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1、Tolstoys Mathematics in “War and Peace” Paul Vit anyi Abstract The nineteenth century Russian author Leo Tolstoy based his egalitarian views on sociology and history on the then newfangled statistical physics and also proposed a mathematical theory of waging war. Contents 1Introduction2 2Mathematica
2、l Sociology3 3Mathematics of War5 4Conclusion6 1 1Introduction It is interesting to consider the excursions of mathematicians and scientists into prose and poetry, and conversely and less known, the explorations of poets and novelists into mathematics. An example of the fi rst is Luitzen E.J. Brouwe
3、rs excursion into literature and environmentalism 1, an appeal avant la lettre to save the earths natural environment from human polution. In particular he wants to abolish the tech- nology that enables mans supremacy over nature and the physics and mathe- matics that makes this possible. Only pure
4、(intuitionistic) mathematics that by its nature is unapplied and unapplicable for evil purposes, and which is the ultimate creation of the noble mind, should be saved. In another direction, the great Russian mathematician Andrei N. Kolmogorov was particularly interested in the form and structure of
5、the poetry by the Rus- sian author Pushkin 3. He also remarks 4: “what real meaning is there, for example, in asking how much information is contained in War and Peace? Is it reasonable to include this novel in the set of possible novels, or even to postulate some probability distribution for this s
6、et? Or, on the other hand, must we assume that the individual scenes in this book form a random sequence with stochastic relations that damp out quite rapidly over a distance of several pages? The answer to the latter question is decidly no.There is a ubiquitous general theme in War and Peace, namel
7、y, the idea that single individuals cannot infl uence in any sense the course of history (contrary to what is assumed in common history writing), but that the course of history is determined by the confl uence of myriad motions of the infi nitesimally small individual human acts of free will, much a
8、s a fl ock of birds wheels about in unision without any apparent governor. Here we have individual humans as interchangeable atoms of ideal gas that in combination determine eff ects on macroscopic scales such as heat and pressure, following the nineteenth century statistical physics of H. von Helmh
9、oltz. It serves to justify egalitarian doctrine. Helmholtz is also the author of the unrelated whitticism, so true and so unknown to politicians and managers of science: “ Whoever in the pursuit of science, seeks after immediate practical utility may rest assured that he seeks in vain,” 2. The autho
10、r of War and Peace, the great Russian novelist Count Leo Niko- layevich Tolstoy, had an intense interest in mathematical approaches to the sciences, as appears from his proposals to found sociology, history, and the sci- ence of war as a mathematical discipline, much like mathematician John von Neum
11、ann proposed to found the science of economy as a mathematical disci- pline in 6. Tolstoys views on the matter are set forth at great length in War and Peace, by many regarded as the greatest novel in any language. Based on, or perhaps called upon as justifi cation for, Tolstoys egalitarian philosop
12、hy, it is set forth passionately in long interludes littered through the later parts of this great novel. Recall that the book is ostensibly about the doings and adventures of a group of aristocratic people, and in the descriptions of great battles, at the time of Napoleons invasion in the bleak rea
13、ches of great Russia. Closer inspec- 2 tion reveals that one of the main themes of the tale is the insignifi cance and expendability of the particular heroslike Napoleonin the sweep of history: the events would have infolded in the same way irrespective of the so-called main fi gures. We base our tr
14、eatment on Rosemary Edmonds 1957 translation into English published in Penguin Classics5 (part I, 1972 printing; part II, revised 1978 printing). I will refer to the page numbers as WP, xx. 2Mathematical Sociology Tolstoy disagrees with the view of history that ascribes the evolution of events to in
15、dividuals: “One might have supposed that the historians, who ascribe the actions of the masses to the will of one man, would have found it impossible to fi t the fl ight of Napoleons armies into their theory, considering that during this period of the campaign in Russia the French did all they could
16、 to bring about their own ruin, and that not a single movement of that rabble of men . betrayed a hint of rhyme or reason. But no! Mountains of volumes have been written by historians . with accounts of Napoleons masterly arrangements and deeply considered plans .” WP, 1266 Not only that individuals
