scholarly journals Quasi-Empirical Fictionalism as an Approach to the Philosophy of Geometry

2021 ◽  
Author(s):  
◽  
Scott Waygood
Keyword(s):  
Projective Geometry ◽  
Predictive Power ◽  
Philosophy Of Mathematics ◽  
Line Drawing ◽  
Physical Reality ◽  
Empirical Science ◽  
Primitive Operation ◽  
Straight Lines ◽  
Final Question ◽  
Central Claim

<p>The central claim of this thesis is that geometry is a quasi-empirical science based on the idealisation of the elementary physical operations that we actually perform with pen and paper. This conclusion is arrived at after searching for a theory of geometry that will not only explain the epistemology and ontology of mathematics, but will also fit with the best practices of working mathematicians and, more importantly, explain why geometry gives us knowledge that is relevant to physical reality. We will be considering all the major schools of thought in the philosophy of mathematics. Firstly, from the epistemological side, we will consider apriorism, empiricism and quasi-empiricism, finding a Kitcherian style of quasi-empiricism to be the most attractive. Then, from the ontological side, we will consider Platonism, formalism, Kitcherian ontology, and fictionalism. Our conclusion will be to take a Kitcherian epistemology and a fictionalist ontology. This will give us a kind of quasiempirical-fictionalist approach to mathematics. The key feature of Kitcher's thesis is that he placed importance on the operations rather than the entities of arithmetic. However, because he only dealt with arithmetic, we are left with the task of developing a theory of geometry along Kitcherian lines. I will present a theory of geometry that parallels Kitcher's theory of arithmetic using the drawing of straight lines as the most primitive operation. We will thereby develop a theory of geometry that is founded upon our operations of drawing lines. Because this theory is based on our line drawing operations carried out in physical reality, and is the idealisation of those activities, we will have a connection between mathematical geometry and physical reality that explains the predictive power of geometry in the real world. Where Kitcher uses the Peano postulates to develop his theory of arithmetic, I will use the postulates of projective geometry to form the foundations of operational geometry. The reason for choosing projective geometry is due to the fact that by taking it as the foundation, we may apply Klein's Erlanger programme and build a theory of geometry that encompasses Euclidean, hyperbolic and elliptic geometries. The final question we will consider is the problem of conventionalism. We will discover that investigations into conventionalism give us further reason to accept the Kitcherian quasi-empirical-fictionalist approach as the most appealing philosophy of geometry available.</p>

scholarly journals Quasi-Empirical Fictionalism as an Approach to the Philosophy of Geometry

2021 ◽  
Author(s):  
◽  
Scott Waygood
Keyword(s):  
Projective Geometry ◽  
Predictive Power ◽  
Philosophy Of Mathematics ◽  
Line Drawing ◽  
Physical Reality ◽  
Empirical Science ◽  
Primitive Operation ◽  
Straight Lines ◽  
Final Question ◽  
Central Claim

<p>The central claim of this thesis is that geometry is a quasi-empirical science based on the idealisation of the elementary physical operations that we actually perform with pen and paper. This conclusion is arrived at after searching for a theory of geometry that will not only explain the epistemology and ontology of mathematics, but will also fit with the best practices of working mathematicians and, more importantly, explain why geometry gives us knowledge that is relevant to physical reality. We will be considering all the major schools of thought in the philosophy of mathematics. Firstly, from the epistemological side, we will consider apriorism, empiricism and quasi-empiricism, finding a Kitcherian style of quasi-empiricism to be the most attractive. Then, from the ontological side, we will consider Platonism, formalism, Kitcherian ontology, and fictionalism. Our conclusion will be to take a Kitcherian epistemology and a fictionalist ontology. This will give us a kind of quasiempirical-fictionalist approach to mathematics. The key feature of Kitcher's thesis is that he placed importance on the operations rather than the entities of arithmetic. However, because he only dealt with arithmetic, we are left with the task of developing a theory of geometry along Kitcherian lines. I will present a theory of geometry that parallels Kitcher's theory of arithmetic using the drawing of straight lines as the most primitive operation. We will thereby develop a theory of geometry that is founded upon our operations of drawing lines. Because this theory is based on our line drawing operations carried out in physical reality, and is the idealisation of those activities, we will have a connection between mathematical geometry and physical reality that explains the predictive power of geometry in the real world. Where Kitcher uses the Peano postulates to develop his theory of arithmetic, I will use the postulates of projective geometry to form the foundations of operational geometry. The reason for choosing projective geometry is due to the fact that by taking it as the foundation, we may apply Klein's Erlanger programme and build a theory of geometry that encompasses Euclidean, hyperbolic and elliptic geometries. The final question we will consider is the problem of conventionalism. We will discover that investigations into conventionalism give us further reason to accept the Kitcherian quasi-empirical-fictionalist approach as the most appealing philosophy of geometry available.</p>

