Disagreements that resist rational resolution, often termed “deep disagreements”, have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi (...) Mochizuki’s work on the abc conjecture as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice. (shrink)

If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. (...) Their ability to recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, (...) I argue that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

We examine the influence of word choices on mathematical practice, i.e. in developing definitions, theorems, and proofs. As a case study, we consider Euclid’s and Euler’s word choices in their influential developments of geometry and, in particular, their use of the term ‘polyhedron’. Then, jumping to the twentieth century, we look at word choices surrounding the use of the term ‘polyhedron’ in the work of Coxeter and of Grünbaum. We also consider a recent and explicit conflict of approach between Grünbaum (...) and Shephard on the one hand and that of Hilton and Pedersen on the other, elucidating that the conflict was engendered by disagreement over the proper conceptualization, and so also the appropriate word choices, in the study of polyhedra. (shrink)

In his book On What Matters, Derek Parfit defends a version of moral non-naturalism, a view according to which there are objective normative truths, some of which are moral truths, and we have a reliable way of discovering them. These moral truths do not exist, however, as parts of the natural universe nor in Plato’s heaven. While explaining in what way these truths exist and how we discover them, Parfit makes analogies between morality on the one hand, and mathematics (...) and logic on the other. Moral truths “exist” in a way that numbers exist, and we discover these truths in a similar way as we discover truths about numbers. By the end of the second volume, Parfit also responds to a powerful objection against his view, an objection based on the phenomenon of moral disagreement. If people widely and deeply disagree about what is the moral truth, it is doubtful whether we have a reliable way of discovering it. In his reply, he claims that in ideal conditions for thinking about moral questions, we would all have sufficiently similar moral beliefs. However, we often find ourselves in less-than-ideal conditions due to various factors that distort our ability to agree. Therefore, differences in moral opinion can be expected. In this paper, I draw a connection between these parts of Parfit’s theory and comment on them. Firstly, I argue that Parfit’s analogy with mathematics and logic and his answer to the disagreement objection are in tension because there are important epistemic differences between morality and these fields. If one would try to account for the differences, one would have to sacrifice some measure of similarity between morality and them. Secondly, I comment on Parfit’s reply to the disagreement objection itself. I believe that, although his description of ideal conditions has some potential for reaching moral agreement, it may be difficult to tell if ideal conditions prevail. This obscurity spells further trouble for Parfit’s overall theory. (shrink)

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...) and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

A holobiont is a composite organism consisting of a host together with its microbiome, such as a coral with its zooxanthellae. To explain the often intimate integration between hosts and their microbiomes, some investigators contend that selection operates on holobionts as a unit and view the microbiome’s genes as extending the host’s nuclear genome to jointly comprise a hologenome. Because vertical transmission of microbiomes is uncommon, other investigators contend that holobiont selection cannot be effective because a holobiont’s microbiome is an (...) acquired condition rather than an inherited trait. This disagreement invites a simple mathematical model to see how holobiont selection might operate and to assess its plausibility as an evolutionary force. This paper presents two variants of such a model. In one variant, juvenile hosts obtain microbiomes from their parents (vertical transmission). In the other variant, microbiomes of juvenile hosts are assembled from source pools containing the combined microbiomes of all parents (horizontal transmission). According to both variants, holobiont selection indeed causes evolutionary change in holobiont traits. Therefore, holobiont selection is plausibly an effective evolutionary force with either mode of microbiome transmission. The modeling employs two distinct concepts of inheritance, depending on the mode of microbiome transmission: collective inheritance whereby juveniles inherit a sample of the collected genomes from all parents, as contrasted with lineal inheritance whereby juveniles inherit the genomes from only their own parents. A distinction between collective and lineal inheritance also features in theories of multilevel selection. (shrink)

