Università degli Studi di Fiume, Facoltà di Lettere e Filosofia , 2019
Il libro è stato ideato come libro di testo per gli studenti e i futuri docenti, ma anche come te... more Il libro è stato ideato come libro di testo per gli studenti e i futuri docenti, ma anche come testo di consultazione per chi già insegna a scuola o si interessa all’argomento. Nella prima parte del manuale vengono introdotti i concetti fondamentali del ragionamento critico appartenenti al nucleo logico-argomentativo, mentre nella seconda parte vengono discussi gli elementi di pragmatica: la sistematicità, la capacità comunicativa e l’apertura o elasticità mentale. I concetti trattati sono stati introdotti senza richiedere al lettore nessuna conoscenza preliminare.
International Studies in The Philosophy of Science, Dec 1, 2011
The problem of proving the objective truth of mathematical statements and the objective existence... more The problem of proving the objective truth of mathematical statements and the objective existence of mathematical objects is one of the most compelling in philosophy of mathematics. There are several arguments that, more or less successfully or convincingly, offer reasons for believing in the (objective) existence of the mathematical realm. Of course, there are dissenting voices. A new and interesting one is the voice of Mary Leng in her thought-provoking book, Mathematics and Reality. In the book she concentrates her critical attention on the Quine–Putnam indispensability argument—the argument for the objective existence of mathematical items that many, friends and foes, have labelled the best argument for Platonism. Leng critically presents the argument and offers reasons for rejecting it by mainly questioning one of its premises, labelled confirmational holism. The book is structured in ten chapters. In the first chapters (1–4), Leng nicely introduces the subject. After presenting the Quine–Putnam argument itself, she goes on critically to address different replies to it and to the discussions as they have been developed since. After careful and exhaustive presentation of the argument, in the remaining chapters (5–10), Leng concentrates on confirmational holism, one of the premises in the argument, questioning it with the contrary idea to the effect that, notwithstanding the indispensability of mathematics to our best scientific theories, we are not forced to believe the literal truth of the mathematical apparatus indispensable to formulating such theory. In her own words: ‘we can account for our reasons to include mathematical hypotheses in our scientific theories without assuming that we ought to believe that these hypotheses are true’ (102). In this review I shall mainly concentrate upon the contested argument. So let us have a look at it in more detail. The indispensability argument is the argument for the acceptance of the literal truth of mathematical statements and the existence of mathematical objects. It can be formulated in the following way:
In this paper we shall concentrate on the issue of those ways of knowing in mathematics that have... more In this paper we shall concentrate on the issue of those ways of knowing in mathematics that have traditionally been taken to support apriorism. We shall do it by critizing pragmatic naturalism in the philosophy of mathematics, and in particular its historical approach in denying any role to apriority in mathematical epistemology. The version of pragmatic naturalism we shall be analyzing is Kitcher’s. In the paper we shall first set out a brief survey of the relevant features of Kitcher’s pragmatic naturalism in the philosophy of mathematics and then indicate the points that provoke our disagreement.U ovome radu ćemo se koncentrirati na pitanje onih načina spoznaje u matematici za koje se tradicionalno smatralo da podržavaju apriorizam. To ćemo učiniti kritizirajući pragmatički naturalizam u filozofiji matematike, posebice njegov povijesni pristup u negiranju ikakve uloge apriorizma u matematičkoj epistemologiji. Verzija pragmatičkog naturalizma koju ćemo razmatrati je Kitcherova. U...
Reduction and Elimination in Philosophy and the Sciences - Papers of the 21st International Wittgenstein Symposium, Oct 4, 2018
The paper is about Wittgenstein's philosophy of mathematics, more precisely about his remarks... more The paper is about Wittgenstein's philosophy of mathematics, more precisely about his remarks on Cantor's theorem and its proof. In the paper firstly Cantor's theorem is presented, then Wittgenstein criticism and then the author's attempted answers.
The paper is about Wittgenstein's philosophy of mathematics, more precisely about his remarks... more The paper is about Wittgenstein's philosophy of mathematics, more precisely about his remarks on Cantor's theorem and its proof. In the paper firstly Cantor's theorem is presented, then Wittgenstein criticism and then the author's attempted answers.
