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科学哲学ニューズレター |
No. 41, May 17, 2001
1. Euler on Space and Time,
Physics and Metaphysics
Editor: Soshichi Uchii
1. Euler on Space and Time, Physics and Metaphysics
John Stachel emphasized the importance of Euler, as regards the philosophy of space and time, in his Notes at the beginning of Foundations of Space-Time Theories (Minnesota Studies in the Philosophy of Science, vol. 8, 1977, vii-viii). Euler's work at issue is a paper written in French in 1748: "Reflexions sur l'espace et le temps" (Memoir de l'academie des sciences de Berlin, 4, 324-333; included in Leonhardi Euleri Opera Omnia, series tertia, vol. 2, 376-383). Actually, it is Ernst Cassirer who made an illuminating remark a long time ago on Euler's paper, in connection with Kant's indebtedness to Euler, with respect to the relation of (mathematical) physics to metaphysics (Zur Einstein'schen Relativitaetstheorie, 1921).
But what did Euler say, and why do Kant, Cassirer and Stachel give so much importance to this short paper? In a word, Euler defended the Newtonian notion of absolute space and time, against the Leibnizian relational theory of space and time, on the ground that the well-established laws of mechanics cannot be maintained according to the relational theory; in particular, Euler referred to the law of inertia (Newton's first law).
But in order to give a better idea, let me translate (tentatively) the whole text, and add my comments (in brackets) . Euler begins this paper as follows (the paragraph numbers are in the original text):
1. The principles of mechanics are already so well established that it would be wrong, if one still wished to doubt their validity. Even though we are unable to demonstrate these principles by the general principles of Metaphysics, the wonderful conformity of all the consequences drawn from these principles by calculation, with the movements of bodies on the earth, fluid as well as solid, and also with the movements of celestial bodies, should be sufficient for making their truth beyond doubt. Thus it is an indubitable truth, that a body once at rest will remain perpetually at rest, unless it is disturbed by some external forces. Likewise it is certain that a body once in a movement, will continue perpetually the same movement with the same velocity and in the same direction, provided that it does not encounter any obstacles against the conservation of that state.[Presumably this firm confidence in physics (as empirical science) distinguishes Euler from such metaphysicians as Leibniz of older generation. While Leibniz tended to argue from metaphysical principles to physics, Euler's way is just the reverse; and that should have impressed Kant (in the pre-critical period), and Stachel as well. It should be noticed that the law of inertia is specifically referred to here.]
2. These two truths, being so indubitably established, it must be absolutely the case that these are founded on the nature of bodies: and since it is the purpose of Metaphysics to study the nature and the properties of bodies, the knowledge of of these truths will be able to serve as a guide in such an abstruse study. For in this science [of mechanics] one has a right to reject all reasonings and all ideas, however they may appear to be well founded in other respects, that can lead to such conclusions as contrary to these truths; and one is authorized to admit in this science nothing but those principles that can coexist with the same truths. Those primary ideas we form of things in our external world are oridinarily so obscure and so indeterminate that it is extremely dangerous to draw such consequences from them as can be trusted. Thus it is always a great advance, when one already knows by other means some conclusions to which the primary ideas of Metaphysics should lead: and it is on these conclusions that one must regulate and determine the primary ideas of Metaphysics.[In the second sentence, "this science" seems ambiguous. I took it to mean mechanics, but other translators may take it to mean Metaphysics; but what is asserted by Euler seems unproblematic if it is meant for mechanics, but a rather radical assertion if it is meant for Metaphysics. In any case, Euler's point is that physics can serve as a guide for metaphysical research, and not the other way around, as is stated in the following paragraph.]
3. Also, Metaphysicians, being far from denying these principles whose truth is assured by Mechanics, endeavor rather to deduce and demonstrate them by their ideas. But they accuse Mathematicians on account of attaching these principles wrongly to the ideas of space and time, which are nothing but imaginary and devoid of any reality. It is quite possible that a true principle, without losing anything of its truth, may be stated in an inept manner so that it does not correspond to precise ideas we must have of things; but then the Metaphysician will be obliged to remedy this defect, and to substitute real ideas instead of imaginary ideas, in that statement of these principles.
