Far from being unwelcome or impossible in a mathematical setting, indeterminacy in various forms can be seen as playing an important role in driving mathematical research forward by providing “sources of newness” in the sense of Hutter and Farías :434–449, 2017). I argue here that mathematical coincidences, phenomena recently under discussion in the philosophy of mathematics, are usefully seen as inducers of indeterminacy and as put to work in guiding mathematical research. I suggest that to call a pair of mathematical (...) facts a coincidence is roughly to suggest that the investigation of connections between these facts isn’t worthwhile. To say of this pair, “That’s no coincidence!” is to suggest just the opposite. I further argue that this perspective on mathematical coincidence, which pays special attention to what mathematical coincidences do, may provide us with a better view of what mathematical coincidences are than extant accounts. I close by reflecting on how understanding mathematical coincidences as generating indeterminacy accords with a conception of mathematical research as ultimately aiming to reduce indeterminacy and complexity to triviality as proposed in Rota Indiscrete thoughts, Birkhäuser, 1997). (shrink)

This work gives a new argument for ‘Empirical Philosophy of Mathematical Practice’. It analyses different modalities on how empirical information can influence philosophical endeavours. We evoke the classical dichotomy between “armchair” philosophy and empirical/experimental philosophy, and claim that the latter should in turn be subdivided in three distinct styles: Apostate speculator, Informed analyst, and Freeway explorer. This is a shift of focus from the source of the information towards its use by philosophers. We present several examples from philosophy of mind/science (...) and ethics on one side and a case study from philosophy of mathematics on the other. We argue that empirically informed philosophy of mathematics is different from the rest in a way that encourages a Freeway explorer approach, because intuitions about mathematical objects are often unavailable for non-mathematicians. This consideration is supported by a case study in set theory. (shrink)

Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, ready to (...) meet them, who have explicitly invoked virtues in discussing what is necessary for a mathematician to succeed. In both ethics and epistemology, virtue theory tends to emphasize character virtues, the acquired excellences of people. But people are not the only sort of thing whose excellences may be identified as virtues. Theoretical virtues have attracted attention in the philosophy of science as components of an account of theory choice. Within the philosophy of mathematics, and mathematics itself, attention to virtues has emerged from a variety of disparate sources. Theoretical virtues have been put forward both to analyse the practice of proof and to justify axioms; intellectual virtues have found multiple applications in the epistemology of mathematics; and ethical virtues have been offered as a basis for understanding the social utility of mathematical practice. Indeed, some authors have advocated virtue epistemology as the correct epistemology for mathematics (and perhaps even as the basis for progress in the metaphysics of mathematics). This topical collection brings together several of the researchers who have begun to study mathematical practices from a virtue perspective with the intention of consolidating and encouraging this trend. (shrink)

The literature on mathematical explanation contains numerous examples of explanatory, and not so explanatory proofs. In this paper we report results of an empirical study aimed at investigating mathematicians’ notion of explanatoriness, and its relationship to accounts of mathematical explanation. Using a Comparative Judgement approach, we asked 38 mathematicians to assess the explanatory value of several proofs of the same proposition. We found an extremely high level of agreement among mathematicians, and some inconsistencies between their assessments and claims in the (...) literature regarding the explanatoriness of certain types of proofs. (shrink)

There are different narratives on mathematics as part of our world, some of which are more appropriate than others. Such narratives might be of the form ‘Mathematics is useful’, ‘Mathematics is beautiful’, or ‘Mathematicians aim at theorem-credit’. These narratives play a crucial role in mathematics education and in society as they are influencing people’s willingness to engage with the subject or the way they interpret mathematical results in relation to real-world questions; the latter yielding important normative considerations. Our strategy is (...) to frame current narratives of mathematics from a virtue-theoretic perspective. We identify the practice of mathematizing, put forward by Freudenthal’s ‘Realistic mathematics education’, as virtuous and use it to evaluate different narratives. We show that this can help to render the narratives more adequately, and to provide implications for societal organization. (shrink)

In this paper, I propose that applying the methods of data science to “the problem of whether mathematical explanations occur within mathematics itself” (Mancosu 2018) might be a fruitful way to shed new light on the problem. By carefully selecting indicator words for explanation and justification, and then systematically searching for these indicators in databases of scholarly works in mathematics, we can get an idea of how mathematicians use these terms in mathematical practice and with what frequency. The results of (...) this empirical study suggest that mathematical explanations do occur in research articles published in mathematics journals, as indicated by the occurrence of explanation indicators. When compared with the use of justification indicators, however, the data suggest that justifications occur much more frequently than explanations in scholarly mathematical practice. The results also suggest that justificatory proofs occur much more frequently than explanatory proofs, thus suggesting that proof may be playing a larger justificatory role than an explanatory role in scholarly mathematical practice. (shrink)