17、 cannot be the main governors of the making of History, but: “It is beyond the power of the human intellect to encompass all the causes of a phenomenon.” . “the human intellect . snatches at the fi rst compre- hensible approximation to a cause and says: There is the cause.” Tolstoy goes on WP, 1168
18、to explain “in historical events (where the actions of men form the subject of observation) the primeval conception of a case was the will of the gods, succeeded later on by the will of those who stand on the historical foregroundthe heroes of history.” On page WP, 1342 Tolstoy continues to unmask c
19、ommon misconceptions of traditional views of History: “Why did things happen thus, and not otherwise?Because they did so happen. Chance created the situation; genius made use of it, says history. But what is chance? What is genius? The words chance and genius do not denote anything that actually exi
20、sts, and therefore they cannot be defi ned. These two words merely indicate a certain degree of comprehension of the phenomena. I do not know why a certain event occurs; I suppose that I cannot know: therefore I do not try to know, and I talk about chance. I see a force producing eff ects beyond the
21、 scope of ordinary human agencies; I do not understand why this occurs, and I cry genius.” Now we come to the true view of history, in the spirit of the so successful natural sciences. The “unreasonable eff ectiveness of mathematics in science” as phrased by E. Wigner, must be extended avant la lett
22、re to sociology and political history WP, 977: “To elicit the laws of history we must leave aside kings, ministers, and generals, and select for study the homogeneous, infi nitesemal elements which infl uence the masses. No one can say how far it is possible for a man to advance 3 in this way to an
23、understanding of the laws of history; but it is obvious that this is the only path to that end, and that the human intellect has not, so far, applied in this direction one-millionth of the energy which historians have devoted to describing the deeds of various kings, generals and ministers, and prop
24、ounding refl ections of their own concerning those deeds.” How then is this proper view of history obtained? Tolstoy discusses the continuity of motion that was captured in laws by dividing continuity into units. He observes that this can be done in a wrong way, WP, 974: “Take, for instance, the wel
25、l-known sophism of the ancients which set out to prove that Achilles would never catch up with the tortoise that had the start on him, even though Achilles traveled ten times as fast as the tortoise: by the time Achilles has covered the distance that separated him from the tortoise, the tortoise has
26、 advanced one-tenth of that distance ahead of him. While Achilles does this tenth the tortoise gains a hudredth, and so on ad infi nitum. This problem appeared to the ancients insoluble. The absurdity of the fi nding (that Achilles can never overtake the tortoise) follows from arbitrarily separating
27、 the motion into separate units, whereas the motion of Achilles and the tortoise was continuous. By adopting smaller and smaller units of motion we only approximate the solution of the problem but never reach it. It is only by admitting infi nitesimal quantities and their progression up to a tenth,
28、and taking the sum of that geometrical progression, that we arrive at the solution of the problem.” Now we come to the heart of the matter: Tolstoys proposal of a diff erential and integral analysis of history WP, 974975: “A new branch of mathematics, having attained the art of reckoning with infi n
29、itesimal, can now yield solutions to other more complex problems of motion which before seemed insoluble. This new branch of mathematics, which was unknown to the ancients,1by admitting the conception, when dealing with problems of motion, of the infi nitely small and thus conforming to the chief co
30、ndition of motion (absolute continuity), corrects the inivitable error which human intellect cannot but make if it considers separate units of motion instead of continuous motion. In the investigation of the laws of historical movement precisely the same principle operates. The march of humanity, sp
31、ringing as it does from an infi nite multitude of individual wills, is continuous. The discovery of the laws of this continuous movement is the aim of history. But to arrive at these laws of continuous motion resulting from the sum of all those human volitions, human reason postulates arbitrarily, s
32、eparated units. The fi rst proceeding of a historian is to select at random a series of successive events and examine them apart from others, though there is and can be no beginning to any event, for an event fl ows without break in continuity from another. The second method is to study the actions
33、of some one mana king or a commanderas though their actions represented the sum of many individual wills; whereas the sum of the individual wills never fi nds expression in the activity of a single historical personage. . Only by assuming an infi nitesimal small unit for observationa diff er- 1Apart