4. Thus this should be the case for the principles of Mechanics which are involved in the ideas of space and time, which, according to those Metaphysicians, do not have any reality. Hence it must be seen whether it is possible to remove those imaginary ideas, and to substitute in their stead real ideas, where we have formed by abstraction those imaginary ideas: in such a way, however, that the meaning and the force of those principles should not be altered in the least. For, no doubt, the bodies, in following these principles, follow nothing which cannot exist but in our imagination. It is rather certain that the bodies are following those real things for which the laws hold that the bodies continue in the same state.
[Euler now wisely follows this strategy: we have here an established principle of mechanics, which seems quite real. If Metaphysicians wish to accuse that it contains imaginary ideas of space and time, they should reconstruct the principle in terms of their own ideas, without changing its validity.]
5. Therefore, it is certain that, if it be not possible to conceive the two principles asserted by Mechanics without involving in them the ideas of space and time, it should be a sure indication that these ideas are not purely imaginary, as those Metaphysicians allege. One would rather have to conclude that the absolute space as well as time is much like as the Mathematicians believe it to be, and it would be a real thing which exists outside of our imagination. For, it would be absurd to maintain that pure imaginations can serve as foundations for the real principles of Mechanics.
6. In order to enter into this research, I will begin with the first principle related with the state of rest of bodies. In Mechanics, one regards space and place as real things, and by the first principle one supposes that a body in some place without movement will remain there unless it is disturbed by some external force: that is, in that case, that body will remain always in the same place in relation to the absolute space. I may well admit that the ideas of space and place may be nothing but imaginary notions; however, I wish someone teach me the reality in virtue of which the body behaves according to that law, and instead of which the Mathematicians content themselves with using the imaginary ideas of space and place.
[Euler now refers to the first law of motion, the law of inertia, and asks wether this can be reconstructed in terms of the relational theory of space and time.]
7. It may be said in the first place, that the place is nothing but the relation a body has as regards others around it. So let us substitute this idea for the idea of place; then we are obliged to say, by the principle in question, that a body once in a certain relation to other bodies around it, will persist in the same relation. That is to say, we must say that a body A having bodies B, C, D, E etc. around it will remain perpetually in the same neighborhood. And therefore, while the Mathematician says that that body remains in the same place in the absolute space, the Metaphysician will say that that body preserves the same relation with respect to other bodies around it.
8. Let us see whether these two ways of expression are equivalent, and whether one can always substitute, without falling into an error, the metaphysical expression for the mathematical, in the truth of which we are already convinced. Thus, in order to compare these two expressions, let us suppose that the body A, as well as its neighbors B, C, D, E etc., be at rest; then in this case, the body A, in remaining in the same neighborhood of the bodies B, C, D, E etc. according to the metaphysical rule, will also remain in the same place according to the mathematical rule; and in this case one does not err in susbstituting the former for the latter.
9. In order to have a better determination of our ideas, let us suppose that the body A is in the still water, and that when it remains in the same place, it will also remain in the same neighborhood of the particles of water which surround it, and the body follows the mathematical rule equally well as the metaphysical rule. But suppose now the water begins to flow, and according to the mathematical rule the body will nevertheless remains in the same place, unless it is drawn little by little by the force of the water. But according to the metaphysical rule, the body would perfectly follow the movement of the water, in order to preserve the neighborhood of the same particles of water which have surrounded it previously. Thus in this case the rule derived from Metaphysics does not conform to the truth any more.
[Here Euler points out a discrepancy between the relational account and the usual rendering of the first law.]
10. On this point, let us consult experience, which informs us that a body having been at rest in the still water will be in motion, as soon as the water begins to flow; and this seems to favor the metaphysical rule. However, Mechanics makes us see very clearly that the body does not follow the flow of the water unless it is hit by the particles of the water; and that, therefore, it is an external force that puts the body in motion. Thus without that force the body would be at rest in the flowing water as well as in the still water, and therefore the body, in preserving its state of rest, does not depend on the bodies surrounding it immediately. From this, it follows that what is named place in Mechanics does not allow the Metaphysical explication, according to which the place is nothing but the relation of the body with other bodies surrounding it.
[Consulting our experience and the mechanical explanation, Euler argues that the relational account fails; it is not in conformity with the principle of mechanics. But Euler does not satisfy himself with this simple argument; he then proceeds to consider another possibility for the Metaphysicians.]
