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鲁金与绝对无穷

(2020-02-28 06:57:43)

鲁金与绝对无穷

  十九世纪是法国柯西等人极限论

的鼎盛时期,无穷小只是他们的口头说法。

  二十世纪30年代,苏联数学家鲁金提出绝对无穷概念把无穷小引入微积分。

  二十世纪60年代,美国数学家创立非标准分析,奠定无穷小微积分的现代基础。

  进入二十一世纪,我们走在超实数的大道上。

  请见附件1.5,

 1.6l, 1.7

袁萌  陈启清 228

附件:

1.5. B. Bolzano, A. Cauchy, and K. Weierstrass The 19th century was the time of building analysis over the theory of limits. Outstanding contribution to this process belongs to Bolzano, Cauchy, and Weierstrass whose achievements are mirrored in every traditional textbook on dierential and integral calculus. The new canon of rigor by Bolzano, the denition by Cauchy of an innitely small quantity as a vanishing variable and, nally, the ε-δ-technique by Weierstrass are indispensable to the history of mathematical thought, becoming part and parcel of the modern culture. It is worth observing (see [408]) that, giving a verbal denition of continuity, both Cauchy and Weierstrass chose practically the same words:

An innitely small increment given to the variable produces an innitely small increment of the function itself. Cauchy Innitely small variations in the arguments correspond to those of the function. Weierstrass This coincidence witnesses the respectful desire of the noble authors to interrelate the new ideas with the views of their great predecessors. Speculating about signicance of the change of analytical views in the 19th century, we should always bear in mind the important observation by Severi [439, p. 113] who wrote: “This reconsideration, close to completion nowadays, has however no everlasting value most scientists believe in. Rigor itself is, in fact, a function of the amount of knowledge at each historical period, a function that corresponds to the manner in which science handles the truth.”

1.6. N. N. Luzin The beginning of the 20th century in mathematics was marked by a growing distrust of the concept of innitesimal. This tendency became prevailing as mathematics was reconstructed on the set-theoretic foundation whose proselytes gained the key strongholds in the 1930s.

8 Chapter 1

In the rst edition of the Great Soviet Encyclopedia in 1934, Luzin wrote: “As to a constant innitely small quantity other than zero, the modern mathematical analysis, without discarding the formal possibility of dening the idea of a constant innitesimal (for instance, as a corresponding segment in some non-Archimedean geometry), views this idea as absolutely fruitless since it turns out impossible to introduce such an innitesimal into calculus” [335, pp. 293–294]. The publication of the textbook Fundamentals of Innitesimal Calculus by VygodskibecameanoticeableeventinRussiaatthattimeandgaverisetoaserious and sharp criticism. Vygodski tried to preserve the concept of innitesimal by appealing to history and paedeutics. He wrote in particular: “If it were only the problem of creating some logical apparatus that could work by itself then, having eliminated innitesimals from considerations and having driven dierentials out of mathematics, one could celebrate a victory over the diculties that have been impeded the way of mathematicians and philosophers during the last two centuries. Innitesimal analysis originated however from practical needs, its relations with the natural sciences and technology (and, later, with social sciences) becoming increasingly strong and fruitful in the course of time. Complete elimination of innitesimals would hinder these relations or even make them impossible” [515, p. 160]. Discussing this textbook by Vygodski, Luzin wrote in the 1940s: “This course, marked by internal integrity and lit by the great idea the author remains faithful to, falls beyond the framework of the style in which the modern mathematical analysis has been developed for 150 years and which is nearing its termination” [335, p. 398]. Luzin’s attitude to innitesimals deserves special attention as apparent manifestation and convincing evidence of the background drama typical of the history of every profound idea that enchants and inspires the mankind. Luzin had a unique capability of penetration into the essence of the most intricate mathematical problems, and he might be said to possess a remarkable gift of foresight [308, 309, 337]. The concept of actual innitesimals seemed to be extremely appealing to him psychologically, as he emphasized: “The idea about them has never been successfully driven out of my mind. There are obviously some deeply hidden reasons still unrevealed completely that make our minds inclined to looking at innitesimals favorably” [335, p. 396]. In one of his letters to Vygodski which was written in 1934 he predicted that “innitesimals will be fully rehabilitated from a perfectly scientic point of view as kind of ‘mathematical quanta.’” In another of his publications, Luzin sorrowfully remarked: “When the mind starts acquaintance with analysis, i.e., during the mind’s spring season, it is the innitesimals, which deserve to be called the ‘elements’ of quantity, that the mind begins with. However, surfeiting itself gradually with knowledge, theory, abstrac

Excursus into the History of Calculus 9

tion and fatigue, the mind gradually forgets its primary intentions, smiling at their ‘childishness.’ In short, when the mind is in its autumn season, it allows itself to become convinced of the unique sound foundation by means of limits” [504]. This limit conviction was energetically corroborated by Luzin in his textbook Dierential Calculus wherein he particularly emphasized the idea that “to grasp the very essence of the matter correctly, the student should rst of all made it clear that each innitesimal is always a variable quantity by its very denition; therefore, no constant number, however tiny, is ever innitely small. The student should beware of using comparisons or similes of such a kind for instance as ‘One centimeter is a magnitude innitely small as compared with the diameter of the sun.’ This phrase is pretty incorrect. Both magnitudes, i.e., one centimeter and the diameter of the sun, are constant quantities and so they are nite, one much smaller than the other, though. Incidentally, one centimeter is not a small length at all as compared for instance with the ‘thickness of a hair,’ becoming a long distance for a moving microbe. In order to eliminate any risky comparisons and haphazard subjective similes, the student must remember that neither constant magnitude is innitesimal nor any number, however small these might be. Therefore, it would be quite appropriate to abandon the term ‘innitesimal magnitude’ in favor of the term ‘innitely vanishing variable,’ as the latter expresses the idea of variability most vividly” [504, p. 61].

1.7. A. Robinson The seventh (posthumous) edition of this textbook by Luzin was published in 1961 simultaneously with Robinson’s Nonstandard Analysis which laid a modern foundation for the calculus of innitesimals. Robinson based his research on the local theorem by Maltsev, stressing its “fundamental importance for our theory” [421, p. 13] and giving explicit references to Maltsev’s article dated as far back as 1936. Robinson’s discovery elucidates the ideas of the founders of dierential and integral calculus, witnessing the spiral evolution of mathematics.

 


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