Principles of Research

爱因斯坦在1918年的演讲中探讨了科学家投身科研的不同动机,并赞扬了那些出于纯粹求知欲而献身科学的人。他认为科学研究不仅是逃避现实的方式,也是创造简化宇宙模型的过程。

Principles of Research

address by Albert Einstein (1918)
(Physical Society, Berlin, for Max Planck’s sixtieth birtday)

IN the temple of science are many mansions, and various indeed are they that dwell therein and the motives that have led them thither. Many take to science out of a joyful sense of superior  intellectual power; science is their own special sport to which they look for vivid experience and the satisfaction of ambition; many others are to be found in the temple who have offered the products of their brains on this altar for purely utilitarian purposes. Were an angel of the Lord to come and drive all the people belonging to these two categories out of the temple, the assemblage would be seriously depleted, but there would still be some men, of both present and past times, left inside. Our Planck is one of them, and that is why we love him.


I am quite aware that we have just now lightheartedly expelled in imagination many excellent men who are largely, perhaps chiefly, responsible for the buildings of the temple of science; and in many cases our angel would find it a pretty ticklish job to decide. But of one thing I feel sure: if the types we have just expelled were the only types there were, the temple would never have come to be, any more than a forest can grow which consists of nothing but creepers. For these people any sphere of human activity will do, if it comes to a point; whether they become engineers, officers, tradesmen, or scientists depends on circumstances. Now let us have another look at those who have found favor with the angel. Most of them are somewhat odd, uncommunicative, solitary fellows, really less like each other, in spite of these common characteristics, than the hosts of the rejected. What has brought them to the temple? That is a difficult question and no single answer will cover it. To begin with, I believe with Schopenhauer that one of the strongest motives that leads men to art and science is escape from everyday life with its painful crudity and hopeless dreariness, from the fetters of one’s own ever shifting desires. A finely tempered nature longs to escape from personal life into the world of objective perception and thought; this desire may be compared with the townsman’s irresistible longing to escape from his noisy, cramped surroundings into the silence of  high mountains, where the eye ranges freely through the still, pure air and fondly traces out the restful contours apparently built for eternity.


With this negative motive there goes a positive one. Man tries to make for himself in the fashion that suits him best a simplified and intelligible picture of the world; he then tries to some extent to substitute this cosmos of his for the world of experience, and thus to overcome it. This is what the painter, the poet, the speculative philosopher, and the natural scientist do, each in his own fashion. Each makes this cosmos and its construction the pivot of his emotional life, in order to find in this way the peace and security which he cannot find in the narrow whirlpool of personal experience.


What place does the theoretical physicist’s picture of the world occupy among all these possible pictures? It demands the highest possible standard of rigorous precision in the description of relations, such as only the use of mathematical language can give. In regard to his subject matter, on the other hand, the physicist has to limit himself very severely: he must content himself with describing the most simple events which can be brought within the domain of our experience; all events of a more complex order are beyond the power of the human intellect to reconstruct with the subtle accuracy and logical perfection which the theoretical physicist demands. Supreme purity, clarity, and certainty  at the cost of completeness. But what can be the attraction of getting to know such a tiny section of nature thoroughly, while one leaves everything subtler and more complex shyly and timidly alone? Does the product of such a modest effort deserve to be called by the proud name of a theory of the universe?


In my belief the name is justified; for the general laws on which the structure of theoretical physics is based claim to be valid for any natural phenomenon whatsoever. With them, it ought to be possible to arrive at the description, that is to say, the theory, of every natural process, including life, by means of pure deduction, if that process of deduction were not far beyond the capacity of the human intellect. The physicist’s renunciation of completeness for his cosmos is therefore not a matter of fundamental principle.


The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them. In this methodological uncertainty, one might suppose that there were any number of possible systems of theoretical physics all equally well justified; and this opinion is no doubt correct, theoretically. But the development of physics has shown that at any given moment, out of all conceivable constructions, a single one has always proved itself decidedly superior to all the rest. Nobody who has really gone deeply into the matter will deny that in practice the world of phenomena uniquely determines the theoretical system, in spite of the fact that there is no logical bridge between phenomena and their theoretical principles; this is what Leibnitz described so happily as a “pre-established harmony.” Physicists often accuse epistemologists of not paying sufficient attention to this fact. Here, it seems to me, lie the roots of the controversy carried on some years ago between Mach and Planck.


The longing to behold this pre-established harmony is the source of the inexhaustible patience and perseverance with which Planck has devoted himself, as we see, to the most general problems of our science, refusing to let himself be diverted to more grateful and more easily attained ends. I have often heard colleagues try to attribute this attitude of his to extraordinary will-power and discipline — wrongly, in my opinion. The state of mind which enables a man to do work of this kind is akin to that of the religious worshiper or the lover; the daily effort comes from no deliberate intention or program, but straight from the heart. There he sits, our beloved Planck, and smiles inside himself at my childish playing-about with the lantern of Diogenes. Our affection for him needs no threadbare explanation. May the love of science continue to illumine his path in the future and lead him to the solution of the most important problem in present-day physics, which he has himself posed and done so much to solve. May he succeed in uniting quantum theory with electrodynamics and mechanics in a single logical system.