An Easy Proof for Some Classical Theorems in Plane Geometry

1992 ◽  
Vol 35 (4) ◽  
pp. 560-568 ◽  
Author(s):  
C. Thas
Keyword(s):  
Projective Plane ◽  
Projective Geometry ◽  
The Other ◽  
Plane Geometry ◽  
The Real ◽  
Easy Proof ◽  
Special Cases ◽  
Euclidean Planes ◽  
Straight Lines ◽  
Short Method

AbstractThe main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

scholarly journals Determination of the characteristic parameters in the special collinear space in the general case

2007 ◽  
Vol 5 (1) ◽  
pp. 49-59
Author(s):  
Sonja Krasic ◽  
Miroslav Markovic
Keyword(s):  
Projective Space ◽  
Projective Geometry ◽  
Descriptive Geometry ◽  
Characteristic Parameters ◽  
Straight Lines ◽  
Structural Methods

The projective space consists of the finitely and infinitely distant elements. The special collinear spaces in the general case, are set with five pairs of biunivocally associated points, so the quadrangle in the first space obtained by the three principal and one penetration point of the remaining two through the plane of the first three identical or similar to the associated quadrangle obtained in the same way in the second space. In order to associate two special collinear spaces, it is necessary to determine the following characteristic parameters: vanishing planes, space axes (principal normal lines), foci (apexes of the associated identical bundles of straight lines) and directrix plane (associated identical fields of points). The paper is based on constructive processing of the special collinear spaces in the general case. The structural methods which are used are Descriptive Geometry (a pair of Mange's projections) and Projective geometry.

Antirealism in the philosophy of mathematics

2018 ◽  
Author(s):  
A.W. Moore
Keyword(s):  
Philosophy Of Mathematics ◽  
Physical Reality ◽  
Mathematical Statement ◽  
Mathematical Objects ◽  
Mathematical Reality

Realism in the philosophy of mathematics is the position that takes mathematics at face value. According to realists, mathematics is the science of mathematical objects (numbers, sets, lines and so on); mathematicians, to use the old metaphor, are discoverers, not inventors. Moreover, just as there may be truths about physical reality which we can never know, so too, realists say, there may be truths about mathematical reality which we can never know. It is this claim in particular which antirealists find unacceptable. Equating what can be known in mathematics with what can be proved, they insist that only what can be proved is true. (Only what can be proved: different accounts of what this ‘can’ means, facing different difficulties, generate different positions.) This leads antirealists to recoil not only from realism but also from the practice of mathematicians themselves. For the orthodox assumption that every mathematical statement is either true or false would be invalidated, on the antirealist view, by a statement that was neither provable nor disprovable. Not that antirealists themselves can see it in these terms. For if a statement were neither provable nor disprovable, that would itself be an unprovable truth about mathematical reality. Antirealists must learn how to be circumspect even in defence of their own circumspection.

scholarly journals Machines, Logic and Quantum Physics

2000 ◽  
Vol 6 (3) ◽  
pp. 265-283 ◽  
Author(s):  
David Deutsch ◽  
Artur Ekert ◽  
Rossella Lupacchini
Keyword(s):  
Quantum Physics ◽  
A Priori ◽  
Physical Reality ◽  
Physical World ◽  
Empirical Science ◽  
Physical Sciences ◽  
Mathematical Structures ◽  
Eugene Wigner ◽  
The Universe ◽  
Modern Status

