Artículos

Towards a critical epistemology of mathematics

Hacia una epistemología crítica de las matemáticas

Para uma epistemologia crítica da matemática

David Kollosche

(University of Klagenfurt, Austria)

Recibido: 18/07/2023

Aprobado: 18/07/2023

Abstract

This essay addresses a critical epistemology of mathematics as an investigation into the epistemic limitations of mathematical thinking. After arguing for the relevance of a critical epistemology of mathematics, I discuss assumptions underlying standard arithmetic and assumptions underlying standard logic as examples for such epistemic limitations of mathematical thinking. Looking into the work of philosophically interested scholars in mathematics education such as Alan Bishop and Ole Skovsmose, I discuss some early insights for a critical epistemology of mathematics. I conclude that these insights can only be the beginning, that we are yet far away from a proper understanding of the epistemic limitations of mathematics, and that more research is needed.

Keywords: mathematics education. philosophy of mathematics. critical theory. epistemology.

Resumen

Este ensayo aborda una epistemología crítica de las matemáticas como investigación de las limitaciones epistémicas del pensamiento matemático. Tras argumentar a favor de la relevancia de una epistemología crítica de las matemáticas, discuto los supuestos subyacentes a la aritmética estándar y los supuestos subyacentes a la lógica estándar como ejemplos de tales limitaciones epistémicas del pensamiento matemático. Analizando el trabajo de estudiosos de la educación matemática interesados por la filosofía, como Alan Bishop y Ole Skovsmose, discuto algunas de las primeras ideas para una epistemología crítica de las matemáticas. Concluyo que estas ideas sólo pueden ser el principio, que aún estamos lejos de una comprensión adecuada de las limitaciones epistémicas de las matemáticas y que se necesita más investigación.

Palabras clave: educación matemática. filosofía de las matemáticas. la teoría crítica. epistemología.

Resumo

Este ensaio aborda uma epistemologia crítica da matemática como uma investigação das limitações epistêmicas do pensamento matemático. Depois de defender a relevância de uma epistemologia crítica da matemática, discuto os pressupostos subjacentes à aritmética padrão e os pressupostos subjacentes à lógica padrão como exemplos de tais limitações epistêmicas do pensamento matemático. Analisando o trabalho de estudiosos da educação matemática interessados em filosofia, como Alan Bishop e Ole Skovsmose, discuto algumas das primeiras ideias de uma epistemologia crítica da matemática. Concluo que essas ideias podem ser apenas o começo, que ainda estamos longe de uma compreensão adequada das limitações epistêmicas da matemática e que são necessárias mais pesquisas.

Palavras-chave: educação matemática. filosofia da matemática. teoria crítica. epistemologia.

Introduction

According to

All these examples and much of the research in the philosophy of mathematics are concerned with issues internal to mathematics and not with issues of applying mathematics. This is a problem, for, as Wittgenstein (1978) put it, “it is the use outside mathematics […] that makes the sign-game into mathematics” (p. 257); or, in other words, the usefulness of mathematics is its

Quite generally, we might raise the question why we, especially as researchers in mathematics education, should even care about epistemic limitations of mathematics. I argue that an understanding of the mechanisms in which mathematics offers, directs, beguiles, represses and displaces ways of understanding our world

should be an integral part of any mathematical literacy in order to be able to act as a self-determined and critically informed citizen (Skovsmose, 1985),

assists us in understanding cognitive obstacles in learning mathematics (Schneider, 2020),

allows us to understand the interest in or the rejection of mathematics (Kollosche, 2018),

enables us to identify epistemological biases that discriminate specific social groups (e.g, Mendick, 2006)

Given that these dimensions feature critical perspectives on mathematics, on learning mathematics, and on identifying with mathematics, and given that we now put a stronger emphasis on the epistemic

The relevance of a critical epistemology of mathematics can be seen at least in three domains: First, research shows that part of the popular aversion against mathematics is based on a critique of its epistemic premises (Kollosche, 2019). Such critique has already been addressed in philosophy (e.g., Horkheimer & Adorno, 1944/1997) and in mathematics education research (Skovsmose, 2021), yet on the basis of a fragmented theory of a critical epistemology of mathematics. Second, the last decades have seen an increasing awareness for unwanted consequences of the use of mathematics (e.g., O’Neil, 2016; Porter, 1996; Skovsmose, 2005), but these discussions have to remain limited to the consequences alone as long as there is no robust theory of the epistemic limits of mathematics which could explain such consequences. Third, despite the fragmentation of the critical epistemology of mathematics, proponents, especially within mathematics education research, have proposed a reorientation of the epistemology of mathematics. I immediately recall Chapman’s (1993) and Burton’s (1995) call for a “feminist epistemology of mathematics”, but there might be more contemporary examples such as Gutiérrez’ (2012) attempt to reconcile mathematics with indigenous epistemologies. However, what these attempts lack is a reliable account of the epistemological limits of mainstream mathematics, against which such reorientations could be developed.

Two short examples

I want to give two short examples to make my theoretical remarks about epistemological limits of mathematics more concrete. Elsewhere, I discussed how the equation 2+2=4 is taken as an absolute mathematical truth by some, while others try to find examples where 2+2=4 does not hold true (Kollosche, in press). On the one hand, there is no doubt that 2+2=4 is a true statement that can be deduced from any axiom system of the arithmetic of natural numbers. On the other hand, there are limitations to the application of the arithmetic of natural numbers; for example, that any countable object stays one and one object (and does not merge with others, does not destroy others, does not create new objects as offspring, does not itself become less than one).

