In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice. In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education.
What is good mathematics?
Some personal thoughts and opinions on what “good quality mathematics” is and whether one should try to define this term rigorously. As a case study, the story of Szemer´edi’s theorem is presented.
Commemorating Public Figures–In Favour of a Fictionalist Position
In this article, I discuss the commemoration of public figures such as Nelson Mandela and Yitzhak Rabin. In many cases, our commemoration of such figures is based on the admiration we feel for them. However, closer inspection reveals that most (if not all) of those we currently honour do not qualify as fitting objects of admiration. Yet, we may still have the strong intuition that we ought to continue commemorating them in this way. I highlight two problems that arise here: the problem that the expressed admiration does not seem appropriate with respect to the object and the problem that continued commemorative practices lead to rationality issues. In response to these issues, I suggest taking a fictionalist position with respect to commemoration. This crucially involves sharply distinguishing between commemorative and other discourses, as well as understanding the objects of our commemorative practices as fictional objects.
Objectionable Commemorations, Historical Value, and Repudiatory Honouring
Many have argued that certain statues or monuments are objectionable, and thus ought to be removed. Even if their arguments are compelling, a major obstacle is the apparent historical value of those commemorations. Preservation in some form seems to be the best way to respect the value of commemorations as connections to the past or opportunities to learn important historical lessons. Against this, I argue that we have exaggerated the historical value of objectionable commemorations. Sometimes commemorations connect to biased or distorted versions of history, if not mere myths. We can also learn historical lessons through what I call repudiatory honouring: the honouring of certain victims or resistors that can only make sense if the oppressor(s) or target(s) of resistance are deemed unjust, where no part of the original objectionable commemorations is preserved. This type of commemorative practice can even help to overcome some of the obstacles objectionable commemorations pose against properly connecting to the past.
Against Simple Removal: A Defence of Defacement as a Response to Racist Monuments
In recent years, protesters around the world have been calling for the removal of commemorations honouring those who are, by contemporary standards, generally regarded as seriously morally compromised by their racism. According to one line of thought, leaving racist memorials in place is profoundly disrespectful, and doing so tacitly condones, and perhaps even celebrates, the racism of those honoured and memorialized. The best response is to remove the monuments altogether. In this article, I first argue against a prominent offense-based account of the wrong of simply leaving memorials in place, unaltered, before offering my own account of this wrong. In at least some cases, these memorials wrong insofar as they express and exemplify a morally objectionable attitude of race-based contempt. I go on to argue that the best way of answering this disrespect is through a process of expressively “dehonouring” the subject. Removal of these commemorations is ultimately misguided, in many cases, because removal, by itself, cannot adequately dishonour, and simple removal does not fully answer the ways in which these memorials wrong. I defend a more nuanced approach to answering the wrong posed by these monuments, and I argue that public expressions of contempt through defacement have an ineliminable role to play in an apt dishonouring process.
Monuments as commitments: How art speaks to groups and how groups think in art
Art can be addressed, not just to individuals, but to groups. Art can even be part of how groups think to themselves – how they keep a grip on their values over time. I focus on monuments as a case study. Monuments, I claim, can function as a commitment to a group value, for the sake of long-term action guidance. Art can function here where charters and mission statements cannot, precisely because of art’s powers to capture subtlety and emotion. In particular, art can serve as the vessel for group emotions, by making emotional content sufficiently public so as to be the object of a group commitment. Art enables groups to guide themselves with values too subtle to be codified.
Critical commemorations
Drawing on the works of Friedrich Nietzsche, this contribution will examine commemorative practices alongside critical modes of historical engagement. In Untimely Meditations, Friedrich Nietzsche documents three historical methodologies—the monumental, antiquarian and critical—which purposely use history in non-objective ways. In particular, critical history desires to judge and reject historical figures rather than repeat the past or venerate the dead. For instance, in recent protests against racism there have also been calls to decolonize public space through the defacement, destruction, and removal of monuments. There is thus much potential in critical history being used to address ongoing harms.
Colonial monuments as slurring speech acts
In recent years, the removal of monuments which glorify historical figures associated with racism and colonialism has become one of the most visible and contested forms of decolonisation. Yet many have objected that there is educational value in leaving such monuments standing. In this paper, I argue that public monuments can be understood as speech acts which communicate messages to those who live among them. Some of those speech acts derogate particular social groups, contributing to their marginalisation in much the way that slurs do. Comparing derogating monuments to slurs is also productive in suggesting morally appropriate responses to their harms. I explore the limits of the use-mention distinction in relation to the harmfulness of slurs and apply this to show that attempting to recontextualise harmful monuments in situ—by, for example, changing the text on an accompanying plaque in order to retain the monument for its educational value—will not solve the problem in most cases. I conclude that the removal of slurring monuments, or their relocation to museum exhibitions dedicated to presenting a more critical view of history, is a more robust and reliable way of protecting against harm, and that this consideration outweighs any purported educational value in leaving monuments in place.
The Duty to Remove Statues of Wrongdoers
This paper argues that public statues of persons typically express a positive evaluative attitude towards the subject. It also argues that states have duties to repudiate their own historical wrongdoing, and to condemn other people’s serious wrongdoing. Both duties are incompatible with retaining public statues of people who perpetrated serious rights violations. Hence, a person’s being a serious rights violator is a sufficient condition for a state’s having a duty to remove a public statue of that person. I argue that this applies no less in the case of the ‘morally ambiguous’ wrongdoer, who both accomplishes significant goods and perpetrates serious rights violations. The duty to remove a statue is a defeasible duty: like most duties, it can be defeated by lesser-evil considerations. If removing a statue would, for example, spark a violent riot that would risk unjust harm to lots of people, the duty to remove could be outweighed by the duty not to foreseeably cause unjust harm. This would provide a lesser-evil justification for keeping the statue. But it matters that the duty to remove is outweighed, rather than negated, by these consequences. Unlike when a duty is negated, one still owes something in cases of outweighing. And it especially matters that it is outweighed by the predicted consequences of wrongful behaviour by others.
Is it Wrong to Topple Statues and Rename Schools?
In recent years, campaigns across the globe have called for the removal of objects symbolic of white supremacy. This paper examines the ethics of altering or removing such objects. Do these strategies sanitize history, destroy heritage and suppress freedom of speech? Or are they important steps towards justice? Does removing monuments and renaming schools reflect a lack of parity and unfairly erase local identities? Or can it sometimes be morally required, as an expression of respect for the memories of people who endured past injustices; a recognition of this history’s ongoing legacies; and a repudiation of unjust social hierarchies?