11. That property of bodies, in virtue of which they endeavors to preserve their state, rest as well as motion, is named Inertia. Thus this Inertia, as we have seen, does not depend on neighboring bodies; but it surely depends on the idea of place, which the Mathematicians regard as real and the Metaphysicians as imaginary. Since we are not allowed to substitute for the idea of place, the relation of bodies to sorrounding neighbors, it remains only remote bodies, in relation to which one can judge that general principle of inertia. But I strongly doubt that the Metaphysicians would dare to maintain that the bodies in virtue of their inertia are disposed to preserve the same relation to those bodies separated by some distances from them. For, it should be easy to see the falsity of such an explication, in view of similar reflections to those I have made on the bodies immediately in the neighborhood.
[Euler now touches on the possibility, similar to the one later pointed out by Mach, that inertia may be due to remote bodies, maybe fixed stars.]
12. If the Metaphysicians say that it is the relation to fixed stars that is required for explicating the principle of inertia, it may be very difficult to refute that, since fixed stars, being themselves at rest, are so remote from us, that the bodies which are at rest in the absolute space, as so regarded in Mathmamatics, would also be at rest relative to fixed stars. But aside from that it would be a very strange proposition and contrary to a number of other dogmas of Metaphysics, to say that fixed stars control the bodies in their inertia; this rule would be equally false if we be allowed to apply it to those bodies close to some fixed stars. These things remarked, there remain no more real ideas which may be substituted for the presumed imaginary ideas of space and time, in the explication of Inertia.
13. We saw thus that the idea of place, as conceived by the Mathematicians, cannot be explicated by any relation to other bodies, in the neigborhood or remote, and consequently, the metaphysical notions which are believed to correspond to the mathematical idea of place are not appropriate for being introduced into the explication of the mechanical principle in question. That is to say, the conservation of the state of a body depends on the place, as conceived in Mathematics, and not in the least on the relation to other bodies. But one would not want to say that the principle of Mechanics is founded on a thing which does not exist but in our imagination. Therefore, it must be concluded absolutely that the mathematical idea of place is not imaginary, but there is something real in the world which corresponds to that idea. Thus there is in the world, aside from the bodies which constitute the world, some reality which we represent to us by the idea of place.
14. Hence the Metaphysicians are wrong, when they wish to deny the world of space and place entirely by saying that they are nothing but arbitrary and imaginary ideas. Therefore the proofs they put forward to support their intuition, however strong they may appear, should be in fact ill-founded, and it must be the case that some paralogism is hidden there. It is true that our senses cannot provide us with the ideas of space and place, and that it is only by reflection that we form these ideas. From this the Metaphysicians conclude that these ideas are nothing but arbitrary ideas, similar to the ideas of genus and species, which do not exist but in our understanding and have no real objects corresponding. But it seems to me that this conclusion is premature. For, if one reflects a little oneself, one should easily see that the manner by which we form the idea of space and place is quite different from the manner by which we come to the idea of genus and species. And one would make a great mistake if one wishes to maintain that there exist nothing when we have no ideas other than those by reflection.
[Concluding that the Metaphysicians are wrong, Euler then proceeds to point out the difference between space and the usual abstraction of properties.]
15. I agree that all things that exist are perfectly determinate; and if we take away one or more determinations from the idea of such an object, thereby a general idea is generated, and to such an idea no existing object corresponds any more. Thus it is the case when we form the idea of extension in general, by taking away all the determinations except for extension, from the ideas of bodies. But the idea of the place a body occupies is not formed this way, by taking away some determinations of that body; it arises by removing the whole body: thus the place was not one of the determinations of the body, since it still remains after the whole body, together with all of its qualities, is removed. It must be noticed that the place a body occupies is quite different from the body's extension, because the extension belongs to the body and goes along with the body when it moves from one place to another; whereas the place and space are not susceptible to any movement.
16. I do not wish to discuss the objections made against the reality of space and place; because, having demonstrated that this reality can no more be doubted, it necessarily follows that all of these objections are of little importance, even though we do not respond appropriately to them. If one believes it to be absurd that all different places, or parts of space, are similar to each other, which would be contrary to the principle of [the identity of] indiscernibles, still, I do not know whether that principle is so general as one thinks; presumably it is not applicable to other than bodies and minds, in general, where one may well be satisfied. But since space and place are so essentially different from minds and bodies, one may be unable to judge by the same principles.
