Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an... more Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an explanation of cognitive phenomena, one that does not include mental representations or mental content in basic minds. Recently, Zahidi and Myin have presented an account of arithmetical cognition that is consistent with the REC view. In this paper, I first evaluate the feasibility of that account by focusing on the evolutionarily developed proto-arithmetical abilities and whether empirical data on them support the radical enactivist view. I argue that although more research is needed, it is at least possible to develop the REC position consistently with the state-of-the-art empirical research on the development of arithmetical cognition. After this, I move the focus to the question whether the radical enactivist account can explain the objectivity of arithmetical knowledge. Against the realist view suggested by Hutto, I argue that objectivity is best explained through analyzing the way...
One main challenge of non-platonist philosophy of mathematics is to account for the apparent obje... more One main challenge of non-platonist philosophy of mathematics is to account for the apparent objectivity of mathematical knowledge. Cole and Feferman have proposed accounts that aim to explain objectivity through the intersubjectivity of mathematical knowledge. In this paper, focusing on arithmetic, I will argue that these accounts as such cannot explain the apparent objectivity of mathematical knowledge. However, with support from recent progress in the empirical study of the development of arithmetical cognition, a stronger argument can be provided. I will show that since the development of arithmetic is (partly) determined by biologically evolved proto-arithmetical abilities, arithmetical knowledge can be understood as maximally intersubjective. This maximal intersubjectivity, I argue, can lead to the experience of objectivity, thus providing a solution to the problem of reconciling non-platonist philosophy of mathematics with the (apparent) objectivity of mathematical knowledge.
Why would we want to develop artificial human-like arithmetical intelligence, when computers alre... more Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially shed light on the development of human numerical abilities, from the proto-arithmetical abilities of subitizing and estimating to counting procedures. Although the current results are far from conclusive and much more work is needed, I argue that AI research should be included in the interdisciplinary toolbox when we try to explain the development and character of numerical cognition and arithmetical intelligence. This makes it re...
Tutkin tässä artikkelissa Kurt Gödelin epätäydellisyysteoreemojen tulkintoja filosofiassa. Aihepi... more Tutkin tässä artikkelissa Kurt Gödelin epätäydellisyysteoreemojen tulkintoja filosofiassa. Aihepiiri kattaa valtavan määrän eri tulkintoja tekoälystä fysiikkaan ja runouteen asti. Osoitan, että kriittisesti tarkasteltuna kaikki radikaalit epätäydellisyysteoreemojen sovellukset ovat virheellisiä
I take all five texts to be part of one coherent program and will not focus on the possible minor... more I take all five texts to be part of one coherent program and will not focus on the possible minor differences between the ideas in them. There is also a new book out expanding the EPM case (see Löwe & Müller (eds.) 2010) as well as presenting other angles to empirical study of mathematics.
There is no questioning the importance of a book like Naturalizing Logico-Mathematical Knowledge.... more There is no questioning the importance of a book like Naturalizing Logico-Mathematical Knowledge. With volumes like The Oxford Handbook of Numerical Cognition coming out in recent years, it is clear that there is a wealth of empirical data available for philosophical considerations on the cognitive foundations of mathematical knowledge. There have also been important monographs, such as Stanislas Dehaene's The Number Sense and Susan Carey's The Origin of Concepts, that have acquired the status of standard reference works both for empirical scientists and philosophers. What has been lacking, however, is a distinctively philosophical volume that aims to connect the empirical developments to fundamental issues in the philosophy of mathematics. This is what Naturalizing Logico-Mathematical Knowledge sets out to do, and based on a conference held at the University of Bergen in 2015, Sorin Bangu has gathered together a fine group of authors to contribute.