This paper is about structuralism, more precisely about the ante rem structuralism, version of wh... more This paper is about structuralism, more precisely about the ante rem structuralism, version of which can be treated as a sort of (non-traditional) platonism and which tries to solve problems that platonism is concerned with, i.e. it allegedly solves both a) “the plight of the mathematical Platonist arising from the existence of multiple reductions of the major mathematical theories”, and b) the epistemological problem for platonism due to abstract mathematical entities being causally inert. Ante rem structuralism, more precisely, the version endorsed by Shapiro, is the doctrine according to which mathematics is concerned with abstract structures and the elements of the structures have no properties beside the structural ones; that is, they have no non-structural properties. Mathematical objects (numbers, sets, ...) are just places within structures; e.g. real analysis is about the real number structure and everything we can say about real numbers consists in their “structural” prope...
Motivated by the analogy which holds within the context of discovery between mathematics and phys... more Motivated by the analogy which holds within the context of discovery between mathematics and physics, we aim to show that there is a connection between two fi elds within the context of justifi cation too. Based on the careful analysis of examples from science (especially within the domain of physics) we suggest that the logic of scientifi c research, which might appear as enumerative induction, is deduction, and we propose it to be universal generalization inference rule. Our main argument closely follows the analysis of the structure of physical theory proposed by theoretical physicist Eugene P. Wigner.
The aim of this paper is to point to the analogy between mathematical and physical thought experi... more The aim of this paper is to point to the analogy between mathematical and physical thought experiments, and even more widely between the epistemic paths in both domains. Having accepted platonism as the underlying ontology as long as the platonistic path in asserting the possibility of gaining knowledge of abstract, mind-independent and causally inert objects, my widely taken goal is to show that there is no need to insist on the uniformity of picture and monopoly of certain epistemic paths in the epistemic descriptive context. And secondly, to show the analogy with the ways we come to know the truths of (natural) sciences.
The lynch-pin of the structuralist account of logic endorsed by Koslow is the definition of logic... more The lynch-pin of the structuralist account of logic endorsed by Koslow is the definition of logical and modal operators with respect to implication relations, i.e. relative to implication structures. Logical operators are depicted independently of any possible semantic or syntactic limitations. It turns out that it is possible to define conjunction as well as other logical operators much more generally than it has usually been, and items on which the logical operators may be applied need not be syntactic objects and need not have truth values. In this paper I analyse Koslow’s structuralist theory and point out certain objectionable aspects to as well as reasons why such a theory does not fulfil the (possibly unjustified) expectation of getting defined a universal logical structure.
ion principles have the general form: ∀f ∀g ( ψ(f) = ψ(g) ↔ f ≈ g ) Here, f and g are variables r... more ion principles have the general form: ∀f ∀g ( ψ(f) = ψ(g) ↔ f ≈ g ) Here, f and g are variables referring to entities of a certain kind (objects or concepts usually), ψ is an operator which forms singular terms when applied to f and g – so that ψ(f) and ψ(g) are singular terms referring to objects, and ≈ is an equivalence relation on entities denoted by f and g. Frege himself invokes three abstraction principles: The direction principle: The direction of the line a is identical to the direction of the line b if and only if the line a is parallel to the line b. Hume's principle: The number belonging to the concept F is identical to the number belonging to the concept F if and only if the concept F is equinumerous with the concept G. Basic Law V: The extension of the concept F is identical with the extension of the concept G if and only if every object that falls under F falls under G and conversely. The third one notoriously turns out to be inconsistent. Abstraction principles ar...
Define a system to be a collection of objects with certain relations. A basketball defence is a s... more Define a system to be a collection of objects with certain relations. A basketball defence is a system of people under various positioning and defence-role relations; an extended family is a system of people under blood and marital relationships; and a chess configuration is a system of pieces under certain spatial and ‘possible move’ relations. A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations. The slogan of structuralism is that mathematics is the science of structure. The idea is that the subject matter of a given branch of mathematics is a structure, or a kind of structure. Typically, the structures studied in mathematics are free-standing, in the sense that the various positions or places in the structure are characterized only with respect to each other, and anything at all can play the various roles, and stand in the various relations. Define a ‘natural number system’ to be a counta...