When evidence-based policymaking is so often mired in disagreement and controversy, how can we know if the process is meeting its stated goals? We develop a novel mathematical model to study disagreements about adequate knowledge utilization, like those regarding wild horse culling, shark drumlines and facemask policies during pandemics. We find that, when stakeholders disagree, it is frequently impossible to tell whether any party is at fault. We demonstrate the need for a distinctive kind of transparency in evidence-based policymaking, (...) which we call transparency of reasoning. Such transparency is critical to the success of the evidence-based policy movement, as without it, we will be unable to tell whether in any instance a policy was in fact based on evidence. (shrink)

It may be a myth that Plato wrote over the entrance to the Academy “Let no-one ignorant of geometry enter here.” But it is a well-chosen motto for his view in the Republic that mathematical training is especially productive of understanding in abstract realms, notably ethics. That view is sound and we should return to it. Ethical theory has been bedevilled by the idea that ethics is fundamentally about actions (right and wrong, rights, duties, virtues, dilemmas and so on). That (...) is an error like the one Plato mentions of thinking mathematics is about actions (of adding, constructing, extracting roots and so on). Mathematics is about eternal relations between universals, such as the ratio of the diagonal of a square to the side. Ethics too is about eternal verities, such as the equal worth of persons and just distributions. Mathematical and ethical verities do both constrain actions, such as the possibility of walking over the seven bridges of Königsberg once and once only or of justly discriminating between races. But they are not themselves about action. In principle, neither mathematical nor ethical verities are subject to historical forces or disagreement among tribes (though they can be better understood as time goes on). Plato is right: immersion in mathematics induces an understanding of the necessities underpinning reality, an understanding that is essential for distinguishing objective ethics from tribal custom. Equality, for example, is an abstract concept which is foundational for both mathematics and ethics. (shrink)

In this article I present a disagreement between classical and constructive approaches to predicativity regarding the predicative status of so-called generalised inductive definitions. I begin by offering some motivation for an enquiry in the predicative foundations of constructive mathematics, by looking at contemporary work at the intersection between mathematics and computer science. I then review the background notions and spell out the above-mentioned disagreement between classical and constructive approaches to predicativity. Finally, I look at possible ways of defending (...) the constructive predicativity of inductive definitions. (shrink)

