In recent decades, non-representational approaches to mental phenomena and cognition have been gaining traction in cognitive science and philosophy of mind. In these alternative approach, mental representations either lose their central status or, in its most radical form, are banned completely. While there is growing agreement that non-representational accounts may succeed in explaining some cognitive capacities, there is widespread skepticism about the possibility of giving non-representational accounts of cognitive capacities such as memory, imagination or abstract thought. In this paper, I (...) will critically examine the view that there are fundamental limitations to non-representational explanations of cognition. Rather than challenging these arguments on general grounds, I will examine a set of human cognitive capacities that are generally thought to fall outside the scope of non-representational accounts, i.e. numerical cognition. After criticizing standard representational accounts of numerical cognition for their lack of explanatory power, I will argue that a non-representational approach that is inspired by radical enactivism offers the best hope for developing a genuine naturalistic explanatory account for these cognitive capacities. (shrink)

Since the publication of Clark and Chalmers' Extended Mind paper, the central claims of that paper, viz. the thesis that cognitive processes and cognitive or mental states extend beyond the brain and body, have been vigorously debated within philosophy of mind and philosophy of cognitive science. Both defenders and detractors of these claims have since marshalled an impressive battery of arguments for and against “active externalism.” However, despite the amount of philosophical energy expended, this debate remains far from settled. We (...) argue that this debate can be understood as answering two metaphysical questions. Yet prominent voices within the debate have assumed that there is a tight relationship between these two questions such that one question can be answered via the other. We defend an alternative ‘wide’ view, whereby mentality is understood as constituted by wide social and cultural factors. Our wide view entails that the two metaphysical questions are separate and should be kept distinct. This suggests that active externalism as understood by prominent voices within that debate requires dissolution, rather than solution. However, if the debate were instead understood as only focusing on the second of the two questions, then there could be a possible future for this debate. (shrink)

Embodied and extended cognition is a relatively new paradigm within cognitive science that challenges the basic tenet of classical cognitive science, viz. cognition consists in building and manipulating internal representations. Some of the pioneers of embodied cognitive science have claimed that this new way of conceptualizing cognition puts pressure on epistemological and ontological realism. In this paper I will argue that such anti-realist conclusions do not follow from the basic assumptions of radical embodied cognitive science. Furthermore I will show that (...) one can develop a form of realism that reflects rather than just accommodates the core principles of non-representationalist embodied cognitive science. (shrink)

This paper offers an evaluation of Wittgenstein's critique of Cantorian set theory, illustrating his broader philosophical stance on mathematics. By emphasizing the constructed nature of mathematical theories, Wittgenstein encourages a reflective approach to mathematics that acknowledges human agency in its development. His engagement with Cantorian set theory provides valuable insights into the philosophical and practical dimensions of mathematics, urging a reconsideration of its foundations and the nature of mathematical proofs. This perspective aligns closely with the philosophy of mathematical practice, which (...) emphasizes the human side of mathematics by its focus on mathematical practice. (shrink)

This paper critically examines and evaluates Yacin Hamami’s reconstruction of the standard view of mathematical rigor. We will argue that the reconstruction offered by Hamami is premised on a strong and controversial epistemological thesis and a strong and controversial thesis in the philosophy of mind. Secondly, we will argue that Hamami’s reconstruction of the standard view robs it of its original philosophical rationale, i.e. making sense of the notion of rigor in mathematical practice.

Clarke and Beck rightly contend that the number sense allows us to directly perceive number. However, they unnecessarily assume a representationalist approach and incur a heavy theoretical cost by invoking “modes of presentation.” We suggest that the relevant evidence is better explained by adopting a radical enactivist approach that avoids characterizing the approximate number system as a system for representing number.

We show that if R is a nonconstant regular (semi-)local subring of a rational function field over an algebraically closed field of characteristic zero, Hilbert's Tenth Problem for this ring R has a negative answer; that is, there is no algorithm to decide whether an arbitrary Diophantine equation over R has solutions over R or not. This result can be seen as evidence for the fact that the corresponding problem for the full rational field is also unsolvable.

We develop an elimination theory for addition and the Frobenius map over rings of polynomials. As a consequence we show that if F is a countable, recursive and perfect field of positive characteristic p, with decidable theory, then the structure of addition, the Frobenius map x→ xp and the property ‘x∈ F', over the ring of polynomials F[T], has a decidable theory.

Both mainstream cognitive science and analytic philosophy of mind remain wedded to the Cartesian picture of the mind as an isolated, self-sufficient, and constitutively individual phenomenon. However, recently approaches to the mind (e.g. extended mind thesis, enactivism) that depart from the standard view have emerged. Aunifying thread that runs through these approaches can be summed up in the slogan: “to understand mental phenomena one cannot do away with the environment”. Differences between these related views pertain to the strength of the (...) modal operator “cannot”. On the strongest reading the slogan implies that the mind is constituted by the environment. While this interpretation is akin to Marx view on the constitution of consciousness, this link is overlooked in the literature. In this paper, I will argue that Marxists philosophical thinking about the mind, as exemplified by the activity approach, offers a sound philosophical basis for the further development of post-Cartesian views in cognitive science and philosophy of mind. Furthermore, I will argue that the materialistic method proposed by these thinkers is the most promising approach to the problem of naturalizing the mind. (shrink)

I examine to what extend Varela’s remarks on problem-solving can be applied to mathematical problem-solving. I argue that despite similarities between Varela’s epistemological model and recent advances in mathematics education research on problem-solving, trying to fit ideas and concepts from the latter domain in the Varelian mold runs the risk of misconstruing fundamental aspects of mathematical problem-solving.