[The second argument against the relational theory, from the concept of direction.]
17. The reality of space will be established again by another principle of Mechanics, which contains the conservation of uniform movement in the same direction. For, if space and place were nothing but the relation among coexistent bodies, what would the same direction be? One should be quite at a loss for giving it an idea, by mere mutual relation among coexistent bodies without introducing the idea of immovable space. For in what manner the bodies may move and change situations among them, this does not prevent one from preserving a clear enough idea of a fixed direction that the bodies endeavor to follow in their movement, despite all the changes the other bodies may suffer. From this, it is evident that the identity of direction, which is truly an essential condition in the general principles of movement, would be absolutely unexplicated by the relation or the order of coexistent bodies. Therefore, it must be the case that there is again some other real thing, aside from bodies, to which the idea of the same direction is related; and there is no doubt that that is space, for which we have established reality.
[Euler now proceeds to arguments for time.]
18. The ideas of space and of time have almost always been of the same sort, so that those who have denied the reality of the one have also denied of the other, and conversely. Therefore one will not be surprised, in establishing the reality of space we regard also time as something real, which exists not merely in our mind but flows really, serving as a measure for the duration of things. We have a very clear idea of time, and I agree that we form this idea of the successions of changes we recognize: and in this regard I agree that the idea of time does not exist but in our imagination. But we have a reason to ask whether the idea of time and the time itself are not different from each other. And it seems to me that the Metaphysicians, in denying the reality of time, have confounded the time itself with the idea we have of time.
[The third argument against the relational theory, in terms of the equality of space and time, which again is assumed in the principle of movement.]
19. The principle of movement of bodies, according to which a body put in movement must continue the movement with the same velocity and to the same direction, as I said, provides us with new proofs, not only for the reality of space but also for the reality of time. For, since that uniform movement travels an equal amount of space in an equal time, I ask, first, what is this equal amount of space according to the intuition of those who deny the reality of space? I doubt whether the Metaphysicians dare to say that the equality of spaces must be judged by the equality of the number of monades which fill them; for they would have to asert that the monades are equally dispersed by all the bodies. But even if they wanted to stick to this explication, this explication would be reversed as soon as one consider the movement of bodies in relation to which one wishes to determine the equality of spaces. For we understand, and the principle of movement teaches us, that when a body runs equal spaces, the equality of spaces does not at all depend on other bodies around it, and the equality remains the same whatever changes those other bodies may suffer.
20. The situation is the same with respect to the equality of time; for if time is nothing but the order of successions, how can one make the equality of time intelligible? One claims that each object of the world is subject to continual changes, and that it is the succession of these changes that produces time. According to this explication, two intervals of time should be equal if the same number of successions occur during these intervals. But if one considers a body which runs the same spaces in the same times, on what changes, or on what bodies, should we judge the equality of these two times? Or does one want to say that all bodies are subject to changes with an equal frequency so that it would come to the same thing whichever bodies one may choose for measuing the equality of time on the number of changes that occur? But I am sure that as soon as you push that explication a little, you will find so many other inconveniencies that you will easily come to abandon it.
[Finally, Euler comes back to his previous distinction between the time itself and our idea of time.]
21. The question here is not our estimation of the equality of time, which will no doubt depends on the state of our mind; but the question is the equality of time during which a body put in an uniform movement runs equal spaces. Since that equality cannot be explicated in terms of the order of successions, no more than the equality of space in terms of the order of coexistents, and that equality is essentially involved in the principle of movement; one cannot say that the bodies, in continuing their movement, depend on a thing which does not exist but in our imagination. Therefore one is obliged to admit, as was the case with space, that time is something which exists outside of our mind, or time is something real as space was also. Here I address myself to those Metaphysicians who admit some reality to the bodies and the movement. For those who deny this reality absolutely and who allow nothing but phenomena, because they regard the movement itself as well as the laws of movement as chimerical, I do not flatter myself that these reflections should give the least impression on their mind.
