Cover of The Paideia Archive: Twentieth World Congress of Philosophy
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1. The Paideia Archive: Twentieth World Congress of Philosophy: Volume > 34
Maciej Gos

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The general theory of relativity and field theory of matter generate an interesting ontology of space-time and, generally, of nature. It is a monistic, anti-atomistic and geometrized ontology — in which the substance is the metric field — to which all physical events are reducible. Such ontology refers to the Cartesian definition of corporeality and to Plato's ontology of nature presented in the Timaeus. This ontology provides a solution to the dispute between Clark and Leibniz on the issue of the ontological independence of space-time from distribution of events. However, mathematical models of space-time in physics do not solve the problem of the difference between time and space dimensions (invariance of equations with regard to the inversion of time arrow). Recent research on space-time singularities and asymmetrical in time quantum theory of gravitation will perhaps allow for the solution of this problem based on the structure of space-time and not merely on thermodynamics.

2. The Paideia Archive: Twentieth World Congress of Philosophy: Volume > 34
Elaine Landry

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I argue that if we distinguish between ontological realism and semantic realism, then we no longer have to choose between platonism and formalism. If we take category theory as the language of mathematics, then a linguistic analysis of the content and structure of what we say in and about mathematical theories allows us to justify the inclusion of mathematical concepts and theories as legitimate objects of philosophical study. Insofar as this analysis relies on a distinction between ontological and semantic realism, it relies also on an implicit distinction between mathematics as a descriptive science and mathematics as a descriptive discourse. It is this latter distinction which gives rise to the tension between the mathematician qua philosopher. In conclusion, I argue that the tensions between formalism and platonism, indeed between mathematician and philosopher, arise because of an assumption that there is an analogy between mathematical talk and talk in the physical sciences.

3. The Paideia Archive: Twentieth World Congress of Philosophy: Volume > 34
Anna Lemanska

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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?

4. The Paideia Archive: Twentieth World Congress of Philosophy: Volume > 34
Jarosław Mrozek

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This paper is an attempt to explain the structure of the process of understanding mathematical objects such as notions, definitions, theorems, or mathematical theories. Understanding is an indirect process of cognition which consists in grasping the sense of what is to be understood, showing itself in the ability to apply what is understood in other circumstances and situations. Thus understanding should be treated functionally: as acquiring sense. We can distinguish three basic planes on which the process of understanding mathematics takes place. The first is the plane of understanding the meaning of notions and terms existing in mathematical considerations. A mathematician must have the knowledge of what the given symbols mean and what the corresponding notions denote. On the second plane, understanding concerns the structure of the object of understanding wherein it is the sense of the sequences of the applied notions and terms that is important. The third plane-understanding the 'role' of the object of understanding-consists in fixing the sense of the object of understanding in the context of a greater entity, i.e., it is an investigation of the background of the problem. Additionally, understanding mathematics, to be sufficiently comprehensive, should take into account (apart from the theoretical planes) at least three other connected considerations-historical, methodological and philosophical-as ignoring them results in a superficial and incomplete understanding of mathematics.

5. The Paideia Archive: Twentieth World Congress of Philosophy: Volume > 34
Jarmo Pulkkinen

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The publication of Russell's The Principles of Mathematics (1903) and Couturat's Les principes des mathematiques (1905) incited several prominent neo-Kantians to make up their mind about the logicist program. In this paper, I shall discuss the critiques presented by the following neo-Kantians: Paul Natorp, Ernst Cassirer and Jonas Cohn. They argued that Russell's attempt to deduce the number concept from the class concept is a petitio principii. Russell replied that the sense in which every object is 'one' must be distinguished from the sense in which 'one' is a number. I claim that Russell was wrong in dismissing the neo-Kantian argument as an elementary logical error. To accept Russell's distinction would be to accept at least part of Russell's logicist program. The expression 'a class with one member' would presuppose the number 'one' only if one simultaneously accepted the analysis which mathematical logic provides for it (the class u has one member when u is not null and 'x and y are us' implies 'x and y are identical'). My point is that the aforementioned analysis provided by mathematical logic was something that the neo-Kantians were not ready to accept.

6. The Paideia Archive: Twentieth World Congress of Philosophy: Volume > 34
Krassimir D. Tarkalanov

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Russell and Popper are concordant with Plato with respect to the independence of mathematics upon the sensations. Beth shares the opinion of the complete independence between the world of science and mathematics and that of psychology. Essenin-Vol'pin's opinion is of an ascendance of ethics and jurisprudence over mathematics. For the first time, the position of Plato, Russell, and Popper are substantiated in this paper through Hegel's reflexive natural scientific method. The external activation of numbers into interaction through arithmetical operations, adopted by him, has been taken as a basis of this substantion. This is the reason why mathematical rules of reasoning are exact-they represent a pure product of the 'third world.' The rules of ethics and the related humanities are their reflective approximate reverberations. Ascendancy of the rules of such types of science over mathematics is impossible due to the irreversibility of the reflexion.

7. The Paideia Archive: Twentieth World Congress of Philosophy: Volume > 34
Halil Turan

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The view that Descartes called mathematical propositions into doubt as he impugned all beliefs concerning common-sense ontology by assuming that all beliefs derive from perception seems to rest on the presupposition that the Cartesian problem of doubt concerning mathematics is an instance of the problem of doubt concerning existence of substances. I argue that the problem is not 'whether I am counting actual objects or empty images,' but 'whether I am counting what I count correctly.' Considering Descartes's early works, it is possible to see that for him, the proposition '2+3=5' and the argument 'I think, therefore I am,' were equally evident. But Descartes does not found his epistemology upon the evidence of mathematical propositions. The doubt experiment does not seem to give positive results for mathematical operations. Consciousness of carrying out a mathematical proposition, however, unlike putting forth a result of an operation, is immune to doubt. Statements of consciousness of mathematical or logical operations are instances of 'I think' and hence the argument 'I count, therefore I am' is equivalent to 'I think, therefore I am.' If impugning the veridicality of mathematical propositions could not pose a difficulty for Descartes's epistemology which he thought to establish on consciousness of thinking alone, then he cannot be seen to avoid the question. Discarding mathematical propositions themselves on the grounds that they are not immune to doubt evoked by a powerful agent does not generate a substantial problem for Descartes provided that he believes that he can justify them by appeal to God's benevolence.

8. The Paideia Archive: Twentieth World Congress of Philosophy: Volume > 34
Alan Weir

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I outline a variant on the formalist approach to mathematics which rejects textbook formalism's highly counterintuitive denial that mathematical theorems express truths while still avoiding ontological commitment to a realm of abstract objects. The key idea is to distinguish the sense of a sentence from its explanatory truth conditions. I then look at various problems with the neo-formalist approach, in particular at the status of the notion of proof in a formal calculus and at problems which Gödelian results seem to pose for the tight link assumed between truth and proof.

9. The Paideia Archive: Twentieth World Congress of Philosophy: Volume > 34
Roger Wertheimer

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If logical truth is truth due solely to syntactic form, then mathematics is distinct from logic, even if all mathematical truths are derivable from definitions and logical principles. This is often obscured by the plausibility of the Synonymy Substitution Principle that is implicit in the Fregean conception of analyticity: viz., that synonyms are intersubstitutable without altering sentence sense. Now, unlike logical truth, mathematical truth is not due to syntax, so synonym interchange in mathematical truths preserves sentence syntax, sense, and mathematical necessity. Mathematical necessity, therefore, differs from both logical and lexical necessity.