The first example addresses only one limitation of one mathematical theory (although a central and elementary one). The more pressing question for the epistemology of mathematics is whether there are epistemological limitations that affect mathematics as a whole (or at least most of it). This task can quickly lead to the question what mathematics is – in an attempt to describe its epistemic nature and derive its limitations from that description. However, as we lack a consensus on the definition of mathematics (Hacking, 2014), it might be a pragmatic step to pick epistemic qualities that we agree mathematics to have, and to start from there. For example, even though there is a niche for experimental mathematics and for non-standard logics, mathematics is usually expected to prove its statements on the basis of a logic whose basics have been developed in Ancient Greece. This includes the laws of the excluded middle and of the excluded contradiction, which, taken together, hold that any statement is either true or not true (Kollosche, 2013). But not all worldly statements subject to such a dichotomy: For some people, “Are you in a relationship?” is not a question that can easily be answered with “Yes” or “No”. The same might hold true for the question “Are you a woman?”. Note that a dating website might actually ask these questions and use the answers for a mathematical model to facilitate romantic relationships or sexual encounters. The last question also shows that the problem is not that the question is underdefined: We could ask more specifically “Is your biological sex female?”, but medicine teaches us that even this question cannot in all cases simply be answered with “Yes” or “No”. Thus, again, mathematics, for example, that of our dating website, requires very specific epistemic presumptions in order to be applicable. And vice versa, wherever mathematics is applied, such epistemic presumptions have been made – consciously or not, justified or not. The interesting epistemological question then is: What are these epistemic presumptions of using mathematics? And, returning to my initial question: What are the epistemic limits of mathematics?

Some first answers

I have not yet found any comprehensive theory of the epistemic limits of mathematics, and I am afraid that there is none yet. However, there are answers scattered throughout the research literature in mathematics education and beyond. In this section, I will present and discuss some of these.

In his seminal book

Bishop claimed that

Bishop used the term

Bishop described

Bishop listed

The next “value” that Bishop mentioned is

The last “value” that Bishop (1988) addressed was

All in all, Bishop’s (1988) explanations are sometimes too general and do not address mathematics specifically; and if they do, they are not discussed critically. The literature sources for his discussions are not very broad and the discussions stay short. To his defence, we have to admit that it was never Bishop’s intention to write a critical epistemology of mathematics.

The “limitations of the subject” have been in the focus of Skovsmose’s work since the very beginning of his academic work (Skovsmose, 1985, p. 341). Although he has never attempted to provide a comprehensive theory of the epistemic limits of mathematics, he has discussed such limits on several occasions. Here, I will restrict myself to discussing some ideas from his books

In

A new consideration about the epistemic limits of mathematics can be derived from Skovsmose’s (2009) discussion of Frege’s distinction between sense (

Conclusion and outlook

As the scattered answers above may exemplify, the idea of a critical epistemology of mathematics describes a complex and highly relevant field of inquiry. But these answers also show how random it yet seems who discusses which issues in the field on the basis on what references. Concerning the references, just note that Skovsmose does not even reference or discuss Bishop’s earlier ideas in this context. Concerning the randomness of the issues discussed, just consider the following (incomplete) list of additional questions that would fall under a critical epistemology of mathematics:

Under which conditions (or for which price) can phenomena be counted and measured?

Under which conditions (or for which price) do theories subject to the logic of either-or dichotomies and deduction?

Under which conditions (or for which price) can practical questions be answered on the basis of a formalistic algorithm?

Under which conditions (or for which price) can a concept become a concept in a mathematical theory?

How may we proceed? Apart from the fact that any beginning of a critical epistemology of mathematics deserves a more profound and comprehensive study of what we already know in mathematics education research than I can provide within the limits of this paper, and apart from the fact that a critical epistemology would benefit from establishing a line of discussion where relevant publications refer to each other, it may be fruitful to look for inspiration beyond the limits of mathematics education research, maybe in in the philosophy of mathematics, but certainly in general philosophy. For the latter, I want to provide short teasers to three promising sources:

Although Horkheimer and Adorno’s (1944/1997)

While Hegel built his philosophy on such basic concepts as identity and negation, which might be very close to our standard understanding of mathematics, Deleuze (1968/1994) started an attempt to replace these by the basic concepts of difference and repetition, which gives rise to a very different philosophy and has the potential to understand mathematics as a discipline that is built on such basic epistemic practices as repeating an action and marking a difference. I am grateful to Liz de Freitas for pointing my attention in this direction (and I hope that one day I will also have the time to fully explore this direction).

It is due to Felix Lensing that I can also list the work of Husserl, a world-rank philosopher who goes strangely unnoticed by mathematics education research and the philosophy of mathematics, although he had a doctorate in mathematics and published extensively about philosophy and mathematics. In his early work, Husserl (1891/2003) developed an introduction to the natural numbers on the basis of descriptive psychology, thus identifying specific thought processes that precede the idea of the natural numbers and are inherent to the use of arithmetic. It might be able to reinterpret these thought processes as epistemic assumptions of arithmetic, and they could thus demarcate epistemic limits of mathematics.

Exploring these sources will allow to widen the theoretical background of a critical epistemology of mathematics, but will not provide the field with any structure. Therefore, a next step may be to elaborate further on epistemological limits in general, to devise a possible logic or structure for their study, only to then apply it to mathematics.

I hope that my theoretical considerations, my practical examples, and my insights into selected studies, have at least convinced the reader that such an endeavour is important, if not even motivated the reader to contribute to it.

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