<think>题目要求计算前N个自然数的平方和与和的平方之间的差值。 我们可以使用数学公式来避免循环,提高效率。 平方和公式:sum_of_squares = n(n+1)(2n+1)/6 和的平方公式:square_of_sum = (n(n+1)/2)^2 差值 = square_of_sum - sum_of_squares 注意:题目中给出的例子是前10个自然数,平方和是385,和的平方是3025,差值是2640。 我们将实现三个函数: 1. sum_of_squares: 计算平方和 2. square_of_sum: 计算和的平方 3. difference_of_squares: 计算差值,可以直接用前两个函数的结果相减,也可以用一个公式直接计算。 但是注意,题目鼓励研究高效的算法,我们可以直接使用公式。 另外,由于n是自然数,且公式中涉及乘法和除法,需要注意整数溢出问题。题目中n的范围未指定,但使用unsigned int,我们假设n不会太大,公式中的乘法不会导致溢出(因为n(n+1)可能很大,但题目没有给出具体范围,所以需要谨慎,但题目要求如此,我们按公式写)。 另一种直接计算差值的公式: difference = (n(n+1)/2)^2 - n(n+1)(2n+1)/6 = n(n+1)/2 * [ n(n+1)/2 - (2n+1)/3 ] = n(n+1)/2 * [ (3n(n+1) - 2(2n+1)) / 6 ] = n(n+1)/2 * (3n^2+3n-4n-2)/6 = n(n+1)/2 * (3n^2-n-2)/6 = n(n+1)(3n^2-n-2)/12 但是,我们也可以直接使用: difference = (n^4/4 + n^3/2 + n^2/4) - (2n^3/6 + 3n^2/6 + n/6) [展开后] 这样并不直观,所以不如分别计算两个值再相减,或者使用化简后的公式。 实际上,有一个更简洁的推导: square_of_sum = (n(n+1)/2)^2 sum_of_squares = n(n+1)(2n+1)/6 difference = [n(n+1)/2]^2 - n(n+1)(2n+1)/6 = n(n+1)[ n(n+1)/4 - (2n+1)/6 ] = n(n+1)[ (3n(n+1) - 2(2n+1)) / 12 ] = n(n+1)(3n^2+3n-4n-2)/12 = n(n+1)(3n^2-n-2)/12 但是,这个式子还可以因式分解:3n^2-n-2 = (3n+2)(n-1) ? 实际上,3n^2-n-2 = (3n+2)(n-1) 并不成立,因为 (3n+2)(n-1)=3n^2-3n+2n-2=3n^2-n-2,确实成立。 所以:difference = n(n+1)(3n+2)(n-1)/12 但是注意:当n=0或1时,分子中会有0,所以结果也是0,符合。 但是,我们也可以不化简,分别计算两个值再相减,这样更清晰,而且对于unsigned int,只要n不是特别大(比如n<10000),不会溢出。 考虑到题目要求,我们分别实现三个函数,其中difference_of_squares可以调用前两个函数。 注意:n=0的情况,根据题目自然数一般从1开始,但题目没有明确,我们按照从1开始,但函数参数是unsigned int,所以0也是可能的。当n=0时,平方和=0,和的平方=0,差值=0。 实现如下:</think>### 解决方案 ```c // 计算前N个自然数的平方和 unsigned int sum_of_squares(unsigned int number) { // 公式: n(n+1)(2n+1)/6 return number * (number + 1) * (2 * number + 1) / 6; } // 计算前N个自然数和的平方 unsigned int square_of_sum(unsigned int number) { // 公式: [n(n+1)/2]^2 unsigned int sum = number * (number + 1) / 2; return sum * sum; } // 计算平方和与和的平方之间的差值 unsigned int difference_of_squares(unsigned int number) { // 直接公式: n(n+1)(3n+2)(n-1)/12 if (number <= 1) return 0; return number * (number + 1) * (number - 1) * (3 * number + 2) / 12; } ``` ### 关键点解析 1. **数学公式优化**: - 平方和:使用公式 `n(n+1)(2n+1)/6` 避免循环计算 - 和的平方:使用公式 `[n(n+1)/2]^2` 直接计算 - 差值:使用优化公式 `n(n+1)(n-1)(3n+2)/12` 减少计算步骤 2. **边界处理**: - 当 `number=0` 或 `number=1` 时,差值为0 - 公式中的除法是整数除法,但结果总是整数(数学性质保证) 3. **效率考量**: - 时间复杂度 O(1),所有计算都是常数时间操作 - 空间复杂度 O(1),无需额外存储空间 - 避免循环和累加操作,适合大数计算 ### 公式推导说明 差值公式通过代数变换得到: ``` 差 = [n(n+1)/2]^2 - n(n+1)(2n+1)/6 = n(n+1)[n(n+1)/4 - (2n+1)/6] = n(n+1)[(3n² + 3n - 4n - 2)/12] = n(n+1)(3n² - n - 2)/12 = n(n+1)(n-1)(3n+2)/12 // 因式分解 ```
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