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Psychologism and the Cognitive Foundations of Mathematics Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences 9-2 | 2005 Aperçus philosophiques en logique et en mathématiques Psychologism and the Cognitive Foundations of Mathematics Christophe Heintz Electronic version URL: http://journals.openedition.org/philosophiascientiae/519 DOI: 10.4000/philosophiascientiae.519 ISSN: 1775-4283 Publisher Éditions Kimé Printed version Date of publication: 1 November 2005 Number of pages: 41-59 ISBN: 2-84174-379-9 ISSN: 1281-2463 Electronic reference Christophe Heintz, “Psychologism and the Cognitive Foundations of Mathematics”, Philosophia Scientiæ [Online], 9-2 | 2005, Online since 15 June 2011, connection on 16 January 2021. URL: http:// journals.openedition.org/philosophiascientiae/519 ; DOI: https://doi.org/10.4000/ philosophiascientiae.519 Tous droits réservés Psychologism and the Cognitive Foundations of Mathematics Christophe Heintz Institut Jean Nicod 1Introduction In this paper, I will attempt to purport the two following claims: Claim1: Naturalistic theories of the cognition of Mathematics im- ply psychologism, unless one recognises the social nature of Math- ematics. Claim2: A non psychologistic enquiry into the cognitive founda- tions of mathematics should aim at disentangling the social and the cognitive causes at work in the making of mathematical knowledge. My argument unfolds accordingly: In the first part of the paper, I analyse how an enquiry into the cognitive foundations of mathematics can turn out psychologistic, i.e., assert that the truths of mathematics and logic are psychological facts. I then attempt to show what philosoph- ical presumptions on the nature of mathematics lead to psychologism. I argue that the only way out of this aporiais to acknowledge that social normativity is an essential constituent of mathematical practice. In the last part, I briefly sketch what could be a non-psychologistic enquiry into the cognitive foundations of mathematics. I argue that such an enquiry should include — apart from psychological studies — (1) an an- thropological analysis of the role of psychological facts in mathematical Philosophia Scientiæ, 9 (2), 2005, 39–56. practice and (2) a historical analysis of the cognitive constraints on the development of mathematical knowledge. Along these lines, I support the idea that Mathematics is at the same time a cognitive product and a cultural object. Thus, the theoretical position I advocate is, using the current labels, cognitivism plus sociologism; where cognitivism, for my purpose, is a research program that led to assert that there are specific competencies that underlay our knowledge of Mathematics and Logic, and sociologism is the theoretical position that asserts that knowledge is socially constituted and that leads to the study of the social texture of knowledge, such as scientific institutions, processes of communication, deference and, more importantly for my current concern, the normative aspects of knowledge. 2 Psychologism is coming back! Psychologism, in its crude form, is the doctrine that asserts that “logic is a study of the mind” (Macnamara, 1986:10). This doctrine has been fought against and was officially dismissed by the arguments of Frege (1884, 1893 and 1894) and Husserl (1900). Their main argument was that the truths of logic are objective and independent of psycholog- ical empirical and subjective facts. Psychology deals with what people believe to be true while logic deals with what is necessarily true. Since Frege and Husserl the question seemed to be settled. I will argue, how- ever, that psychologism is still a lively philosophical problem. To begin with, one of the present-day avatars of psychologism is the use of logic in psychology. Let us start with the work of cognitive science on rational behaviour. Cognitive science asserts that human behaviour stems from, and can be accounted for by, cognitive processes. Applied to Mathematics, this means that the production of proofs and mathematical concepts should be explained in terms of cognitive processes. Moreover, one of the most important paradigms in cognitive science asserts that cognitive events are performances rendered possible thanks to some cognitive compe- tences. This paradigm was first initiated by Chomsky’s theory of linguis- tic competence. The modular theory, first enounced by Fodor, specifies the functioning of some cognitive competencies, which are described as mental devices, or modules, which perform specific tasks. Modules are cognitive organs; their function is to perform specified computations on mental representations. Within this framework, it is natural and fruitful to hypothesise the existence of cognitive competencies such as a ‘logic module’ and/or an ‘arithmetical module’, which are mental devices that produce/perform logic and/or arithmetic. Yet, once this assumption is made, the threat of psychologism is not far. Here is the path that leads to it: (1) Producing Mathematics is using our cognitive mental de- vices. (2) Thus Mathematics is the product of mental devices. (3) Mathematics