Bergen Workshop: The Cognitive Basis of Logico-Mathematical Knowledge

The Cognitive Basis of Logico-Mathematical Knowledge

Department of Philosophy, University of Bergen

November 16-17, 2015. Bergen, Norway

Seminar room 1st floor

The objective of this workshop is to reflect on the cognitive basis of the human abilities to reason abstractly, in mathematics and logic. While looked down on by some major philosophers, these abilities have been extensively studied by psychologists and cognitive scientists; our assumption is that the issue can still benefit from philosophical analysis.

Program

Monday 16 Nov

9.55-10.00 Welcome from the Dept. Chair, Prof. Reidar Lie

10.00-11.00 Rolf Reber (Oslo, Psychology)

The Aesthetic of Truth in Mathematics

11.05-11.50 Mario Santos-Sousa (Univ. College London, Philosophy)

Understanding Basic Arithmetic

12.00-13.00 Helen De Cruz (Amsterdam, Philosophy)

Testimony and the Acquisition of Number Concepts

14.00-14.45 Karim Zahidi (Antwerp, Philosophy)

Numerical Cognition from a Radical Enactive Point of View

15.00-16.00 Catarina Dutilh Novaes (Groningen, Philosophy)

The Phylogeny and Ontogeny of Deductive Reasoning: a Cultural Story \

16.15-17.00 Sorin Costreie (Bucharest, Philosophy)

The Geometrical Roots of Arithmetical Cognition: Frege and Dehaene

17.15-18.00 Joe Morrison (Belfast, Philosophy)

Second Philosophy and Logical Contingentism

Tuesday 17 Nov

9.10-9.55 Fabio Sterpetti (La Sapienza Rome, Philosophy)

Mathematical Knowledge, the Analytic Method and the Naturalization of Mathematics

10.00-11.00 Andrea Bender, Sieghard Beller (Bergen, Psychology)

Numeration Systems as Cultural Tools

11.15-12.00 Max Jones (Bristol, Philosophy)

Active Numerical Perception

12.30-13.15 Jean-Charles Pelland (Montreal / London, Philosophy)

On the Internalist Origin of Numerical Cognition

13.30-14.30 Sorin Bangu (Bergen, Philosophy)

Methodological Remarks on the Experiments on Infants' Arithmetical Abilities

Registration free. If interested to attend, please contact [email protected]

Abstracts:

The Aesthetic of Truth in Mathematics * Rolf Reber

The processing fluency theory of aesthetic pleasure has influenced research in empirical aesthetics, cognitive science, social psychology, and marketing. After summarizing research of fluency effects on aesthetic pleasure and truth, I present recent research from our laboratory that reveals how beauty and truth interact in mathematics. Experiments on problem solving show that the beauty of elements in geometrical problems determines the judged correctness of the presented solution. Finally, studies on aha-experiences suggest that a sudden increase in processing fluency leads to more positive affect (which Poincaré called an “aesthetic emotion”) and the certainty that the solution is correct.

Understanding Basic Arithmetic * Mario Santos-Sousa

Any account of our knowledge of a particular subject matter presupposes some account of our understanding of that subject matter. Accordingly, any account of our knowledge of basic cardinal arithmetic presupposes some account of our grasp of finite cardinal numbers and of simple arithmetic operations on them. That is why one can find such accounts in the writings of Kant or Mill. And yet, they did not have the benefit of systematic empirical research in numerical cognition. My goal is to give such an account — one that draws on, and is consistent with, current empirical studies of numerical cognition.

Testimony and the Acquisition of Number Concepts * Helen De Cruz

Cognitive scientists have developed various models of how young children transition from an evolved, imprecise system of estimating cardinalities to the use of exact number concepts, such as natural numbers. These models emphasize individual learning processes, such as induction and the discovery of principles underlying counting and cardinality. However, convergent lines of evidence (such as the influence of parental use of counting words and counting routines) indicate that testimony plays an important role in children's acquisition of number concepts. An important element in this is mastery of the counting routine. To explain how testimony can help to shape children's learning from counting words, I develop a philosophical account of the testimonial transmission of knowledge-how. I show that counting routines require deference and the adoption of practices that are semantically opaque, just like testimony to propositions does.

Numerical Cognition from a Radical Enactive Point of View * Karim Zahidi

Contemporary theorising about numerical cognition is characterised by two assumptions. Firstly it is assumed that numerical competence is best explained by invoking internal (i.e. neural) representations that code for numbers. And, secondly, at least part of our internal numerical machinery is innate. The dominant research paradigm in the area of numerical cognition is thus in line with the dominant cognitivism in contemporary cognitive science and the underlying philosophy of mind. There are however non-representational ways of conceiving of cognition. In this talk I want to propose an alternative account of numerical cognition in line with the radical enactive approach to cognition as put forward by Hutto and Myin (2012).