Linnebo in 2018 argues that abstract objects like numbers are "thin" because they are only requir... more Linnebo in 2018 argues that abstract objects like numbers are "thin" because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the account of arithmetic based on Hume's principle because we are specifying numbers in terms of the concept of equinumerosity, or its ordinal equivalent. But far from being only a matter for philosophy, this implies a distinct empirical prediction: in cognitive development, the principle of equinumerosity is primary to number concepts. However, by analysing and expanding on the bootstrapping theory of Carey in 2009, I argue in this paper that there are good reasons to think that the development could be the other way around: possessing numerosity concepts may precede grasping the principle of equinumerosity. I propose that this analysis of early numerical cognition can also help us understand what numbers as thin objects are like, moving away from Platonist interpretations.
Linnebo in 2018 argues that abstract objects like numbers are "thin" because they are only requir... more Linnebo in 2018 argues that abstract objects like numbers are "thin" because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the account of arithmetic based on Hume's principle because we are specifying numbers in terms of the concept of equinumerosity, or its ordinal equivalent. But far from being only a matter for philosophy, this implies a distinct empirical prediction: in cognitive development, the principle of equinumerosity is primary to number concepts. However, by analysing and expanding on the bootstrapping theory of Carey in 2009, I argue in this paper that there are good reasons to think that the development could be the other way around: possessing numerosity concepts may precede grasping the principle of equinumerosity. I propose that this analysis of early numerical cognition can also help us understand what numbers as thin objects are like, moving away from Platonist interpretations.
Boston Studies in the Philosophy and History of Science, 2016
In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is... more In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the third one: why arithmetical knowledge appears to be necessary. A Kripkean analysis of necessity is used as an example to show that a proper analysis of the relevant possible worlds can explain arithmetical necessity in a suciently strong form.
In computational complexity theory, decision problems are divided into complexity classes based o... more In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only firstorder (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes.
Although writing a monograph on philosophy of mathematics is largely a solitary pursuit, this pro... more Although writing a monograph on philosophy of mathematics is largely a solitary pursuit, this project could not have been completed without the help of various people. I am very grateful to my supervisor Professor Gabriel Sandu, whose expertise was invaluable in leading my research down the right paths, and whose comments on various drafts of this work have been essential in giving it the current form. I also owe gratitude to various members of the Department of Philosophy in the University of Helsinki for their help along the years. Professors Matti Sintonen and Ilkka Niiniluoto in particular deserve special thanks. With great gratitude I want to acknowledge the help of everybody who has at one point or another used his time to answer my questions. These include Professor Jaakko Hintikka, Dr. Philippe de Rouilhan and Professor Jouko Väänänen. I also want to thank my pre-examiners Professors Jan Woleński and Sten Lindström for their valuable comments, which helped improve the book a great deal at the final stage.
Beck (Cognition 158:110–121, 2017) presents an outline of the procedure of bootstrapping of integ... more Beck (Cognition 158:110–121, 2017) presents an outline of the procedure of bootstrapping of integer concepts, with the purpose of explicating the account of Carey (The Origin of Concepts, 2009). According to that theory, integer concepts are acquired through a process of inductive and analogous reasoning based on the object tracking system (OTS), which allows individuating objects in a parallel fashion. Discussing the bootstrapping theory, Beck dismisses what he calls the "deviant-interpretation challenge"—the possibility that the bootstrapped integer sequence does not follow a linear progression after some point—as being general to any account of inductive learning. While the account of Carey and Beck focuses on the OTS, in this paper I want to reconsider the importance of another empirically well-established cognitive core system for treating numerosities, namely the approximate number system (ANS). Since the ANS-based account offers a potential alternative for integer c...
Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an... more Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an explanation of cognitive phenomena, one that does not include mental representations or mental content in basic minds. Recently, Zahidi and Myin have presented an account of arithmetical cognition that is consistent with the REC view. In this paper, I first evaluate the feasibility of that account by focusing on the evolutionarily developed proto-arithmetical abilities and whether empirical data on them support the radical enactivist view. I argue that although more research is needed, it is at least possible to develop the REC position consistently with the state-of-the-art empirical research on the development of arithmetical cognition. After this, I move the focus to the question whether the radical enactivist account can explain the objectivity of arithmetical knowledge. Against the realist view suggested by Hutto, I argue that objectivity is best explained through analyzing the way...
