Klein, Hilbert, and the Gottingen Mathematical Tradition

Rowe, David E.

doi:10.1086/368687

Cited by 75 publications

(16 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…On the history of Klein -Hilbert partnership see [208]. Klein describes the projective structure as the core geometrical structure, which unifies geometry (or at least its fragment) as described above.…”

Section: Erlangen Program and Axiomatic Methodsmentioning

confidence: 99%

Axiomatic Method and Category Theory

Rodin

2014

Synthese Library

View full text Add to dashboard Cite

PrefaceI first learned about category theory about 20 years ago from Yuri I. Manin's course on algebraic geometry [180] when I was preparing my dissertation on Euclid's Elements and was focused on studying Greek mathematics and classical Greek philosophy. Then I convinced myself that the mathematical category theory is philosophically relevant not only because of its name but also because of its content and because of its special role in the contemporary mathematics, which I privately compared to the role of the notion of figure in Euclid's geometry. Today I have more to say about these matters. The broad historical and philosophical context, in which I studied category theory, is made explicit throughout the present book. My interest to the Axiomatic Method stems from my work on Euclid and extends through Hilbert and axiomatic set theories to Lawvere's axiomatic topos theory to the Univalent Foundations of mathematics recently proposed by Vladimir Voevodsky. This explains what the two subjects appearing in the title of this book share in common.The next crucial biographical episode took place in 1999 when I was a young scholar visiting Columbia University on the Fulbright grant working on ontology of events under the supervision of Achille Varzi. As a part of my Fulbright program I had to make a presentation in a different American university, and I decided to use this opportunity for talking about the philosophical significance of category theory (I cannot now remember how exactly I married then this subject with the event ontology). Achille Varzi kindly arranged for me the invitation from Barry Smith to give a talk at his seminar on formal ontology in the SUNY in Buffalo. When I sent to Barry Smith my abstract he replied that nobody except probably Bill Lawvere will be able to understand my paper, and suggested to make the paper more accessible to the general audience.By that time I had already read some of Lawvere's papers but was wholly unaware about the fact that Lawvere worked in the same university and could attend my planned talk. So I took Smith's words for a joke. When I realized that this was not a joke I was very excited and, as Vladimir ArnoldThe main motivation of writing this book is to develop the view on mathematics described in the above epigraphs. Some 200 years ago this view used to be by far more common and easier to justify than today. It is sufficient to say that it made part of Kant's view on mathematics, and that Kant's view on mathematics remained extremely influential until the very end of the 19th century. When Cassirer defended this Kantian view in the beginning of the 20th century it was already polemical. When Arnold defended it in the end of the 20th century and in the beginning of this current century it already sounded as an intellectual provocation, and so his words sound today. Kant, Cassirer and Arnold do not speak about the same mathematics: each speaks about mathematics of his own time. So the growing polemical attitude to their shared view reflects not only a change of...

show abstract

Section: Erlangen Program and Axiomatic Methodsmentioning

confidence: 99%

Axiomatic Method and Category Theory

Rodin

2014

Synthese Library

View full text Add to dashboard Cite

show abstract

“…One could even go further and speculate whether the shift from an integrative standard of mathematical rationality as expressed by Klein toward a differentiative standard as implicit in (at least the reception of) Hilbert's Grundlagen der Geometrie reflects the general trend of modern culture toward differentiation noticed by Max Weber and others. For further information on Klein, Hilbert and general matters, I refer to work by Herbert Mehrtens [52] and David Rowe [57]. It will be clear that my perspective on the developments presented in this article owes much to both of them, a debt which I gratefully acknowledge.…”

