排序方式: 共有5条查询结果,搜索用时 0 毫秒
1
1.
In the first two decades of the century Vvedenskij developed and defended what he took to be an original argument in support of the impossibility of metaphysical knowledge. This argument, which he hailed as a proof, involved an examination of the four laws of thought alone. As it made no appeal to the highly technical analyses found in Kant's first Critique, Vvedenskij considered it to be more efficient and thereby effective than Kant's own arguments. Although Vvedenskij's estimation of his accomplishment actually increased with the passage of time, the proof rested on highly dubious assumptions. 相似文献
2.
Markus Schmitz 《Journal for General Philosophy of Science》2001,32(2):271-305
The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between
Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for
construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of
the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such a definition is found in the arithmetic in Euclid's Elements. To draw this conclusion a profound reflection on the role of epistemology for the foundation of mathematics, especially
for the method of definition of number, is indispensable; a reflection not to be found in the contemporary debate (the predominate
‘pragmaticformalism’ in current mathematics just shirks from trying to solve the epistemological problems raised by the debate
between logicism, intuitionism, and formalism). Frege's definition of number, ‘The number of the concept F is the extension
of the concept ‘numerically equal to the concept F”, which is still substantial for contemporary mathematics, does not fulfil
the requirements of logical and methodological correctness because the definiens in a double way (in the concepts ‘extension
of a concept’ and ‘numerically equal’) implicitly presupposes the definiendum, i.e. number itself. Number itself, on the contrary,
is defined adequately by Euclid as ‘multitude composed of units’, a definition which is even, though never mentioned, an implicit
presupposition of the modern concept ofset. But Frege rejects this definition and construes his own - for epistemological
reasons: Frege's definition exactly fits the needs of modern epistemology, namely that for to know something like the number
of a concept one must become conscious of a multitude of acts of producing units of ‘given’ representations under the condition
of a 1:1 relationship to obtain between the acts of counting and the counted ‘objects’. According to this view, which has
existed at least since the Renaissance stoicism and is maintained not only by Frege but also by Descartes, Kant, Husserl,
Dummett, and others, there is no such thing as a number of pure units itself because the intellect or pure reason, by itself
empty, must become conscious of different units of representation in order to know a multitude, a condition not fulfilled by Euclid's conception. As this is Frege's
main reason to reject Euclid's definition of number (others are discussed in detail), the paper shows that the epistemological
reflection in Neoplatonic mathematical philosophy, which agrees with Euclid's definition of number, provides a consistent
basement for it. Therefore it is not progress in the history of science which hasled to the a poretic contemporary state of
affairs but an arbitrary change of epistemology in early modern times, which is of great influence even today.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
3.
4.
Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator
governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the
Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true
of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in
this extended system. 相似文献
5.
For each regular cardinal , we set up three systems of infinitary type logic, in which the length of the types and the length of the typed syntactical constructs are < . For a fixed , these three versions are, in the order of increasing strength: the local system (), the global system g() (the difference concerns the conditions on eigenvariables) and the -system () (which has anti-selection terms or Hilbertian -terms, and no conditions on eigenvariables). A full cut elimination theorem is proved for the local systems, and about the -systems we prove that they admit cut-free proofs for sequents in the -free language common to the local and global systems. These two results follow from semantic completeness proofs. Thus every sequent provable in a global system has a cut-free proof in the corresponding -systems. It is, however, an open question whether the global systems in themselves admit cut elimination. 相似文献
1