In a lot of domains in metaphysics the tacit assumption has been that whichever metaphysical principles turn out to be true, these will be necessarily true. Let us call necessitarianism about some domain the thesis that the right metaphysics of that domain is necessary. Necessitarianism has flourished. In the philosophy of maths we find it held that if mathematical objects exist, then they do of necessity. Mathematical Platonists affirm the necessary existence of mathematical objects (see for instance Hale and Wright (...) 1992 and 1994; Wright 1983 and 1988; Schiffer 1996; Resnik 1997; Shapiro 1997 and Zalta 1988) while mathematical nominalists, usually in the form of fictionalists, hold that necessarily such objects fail to exist (see for instance Balaguer 1996 and 1998; Rosen 2001 and Yablo 2005). In metaphysics more generally, until recently it was more or less assumed that whatever the right account of composition—the account of under what conditions some xs compose a y—that account will be necessarily true (for a discussion of theories of composition see Simons 1987 and van Inwagen 1987 and 1990; the modal status of the composition relation is explicitly addressed in Schaffer 2007; Parsons 2006 and Cameron 2007). Similarly, it has generally been assumed that whatever the right account is of the nature of properties, whether they be universals, tropes, or whether nominalism is true, that account will be necessarily true (though see Rosen 2006 for a recent suggestion to the contrary). In considering theories of persistence it has been widely held that whether objects endure or perdure through time is a matter of necessity (Sider 2001; though see Lewis 1999 p227 who defends contingent perdurantism). And with respect to theories of time it is frequently held that whichever of the A- or B-theory is true is necessarily true. A-theorists often argue that there is time in a world only if the A-theory is true at that world (see for instance McTaggart 1903; Markosian 2004; Bigelow 1996; Craig 2001) while B-theorists often argue that the A-theory is internally inconsistent (Smart 1987; Mellor 1998; Savitt 2000 and Le Poidevin 1991). Once again, we find a few recent contingentist dissenters. Bourne (2006) suggests that it is a contingent feature of time that it is tensed, and thus that the A-theory is contingently true. Worlds in which there exist only B-theoretic properties are worlds with time, it is just that time in those worlds time is radically different to the way it is actually. Other defenders of the B-theory, though not expressly contingentists, do offer arguments against versions of the A-theory that try to show that such A-theories theories are inconsistent with the actual laws of nature (see for instance Saunders 2002 and Callender 2000); these arguments, at least, leave room for the possibility that the A-theories in question are contingently false (at least on the assumption that the laws of nature are themselves contingent, an assumption that not everyone accepts). Despite some notable exceptions, necessitarianism has flourished in many, if not most, domains in metaphysics. One such exception is Lewis’ famous defence of Humean supervenience as a contingent claim about our world. Lewis does not argue that necessarily, the supervenience base for all matters of fact in a world is nothing but a vast mosaic of local matters of particular fact. Rather, he thinks that we have reason to think that our world is one in which Humean supervenience holds (see Lewis 1986 p9-10 and 1994). Another exception to the necessitarian orthodoxy is to be found in the lively debate about the modal status of the laws of nature. Here, if anything, contingentism has been the dominating force, with it generally being held that there are possible worlds in which different laws of nature hold (this view is defended by, among others, Lewis 1986 and 2010; Schaffer 2005 and Sidelle 2002). Necessitarian dissenters hold that the laws of nature are necessary, frequently because they think it is necessary that fundamental properties have the causal or nomic profiles they do (see for instance Shoemaker 1980 and 1988; Swoyer 1982; Bird 1995; Ellis and Lierse 1994). Nevertheless, when it comes to thinking about the nature of the laws themselves, the necessitarian presumption is back on firm footing. Though there is disagreement about whether the laws are generalisations that feature in the most virtuous true axiomatisation of all the particular matters of fact (often known as the Humean view of laws and defended by Ramsey 1978; Lewis 1986 and Beebee 2000) or whether laws are relations of necessity that hold between universals (a view defended by Armstrong 1983; Dretske 1977; Tooley 1977 and Carroll 1990) no one has seriously suggested that it might be a contingent matter which of these is the right account of laws. The necessitarian orthodoxy is not surprising since metaphysics is largely an a priori process. While a priori reasoning may be used to determine whether a proposition is necessary or contingent, it is not well placed (in the absence of a posteriori evidence) to determine whether a contingent proposition is actually true or false. Since metaphysicians aim to tell us which principles are true in which worlds, on the face of it the discovery that metaphysical principles are contingent seems to make part of the task of metaphysics epistemically intractable. In what follows I consider two reasons one might end up embracing contingentism and whether this would lead one into epistemic difficulty. The following section considers a route to contingent metaphysical truths that proceeds via a combination of conceptual necessities and empirical discoveries. Section 3 considers whether there might be synthetic contingent metaphysical truths, and the final section raises the question of whether if there were such truths we would be well placed to come to know them. (shrink)

This essay is concerned with Graham Bird’s treatment, in The Revolutionary Kant, of Kant’s mathematical antinomies. On Bird’s interpretation, our error in these antinomies is to think that we can settle certain issues about the limits of physical reality by pure reason whereas in fact we cannot settle them at all. On the rival interpretation advocated in this essay, it is not true that we cannot settle these issues. Our error is to presuppose that the concept of the unconditioned has (...) application to physical reality. Once this presupposition has been abandoned, we can retrieve sound arguments from the antinomies, not indeed to demonstrate that the views originally being defended are correct, but to demonstrate that the views originally being attacked are incorrect. The essay concludes with some comments concerning how this disagreement relates to a broader disagreement about the best way to understand Kant. (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)