Many of the methods commonly used to research mathematical practice, such as analyses of historical episodes or individual cases, are particularly well-suited to generating causal hypotheses, but less well-suited to testing causal hypotheses. In this paper we reflect on the contribution that the so-called hypothetico-deductive method, with a particular focus on experimental studies, can make to our understanding of mathematical practice. By way of illustration, we report an experiment that investigated how mathematicians attribute aesthetic properties to mathematical proofs. We demonstrate (...) that perceptions of the aesthetic properties of mathematical proofs are, in some cases at least, subject to social influence. Specifically, we show that mathematicians’ aesthetic judgements tend to conform to the judgements made by others. Pedagogical implications are discussed. (shrink)

В монографии представлены несколько смысловых блоков, связанных с восприятием и интерпретацией человеком исторического бытия. Ранние греческие мыслители пытались получить доступ к исходникам (началам) бытия, и эти интенции легли в основу научного знания, а также привели к появлению метафизики. В классической (и в неклассической) метафизике за основу была принята догма Пифагора и Платона о неизменности подлинной реальности, из чего следовало отрицание бытийного характера времени. Автор монографии отказывается от этой догмы и предлагает стратегию обновления метафизики и перехода ее к новому — постнеклассическому (...) — этапу. Поскольку трансформационный потенциал реальности наиболее явно представлен в историческом становлении, авторская стратегия реализуется в метафизике истории. При этом выявляются начала истории, в каком качестве выступают «дух», «культурно-историческая матрица» и «метафизический проект организации бытия» («онтологический проект»). Исследуются формы и механизмы «работы» человека с прошлым, обсуждаются вопросы, связанные с постижением логики исторического процесса, а также с ролью (творческой) личности в истории. Анализируются причины цивилизационных кризисов и очерчиваются возможные пути их разрешения, предлагается метафизическая интерпретация социальных и военных конфликтов. Завершает исследование рассмотрение трансформации человеческого естества под влиянием информационных, био- и нанотехнологий. (shrink)

I offer a set of sufficient conditions for beauty, drawing on Parsons and Carlson’s account of ‘functional beauty’. First, I argue that their account is flawed, whilst falling short of...

This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and assessing (...) the relation between cognitive processes and certain properties of the stimuli at which they are directed. (shrink)

ABSTRACT This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call ‘aesthetic-epistemic feelings’ that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and (...) assessing the relation between cognitive processes and certain properties of the stimuli at which they are directed. (shrink)

Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, how do (...) they relate to the sorts of explanation encountered in philosophy of science and metaphysics? (shrink)

We investigated whether mathematicians typically agree about the qualities of mathematical proofs. Between-mathematician consensus in proof appraisals is an implicit assumption of many arguments made by philosophers of mathematics, but to our knowledge the issue has not previously been empirically investigated. We asked a group of mathematicians to assess a specific proof on four dimensions, using the framework identified by Inglis and Aberdein (2015). We found widespread disagreement between our participants about the aesthetics, intricacy, precision and utility of the proof, (...) suggesting that a priori assumptions about the consistency of mathematical proof appraisals are unreasonable. (shrink)

Philosophers of science discuss whether theory selection depends on aesthetic judgments or criteria, and whether these putatively aesthetic features are genuinely extra-epistemic. As examples, judgments involving criteria such as simplicity and symmetry are often cited. However, other theory selection criteria, such as fecundity, coherence, internal consistency, and fertility, more closely match those criteria used in art contexts and by scholars working in aesthetics. Paying closer attention to the way these criteria are used in art contexts allows us to understand some (...) evaluative and developmental practices in scientific theory selection as genuinely aesthetic, enlarging the scope of the goals of science. (shrink)

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. (shrink)

This is a teaching and learning guide to accompany "Explanation in Mathematics: Proofs and Practice".