Section: Wirtinger Dehn and The Memory Of The Mathematical Communitymentioning

confidence: 93%

Branch Points of Algebraic Functions and the Beginnings of Modern Knot Theory

Epple

1995

Historia Mathematica

View full text Add to dashboard Cite

Many of the key ideas which formed modern topology grew out of "normal research" in one of the mainstream fields of 19th-century mathematical thinking, the theory of complex algebraic functions. These ideas were eventually divorced from their original context. The present study discusses an example illustrating this process. During the years 1895-1905, the Austrian mathematician, Wilhelm Wirtinger, tried to generalize Felix Klein's view of algebraic functions to the case of several variables. An investigation of the monodromy behavior of such functions in the neighborhood of singular points led to the first computation of a knot group. Modern knot theory was then formed after a shift in mathematical perspective took place regarding the types of problems investigated by Wirtinger, resulting in an elimination of the context of algebraic functions. This shift, clearly visible in Max Dehn's pioneering work on knot theory, was related to a deeper change in the normative horizon of mathematical practice which brought about mathematical modernity. © 1996 Academic Press, Inc.Viele der Ideen, die die moderne Topologie geprigt haben, stammen aus 'normaler Forschung' in einer der Hauptstr6mungen des mathematischen Denkens des 19. Jahrhunderts, der Theorie komplexer algebraischer Funktionen, und wurden erst allmihlich yon ihrem urspringlichen Kontext abgelOst. Ein Beispiel dieser Entwicklung wird diskutiert. In den Jahren [1895][1896][1897][1898][1899][1900][1901][1902][1903][1904][1905] INTRODUCTIONDue to the rapid development and application of new knot invariants in mathematics and physics following Vaughan Jones' discovery of a new knot polynomial, knot theory has received growing attention within and even outside the mathematical community. 2 In this context, it has often been asked why the knot problem--of all topological problems--was among the first to be studied by early topologists of our century such as Heinrich Tietze, Max Dehn, James W. Alexander, and Kurt Reidemeister. This question appears all the more puzzling since 19-century work on knots had certainly not been at the cutting edge of mainstream mathematical research--unlike, for instance, the topological problems that arose in connection with the theory of algebraic functions or algebraic geometry. In the following, an answer to this question will be given. Using hitherto unpublished correspondence between the Austrian mathematician, Wilhelm Wirtinger, and Felix Klein, it will be shown that modern knot theory did in fact originate from these latter fields. Furthermore, while they were familiar to the pioneers of modern topology, a series of events gradually left these roots forgotten by their followers.In considering the origins of modern knot theory, I will do more than merely retrace the technical developments. Rather, this formation of a new field of mathematical research illustrates a certain pattern which, in a nutshell, may be characterized as follows:Thesis. What appears, at first sight, to be the invention of a new mathematical discipline, turns o...

show abstract

“…Then, according to the picture, the philosophers and philosophically-minded mathematicians scrambled to find the proper philosophical interpretation of these results and produce logical duplicates: thus we had the work of Frege, Russell, and so on. 56 This caricature is so crude that perhaps no one would accept it exactly as it stands. But it is close enough to attitudes that are occasionally encountered that a warning can be seen to have some point: it should not be assumed that the mathematical work Frege was discussing was mathematically above reproach.…”

Section: Possibility Iii-the Jourdain Sentence Rejects the 1906 Test mentioning

confidence: 99%

Frege on Axioms, Indirect Proof, and Independence Arguments in Geometry: Did Frege Reject Independence Arguments?

Tappenden¹

2000

Notre Dame J. Formal Logic

View full text Add to dashboard Cite

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege's mathematical environment. This feeds into a currently active scholarly debate because Frege's supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege's attitude toward metatheory in general. I show that this thesis gains no support from Frege's puzzling remarks about independence arguments.

show abstract

Klein, Hilbert, and the Gottingen Mathematical Tradition

Cited by 75 publications

References 0 publications

Axiomatic Method and Category Theory

Axiomatic Method and Category Theory

Branch Points of Algebraic Functions and the Beginnings of Modern Knot Theory

Frege on Axioms, Indirect Proof, and Independence Arguments in Geometry: Did Frege Reject Independence Arguments?

Contact Info

Product

Resources

About