Translation (c) Soshichi Uchii
Reference
カッシーラー『アインシュタインの相対性理論』（山本義隆訳）河出書房新社、1996。
Earman, J., Glymour, C., and Stachel, J., eds., Foundations of Space-Time Theories, Minnesota Studies in the Philosophy of Science 8, 1977.
2. Our Avtivities in 2000
新年度になり、学部新3回生が3名、杉本舞、田中泉吏、山口健太郎（大阪府大工学部より学士入学）、大学院進学3名、修士、山下幸宏、博士、網谷祐一、瀬戸口明久、となった。なお、松王政浩博士は本年4月より静岡大学（浜松）に助教授として赴任した。
昨年度の当研究室の活動実績を以下にまとめておく。国立大学で不祥事が相次ぎ、われわれもいっそう言動に気をつけなければならないが、だからといってナマケモノ学生に甘くするわけにはいかない。某教授の「イラチ」にもますます磨きがかかってきたが、それも余命が少なくなってきたからのことと察せられる！彼いわく、次に学位を取るのは誰じゃ？朝の紅顔、夕べの白骨、チーン、エイメン・・・ピッツバーグではWes Salmon 基金が設けられるとのこと。
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内井惣七 （教授）
道徳起源論から進化倫理学へ。第二部 規範倫理学における還元主義（続）。 『哲学研究』569号, 2000年4月。
書評 島尾永康『ファラデー』岩波書店、2000。PHS Newsletter 32, April.
Wes and Merrilee Salmon in Kyoto University, PHS Newsletter 33, June.
セイント・ピータースバーグのパラドックス、林晋編『パラドックス！』日本評論社、2000年7月。
書評 アインシュタインの裏側、PHS Newsletter 36, September 2000.
道徳起源論、松沢哲郎・長谷川寿一編『心の進化』、岩波書店、2000年11月。
Review: Ian Hacking, The Social Construction of What? PHS Newsletter 37, November 2000.
「適応主義の構造」（発表）シンポジウム「進化的視点と社会科学」日本科学哲学会、名古屋大学、2000年12月2日。
本年度の卒業論文・修士論文特集、PHS Newsletter 38, January 2001.
伊藤和行（助教授）
論文「ルネサンスにおける実践的学問観」，『科学の文化的基底(I)』（国際高等研 究所報告書1999-005），2000年4月発行，pp.123-137．
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論文「古典力学における運動法則の歴史性−ニュートンの第二法則をめぐって−」， 『哲学研究』，570号，2000年10月発行，pp.53-78．
論文「ガリレオの数学的原子論」，『ルネサンスにおける自然観の総合的研究』（平 成12年度科学研究費補助金：基盤研究(B)(1)研究成果報告書），2001年3月発行， pp.65-74．
報告書『17世紀力学理論における物理学的基礎概念の発展に関する歴史的研究』（平 成9年度〜12年度科学研究費補助金：基盤研究(C)(2)研究成果報告書），2001年3月発 行．（「17世紀力学関係著作コンコーダンス」を含む）
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研究発表：「非還元的物理主義について」，日本科学基礎論学会（慶應義塾大学）， 2000年6月17日
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井上和子（D3)
研究発表：「ランキンの熱力学関数とクラウジウスのエントロピー」，日本物理学会 第56回年次大会（中央大学多摩キャンパス），2001年3月29日
澤井 直（D2)
研究発表：「ウィリアム・ハーヴィの方法論 -類推の正当化をめぐって-」，日本医 史学会第101回総会（京都府立医大）２000年10月14日
瀬戸口明久（D1)
論文：「生態系生態学から保全生物学へ−生態学と環境問題，1960-1990−」，『生 物学史研究』65:1-13（2000年4月発行）
小野田波里（M2)
研究発表：「EinsteinとMach原理：一般相対性理論の形成」，科学基礎論夏のセミ ナー（北海道大学），2000年8月28日
「Einstein の宇宙論―慣性の相対性をめぐって」，日本科学 哲学会第33回大会（名古屋大学），2000年12月3日
岸田功平（M2)
研究発表：「義務論理のパラドクス再考」， 日本科学哲学会第33回大会（名古屋大 学），2000年12月2日
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May 17, 2001; last modified October 18, 2001. (c) Soshichi Uchii