Markus Pantsar

RWTH Aachen University · Human Technology Centre

The Turing test has a peculiar status in the artificial intelligence (AI) research community. On the one hand, it is presented as an important topic in virtually every AI textbook, and the research direction focused on developing AI systems that behave in human-like fashion is standardly called the “Turing test approach”. On the other hand, reports...

How do we acquire the notions of cardinality and cardinal number? In the (neo-)Fregean approach, they are derived from the notion of equinumerosity. According to some alternative approaches, defended and developed by Husserl and Parsons among others, the order of explanation is reversed: equinumerosity is explained in terms of cardinality, which, i...

Computer assisted theorem proving is an increasingly important part of mathematical methodology, as well as a long-standing topic in artificial intelligence (AI) research. However, the current generation of theorem proving software have limited functioning in terms of providing new proofs. Importantly, they are not able to discriminate interesting...

In Chapter 9, I move the focus from epistemology to ontology. I ask what kind of objects natural numbers are and show why different types of realist answers are problematic. I then endorse a constructivist account, according to which natural numbers exist as social constructs through our shared number concepts. This account is compatible with the s...

In Chapter 5, the focus is on the development of arithmetical operations. I review empirical literature and argue that addition has plausibly developed as a direct continuation of counting procedures, which in turn made it possible to develop other arithmetical operations (multiplication, subtraction, and division). After that, reviewing literature...

In the Introduction, I presented the question of how arithmetical knowledge can be acquired as the most fundamental question in the epistemology arithmetic. In Part I, I have presented an answer to this question in terms of individual ontogeny. Arithmetical knowledge can be acquired by individuals through a process of enculturation in which they em...

In Chapter 7, I focus on the ‘conventionalist threat’ against the account presented in this book. I present Warren’s conventionalist account of mathematics and discuss Wittgenstein’s philosophy of mathematics and its debated conventionalism. I identify strict conventionalism according to which mathematical truths are fundamentally arbitrary as the...

In Chapter 2, I focus on the acquisition of number concepts related to natural numbers. I review nativist views, as well as Dehaene’s early view that number concepts arise from estimations due to the approximate numbers system. I end up focusing in most detail on the bootstrapping account of Carey and Beck, according to which the object tracking sy...

In Chapter 1, I present empirical research on proto-arithmetical abilities, that is, subitising and estimating, and emphasise the importance of distinguishing them from arithmetical abilities. I review the empirical literature on the cognitive basis of proto-arithmetical abilities, focusing mostly on the core cognitive theory of the object tracking...

In Chapter 6, I discuss the way knowledge and skills evolve culturally. Starting from the notion of cumulative cultural evolution, I show how the emergence of arithmetical knowledge and skills can be understood as the product of trans-generational cultural evolution. Cumulative cultural evolution, however, requires a particular type of social learn...

In Chapter 4, I move the focus from ontogeny (the level of the individual) to phylogeny and cultural history (the level of communities). Starting from the question of which came first, numerals or number concepts, I review literature (e.g., Overmann) applying the material engagement theory of Malafouris, according to which numeral systems developed...

In Chapter 3, I focus on the cultural influences in the ontogenetic acquisition of number concepts and arithmetical knowledge and skills. Using the notion of enculturation as specified by Menary, I provide an account of how both cultural and evolutionarily developed biological aspects influence the acquisition of numbers concepts and arithmetic in...

In Chapter 8, I deal with the threat that the present account strips arithmetical knowledge of all the important characteristics traditionally associated with it: apriority, objectivity, necessity and universality. I argue that apriority can be saved in the strong sense of arithmetical knowledge being contextually a priori in the context set by our...

In this paper, we connect two research directions concerning numeral symbol systems and their epistemological significance. The first direction concerns the cognitive processes involved in acquiring and applying different numeral symbols, e.g. the Indo-Arabic or Roman numeral systems. The second direction is a semiotic one, with focus on Charles Pe...

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...

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 thi...

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 e...

How is knowledge of geometry developed and acquired? This central question in the philosophy of mathematics has received very different answers. Spelke and colleagues argue for a “core cognitivist”, nativist, view according to which geometric cognition is in an important way shaped by genetically determined abilities for shape recognition and orien...

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), w...

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;...

Following Marr’s famous three-level distinction between explanations in cognitive science, it is often accepted that focus on modeling cognitive tasks should be on the computational level rather than the algorithmic level. When it comes to mathematical problem solving, this approach suggests that the complexity of the task of solving a problem can...

I identify two reasons for believing in the objectivity of mathematical knowledge: apparent objectivity and applications in science. Focusing on arithmetic, I analyze platonism and cognitive nativism in terms of explaining these two reasons. After establishing that both theories run into difficulties, I present an alternative epistemological accoun...

Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than...

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 pol...

An important paradigm in modeling the complexity of mathematical tasks relies on computational complexity theory, in which complexity is measured through the resources (time, space) taken by a Turing machine to carry out the task. These complexity measures, however, are asymptotic and as such potentially a problematic fit when descriptively modelin...

The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically...

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...

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez (2000), I propose one particular conceptual metaphor, the Process → Object Metaphor (POM), as a key element in understanding the development of mathematical thinking.

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 o...

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psy...

In this paper, I study how mathematicians are presented in western popular culture. I identify five stereotypes that I test on the best-known modern movies and television shows containing a significant amount of mathematics or important mathematician characters: (1) Mathematics is highly valued as an intellectual pursuit. (2) Little attention is gi...

In this paper I develop a philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions. The account is based on a simple metaphor in which we think of indefinitely continuing processes as defining objects. It is shown that such a metaphor is valid in terms of mathematical prac...

Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical...

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theo...