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Added by: Fenner Stanley TanswellAbstract:
According to the received view, genuine mathematical justification derives from proofs. In this article, I challenge this view. First, I sketch a notion of proof that cannot be reduced to deduction from the axioms but rather is tailored to human agents. Secondly, I identify a tension between the received view and mathematical practice. In some cases, cognitively diligent, well-functioning mathematicians go wrong. In these cases, it is plausible to think that proof sets the bar for justification too high. I then propose a fallibilist account of mathematical justification. I show that the main function of mathematical justification is to guarantee that the mathematical community can correct the errors that inevitably arise from our fallible practices.Hamami, Yacin, Morris, Rebecca Lea. Philosophy of mathematical practice: a primer for mathematics educators2020, ZDM, 52(6): 1113-1126.-
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Added by: Fenner Stanley TanswellAbstract:
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.Comment (from this Blueprint): While this paper by Hamami & Morris is not a necessary reading, it provides a fairly broad overview of the practical turn in mathematics. Since it was aimed at mathematics educators, it is a very accessible piece, and provides useful directions to further reading beyond what is included in this blueprint.
Tao, Terence. What is good mathematics?2007, Bulletin of the American Mathematical Society, 44(4): 623-634.-
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Added by: Fenner Stanley TanswellAbstract:
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.Comment (from this Blueprint): Tao is a mathematician who has written extensively about mathematics as a discipline. In this piece he considers what counts as “good mathematics”. The opening section that I’ve recommended has a long list of possible meanings of “good mathematics” and considers what this plurality means for mathematics. (The remainder details the history of Szemerédi’s theorem, and argues that good mathematics also involves contributing to a great story of mathematics. However, it gets a bit technical, so only look into it if you’re particularly interested in the details of the case.)
Cheng, Eugenia. Mathematics, Morally2004, Cambridge University Society for the Philosophy of Mathematics.-
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Added by: Fenner Stanley TanswellAbstract:
A source of tension between Philosophers of Mathematics and Mathematicians is the fact that each group feels ignored by the other; daily mathematical practice seems barely affected by the questions the Philosophers are considering. In this talk I will describe an issue that does have an impact on mathematical practice, and a philosophical stance on mathematics that is detectable in the work of practising mathematicians. No doubt controversially, I will call this issue ‘morality’, but the term is not of my coining: there are mathematicians across the world who use the word ‘morally’ to great effect in private, and I propose that there should be a public theory of what they mean by this. The issue arises because proofs, despite being revered as the backbone of mathematical truth, often contribute very little to a mathematician’s understanding. ‘Moral’ considerations, however, contribute a great deal. I will first describe what these ‘moral’ considerations might be, and why mathematicians have appropriated the word ‘morality’ for this notion. However, not all mathematicians are concerned with such notions, and I will give a characterisation of ‘moralist’ mathematics and ‘moralist’ mathematicians, and discuss the development of ‘morality’ in individuals and in mathematics as a whole. Finally, I will propose a theory for standardising or universalising a system of mathematical morality, and discuss how this might help in the development of good mathematics.
Comment (from this Blueprint): Cheng is a mathematician working in Category Theory. In this article she complains about traditional philosophy of mathematics that it has no bearing on real mathematics. Instead, she proposes a system of “mathematical morality” about the normative intuitions mathematicians have about how it ought to be.
Berninger, Anja. Commemorating Public Figures–In Favour of a Fictionalist Position2020, Journal of Applied Philosophy-
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Added by: Ten-Herng LaiAbstract:
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.Comment (from this Blueprint): This is a persuasive article arguing for a somewhat counter-intutive conclusion. The fictionalist approach, that what we honour is not the historical figure, but some idealised version of them, seems to capture what we actually do in the real world, even if we think we are not doing this. Do compare the position on eliminativism with Frowe's paper.
Lai, Ten-Herng. Objectionable Commemorations, Historical Value, and Repudiatory Honouring, Australasian Journal of Philosophy-
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Added by: Ten-Herng LaiAbstract:
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.Comment (from this Blueprint): Many scholars in this debate have been too charitable to racists, colonialists, oppressors, and their sympathisers. While admirable, I think it is important to expose the flaws of preservationism: there is simply not much value in preservation.
Bell, Macalester. Against Simple Removal: A Defence of Defacement as a Response to Racist Monuments, Journal of Applied Philosophy-
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Added by: Ten-Herng LaiAbstract:
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.Comment (from this Blueprint): Two things should be noted in this paper. First, many have discussed the importance of stopping or blocking the harm of objectionable commemorations. This paper goes a step further and discusses the importance of “answering” the wrong done by these monuments. Second, the paper engages with a “negative” emotion, namely, contempt, that is present at both racist monuments and the effort to confront them. It allows us to see the legitimate role this negative emotion may play in the struggle for equality: contempt can be apt towards inapt contempt expressed through racist monuments. It also nicely spells out the potential practical implications of taking this negative emotion seriously.
Nguyen, C. Thi. Monuments as commitments: How art speaks to groups and how groups think in art2019, Pacific Philosophical Quarterly 100(4), 971-994-
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Added by: Ten-Herng LaiAbstract:
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.Comment (from this Blueprint): This paper highlights the role monuments can play as groups attempt to speak to itself to solidify its own commitment. As a form of art, it can publicly reinforce the commitments, especially through carrying the emotions, attitudes that cannot be easily expressed in propositions, towards certain individuals or ideals. The commitments can be something great, evil, or mediocre. Also consider the fact that art engages with our emotions rather than our rational capacity.
Miranda, Dana Francisco. Critical commemorations2020, Journal of Global Ethics 16(3): 422-430-
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Added by: Ten-Herng LaiAbstract:
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.Comment (from this Blueprint): This paper brings out nicely doubts on the objectivity of history as it is presented to us. The pretence of objective history can be used as an oppressive tool to delegitimise the critical reflection of the history of the marginalised. A particular point of interest is objecting to the standards of "greatness," which could be found very plausible. It seems that we have indeed been honouring people who have done great (from a certain point of view) but terrible things.
Frowe, Helen. The Duty to Remove Statues of Wrongdoers2019, Journal of Practical Ethics 7(3):1-31-
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Added by: Ten-Herng LaiAbstract:
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.Comment (from this Blueprint): This paper highlights several important things. First, statues are blunt tools and express pro-attitudes to the persons they represent as a whole. Second, it sets out a clear standard for removal, and defends the conclusion that we should remove many or even most existing statues. Third, to the question “what if removal incites violence?” this paper provides a good answer. Fourth, a legitimate question is what we should do about statues of wrongdoers of the distant past? The discussion on this here is insightful.
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De Toffoli, Silvia. Groundwork for a Fallibilist Account of Mathematics
2021, The Philosophical Quarterly, 71(4).
Comment (from this Blueprint): De Toffoli makes a strong case for the importance of mathematical practice in addressing important issues about mathematics. In this paper, she looks at proof and justification, with an emphasis on the fact that mathematicians are fallible. With this in mind, she argues that there are circumstances under which we can have mathematical justification, despite a possibility of being wrong. This paper touches on many cases and questions that will reappear later across the Blueprint, such as collaboration, testimony, computer proofs, and diagrams.