This paper aims to show that Frege’s and Hilbert’s mutual disagreement results from different notions of Anschauung and their relation to axioms. In the first section of the paper, evidence is provided to support that Frege and Hilbert were influenced by the same developments of 19th-century geometry, in particular the work of Gauss, Plücker, and von Staudt. The second section of the paper shows that Frege and Hilbert take different approaches to deal with the problems that the developments in 19th-century (...) geometry posed for the traditional Kantian philosophy of mathematics. Frege maintains that Anschauung is a source of knowledge by which we acknowledge geometrical axioms as true. For Hilbert, in contrast, axioms provide one of several correct “pictures” of reality. Hilbert’s position is thereby deeply influenced by epistemological ideas from Hertz and Helmholtz, and, in turn, influenced the neo-Kantian Cassirer. (shrink)

The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are (...) postulated to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus over observations, but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles stereotypical philosophy. We conclude by distinguishing two ideas that have long been associated -- realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, these domains fail to be objective. One upshot of the discussion is that even questions of fundamental physics may fail to be objective in roughly the sense that the question, ‘Is the Parallel Postulate is true?’, understood as one of pure mathematics, fails to be. Another is a kind of pragmatism. Factual questions in mathematics, modality, logic, and evaluative areas go proxy for non-factual practical ones. -/- . (shrink)

Wright's _Truth and Objectivity_ seeks to systematise a variety of anti-realist positions. I argue that many objections to the system are avoided by transposing its talk of truth into talk of warrant. However, a problem remains about debates involving 'direction-of-fit'. -/- Dummett introduced 'anti-realism' as a philosophical view informed by mathematical intuitionism. Subsequently, the term has been associated with many debates, ancient and modern. _Truth and Objectivity_ proposes that truth admits of different characteristics; these various debates then concern which characteristics (...) truth has, in a given area. This pluralism of truth is at odds with deflationism. I find fault with Wright's argument against deflationism. However, transmission of warrant across the Disquotational Schema suffices to ground Wright's proposal, which survives as a pluralism of classes of warrant. -/- The two main debates concern whether truths are always knowable (Epistemic Constraint) and whether disagreements in an area must be down to some fault of one of those involved (Cognitive Command). I introduce Assertoric Constraint, relating to Epistemic Constraint, where truths cannot outstrip the availability of warrant for their assertion. I solve a structural problem by a comparison with a constitutive analysis of Moore's Paradox. The relativism of blameless disagreement is problematic. Wright's response invokes a sort of ignorance which he calls `Quandary'. I criticise this before proposing an alternative. -/- I agree with Wright that Dummett's original anti-realism does not belong among the positions which Wright seeks to systematise. However, two candidates show that the proposal suffers a weakness. Wright thinks Expressivism misguided, and implicitly rules out his earlier non-cognitivism about necessity. I argue that Expressivism has promise, and I endorse Wright's Cautious Man argument for non-cognitivism about necessity; both involve play with `direction-of-fit'. I conclude that this sort of anti-realist debate needs to be accommodated by the proposal. (shrink)

This paper examines Hobbes’s criticisms of Robert Boyle’s air-pump experiments in light of Hobbes’s account in _De Corpore_ and _De Homine_ of the relationship of natural philosophy to geometry. I argue that Hobbes’s criticisms rely upon his understanding of what counts as “true physics.” Instead of seeing Hobbes as defending natural philosophy as “a causal enterprise … [that] as such, secured total and irrevocable assent,” 1 I argue that, in his disagreement with Boyle, Hobbes relied upon his understanding of natural (...) philosophy as a mixed mathematical science. In a mixed mathematical science one can mix facts from experience with causal principles borrowed from geometry. Hobbes’s harsh criticisms of Boyle’s philosophy, especially in the _Dialogus Physicus, sive De natura aeris_, should thus be understood as Hobbes advancing his view of the proper relationship of natural philosophy to geometry in terms of mixing principles from geometry with facts from experience. Understood in this light, Hobbes need not be taken to reject or diminish the importance of experiment/experience; nor should Hobbes’s criticisms in _Dialogus Physicus_ be understood as rejecting experimenting as ignoble and not befitting a philosopher. Instead, Hobbes’s viewpoint is that experiment/experience must be understood within its proper place – it establishes the ‘that’ for a mixed mathematical science explanation. (shrink)

