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.
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.
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?
Philosophy En Route to Reality: A Bumpy Ride
My intellectual journey in philosophy proceeded along two mountainous paths that coincided at their base, but forked less than halfway up the incline. The first is that of my philosophical development, a steep but steady and continuous ascent. It began in my family, and accelerated in high school, art school, college, and graduate school. Those foundations propelled my philosophical research into the nature of rationality and its relation to the structure of the self, a long-term project focused on the Kantian and Humean metaethical traditions in Anglo-American analytic philosophy. It would have been impossible to bring this project to completion without the anchor, compass, and conceptual mapping provided by my prior, longstanding involvement in the practice and theory of Vedic philosophy. The second path is that of my professional route through the field of academic philosophy, which branched onto a rocky detour in graduate school, followed by a short but steep ascent, followed next by a much steeper, sustained descent off that road, into the ravine, down in flames, and out of the profession. In order to reach the summit of the first path, I had to reach the nadir of the second. It was the right decision. My yoga practice cushioned my landing.
Hatata [II] (~1692)
Translating to ‘an investigation’, this is the second of two 17th century ethical and rational treatises from present-day Ethiopia. Walda Heywat (Wäldä Hewat) continued the work of his mentor, Zera Yacob (Zär’a Ya’eqob, Wärqe), and expanded on it, turning it into more of a practical guide. Hatata (II) is considered to be more in line with more traditional views in its approach to topics such as marriage and abortion. However, where as Zara Yacob’s ideas were relatively individualistic, Walda Heywat was particularly known for his social ethics. In his writing, he states, “God did not create me only for myself, but placed me in the midst of other created [men] who are equal to me.” He also adds, “Man cannot come to existence, grow and serve by himself without the help of other men.”
An Introduction to Africana Philosophy
In this undergraduate textbook Lewis R. Gordon offers the first comprehensive treatment of Africana philosophy, beginning with the emergence of an Africana (i.e. African diasporic) consciousness in the Afro-Arabic world of the Middle Ages. He argues that much of modern thought emerged out of early conflicts between Islam and Christianity that culminated in the expulsion of the Moors from the Iberian Peninsula, and from the subsequent expansion of racism, enslavement, and colonialism which in their turn stimulated reflections on reason, liberation, and the meaning of being human. His book takes the student reader on a journey from Africa through Europe, North and South America, the Caribbean, and back to Africa, as he explores the challenges posed to our understanding of knowledge and freedom today, and the response to them which can be found within Africana philosophy.
The Racism of Philosophy’s Fear of Cultural Relativism
By looking at a canonical article representing academic philosophy’s orthodox view against cultural relativism, James Rachels’ “The Challenge of Cultural Relativism,” this paper argues that current mainstream western academic philosophy’s fear of cultural relativism is premised on a fear of the racial Other. The examples that Rachels marshals against cultural relativism default to the persistent, ubiquitous, and age-old stereotypes about the savage/barbarian Other that have dominated the history of western engagement with the non-western world. What academic philosophy fears about cultural relativism, it is argued, is the barbarians of the western imagination and not fellow human beings. The same structure that informs fears of cultural relativism, whereby people with different customs are reduced to the barbarian/savage of the western imagination, can be seen in the genesis of international law which arose as a justification for the domination of the Amerindian (parsed as “barbarians”). It is argued that implicit in arguments against cultural relativism is the preservation of the same right to dominate the Other. Finally, it is argued that the appeal of the fear of cultural relativism is that, in directing moral outrage at others, one can avoid reflecting on the failures of one’s own cultural tradition.
Hatata [I] (1667)
Translating to ‘an investigation’, this is the first of two 17th century ethical and rational treatises from present-day Ethiopia. Zera Yacob (Zär’a Ya’eqob, Wärqe) is most noted for his philosophy surrounding the principle of harmony. He asserted that an action’s morality is decided by whether it advances or degrades overall harmony in the world. While he did believe in a deity, whom he referred to as God, he criticised several sets of religious beliefs. Rather than deriving beliefs from any organized religion, Yacob sought the truth in observing the natural world. In Hatata, Zera Yacob applied the idea of a first cause to produce a proof for the existence of God, thus proposing a cosmological argument. “If I say that my father and my mother created me, then I must search for the creator of my parents and of the parents of my parents until they arrive at the first who were not created as we [are] but who came into this world in some other way without being generated.” However, the knowability of God does not depend on human intellect, but “Our soul has the power of having the concept of God and of seeing him mentally. God did not give this power purposelessly; as he gave the power, so did he give the reality.” Yacob’s work was continued in a second Hatata by his pupil and patron’s son, Walda Heywat (Wäldä Hewat).