BR_MPD_16-210.HTM
繰り返して述べるが、これらの要請(注:非論証的推論を擁護するために必要な5つの要請)は、それらの要請が我々がみな妥当であると容認する推論の中に暗に含まれているという事実によって正当化される。また、それらの要請は形式的意味では証明することはできないけれども、それらが抽出されたもとであるところの科学の全体系及び日常的知識は、適度に(within limits ほどほどに)、自己確証的である(訳注:self-confirmatory)という事実によって正当化される(のである)。私は真理の整合説(coherence theory of truth ある命題が真であるかどうかはその命題と他の命題群との整合性によって決まるとする立場)を受けいれない。しかし、確率の整合説(oherence theory of probability)というものがあり、それは重要であって、妥当だと私は考える。(つまり)二つの事実とそれらを結合する一つの因果律(a causal principle 因果の原理)があると仮定すると、(一つの因果律を含めた)これら3つ全ての確率がどれか1つの確率よりも高くなることがあり(高くなる可能性があり、)相互に結びついた事実と因果律が多数で複合的(複雑)になればなるほど、それらの相互の整合性から生ずる確率の増加は大きい(のである)。(そうして)原理の導入なくしては、事実または事実と思われるもの(想定されるもの)は(それらをいかに多く集めても)整合的とも 矛盾している(inconsistent 一貫性がない)とも言えないことに注意すべきである(to be observed)。なぜなら、いかなる(任意の)2つの事実も、何か論理外の原理(extralogical principle)によってでなければ、互いに他を含意したり(含んだり)互いに他に矛盾したりすることはありえないからである。 そうして、上記の5つの原理(要請)あるいはそれらに類似したものが、これまで関心を寄せてきた確率の増加を生じさせるところの整合性(首尾一貫姓)の基礎を形成することができると私は信じている。 「因果関係(causality)、あるいは「自然の一様性(the uniformity of nature「自然の斉一性」とも言う)と曖昧に呼ばれているものが、科学的方法の多くの議論の中に現われる。 私の(5つの)要請の目的は、そういったかなり漠然とした原理の代りに、もっと精確かつ有効な原理に置き換えることである。私は上記に列挙した要請に大きな自信を持っているわけではないが、もし我々が事実上(実際上)いかなる疑問もを感じることができないことについての(concerning)非論証的推論を正当化しようとするならば、同じ種類の何ものかが必要であるという点には、かなり自信があると感じている。
Chapter 16: Non-Demonstrative Inference , n.20 The above postulates, I repeat, are justified by the fact that they are implied in inferences which we all accept as valid, and that, although they cannot be proved in any formal sense, the whole system of science and everyday knowledge, out of which they have been distilled, is, within limits, self-confirmatory. I do not accept the coherence theory of truth, but there is a coherence theory of probability which is important and I think valid. Suppose you have two facts and a causal principle which connects them, the probability of all three may be greater than the probability of any one, and the more numerous and complex the inter-connected facts and principles become, the greater is the increase of probability derived from their mutual coherence. It is to be observed that, without the introduction of principles, no suggested collection of facts, or supposed facts, is either coherent or inconsistent, since no two facts can either imply or contradict each other except in virtue of some extralogical principle. I believe that the above five principles, or something analogous to them, can form the basis for the kind of coherence which gives rise to the increased probability with which we have been concerned. Something vaguely called ‘causality’ or ‘the uniformity of nature’ appears in many discussions of scientific method. The purpose of my postulates is to substitute something more precise and more effective in place of such rather vague principles. I feel no great confidence in the precise postulates above enumerated, but I feel considerable confidence that something of the same sort is necessary if we are to justify the non-demonstrative inferences concerning which none of us, in fact, can feel any doubt. Source: My Philosophical Development, 1959, by Bertrand Russell More info.: https://russell-j.com/beginner//BR_MPD_16-200.HTM