In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglorious' revolutions--only the former preserves all 'the key components of a theory' [1]. A widespread view, expressed in these terms, is that empirical science characteristically exhibits inglorious revolutions but that revolutions in mathematics are at most glorious [2]. Here are three possible responses: 0. Accept that empirical science and mathematics are methodologically discontinuous; 1. Argue that mathematics can exhibit inglorious revolutions; 2. Deny that (...) inglorious revolutions are characteristic of science. Where Aberdein and Read take option 1, option 2 is preferred by Mizrahi [3]. This paper seeks to resolve this disagreement through consideration of some putative mathematical revolutions. [1] Andrew Aberdein and Stephen Read, The philosophy of alternative logics, The Development of Modern Logic (Leila Haaparanta, ed.), Oxford University Press, Oxford, 2009, pp. 613-723. [2] Donald Gillies (ed.), Revolutions in Mathematics, Oxford University Press, Oxford, 1992. [3] Moti Mizrahi, Kuhn's incommensurability thesis: What's the argument?, Social Epistemology 29 (2015), no. 4, 361-378. (shrink)

Introduction to the Synthese Special Issue on Disagreement in Science.

Clinical psychology is characterized by persistent disagreement about fundamental aspects of the discipline ranging from what mental disorders are to what constitutes effective treatment. Attempts to address the problem of epistemic disagreement have been frequently based on establishing the correct answer by fiat without identifying and addressing the sources of the disagreement. We argue that this strategy has not worked very well and the result is frequently ongoing and intractable disagreement, with each side in an argument convinced they are correct. (...) In this paper, we outline an epistemic disagreement procedural model intended to assist researchers and clinicians in the field of clinical psychology to identify, explore, and develop inquiry strategies that capitalize on situations where competing knowledge claims are made. The result is a flexible conceptual framework committed to a defensible epistemic pluralism, fallibilism, and contextualism, across the various research tasks constituting the field of clinical psychology. (shrink)

Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of mathematical reasoning closely resemble patterns of reasoning in nonmathematical domains. Hence the tools developed to understand informal reasoning, collectively known as argumentation theory, are also applicable to much mathematical argumentation. This chapter investigates some of the details of that application. Consideration is given to the many contrasting meanings (...) of the word “argument”; to some of the specific argumentation-theoretic tools that have been applied to mathematics, notably Toulmin layouts and argumentation schemes; to some of the different ways that argumentation is implicated in mathematical practices; and to the social aspects of mathematical argumentation. (shrink)

I reply to criticisms from Duetz and Dentith, Basham, and Hewitt. I argue that the central disputes on this topic concern how ordinary people understand conspiracy theories and how to evaluate concrete conspiracy theories and conspiracy theorists.

The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main problem (...) facing Platonists—the problem of explaining how our experiences make contact with mathematical reality. (shrink)

In his influential book, The Nature of Morality, Gilbert Harman writes: “In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles.” What is the epistemological relevance of this contrast, if genuine? This chapter argues that ethicists and philosophers of mathematics have misunderstood it. They have confused what the chapter calls the justificatory challenge for realism about an area, (...) D—the challenge to justify our D-beliefs—with the reliability challenge for D-realism—the challenge to explain the reliability of our D-beliefs. Harman’s contrast is relevant to the first, but not, evidently, to the second. One upshot of the discussion is that genealogical debunking arguments are fallacious. Another is that indispensability considerations cannot answer the Benacerraf–Field challenge for mathematical realism. (shrink)