The aim of the paper is to show that the objection to non-structuralist platonism that derives fr... more The aim of the paper is to show that the objection to non-structuralist platonism that derives from Benacerraf's problem of indeterminacy is flawed: this problem does not refuse non-structuraslit platonism. The author presents and defends a solution to the problem of indeterminacy that non-structuralist platonists can accept.
In this paper I will try to say something about how we can know anything about mathematics from a... more In this paper I will try to say something about how we can know anything about mathematics from a Platonist’s point of view. For Platonism the problem of how we know things about mathematics seems to be particularly acute. So, I will firstly present the epistemological problem and secondly see how it could be solved. I will not try to argue that Platonism is true; I will just try to protect Platonism from what is regarded to be the best attack on Platonism Benacerraf's epistemological attack.
Cilj je ovoga teksta predstavljanje osnovnih pravaca u filozofiji matematike i najplauzibilnijih ... more Cilj je ovoga teksta predstavljanje osnovnih pravaca u filozofiji matematike i najplauzibilnijih odgovora na pitanja o postojanju i prirodi matematickih objekata kao i mogucnostima njihove spoznaje. Predstavljeni pravci podijeljeni su u dvije osnovne skupine: realizam i antirealizam. U tekstu su predstavljene glavne teze tih dvaju skupina i njihovih predstavnika. U dijelu o realizmu predstavljen je platonizam i problemi koji se javljaju kod platonizma kao i modalni i 'skupovno-teorijski' realizam. Osnovne antirealisticki pravci predstavljeni u tekstu su: intuicionizam, nominalizam i formalizam. Koji je pravac najplauzibilniji jos uvijek ostaje otvoreno pitanje ; problemi u filozofiji matematike jos uvijek traže odgovore i rjesenja a rasprava danas nije nista manje zanimljiva od one koja se vodila kroz stoljeca.
Università degli Studi di Fiume, Facoltà di Lettere e Filosofia , 2019
Il libro è stato ideato come libro di testo per gli studenti e i futuri docenti, ma anche come te... more Il libro è stato ideato come libro di testo per gli studenti e i futuri docenti, ma anche come testo di consultazione per chi già insegna a scuola o si interessa all’argomento. Nella prima parte del manuale vengono introdotti i concetti fondamentali del ragionamento critico appartenenti al nucleo logico-argomentativo, mentre nella seconda parte vengono discussi gli elementi di pragmatica: la sistematicità, la capacità comunicativa e l’apertura o elasticità mentale. I concetti trattati sono stati introdotti senza richiedere al lettore nessuna conoscenza preliminare.
International Studies in The Philosophy of Science, Dec 1, 2011
The problem of proving the objective truth of mathematical statements and the objective existence... more The problem of proving the objective truth of mathematical statements and the objective existence of mathematical objects is one of the most compelling in philosophy of mathematics. There are several arguments that, more or less successfully or convincingly, offer reasons for believing in the (objective) existence of the mathematical realm. Of course, there are dissenting voices. A new and interesting one is the voice of Mary Leng in her thought-provoking book, Mathematics and Reality. In the book she concentrates her critical attention on the Quine–Putnam indispensability argument—the argument for the objective existence of mathematical items that many, friends and foes, have labelled the best argument for Platonism. Leng critically presents the argument and offers reasons for rejecting it by mainly questioning one of its premises, labelled confirmational holism. The book is structured in ten chapters. In the first chapters (1–4), Leng nicely introduces the subject. After presenting the Quine–Putnam argument itself, she goes on critically to address different replies to it and to the discussions as they have been developed since. After careful and exhaustive presentation of the argument, in the remaining chapters (5–10), Leng concentrates on confirmational holism, one of the premises in the argument, questioning it with the contrary idea to the effect that, notwithstanding the indispensability of mathematics to our best scientific theories, we are not forced to believe the literal truth of the mathematical apparatus indispensable to formulating such theory. In her own words: ‘we can account for our reasons to include mathematical hypotheses in our scientific theories without assuming that we ought to believe that these hypotheses are true’ (102). In this review I shall mainly concentrate upon the contested argument. So let us have a look at it in more detail. The indispensability argument is the argument for the acceptance of the literal truth of mathematical statements and the existence of mathematical objects. It can be formulated in the following way:
In this paper we shall concentrate on the issue of those ways of knowing in mathematics that have... more In this paper we shall concentrate on the issue of those ways of knowing in mathematics that have traditionally been taken to support apriorism. We shall do it by critizing pragmatic naturalism in the philosophy of mathematics, and in particular its historical approach in denying any role to apriority in mathematical epistemology. The version of pragmatic naturalism we shall be analyzing is Kitcher’s. In the paper we shall first set out a brief survey of the relevant features of Kitcher’s pragmatic naturalism in the philosophy of mathematics and then indicate the points that provoke our disagreement.U ovome radu ćemo se koncentrirati na pitanje onih načina spoznaje u matematici za koje se tradicionalno smatralo da podržavaju apriorizam. To ćemo učiniti kritizirajući pragmatički naturalizam u filozofiji matematike, posebice njegov povijesni pristup u negiranju ikakve uloge apriorizma u matematičkoj epistemologiji. Verzija pragmatičkog naturalizma koju ćemo razmatrati je Kitcherova. U...
