Formalizing Darwinism, Naturalizing Mathematics

Foundations of science, 2006

In the past decades, recent paradigm shifts in ethology, psychology, and the social sciences have given rise to various new disciplines like cognitive ethology and evolutionary psychology. These disciplines use concepts and theories of evolutionary biology to understand and explain the design, function and origin of the brain. I shall argue that there are several good reasons why this approach could also apply to human mathematical abilities. I will review evidence from various disciplines (cognitive ethology, cognitive psychology, cognitive archaeology and neuropsychology) that suggests that the human capacity for mathematics is a category specific domain of knowledge, which can be explained as the result of natural selection.

Philosophia. Philosophical Quarterly of Israel, 2019

How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when one tries to naturalize in this way the traditional view of mathematics, according to which mathematical knowledge is certain and the method of mathematics is the axiomatic method. This paper suggests that, in order to naturalize mathematics through Darwinism, it is better to take the method of mathematics not to be the axiomatic method.

We propose a simple cognitive model where qualitative and quantitative comparisons enable animals to identify objects, associate them with their properties held in memory and make naive inference. Simple notions like equivalence relations, order relations are used. We then show that such processes are at the root of human mathematical reasoning by showing that the elements of totally ordered sets satisfy the Peano axioms. The process through which children learn counting is then formalized. Finally association is modeled as a Markov process leading to a stationary distribution.

2012

Biological evolution is perhaps the most revolutionary discovery ever made. For many of us it eliminates questions about our origin which otherwise would have been troubling. During the last decades a deeper understanding of the mechanisms of evolution, like the gene-perspective, has broadened the explanations to include, for instance, altruism and social structures. Edward O. Wilson, one of the leading evolutionary biologists, has even proposed a unification of all knowledge where the social sciences and humanities are integrated with the natural sciences by the theory of biological evolution (Wilson 1998). But still, does evolution help us in answering the fundamental questions of philosophy like “What can we know?” and “What exists?”; more specifically, can evolution say something about the foundations of mathematics and logic? I will here give some personal thoughts on the subject and for me the starting point is David Hume’s sceptical empiricism and in particular his view on ca...

Southern Journal of Philosophy, 1989

Dazzled by the rich particularity of individual cultures, anthropology in the early 20th century had bogged down in "a programmatic avoidance of theoretical syntheses."' It was the cultural evolutionist Leslie A. White (1900White ( -1975 who, more than anyone else, called anthropology back to the synoptic concerns that had quickened the discipline in the late 19th century. In this paper, we shall argue that one of White's lesser known essays makes a timely contribution toward a viable ontology for Philip Kitcher's "naturalistic" philosophy of mathematics. We shall also briefly sketch one contribution which anthropology (especially anthropology of a Whitean bent) might make to Kitcher's analysis of rationality.

International Electronic Journal of Mathematics Education, 2007

This article provides a philosophical conceptualization of mathematics given the particular tasks of its teaching and learning. A central claim is that mathematics is a discipline that has been largely untouched by the Darwinian revolution; it is a last bastion of certainty. Consequently, mathematics educators are forced to draw on overly absolutist or constructivist accounts of the discipline. The resulting "math wars" often impede genuine reform. I suggest adopting an evolutionary metaphor to help explain the epistemology/nature of mathematics. In order to use this evolutionary metaphor to its fullest effect in overcoming the polarization of the math wars, mathematical empiricism is presented as a means of constraint on the development of mathematics. This article sketches what an evolutionary philosophy of mathematics might look like and provides a detailed descriptive account of mathematical empiricism and its potential role in this novel way of thinking about mathematical enterprises.

Springer, Cham, 2013

Despite strenuous efforts by its proponents, the contemporary form of logic, mathematical logic, has generally failed to convince mathematicians, natural scientists and human scientists of its relevance to their work, increasingly so in the last few decades. This contrasts with the reputation logic enjoyed in antiquity, not only as one of the main parts of philosophy, but also as a supplier of instruments for the sciences. The purpose of this book is to explain how the present condition of logic came about and to propose an alternative to it. To this end, the book first gives an overview of how logic and its relation to the scientific method have been conceived in antiquity and in the modern age, because this provides indications for a new approach to the subject. Then the book proposes a new view of logic and its relation to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also proposes a new view of philosophy and its relation to knowledge, because seeing logic in a wider context helps to place it on a more satisfactory basis. In terms of the proposed new view, logic is primarily a logic of discovery. Accordingly, the book deals with the rules of discovery.

Epistemologia, Vol. 38, n2. 2/2015, pp. 327-329.

Logic Across the University: Foundations and Applications, Proceedings of the Tsinghua Logic Conference, Beijing 2013, pp.99-117, College Publications, 2013

This article introduces my research project exploring a nominalistic, strictly finitistic, and truly naturalistic philosophy of mathematics. There are two distinctive features of my approach compared with other contemporary naturalistic and/or nominalistic philosophies of mathematics. First, it is strict finitism, which means a position that does not assume reality of infinity in any sense, any format. Second, it takes physicalism as its philosophical basis, and in particular, it emphasizes physicalism about cognitive subjects and processes. The technical work in this research tries to propose a logical explanation of the applicability of classical mathematics in the sciences consistent with nominalism and strict finitism [29]. The philosophical work starts from methodological naturalism and argues that a coherent naturalist should adopt physicalism about cognitive subjects and that this should imply nominalism and strict finitism in philosophy of mathematics. The research is still in progress, but I hope it has accomplished enough to make it attractive.

Naturalized epistemology and philosophy of …, 2007