Arguments from disagreement often take centre stage in debates between competing semantic theories. This paper explores the theoretical basis for arguments from disagreement and, in so doing, proposes methodological principles which allow us to distinguish between legitimate arguments from disagreement and dialectically ineffective arguments from disagreement. In the light of these principles, I evaluate Cappelen and Hawthorne's [2009] argument from disagreement against relativism, and show that it fails to undermine relativism since it is dialectically ineffective. Nevertheless, I argue that an (...) alternative challenge to relativism based on disagreement is available. More generally, I argue that semantic theory is not answerable to data stemming from ‘loaded’ philosophical principles regarding the nature of disagreement. Rather, semantic theorists will exhaust their dialectical responsibilities regarding disagreement if they can demonstrate consistency with a minimal account of the concept. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, (...) but have been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

Epistemic peer disagreement raises interesting questions, both in epistemology and in philosophy of science. When is it reasonable to defer to the opinion of others, and when should we hold fast to our original beliefs? What can we learn from the fact that an epistemic peer disagrees with us? A question that has received relatively little attention in these debates is the value of epistemic peer disagreement—can it help us to further epistemic goals, and, if so, how? We investigate this (...) through a recent case in paleoanthropology: the debate on the taxonomic status of Homo floresiensis remains unresolved, with some authors arguing the fossils represent a novel hominin species, and others claiming that they are Homo sapiens with congenital growth disorders. Our examination of this case in the recent history of science provides insights into the value of peer disagreement, indicating that it is especially valuable if one does not straightaway defer to a peer’s conclusions, but nevertheless remains open to a peer’s evidence and arguments. (shrink)

I identify two reasons for believing in the objectivity of mathematical knowledge: apparent objectivity and applications in science. Focusing on arithmetic, I analyze platonism and cognitive nativism in terms of explaining these two reasons. After establishing that both theories run into difficulties, I present an alternative epistemological account that combines the theoretical frameworks of enculturation and cumulative cultural evolution. I show that this account can explain why arithmetical knowledge appears to be objective and has scientific applications. Finally, I will argue (...) that, while this account is compatible with platonist metaphysics, it does not require postulating mind-independent mathematical objects. (shrink)

This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.

Default logic has been a very active research topic in artificial intelligence since the early 1980s, but has not received as much attention in the philosophical literature thus far. This paper shows one way in which the technical tools of artificial intelligence can be applied in contemporary epistemology by modeling a paradigmatic case of deep disagreement using default logic. In §1 model-building viewed as a kind of philosophical progress is briefly motivated, while §2 introduces the case of deep disagreement we (...) aim to model. On the heels of this, §3 defines our formal framework, viz., a refined Horty-style default logic. §4 then uses the framework to model deep disagreement, and finally §5 provides a critical discussion of the result. (shrink)

When members of a group doxastically disagree with each other, decisions in the group are often hard to make. The members are supposed to find an epistemic compromise. How do members of a group reach a rational epistemic compromise on a proposition when they have different (rational) credences in the proposition? I answer the question by suggesting the Fine-Grained Method of Aggregation, which is introduced in Brössel and Eder 2014 and is further developed here. I show how this method faces (...) challenges of the standard method of aggregation, Weighted Straight Averaging, in a successful way. One of the challenges concerns the fact that Weighted Straight Averaging does not respect the evidential states of agents. Another challenge arises because Weighted Straight Averaging does not account for synergetic effects. (shrink)

