π. ππ¦πͺπ° β° ππ π¬ππ’
ππ₯π’ traditional historiography of set theory has reinforced several miscoceptions as regards the development of modren mathematics. Excessive concentration upon the work of Cantor has led to the conclusion that set theory originated in the needs of analysis, a conclusion embodied in the very title of [Grattan-Guinness 1989]. From this start point, it would seem that the successful application of set-theoratical approach in algebra, geometry, and all other branches of mathematics came after-wards, in the early 20th century. These novel developments thus appear as unforseen successes of Cantor's brain child, whose most explicit expression would be found in Bourbaki [Meschkowski 1967, 232–33].
ππ₯π’ present work will show, on the contrary, that during the second half of the 19th century the notation of sets was crucial for emerging new conceptions of algebra, the foundations of arithmetic, and even geometry. Moreover, all of these developments antedate Cantor's earliest investigations in set-theory, and it is likely that some may have motivated his work. The set-theoratical conception of different branches of mathematics is thus in scribed in the very origins of set-theory. It is the purpose of Part One of the present work to describe the corresponding process. That will lead us to consider whether there was a flux of ideas between the different domains, trespassing disciplinary boundaries.
πππ―π± Two analyzes the crucial contributions to abstract set theory made in the last quarter of the 19th century. This means, above all, Cantor's exploration of the transfinite realm - the labyrinth of infinity and the continuum - which started with his radical discovery of the non-denumerability of β in December 1873. It also means Dedekind's work on sets and mapping as a basis for pure mathematics, elaborated from 1872. And, as a natural consequence, the interaction between both authors, who maintained some personal contacts and an episodic but extremely interesting correspondence. Particular attention will be given to the reception of their new ideas among mathematicians and logicians, ranging from the well-known opposition of Kronecker and his followers, to the employment of transfinite numbers in function theory, the rise of modern algebra and topology, and the expansion of logicism.
 |
| Leopold Kronecker 1823―1891 |
βπ« Part Three, a synthetic account of the further evolution of set theory upto 1950 is offered, concentrating on foundational questions and the gradual emergence of a modern axiomatization. Apart from the attempt to offer, perhaps for the first time, a comprehensive overview, a novel feature that deserve special mention is the attention given to a frequently forgotten aspect of this period - the bifurcation of two alternative systems, Russellian type theory and Zermelian set theory, and their subsequent convergence. Along the way we shall review a wide range of topics, including aspects of the so–called 'foundational crisis', constructivist alternatives to set theory, the main axiomatic systems for set theory, metatheoretical work on them (in particular Godel's results), and the formation of modern first–order logic in interaction with formal systems of set theory.
ππ«π’ can see that the topics dealt with concentrate gradually from part to part. Part one studies the emergence of the set–theoretical approach against the background of more traditional viewpoints. Without abandoning the question of how the language of sets became dominant, Part Two concentrates on abstract set theory, and Part Three has an event more restricted focus on the Ζoundations of abstract set theory.¹
ππ₯π’ reader must have noticed that we shall not just focus on set theory in the strict sense of the word. If by those words the reader understands abstract set theory as it is presently studied by authors who are classified as mathematical logicians, he or she should turn to Parts Two and Three.² Such a conception over looks the question of how the set‐theoretical appraoch to mathematics arose and why it came to play a central role in modern mathematics. These questions are certainly of much wider interest, and they seem no less important for the historian. Thus I have decided not to deal exclusively with (transfinite) set theory, but to pay careful attention to the set-theoretical approach too, asking questions like the following; How did mathematicians arrive at the notion of set? How did they became convinced that it offered an adequate basis and language for their discipline? How could a mathematician (like Cantor) convince himself that it is important to develop a theory of sets?
ππ₯π’ whole historical development that we shall analyze is the best described as a progressive differentiation of subdisciplines within the historical context of research programs that originally were unitary. We shall observe that, at first, the notion of set was employed in several different domains and ways, with no recognition of subdisciplinary boundaries. Gradually, mathematicians recognized differences between several aspects that originally appeared intertwined in the traditional objects of mathematics ― matric properties, topological features, algebraic structures, measuric-theoretic properties, and finally abstract set-theoretical aspects, In the process, new subdisciplines emerged.
β realize that, in following this path, i risk the danger of being sharply criticized by lovers of neat conceptual distinctions. In general, it is not commendable to project present-day disciplinary boundaries on the past. In this connection, it is interesting to consider the rather negative review that K. O. May wrote [1969] of a book by the late Medvedev [1965] of the development of set theory (a book that I have not read, since it has never been translated from Russian, but which may be similar to the present work in some respects). May argued that, in order to counter the opinion that Cantor was the creator of set theory, Medvedev confused abstract set theory with the topological theory of point-sets. But Cantor himself did not differentiate the theory of point-sets from abstract set theory until as late as 1885, fifteen years after he had started to do original work involving sets (see chapter VIII)³. May also critisized Medvedev's search for 19-th‐century precedents of the notion of set, like Gauss, since they never went beyond an implicit use of actual infinity, this is a more serious criticism, yet one may reflect on the fact that Dedekind, who indulged in a very explicit use of sets and actual infinity, mentioned Gauss's work in order to justify his view point (see §111.5).⁴
 |
| Carl Friedrich Gauss 1777 – 1855 |
π third important distinction is that between set theory as an approach to or a branch of mathematics and set theory as a Ζoundation for mathematics. In our present picture of set theory, and a (purported) foundation for mathematics, are so intertwined that it may seem artificial to distinguish them. Nevertheless, within the context of the early history of sets it is essential to take these distinctions into account.