34、 from Archimedes and Eudoxos PV. 4 ential of history (that is, the common tendencies of men)and arriving at the art of integration (fi nding the sum of the infi nitesimals) can we hope to discover the laws of history.” 3Mathematics of War The causality involved in war defi es simple analysis, Tolsto
35、y says, but is reached by the integration of the infi nitesimal individual causes, WP, 1184: “An infi nite amount of freely acting forces (and nowhere is a man freer than during a life and death struggle) infl uence the course taken by a battle, and that course can never be known beforehand and neve
36、r coincides with the direction it would have taken under the impulsion of any single force. If simultaneously and variously directed forces act on a given body, the direction which that body will take cannot be the course of anyone of the forces individuallyit will always follow an intermediate, as
37、it were, shortest path, or what is presented in mechanics by the diagonal of a parallelogram of forces.” In WP, 12231224 Tolstoy outlines the mathematics of war and goes into an explicit calculation that is patently false: “Military science says, the greater the numbers of an army the greater the st
38、rength. . For militray science to make this assertion is like defi ning energy in mechanics by reference to the mass only. It is like saying that the momenta of moving bodies will be equal or unequal according to the equality or inequality of their masses. But momentum (or quantity of motion) is the
39、 product of mass and velocity. So in warfare the strength of an army is the product of its mass and of something else, some unknown factor x.” He goes on to debate what this unknown x may stand for and rejects the common explanations, especially the interpretation of x as the amount of genius of the
40、 commanding general. He goes on to say that WP, 1224: “We must accept the unknown and see it for what it is: the more or less ac- tive desire to fi ght and face danger. Only then, expressing the known historical facts by means of equations, shall we be able to compare the relative values of the unkn
41、own factor; only then may we hope to arrive at the unknown itself. If ten men, batalions or divisions, fi ghting fi fteen men, batallions or di- visions, beat the fi fteenthat is, kill or capture them all while losing four themselves, the loss will have been four on one side and fi fteen on the othe
42、r. Therefore, the four are equal to the fi fteen, and we may write 4x = 15y. In other words, x is to y as 15 is to 4. Though this equation does not yet give us the absolute value of the unknown factor, it does give us a ratio between two unknowns. And by putting a whole variety of historical data (b
43、attles, cam- paigns, periods of warfare, and so on) into the form of such equations, a series of fi gures will be obtained which must involve the laws inherent in equations and will in time reveal them.” This argument of Tolstoy is remarkable. He compares the loss of the con- quering army with the t
44、otal of the vanquished armyperhaps on the grounds that the vanquished army is totally lost. Testing the idea by inserting more extreme fi gures, such as that an army of 1.000.000 men beats a small army of 5 10 men, while the conquering army looses one man, we obtain the equation x = 10y. This means
45、that the fi ghting spirit of the million-men army exceeded necessarily the fi ghting spirit of the minuscule ten-men army tenfold. The prob- lem with Tolstoys reasoning here is that he equates the ratio of the loss of the conquering army (irrespective of the size of the total army) and the total of
46、the beaten army (however small) with the ratio of the fi ghting spirit of the beaten army and that of the conquering army. In our opinion this reasoning is hard to defend in general as is shown by substituting extrem numbers as above. The general drift of the argument is of course reasonable. Note t
47、hat (contrary to the intention of the author) the variables x and y may contain the quality of the commanders (much as the quality of performance of a good symphony orchestra greatly depends on the quality of the conductor). 4Conclusion It is seldom the case that a great author deems fi t to incorpo
48、rate extensive discussions about mathematical foundations of social sciences in a major literary novel. It is much more common that scientists strive for literary redemption. Tolstoy is one of the rare examples of the former. In fact, he gives defi nite proposals to mathematize history, sociology, a
49、nd the sciences of war in line with the rational inclination of the nineteenth century. References 1 L.E.J. Brouwer, Leven, Kunst en Mystiek, Waltman, Delft, 1905. Enlish translation by W.P. van Stigt, Notre Dame J. Formal Logic, 37:3(1996), 391431. 2 H.L.F. von Helmholtz, Academic Discourse, Heidelberg, 1862. 3 A.N. Kolmogorov, Statistics and probability theory in research into Rus- sian poetry, Proc. Symp. on complex investigation of artistic creation, Nauka, Leningrad, 1964, 23. 4 A.N. Kolmogorov,