The study aimed to determine if differentiated instruction effectively addresses learning gaps in mathematics. In particular, it explored how it can improve the student’s learning gaps concerning mathematical performance and confidence. The study employed a quasi-experimental design with 30 purposively-selected Grade 10 participants divided into differentiated (n = 15) and control groups (n = 15), ensuring the utmost ethical measures. The mean and standard deviation were used to describe the participants’ performance and confidence. Independent samples t-tests were used to (...) determine the significant differences in the performance and confidence between the two groups. In contrast, dependent samples t-tests were used to determine the significant differences in each group’s pre and posttest performance and confidence. Findings bared that the differentiated instruction successfully addressed students’ performance in mathematics even in a short period. It also increased the participants’ confidence when answering fundamental problems. Continuing differentiated instruction activities are recommended since it benefits students who struggle in mathematics, particularly in answering fundamental operations. Differentiated teaching activities in mathematics can boost academic achievement and engagement and prepare students for future success while fostering a positive and inclusive classroom culture that values individual learning needs and preferences. (shrink)

Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s (...) random model of the prime numbers and the function field model of the integers. These cases show that mathematicians, like empirical scientists, rely on unrealistic models to gain understanding of complex phenomena. They also have important implications for some much-discussed theses about scientific understanding. First, modeling practices in mathematics confirm that one can gain understanding without obtaining an explanation. Second, these cases undermine the popular thesis that unrealistic models confer understanding by imparting counterfactual knowledge. (shrink)

A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. The book expounds the traditional view of proof as deduction of theorems from evident premises via obviously valid steps. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.

Many philosophers regard the persistence of philosophical disputes as symptomatic of overly ambitious, ill-founded intellectual projects. There are indeed strong reasons to believe that persistent disputes in philosophy (and more generally in the discourse at large) are pointless. We call this the pessimistic view of the nature of philosophical disputes. In order to respond to the pessimistic view, we articulate the supporting reasons and provide a precise formulation in terms of the idea that the best explanation of persistent disputes entails (...) that they are pointless. We then show how to answer the pessimistic argument. Taking a well-known mathematical controversy as our paradigm example, we argue that some persistent disputes reflect substantive disagreements at the “meta-analytic” level, i.e., disagreements about the best way, among quite different candidates, to understand the topic at issue, and the best associated cluster of analytic truths one should accept concerning it. Moreover, our concrete example shows that such meta-analytic disagreements can in principle be settled and yield a genuine theoretical (as opposed to merely pragmatic) breakthrough. We conclude optimistically that persistent disputes can be an important means of fostering epistemic progress. (shrink)

The term ‘continuous’ in real analysis wasn’t given an adequate formal definition until 1817. However, important theorems about continuity were proven long before that. How was this possible? In this paper, I introduce and refine a proposed answer to this question, derived from the work of Frank Jackson, David Lewis and other proponents of the ‘Canberra plan’. In brief, the proposal is that before 1817 the meaning of the term ‘continuous’ was determined by a number of ‘platitudes’ which had some (...) special epistemic status. (shrink)

-/- Species lists play an important role in biology and practical domains like conservation, legislation, biosecurity and trade regulation. However, their effective use by non-specialist scientific and societal users is sometimes hindered by disagreements between competing lists. While it is well-known that such disagreements exist, it remains unclear how prevalent they are, what their nature is, and what causes them. In this study, we argue that these questions should be investigated using methods based on taxon concept rather than (...) methods based on Linnaean names, and use such a concept-based method to quantify disagreement about bird classification and investigate its relation to research effort. We found that there was disagreement about 38% of all groups of birds recognized as a species, more than three times as much as indicated by previous measures. Disagreement about the delimitation of bird groups was the most common kind of conflict, outnumbering disagreement about nomenclature and disagreement about rank. While high levels of conflict about rank were associated with lower levels of research effort, this was not the case for conflict about the delimitation of bird groups. This suggests that taxonomic disagreement cannot be resolved simply by increasing research effort. (shrink)

