79 research outputs found
Doing and Showing
The persisting gap between the formal and the informal mathematics is due to
an inadequate notion of mathematical theory behind the current formalization
techniques. I mean the (informal) notion of axiomatic theory according to which
a mathematical theory consists of a set of axioms and further theorems deduced
from these axioms according to certain rules of logical inference. Thus the
usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure
On Constructive Axiomatic Method
In this last version of the paper one may find a critical overview of some
recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure
Did Lobachevsky Have A Model Of His "imaginary Geometry"?
The invention of non-Euclidean geometries is often seen through the optics of
Hilbertian formal axiomatic method developed later in the 19th century. However
such an anachronistic approach fails to provide a sound reading of
Lobachevsky's geometrical works. Although the modern notion of model of a given
theory has a counterpart in Lobachevsky's writings its role in Lobachevsky's
geometrical theory turns to be very unusual. Lobachevsky doesn't consider
various models of Hyperbolic geometry, as the modern reader would expect, but
uses a non-standard model of Euclidean plane (as a particular surface in the
Hyperbolic 3-space). In this paper I consider this Lobachevsky's construction,
and show how it can be better analyzed within an alternative non-Hilbertian
foundational framework, which relates the history of geometry of the 19th
century to some recent developments in the field.Comment: 31 pages, 8 figure
Categories without structures
The popular view according to which Category theory provides a support for
Mathematical Structuralism is erroneous. Category-theoretic foundations of
mathematics require a different philosophy of mathematics. While structural
mathematics studies invariant forms (Awodey) categorical mathematics studies
covariant transformations which, generally, don t have any invariants. In this
paper I develop a non-structuralist interpretation of categorical mathematics
and show its consequences for history of mathematics and mathematics education.Comment: 28 page
On Constructive Axiomatic Method
The formal axiomatic method popularized by Hilbert and recently defended by Hintikka is not fully adequate to the recent practice of axiomatizing mathematical theories. The axiomatic architecture of Topos theory and Homotopy type theory do not fit the pattern of the formal axiomatic theory in the standard sense of the word. However these theories fall under a more general and in some respects more traditional notion of axiomatic theory, which I call after Hilbert constructive. I show that the formal axiomatic method always requires a support of some more basic constructive method
Models of HoTT and the Constructive View of Theories
Homotopy Type theory and its Model theory provide a novel formal semantic framework for representing scientific theories. This framework supports a constructive view of theories according to which a theory is essentially characterised by its methods.
The constructive view of theories was earlier defended by Ernest Nagel and a number of other philosophers of the past but available logical means did not allow these people to build formal representational frameworks that implement this view
Univalent Foundations and the Constructive View of Theories
Univalent Foundations and the Constructive View of Theorie
On Categorical Theory-Building: Beyond the Formal
I propose a notion of theory motivated by Category theory.Comment: 28 pages, no image
Venus Homotopically
The identity concept developed in the Homotopy Type theory (HoTT) supports an analysis of Frege's famous Venus example, which explains how empirical evidences justify judgements about identities. In the context of this analysis we consider the traditional distinction between the extension and the intension of concepts as it appears in HoTT, discuss an ontological significance of this distinction and, finally, provide a homotopical reconstruction of a basic kinematic scheme, which is used in the Classical Mechanics, and discuss its relevance in the Quantum Mechanics
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