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Three Arguments Against Mathematical Constructivism

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Chlastawa D.
Filozofia Nauki
|
2010
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vol. 18
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issue 4
77-95
PL
This paper contains a criticism of mathematical constructivism, i.e. the class of views in the philosophy and foundations of mathematics according to which only constructive notions and methods of proof should be allowed in mathematics. Three main arguments are deployed against such view and its philosophical background. Firstly, an argument from pluralism: constructivism often appeals to intuitive evidence as the root of mathematics, effectively excluding large parts of classical, ab-stract mathematics. But appeals to «intuition» are utterly subjective and unstable, which results in multitude of incompatible constructivist systems of mathematics and makes any criticism toward classical mathematics as «non-constructive» unsubstanti-ated. Secondly, an argument which shows that epistemological arguments, deployed by many constructivists against intelligibility of classical mathematics, are unsound, and moreover, consistent appeal to such arguments leaves constructivists in no position to avoid the menace of ultrafinitism. Thirdly, it is argued that constructivism faces a dilemma whether to consider mathematical truth as what is actually proved, or what is provable, and that this dilemma is unsolvable in a satisfactory way, because the first horn of the dilemma is highly counter-intuitive or absurd, and the second one is impossible to square with constructivist views.
2
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Bertrand Russell and Universals

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Chlastawa D.
Filozofia Nauki
|
2011
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vol. 19
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issue 3
127-149
PL
Bertrand Russell paid considerable attention to the problem of universals throughout his long life. One of main factors which contributed to Russell’s rejection of Hegelian philosophy (which is commonly viewed as a beginning of analytic philosophy) was rejection of so-called internal relations theory, according to which relations reduce to properties of relata or of the whole composed of them. For Russell relations were examples of indispensable universals. Russell is also famous for developing the similarity argument for realism: if we want to get rid of universals by reducing them to sets of objects similar in a certain respect, we have to accept similarity as a genuine universal, for otherwise we are threatened by a vicious regress. The paper contains a presentation of evolution of Russell’s thought regarding universals and a defense of similarity argument against criticism of Michał Hempoliński, according to which causal explanations are sufficient to explain the similarity of objects. It is argued that such explanations are insufficient, as they do not apply to, for example, fundamental particles, like electrons, and the view that there are no fundamental (indivisible) particles is threatened by another vicious regress.
3
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Uwagi krytyczne o Kazimierza Ajdukiewicza semantycznej krytyce idealizmu subiektywnego

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Chlastawa D.
Analiza i Egzystencja. Czasopismo Filozoficzne
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2012
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issue nr 20
15-36
EN
The paper contains objections to Kazimierz Ajdukiewicz’s semantic argumentagainst subjective idealism of George Berkeley, formulated in Ajdukiewicz’s articleEpistemology and Semiotics. It is argued that this argument is question-begging,as it presupposes a realistic conception of intentionality. It is analyzed whetherAjdukiewicz’s argument against idealism fromOn the Notion of Existencecould beused to support the former reasoning, and I argue that it could not. I also evaluateRoman Suszko’s modification of Ajdukiewicz argument and show that this modifi-cation is superficial, as it lays on the very same question-begging assumption thatAjdukiewicz made. In the conclusion I attempt a general defense of Ajdukiewicz’sphilosophical method, regardless of its misapplications.
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Modalny status zdań matematycznych

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Chlastawa D.
Diametros
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2011
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issue 30
2-12
PL
W artykule przedstawione i analizowane są trzy kontrprzykłady dla tezy, że wszystkie zdania matematyczne są konieczne, tzn. koniecznie prawdziwe lub koniecznie fałszywe: argument z przygodnych relacyjnych własności empirycznych, argument z własności wynikających z konwencjonalnych reprezentacji i argument z relatywizacji do modelu. Pierwsze dwa argumenty poddają się łatwemu podważeniu, do zakwestionowania trzeciego natomiast potrzebne jest przyjęcie dość silnego stanowiska realistycznego w teorii mnogości.
EN
The paper contains an analysis of three counterexamples to the view that all mathematical statements are necessary, i.e. necessarily true or necessarily false: an argument from contingent, relational, empirical properties, an argument from properties based on conventional representations and an argument from model relativity. The first and the second argument can be rejected easily, while to answer the third argument one has to adopt a quite strong set-theoretic realism.
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Czy konceptualizm jest wystarczającą podstawą dla odrzucenia niekonstruktywnych dowodów istnienia w matematyce?

100%
Chlastawa D.
Zagadnienia Filozoficzne w Nauce
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2012
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issue 51
116-130
PL
Non-constructive existence proofs (which prove the existence of mathematical objects of a certain kind without giving any particular examples of such objects) are rejected by constructivists, who hold a conceptualist view that mathematical objects exist only if they are constructed. In the paper it is argued that this conceptualist argument against non-constructive proofs is fallacious, because those proofs establish the existence of objects belonging to certain kinds rather than the existence of those objects per se. Moreover, to engage in proving existence theorems in a given mathematical theory one has to define all of the objects of this theory at the very beginning, which can be interpreted as establishing the existence of these objects before any theorem about them is proven. It is also argued that the constructivist may escape these objections by adopting the actualistic view, according to which a mathematical sentence is true if and only if it is established as true, but this view is very implausible, as it seems unable to explain the strictness and objectiveness of mathematics and the fact that it differs so fundamentally from, for example, fictional discourse.
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