• Corpus ID: 5869812

Axiomatizing changing conceptions of the geometric continuuum

@inproceedings{Baldwin2014AxiomatizingCC,
  title={Axiomatizing changing conceptions of the geometric continuuum},
  author={John T. Baldwin},
  year={2014}
}
We begin with a general account of the goals of axiomatization, introducing a variant (modest) on Detlefsen’s notion of ‘complete descriptive axiomatization’. We examine the distinctions between the Greek and modern view of number, magnitude and proportion and consider how this impacts the interpretation of Hilbert’s axiomatization of geometry. We argue, as indeed did Hilbert, that Euclid’s propositions concerning polygons, area, and similar triangles are derivable (in their modern… 
Bridging the gap between analytic and synthetic geometry: Hilbert’s axiomatic approach
TLDR
It is argued that Hilbert not only sought to show that geometry could be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the construction of such an axIomatic system one could proceed purely geometrically, avoiding concept formations borrowed from other mathematical disciplines.

References

SHOWING 1-10 OF 63 REFERENCES
Bridging the gap between analytic and synthetic geometry: Hilbert’s axiomatic approach
TLDR
It is argued that Hilbert not only sought to show that geometry could be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the construction of such an axIomatic system one could proceed purely geometrically, avoiding concept formations borrowed from other mathematical disciplines.
From Geometry to Algebra
Our aim is to see which practices of Greek geometry can be expressed in various logics. Thus we refine Detlefsen’s notion of descriptive complexity by providing a scheme of increasing more
Axiomatizations of arithmetic and the first-order/second-order divide
TLDR
It is argued that the first-order versus second-order divide may be too crude to investigate what an adequate axiomatization of arithmetic should look like, and that, insofar as there are different, equally legitimate projects one may engage in when working on the foundations of mathematics, there is no such thing as the One True Logic.
Philosophy of mathematics and deductive structure in Euclid's Elements
Eudoxos and Dedekind: On the ancient Greek theory of ratios and its relation to modern mathematics
According to Aristotle, the objects studied by mathematics have no independent existence, but are separated in thought from the substrate in which they exist, and treated as separable i.e., are
Is There Completeness in Mathematics after Gödel
What do mathematicians do? There are many ways of approaching this question, but one kind of answer is easy to give. Mathematicians study structures of different kinds: for instance, natural numbers,
COMPLETENESS AND CATEGORICITY (IN POWER): FORMALIZATION WITHOUT FOUNDATIONALISM*
  • J. Baldwin
  • Philosophy, Mathematics
    The Bulletin of Symbolic Logic
  • 2014
TLDR
The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics.
The emergence of first-order logic
To most mathematical logicians working in the 1980s, first-order logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician
What is Elementary Geometry
...
1
2
3
4
5
...