§1. Mathematics and the physical world. Genuine scientific knowledge cannot be certain, nor can it be justified a priori. Instead, it must be conjectured, and then tested by experiment, and this requires it to be expressed in a language appropriate for making precise, empirically testable predictions. That language is mathematics.This in turn constitutes a statement about what the physical world must be like if science, thus conceived, is to be possible. As Galileo put it, “the universe is written in the language of mathematics”. Galileo's introduction of mathematically formulated, testable theories into physics marked the transition from the Aristotelian conception of physics, resting on supposedly necessary a priori principles, to its modern status as a theoretical, conjectural and empirical science. Instead of seeking an infallible universal mathematical design, Galilean science usesmathematics to express quantitative descriptions of an objective physical reality. Thus mathematics became the language in which we express our knowledge of the physical world — a language that is not only extraordinarily powerful and precise, but also effective in practice. Eugene Wigner referred to “the unreasonable effectiveness of mathematics in the physical sciences”. But is this effectiveness really unreasonable or miraculous?Numbers, sets, groups and algebras have an autonomous reality quite independent of what the laws of physics decree, and the properties of these mathematical structures can be just as objective as Plato believed they were (and as Roger Penrose now advocates).

scholarly journals Una teoría combinatoria de las representaciones científicas

2000 ◽  
Vol 32 (95) ◽  
pp. 3-46
Author(s):  
Andoni Ibarra ◽  
Thomas Mormann
Keyword(s):  
Category Theory ◽  
Theory Of Measurement ◽  
Common Type ◽  
Combinatorial Theory ◽  
Empirical Science ◽  
Scientific Representation ◽  
Structure Preserving ◽  
Scientific Representations ◽  
Central Claim ◽  
Natural Way

The aim of this paper is to introduce a new concept of scientific representation into philosophy of science. The new concept -to be called homological or functorial representation- is a genuine generalization of the received notion of representation as a structure preserving map as it is used, for example, in the representational theory of measurement. It may be traced back, at least implicitly, to the works of Hertz and Duhem. A modern elaboration may be found in the foundational discipline of mathematical category theory. In contrast to the familiar concepts of representations, functorial representations do not depend on any notion of similarity, neither structural nor objectual one. Rather, functorial representation establish correlations between the structures of the representing and the represented domains. Thus, they may be said to form a class of quite "non-isomorphic" representations. Nevertheless, and this is the central claim of this paper, they are the most common type of representations used in science. In our paper we give some examples from mathematics and empirical science. One of the most interesting features of the new concept is that it leads in a natural way to a combinatorial theory of scientific representations, i.e. homological or functorial representations do not live in insulation, rather, they may be combined and connected in various ways thereby forming a net of interrelated representations. One of the most important tasks of a theory of scientific representations is to describe this realm of combinatorial possibilities in detail. Some first tentative steps towards this endeavour are done in our paper.

A numerical disproof of a conjecture in projective geometry

1971 ◽  
Vol 323 (1555) ◽  
pp. 443-454 ◽  
Keyword(s):  
Projective Geometry ◽  
Three Dimensions ◽  
Lines In Space ◽  
Straight Lines

A longstanding conjecture concerning the intersection of straight lines in space of three dimensions is disproved by a counterexample. At the same time a related conjecture due to Babbage is confirmed.

The Issue of Experiment in Mathematics

1998 ◽  
pp. 13-16
Author(s):  
Anna Lemanska
Keyword(s):  
A Priori ◽  
Philosophy Of Mathematics ◽  
Empirical Science ◽  
Experimental Science ◽  
Fractal Sets ◽  
The Everyday ◽  
Everyday Work ◽  
The Status ◽  
Quasi Experimental ◽  
Deductive Science

The issue of the status of mathematical knowledge a priori or a posteriori has been repeatedly considered by the philosophy of mathematics. At present, the development of computer technology and their enhancement of the everyday work of mathematicians have set a new light on the problem. It seems that a computer performs two main functions in mathematics: it carries out numerical calculations and it presents new areas of research. Thanks to cooperation with the computer, a mathematician can gather different data and facts concerning the issue of interest. Moreover, he or she can carry out different "tests" with the aid of a computer. For instance, one can study strange attractors, chaotic dynamics, and fractal sets. By this we may talk about a specific experimentation in mathematics. The use of this kind of testing in mathematical research results in describing it as an experimental science. The goal of the paper is to attempt to answer the questions: does mathematics really transform into experimental or quasi-experimental science and does mathematics vary from axiomatic-deductive science into empirical science?

Sign in / Sign up

Export Citation Format

Share Document