Mathematicians often conduct aesthetic judgements to evaluate mathematical objects such as equations or proofs. But is there a consensus about which mathematical objects are beautiful? We used a comparative judgement technique to measure aesthetic intuitions among British mathematicians, Chinese mathematicians, and British mathematics undergraduates, with the aim of assessing whether judgements of mathematical beauty are influenced by cultural differences or levels of expertise. We found aesthetic agreement both within and across these demographic groups. We conclude that judgements of mathematical beauty (...) are not strongly influenced by cultural difference, levels of expertise, and types of mathematical objects. Our findings contrast with recent studies that found mathematicians often disagree with each other about mathematical beauty. (shrink)

Our meta-argumentative vocabulary supplies the conceptual tools used to reflectively analyse, regulate, and evaluate our argumentative performances. Yet, this vocabulary is susceptible to misunderstanding and abuse in ways that make possible new discursive mistakes and pathologies. Thus, our efforts to self-regulate our reason-transacting practices by articulating their norms makes possible new ways to violate and flout those very norms. Scott Aikin identifies the structural possibility of this vicious feedback loop as the Owl of Minerva Problem. In the spirit of a (...) shared concern for the flourishing or our rational, argumentative practices, this paper approaches the Owl of Minerva Problem from a vantage point that, by comparison with Aikin’s, affords perspectives that are more pessimistic in some aspects and more optimistic in others. Pessimistically, the problem at the root of the weaponization of our meta-argumentative vocabulary is motivational, not structural. Its motivational nature explains its resistance to the normal repertoire of reparative argumentative maneuvers, as well as revealing a profound and deeply entrenched misunderstanding of the connection between our reasons-transacting practices and the goods achievable within them. Optimistically, in the absence of this motivational problem, the misunderstandings and errors made possible by our meta-argumentative vocabularies are amenable to remedy by familiar techniques of discursive instruction and repair. More optimistically, even though our meta-argumentative vocabularies are generated only retrospectively, they can be used prospectively, thereby making possible an aspirational motivation resulting in a virtuous cycle of increasingly autonomous normative self-regulation. Properly harnessed, the Owl of Minerva releases the Lark of Arete. (shrink)

In recent years, substructural approaches to paradoxes have become quite popular. But whatever restrictions on structural rules we may want to enforce, it is highly desirable that such restrictions be accompanied by independent philosophical motivation, not directly related to paradoxes. Indeed, while these recent developments have shed new light on a number of issues pertaining to paradoxes, it seems that we now have even more open questions than before, in particular two very pressing ones: what (independent) motivations do we have (...) (if any) for restrictions on structural rules, and what to make of the plurality of new logics emerging from these restrictions, i.e. how to ‘choose’ among the different options. In this paper, we address these two questions from the perspective of a dialogical conception of logic that we've been advocating in recent years. We will argue that dialogical interpretations of structural rules, that is, as rules determining specific properties of the dialogues in question, provide a conveniently neutral framework to adjudicate between the different substructural proposals that have been made in the literature on paradoxes. (shrink)

We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept (...) to highlight a potential danger of intellectual enculturation. (shrink)

Over the last century, there have been considerable variations in the frequency of use and types of diagrams used in mathematical publications. In order to track these changes, we developed a method enabling large-scale quantitative analysis of mathematical publications to investigate the number and types of diagrams published in three leading mathematical journals in the period from 1885 to 2015. The results show that diagrams were relatively common at the beginning of the period under investigation. However, beginning in 1910, they (...) were almost completely unused for about four decades before reappearing in the 1950s. The diagrams from the 1950s, however, were of a different type than those used earlier in the century. We see this change in publication practice as a clear indication that the formalist ideology has influenced mathematicians’ choice of representations. Although this could be seen as a minor stylistic aspect of mathematics, we argue that mathematicians’ representational practice is deeply connected to their cognitive practice and to the contentual development of the discipline. These changes in publication style therefore indicate more fundamental changes in the period under consideration. (shrink)

A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.

Aesthetics in philosophy of mathematics is too narrowly construed. Beauty is not the only feature in mathematics that is arguably aesthetic. While not the highest aesthetic value, being interesting is a sine qua non for publishability. Of the many ways to be interesting, being explanatory has recently been discussed. The motivational power of what is interesting is important for both directing research and stimulating education. The scientific satisfaction of curiosity and the artistic desire for beautiful results are complementary but both (...) aesthetic. (shrink)

Additional theorizing about mathematical practice is needed in order to ground appeals to truly useful notions of the virtues in mathematics. This paper aims to contribute to this theorizing, first, by characterizing mathematical practice as being epistemic and “objectual” in the sense of Knorr Cetina The practice turn in contemporary theory, Routledge, London, 2001). Then, it elaborates a MacIntyrean framework for extracting conceptions of the virtues related to mathematical practice so understood. Finally, it makes the case that Wittgenstein’s methodology for (...) examining mathematics and its practice is the most appropriate one to use for the actual investigation of mathematical practice within this MacIntyrean framework. At each stage of thinking through mathematical practice by these means, places where new virtue-theoretic questions are opened up for investigation are noted and briefly explored. (shrink)