depends on the mental, its truth and content are psychological facts. Let me illustrate my argument with an analysis of Macnamara’s re- search program. Macnamara was a leader cognitive psychologist who studied the cognitive foundations of reasoning while at the same time being conscious about the problem of psychologism, so I think he is a good representative of the theories that I intend to criticise. In “A Bor- der dispute” (1986) Macnamara called for a research program based on the idea that the mind contains some innate devices from which origi- nate our reasoning and that that constitute our basic logical skills. The goal of the program was therefore to discover those devices. He further- more claimed that those devices essentially amount to some “basic logical skills”. As a consequence, logic is the appropriate, and even essential, mathematical tool for the psychology of reasoning. This call was taken seriously, for in 1994, Macnamara edited, together with the G. Reyes, a book called “The Logical Foundation of Cognition” containing articles responding to Macnamara’s program with the logic of types. Our ba- sic logical skills, according to Macnamara, constitute a “mental logic”, which accounts for our linguistic resources of expression and understand- ing and our ability to grasp inference. More generally, “The mind in part of its functioning applies the principles of that [mental] logic”. The men- tal logic is in correspondence with “each ideal logic (true to intuition)” [Macnamara 1986, 22] and includes fundamental principles such as the principle of contradiction. Logical competence is error free and “gives rise to intuition of absolute necessity” [Macnamara 1986, 28]. It consti- tutes a ‘competence’, as opposed to ‘performance’, following Chomsky’s distinction in his theory of universal grammar. That is to say that the mental logic, or logical competence, is not framing all our thoughts as in a Kantian theory (sometimes called transcendental psychologism). The mental logic constitutes an aptitude that we can, and must, call on in order to perform good reasoning. The main purpose of this distinction is to allow the possibility of logical errors in the performance of logical tasks. In that way Macnamara wants to account for the intuitions that: – The layman has the same ideals (in his behaviour) as logic (e.g. consistency). – The building of formal logic is based /founded on basic logical intuitions. – The learning of formal logic is necessarily based on basic logical intuitions. The logical competence “abstracts from logical error, from other psy- chological functioning that accompanies logical thought, and from the specifics of the many devices that could apply the competence” [Macna- mara 1986, 27]. Macnamara defends his theory against the accusation of being psy- chologistic. His claims, he says, “have to do with access to logical prin- ciples, not with justifying them” (1986:42), the latter being the work of logicians. But what does Macnamara mean by “access to logical princi- ples”? It seems that mental logic is a kind of ladder which gives access to the objective realm of logic. In that case the truth is already there; mathematicians describe it and psychologists describe how and why the description is possible. Macnamara, however, explains the human possi- bility of doing logic with the basic principles of logic. When he considers our access to logical connectors he merely asserts that logical connectors are already in our minds. The use we make of connectors is the result of the activation of our logical competence, the mental logic. Hence the description of the access is a generation of the truths of logic plus the assertion that this generation stems from psychological facts. The ladder is exactly our making of logic. Still, Macnamara continues his defence by softening the meaning of ‘access to logical principles’. Mental logic, he says, does not generate logic; it only assesses its validity. So, let us consider how the assessment procedure works. If the basic logical com- petence is just the cause of our conviction, then the assertion is just that we have a feeling of certainty because we have an innate feeling of cer- tainty. This does not provide, as Macnamara claims, “the key element in the psychology of human reasoning”. So the assessment procedure includes some generative consequences, and in particular, the making of logic. It is as if we produced some logical-like propositions, and only the truly logical ones were passing the tests. So here again logic finds its justification in psychological facts, namely passing the assessment tests of mental logic. Logic is in Macnamara’s theory, the very result of our (ideal) performing of the logical competence. The truths and the laws of logic can be reduced to laws of psychology because the formers are just the expression of some characteristics of our mind. But such character- isations of our mind are actually laws of psychology. In brief, to give a mental reality to the laws of logic implies that the objectivity and the normative character of logic stem from the laws of thought.
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