One main challenge of non-platonist philosophy of mathematics is to account for the apparent obje... more One main challenge of non-platonist philosophy of mathematics is to account for the apparent objectivity of mathematical knowledge. Cole and Feferman have proposed accounts that aim to explain objectivity through the intersubjectivity of mathematical knowledge. In this paper, focusing on arithmetic, I will argue that these accounts as such cannot explain the apparent objectivity of mathematical knowledge. However, with support from recent progress in the empirical study of the development of arithmetical cognition, a stronger argument can be provided. I will show that since the development of arithmetic is (partly) determined by biologically evolved proto-arithmetical abilities, arithmetical knowledge can be understood as maximally intersubjective. This maximal intersubjectivity, I argue, can lead to the experience of objectivity, thus providing a solution to the problem of reconciling non-platonist philosophy of mathematics with the (apparent) objectivity of mathematical knowledge.
Why would we want to develop artificial human-like arithmetical intelligence, when computers alre... more Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially shed light on the development of human numerical abilities, from the proto-arithmetical abilities of subitizing and estimating to counting procedures. Although the current results are far from conclusive and much more work is needed, I argue that AI research should be included in the interdisciplinary toolbox when we try to explain the development and character of numerical cognition and arithmetical intelligence. This makes it re...
Tutkin tässä artikkelissa Kurt Gödelin epätäydellisyysteoreemojen tulkintoja filosofiassa. Aihepi... more Tutkin tässä artikkelissa Kurt Gödelin epätäydellisyysteoreemojen tulkintoja filosofiassa. Aihepiiri kattaa valtavan määrän eri tulkintoja tekoälystä fysiikkaan ja runouteen asti. Osoitan, että kriittisesti tarkasteltuna kaikki radikaalit epätäydellisyysteoreemojen sovellukset ovat virheellisiä
I take all five texts to be part of one coherent program and will not focus on the possible minor... more I take all five texts to be part of one coherent program and will not focus on the possible minor differences between the ideas in them. There is also a new book out expanding the EPM case (see Löwe & Müller (eds.) 2010) as well as presenting other angles to empirical study of mathematics.
There is no questioning the importance of a book like Naturalizing Logico-Mathematical Knowledge.... more There is no questioning the importance of a book like Naturalizing Logico-Mathematical Knowledge. With volumes like The Oxford Handbook of Numerical Cognition coming out in recent years, it is clear that there is a wealth of empirical data available for philosophical considerations on the cognitive foundations of mathematical knowledge. There have also been important monographs, such as Stanislas Dehaene's The Number Sense and Susan Carey's The Origin of Concepts, that have acquired the status of standard reference works both for empirical scientists and philosophers. What has been lacking, however, is a distinctively philosophical volume that aims to connect the empirical developments to fundamental issues in the philosophy of mathematics. This is what Naturalizing Logico-Mathematical Knowledge sets out to do, and based on a conference held at the University of Bergen in 2015, Sorin Bangu has gathered together a fine group of authors to contribute.
Linnebo in 2018 argues that abstract objects like numbers are "thin" because they are only requir... more Linnebo in 2018 argues that abstract objects like numbers are "thin" because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the account of arithmetic based on Hume's principle because we are specifying numbers in terms of the concept of equinumerosity, or its ordinal equivalent. But far from being only a matter for philosophy, this implies a distinct empirical prediction: in cognitive development, the principle of equinumerosity is primary to number concepts. However, by analysing and expanding on the bootstrapping theory of Carey in 2009, I argue in this paper that there are good reasons to think that the development could be the other way around: possessing numerosity concepts may precede grasping the principle of equinumerosity. I propose that this analysis of early numerical cognition can also help us understand what numbers as thin objects are like, moving away from Platonist interpretations.