The Phylogeny and Ontogeny of Deductive Reasoning: a Cultural Story * Catarina Dutilh Novaes

Does ontogeny recapitulate phylogeny when it comes to mathematical thinking? A number of authors (Poincaré, Polya, Lakatos) have suggested that it does, at least in some respects. Drawing on historical sources, on the literature on deductive reasoning from psychology, and on findings from mathematics education, in my talk I explore this idea with respect to deductive reasoning specifically -- which would translate, among others, into the ability of producing and understanding mathematical proof. The key idea will be that both for phylogeny and for ontogeny, proof is best understood as an inherently dialogical notion.

The Geometrical Roots of Arithmetical Cognition: Frege and Dehaene * Sorin Costreie

Frege writes in Numbers and Arithmetic about “an a priori mode of cognition” that it may have “a geometrical source.” This resembles somehow the recent findings about mathematical cognition known as theSNARC effect described by Stanislav Dehaene in The Number Sense. In my presentation, I’ll explore this resemblance between the two positions, trying to see in what sense new findings in mathematical cognition could endorse Frege’s later ideas. Could we really say, with Frege, that geometrical knowledge lies at the bottom of all mathematical knowledge?

Second Philosophy and Logical Contingentism * Joe Morrison

Penelope Maddy argues that logical truths are only contingently true. Her premises include: (1) logical truths are truths about stable features of the world, (2) while humans may struggle to detect worldly features which don’t exhibit such structuring (and struggle to reason non-classically about the world), this is not because such structures necessarily obtain, but because (3) our cognitive abilities have developed in response to these (relatively abundant) structures in our environments. I’ll argue that the relationship between the types of inferential habits that humans adopt (2 and 3) and the kinds of structures that might exist in the world (1) is weaker than Maddy requires for her argument to work.

Mathematical Knowledge, the Analytic Method and the Naturalization of Mathematics * Fabio Sterpetti

Mathematical knowledge is considered to be the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, but mathematics seems to provide a counterexample both to methodological and to ontological naturalism. To face this difficulty, some authors tried to naturalize mathematics relying on evolutionism. But several difficulties arise if we try to naturalize the traditional view of mathematics. This paper suggests that in order to naturalize mathematics it is better to take the method of mathematics to be the analytic method, and conceive mathematical knowledge as plausible knowledge.

Numeration Systems as Cultural Tools * Andrea Bender, Sieghard Beller

Numerical competencies are considered a core domain of knowledge, and yet, the development of specifically human abilities seems to presuppose cultural and linguistic input by way of counting sequences. These sequences may be realized in different modalities (verbal, notational, or body-based) and constitute systems with distinct structural properties, the cross-linguistic variability of which has implications for number representation and processing. Here we contrast various numeration systems across languages and modalities, and analyze their representational effects. In doing so, we will also draw more general conclusions on the relative relevance of culture and language for numerical cognition.

Active Numerical Perception * Max Jones

The general consensus in the philosophy of mathematics is that we do not access number through perception. Our access to number is primarily cognitive as opposed to perceptual. This position is supported both by Frege’s critique of Mill’s Empiricism and by traditional cognitivist models of the mind. Despite this, much of the psychological and neurological evidence points towards numerical perception. This can be explained by rejecting cognitivism and adopting a more active, ecological and embodied view of perception. This has potentially significant consequences for providing a naturalised account of the origins of numerical knowledge.

On the Internalist Origin of Numerical Cognition * Jean-Charles Pelland

Many popular proposals explain the origin of numerical cognition by relying on extended cognition involving pre-existing numerical symbols and artifacts in the environment (e.g. Dehaene 2011; Carey 2009; DeCruz 2008). In this talk, I hope to show that there is a problem with such an externalist view. I will argue that explaining how we transcend the limitations of our innate cognitive machinery by relying on external numerical symbols is akin to putting the cart before the ox, since such symbols cannot emerge without there being number concepts built by individual psychological processes in the first place.

Methodological Remarks on the Experiments on Infants' Arithmetical Abilities * Sorin Bangu

Although I agree that the experiments on infants’ arithmetical abilities performed by Spelke, Starkey, Gellman, Wynn and others make a major contribution to our understanding of the cognitive basis of mathematics, I discuss several problems, of methodological nature, that these experiments face if they are to fulfill their original purpose.