ππ±π₯π’π―π΄π¦π°π’ it becomes impossible to produce a clear picture of the development, which includes a rather complex interaction between the three aspects. The idea that sets constitute the foundation of mathematics emerged very early, and Cantor was by no means the leading exponent of this view. Its strongest proponent as of 1890 was Dedekind, but i shall argue that Riemann was also an important voice advocating this position.⁶ Riemann took a significant step in the direction of introducing the language of sets, coupling this with the conception that sets are the basic objects of mathematics. There are good reasons to regard his early contribution as a significant influence on the early set-theoretical attempts of both Dedekind and Cantor.
 |
| Bernhard Riemann 1826 ― 1866 |
ππ₯π²π°, Part One and Two deal mostly with a small group of mathematicians ― above all Riemann, Dedekind and Cantor ― whose work was closely interconnected. This made it possible to present quiet a clear picture of the evolution from some initial set–theoretical glimpses, to what we can presently recognize as abstract set theory. To judge from May's review, this may be what Medvedev's work missed, making his research on the immediate precedents of set theory seem irrelevant.
ππ₯π’ notion of set seems to be, to some extent, natural for the human mind. After all, we employ common names (like 'book' or 'mathematician') and one is easily led to consider sets of object as underlying that linguistic practice. For this simple reason, it is too easy to find historical precedents for the notion of set, and an incursion into such marshy terrian can easily become arbitrary or irrelevant. It is precisely to avoid this risk that I have have concentrated on contributions which can be shown to have been directly linked to the work of the early set theorists. Since all of these men were German mathematicians, the first two parts of the book concentrate on mathematical work written in the German language.
βπ« this connection, I would like to warn the reader that a study of the origins of the notion of set in Germany is immediately confronted with terminological difficulties. In contrast to romance languages and English, where there were rather clear candidates for denoting the concept (ensemble, insieme, conjunto, set), the German language did not suggest a best choice. To give an example, both Dedekind and Cantor accepted that 'ensemble' was an ideal translation into French,⁷ but they used different German terms, Dedekind chose the word 'System,' Cantor changed his choice several times, but mostly used the words 'Manningfaltigkiet' and 'Menge'⁶ In the early period, each mathematician made his own selection, and one must carefully establish whether they are talking about sets, as commonly understood, or something else. The terminological question become particularly critical in the case of Riemann ― the whole issue of his role in the early history of sets depends on how we interpret his notion of 'Manningfaltigkiet' (see Chapter 11). I have decided to translate the relevant terms rigidly throughout the text, but it is important that the reader keep the problem of terminological ambiguity in mind. The mere fact that one finds the word 'Menge' [set] in a text does not in itself mean that the author is employing the right notion. And the mere fact that an author uses another word does not mean that he is not employing the notion of set.⁸
ππ¬ end this section, I would like to comment on the opinion of some historians who have emphasized the difference between early notions resembling that of set (e.g., in Bolzano or in Riemann) and the modren notion, which has commonly been attributed to Cantor.⁹ As a result of taking sets to be determined by concepts, Riemann and Bolzano regard them as endowed with an antecendently given structure. Today one considers an abstract set and then freely imposes structures on it. This contrast might be turned into an argument against the thesis that any of these 19th century authors were important in the development of set theory. My answer is that, in such a criterion were rigidly applied, even Cantor's work would not belong to the history of the set theory. All earlier authors, including Cantor and Dedekind, started with conceptions akin to those of Riemann and Bolzano, and it was only gradually that they (eventually) arrived at an abstract approach. They began with the concrete, complex objects of 19th-century mathematics, and in the course of their work they realized the possibility of distinguishing several different kinds of features or properties (that we would label as metric, or topological, or algebraic, or abstract properties). The abstract, extensional notion of set developed gradually out of the older idea of concept-extension (see § 11.2). Therefore, it is historically inadequate to contrast 'concrete' with an abstract approach as exclusive alternatives; rather they should be regarded as initial and final stage in a complex historical process.¹⁰
 |
| Bernard Bolzano 1781 ― 1848 |
1. An attempt to threat the development of abstract and descriptive set theory in full until about 1940 would have required another volume, and of course a long period of preparation.
2. And complement them with other works, like [Moore 1982], [Kanamori 1996].
3. The mathematical community as a whole began to assimilate the distinction only in the 1900s; abstract set theory appears in full clarity with the work of Zermelo in the same decade.
4. The style of internal refrences is as follows; '§ 2.1' indicates section 2.1 with in the same chapter where the refrences is found '§VI.4' indicates section 4 in chapter VI.
5. The criterion of actual infinity justifies to some extent the exclusion of Weierstrass (§ IV. 2); also important in this connection are methodological questions (§1.4 and §IV.2).
6. See Chapter 11 and also [FerrerirΓ³s 1996].
7. See Dedkind [1877] and Cantor's 1885 paper in [Grattan-Guinness 1970].
8. They were many more linguistic variants: 'Klasse,' 'Inbegriff,' 'Gebiet,' 'Complex,' 'Vielheit', 'Gesamtheit', 'Schaar,' and so on. We shall encounter them along the way.
9. See, for example, [Scholz 1990a, 2] and [Spalt 1990, 192-93].
10. It is quite obvious that, in general, the process of invention/discovery will go from the familiar and concrete to the abstract.
Urdu Translation Ψ§ُΨ±Ψ―Ω ΨͺΨ±Ψ¬Ω Ϋ
by Sohail Tahir
πππ
ReplyDeleteπππ
Delete