Linnebo in 2018 argues that abstract objects like numbers are "thin" because they are only requir... more Linnebo in 2018 argues that abstract objects like numbers are "thin" because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the account of arithmetic based on Hume's principle because we are specifying numbers in terms of the concept of equinumerosity, or its ordinal equivalent. But far from being only a matter for philosophy, this implies a distinct empirical prediction: in cognitive development, the principle of equinumerosity is primary to number concepts. However, by analysing and expanding on the bootstrapping theory of Carey in 2009, I argue in this paper that there are good reasons to think that the development could be the other way around: possessing numerosity concepts may precede grasping the principle of equinumerosity. I propose that this analysis of early numerical cognition can also help us understand what numbers as thin objects are like, moving away from Platonist interpretations.
Boston Studies in the Philosophy and History of Science, 2016
In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is... more In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the third one: why arithmetical knowledge appears to be necessary. A Kripkean analysis of necessity is used as an example to show that a proper analysis of the relevant possible worlds can explain arithmetical necessity in a suciently strong form.
In computational complexity theory, decision problems are divided into complexity classes based o... more In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only firstorder (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes.
Although writing a monograph on philosophy of mathematics is largely a solitary pursuit, this pro... more Although writing a monograph on philosophy of mathematics is largely a solitary pursuit, this project could not have been completed without the help of various people. I am very grateful to my supervisor Professor Gabriel Sandu, whose expertise was invaluable in leading my research down the right paths, and whose comments on various drafts of this work have been essential in giving it the current form. I also owe gratitude to various members of the Department of Philosophy in the University of Helsinki for their help along the years. Professors Matti Sintonen and Ilkka Niiniluoto in particular deserve special thanks. With great gratitude I want to acknowledge the help of everybody who has at one point or another used his time to answer my questions. These include Professor Jaakko Hintikka, Dr. Philippe de Rouilhan and Professor Jouko Väänänen. I also want to thank my pre-examiners Professors Jan Woleński and Sten Lindström for their valuable comments, which helped improve the book a great deal at the final stage.
Beck (Cognition 158:110–121, 2017) presents an outline of the procedure of bootstrapping of integ... more Beck (Cognition 158:110–121, 2017) presents an outline of the procedure of bootstrapping of integer concepts, with the purpose of explicating the account of Carey (The Origin of Concepts, 2009). According to that theory, integer concepts are acquired through a process of inductive and analogous reasoning based on the object tracking system (OTS), which allows individuating objects in a parallel fashion. Discussing the bootstrapping theory, Beck dismisses what he calls the "deviant-interpretation challenge"—the possibility that the bootstrapped integer sequence does not follow a linear progression after some point—as being general to any account of inductive learning. While the account of Carey and Beck focuses on the OTS, in this paper I want to reconsider the importance of another empirically well-established cognitive core system for treating numerosities, namely the approximate number system (ANS). Since the ANS-based account offers a potential alternative for integer c...
In the new millennium, there have been important empirical developments in the philosophy of math... more In the new millennium, there have been important empirical developments in the philosophy of mathematics. One of these directions is the so-called " empirical philosophy of mathematics " (EPM) of Buldt, Löwe, Müller and Müller-Hill, which aims to complement the methodology in philosophy of mathematics with empirical work. Among other things, this includes surveys of mathematicians, which EPM believes to give philosophically important results. In this paper I take a critical look at the sociological part of EPM as a case study of sociological approaches to the philosophy of mathematics, focusing on the most concrete development of EPM so far: a questionnaire-based study by Müller-Hill. I will argue that the study has many problems and the EPM conclusion of context-dependency of mathematical knowledge is unwarranted by the evidence. In addition, I will consider the general justification and criteria for introducing sociological methods in the philosophy of mathematics. While surveys can give us important data about the philosophical views of mathematicians, there is no reason to believe that mathematicians have a privileged access to philosophical questions concerning mathematics. In order to be philosophically relevant in the way EPM claim, the philosophical views of mathematicians cannot be assessed without considering the argumentation behind them.
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