Reduction and Elimination in Philosophy and the Sciences - Papers of the 21st International Wittgenstein Symposium, Oct 4, 2018
The paper is about Wittgenstein's philosophy of mathematics, more precisely about his remarks... more The paper is about Wittgenstein's philosophy of mathematics, more precisely about his remarks on Cantor's theorem and its proof. In the paper firstly Cantor's theorem is presented, then Wittgenstein criticism and then the author's attempted answers.
The paper is about Wittgenstein's philosophy of mathematics, more precisely about his remarks... more The paper is about Wittgenstein's philosophy of mathematics, more precisely about his remarks on Cantor's theorem and its proof. In the paper firstly Cantor's theorem is presented, then Wittgenstein criticism and then the author's attempted answers.
This paper is about structuralism, more precisely about the ante rem structuralism, version of wh... more This paper is about structuralism, more precisely about the ante rem structuralism, version of which can be treated as a sort of (non-traditional) platonism and which tries to solve problems that platonism is concerned with, i.e. it allegedly solves both a) “the plight of the mathematical Platonist arising from the existence of multiple reductions of the major mathematical theories”, and b) the epistemological problem for platonism due to abstract mathematical entities being causally inert. Ante rem structuralism, more precisely, the version endorsed by Shapiro, is the doctrine according to which mathematics is concerned with abstract structures and the elements of the structures have no properties beside the structural ones; that is, they have no non-structural properties. Mathematical objects (numbers, sets, ...) are just places within structures; e.g. real analysis is about the real number structure and everything we can say about real numbers consists in their “structural” prope...
Motivated by the analogy which holds within the context of discovery between mathematics and phys... more Motivated by the analogy which holds within the context of discovery between mathematics and physics, we aim to show that there is a connection between two fi elds within the context of justifi cation too. Based on the careful analysis of examples from science (especially within the domain of physics) we suggest that the logic of scientifi c research, which might appear as enumerative induction, is deduction, and we propose it to be universal generalization inference rule. Our main argument closely follows the analysis of the structure of physical theory proposed by theoretical physicist Eugene P. Wigner.
The aim of this paper is to point to the analogy between mathematical and physical thought experi... more The aim of this paper is to point to the analogy between mathematical and physical thought experiments, and even more widely between the epistemic paths in both domains. Having accepted platonism as the underlying ontology as long as the platonistic path in asserting the possibility of gaining knowledge of abstract, mind-independent and causally inert objects, my widely taken goal is to show that there is no need to insist on the uniformity of picture and monopoly of certain epistemic paths in the epistemic descriptive context. And secondly, to show the analogy with the ways we come to know the truths of (natural) sciences.
The lynch-pin of the structuralist account of logic endorsed by Koslow is the definition of logic... more The lynch-pin of the structuralist account of logic endorsed by Koslow is the definition of logical and modal operators with respect to implication relations, i.e. relative to implication structures. Logical operators are depicted independently of any possible semantic or syntactic limitations. It turns out that it is possible to define conjunction as well as other logical operators much more generally than it has usually been, and items on which the logical operators may be applied need not be syntactic objects and need not have truth values. In this paper I analyse Koslow’s structuralist theory and point out certain objectionable aspects to as well as reasons why such a theory does not fulfil the (possibly unjustified) expectation of getting defined a universal logical structure.
ion principles have the general form: ∀f ∀g ( ψ(f) = ψ(g) ↔ f ≈ g ) Here, f and g are variables r... more ion principles have the general form: ∀f ∀g ( ψ(f) = ψ(g) ↔ f ≈ g ) Here, f and g are variables referring to entities of a certain kind (objects or concepts usually), ψ is an operator which forms singular terms when applied to f and g – so that ψ(f) and ψ(g) are singular terms referring to objects, and ≈ is an equivalence relation on entities denoted by f and g. Frege himself invokes three abstraction principles: The direction principle: The direction of the line a is identical to the direction of the line b if and only if the line a is parallel to the line b. Hume's principle: The number belonging to the concept F is identical to the number belonging to the concept F if and only if the concept F is equinumerous with the concept G. Basic Law V: The extension of the concept F is identical with the extension of the concept G if and only if every object that falls under F falls under G and conversely. The third one notoriously turns out to be inconsistent. Abstraction principles ar...
Define a system to be a collection of objects with certain relations. A basketball defence is a s... more Define a system to be a collection of objects with certain relations. A basketball defence is a system of people under various positioning and defence-role relations; an extended family is a system of people under blood and marital relationships; and a chess configuration is a system of pieces under certain spatial and ‘possible move’ relations. A structure is the abstract form of a system, which ignores or abstracts away from any features of the objects that do not bear on the relations. The slogan of structuralism is that mathematics is the science of structure. The idea is that the subject matter of a given branch of mathematics is a structure, or a kind of structure. Typically, the structures studied in mathematics are free-standing, in the sense that the various positions or places in the structure are characterized only with respect to each other, and anything at all can play the various roles, and stand in the various relations. Define a ‘natural number system’ to be a counta...
The aim of the paper is to show that the objection to non-structuralist platonism that derives fr... more The aim of the paper is to show that the objection to non-structuralist platonism that derives from Benacerraf's problem of indeterminacy is flawed: this problem does not refuse non-structuraslit platonism. The author presents and defends a solution to the problem of indeterminacy that non-structuralist platonists can accept.
In this paper I will try to say something about how we can know anything about mathematics from a... more In this paper I will try to say something about how we can know anything about mathematics from a Platonist’s point of view. For Platonism the problem of how we know things about mathematics seems to be particularly acute. So, I will firstly present the epistemological problem and secondly see how it could be solved. I will not try to argue that Platonism is true; I will just try to protect Platonism from what is regarded to be the best attack on Platonism Benacerraf's epistemological attack.
Cilj je ovoga teksta predstavljanje osnovnih pravaca u filozofiji matematike i najplauzibilnijih ... more Cilj je ovoga teksta predstavljanje osnovnih pravaca u filozofiji matematike i najplauzibilnijih odgovora na pitanja o postojanju i prirodi matematickih objekata kao i mogucnostima njihove spoznaje. Predstavljeni pravci podijeljeni su u dvije osnovne skupine: realizam i antirealizam. U tekstu su predstavljene glavne teze tih dvaju skupina i njihovih predstavnika. U dijelu o realizmu predstavljen je platonizam i problemi koji se javljaju kod platonizma kao i modalni i 'skupovno-teorijski' realizam. Osnovne antirealisticki pravci predstavljeni u tekstu su: intuicionizam, nominalizam i formalizam. Koji je pravac najplauzibilniji jos uvijek ostaje otvoreno pitanje ; problemi u filozofiji matematike jos uvijek traže odgovore i rjesenja a rasprava danas nije nista manje zanimljiva od one koja se vodila kroz stoljeca.
The lynch-pin of the structuralist account of logic endorsed by Koslow has two components: one is... more The lynch-pin of the structuralist account of logic endorsed by Koslow has two components: one is the definition of an implication structure, while the other amounts to the definition of the logical operators as functions defined relative to an implication structure. In this paper I first present the basic tenets of the structuralist account of logic. Then, in the discussion which follows I give reasons for rejecting certain definitions which form part of the theory and thirdly I point out some general difficulties arising from such an account of logic.
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