Legal philosophers and property scholars sometimes disagree over one or more of the following: the meaning of the word 'property,' the concept of property, and the nature of property. For much of the twentieth century, the work of W.N. Hohfeld and Tony Honoré represented a consensus around property. The consensus often went under the heading of property as bundle of rights, or more accurately as a set of normative relations between persons with respect to things. But by the mid-l 990s, (...) the consensus was under attack. Key figures in the attack were James Penner, a legal philosopher, and Thomas W. Merrill and Henry E. Smith, two highly regarded professors of property law. This article aims to repel the attack and argues for property as a set of normative relations between persons with respect to things. The positive case for this view of property pays special attention to the philosophy of language and the analysis of concepts. The positive case also maintains that the right to use and the power to transfer are as central to property as the right to exclude. It is possible that the virtues of Smith's modular theory of property differ from the virtues of a well-crafted bundle theory. Indeed, it may be the case that these two theories throw light on different features of property law and are not, save at the margin, competitors with each other. The label 'new essentialism' sometimes applied to the work of Penner, Merrill, and Smith seems inapt if property does not have an essence. Of course, they might refuse the label. (shrink)

Symmetry is not an inherent characteristic of mathematical proofs; instead, it is a property that arbitrarily manifests in different modes of presentation. This arbitrariness leads to the conclusion that symmetry cannot be part of the defining or essential properties that characterize proofs. Consequently, contrary to some authors’ claims, symmetry does not significantly contribute to the validity, accuracy, or soundness of mathematical proofs. What is more, it does not even play any critical role in heuristic aspects such as explanatory power. The (...) examples developed in this paper constitute compelling evidence supporting these claims. (shrink)

This book is inspired by a German theoretical physicist, Sabine Hossenfelder’s publication: “Lost in Mathematics”. Her book seems to question highly mathematical and a lot of abstraction in the development of physics and cosmology studies nowadays. There is clear tendency that in recent decades, the physics science has been predominated by such an advanced mathematics, which at times sounding more like acrobatics approach to a reality. Through books by senior mathematical-physicists like Unzicker and Peter Woit, we know that (...) the answer of TOE is not in superstring theories or other variations of such 26 dimensional bosonic string theory, of which none of those theories survived experimental test, but perhaps in low dimensional physics. As Alexander Unzicker suggests, perhaps it is more advisable to consider rotation in 3D space (known as SO3), or a kind of superfluid vortices version of gravitation theory. We can also reconsider proposition by the late Prof F. Winterberg (formerly professor at Univ. Nevada, Reno), that it is most likely that superfluid phonon roton theory in 3D can replace the entire superstring theories. While we don’t explore yet implications of his model to particle physics, we discuss here some published papers at several journals in the past few years. (shrink)

Disagreement among experts is a central topic in social epistemology. What should an expert do when confronted with the different opinion of an epistemic peer? Possible answers include the steadfast view (holding to one’s belief), the abstemious view (suspending one’s judgment), and moderate conciliatory views, which specify criteria for belief change when a peer’s different opinion is encountered. The practice of Delphi techniques in healthcare, medicine, and social sciences provides a real-life case study of expert disagreement, where disagreement is gradually (...) transformed into consensus. An analysis of Delphi shows that moderate conciliatory views are descriptively more adequate than rival views. However, it also casts doubt on whether the debate in social epistemology is explanatory relevant vis-à-vis real life cases of expert disagreement, where consensus replaces truth, and acceptance is more explanatorily relevant than belief. (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)

In this paper I defend an intermediate position between the ‘bare mathematical results’ view and the ‘transmission’ view of mathematical explanations of physical phenomena (MEPPs). I argue that, in MEPPs, it is not enough to deduce the explanandum from the generalizations cited in the explanans. Rather, we must add information regarding why those generalizations obtain. However, I also argue that it is not necessary to provide explanatory proofs of the mathematical theorems that represent those generalizations. I illustrate this with the (...) bridges of Königsberg case. -/- . (shrink)

In mathematics, the deductive method reigns. Without proof, a claim remains unsolved, a mere conjecture, not something that can be simply assumed; when a proof is found, the problem is solved, it turns into a “result,” something that can be relied on. So mathematicians think. But is there more to mathematical justification than proof? -/- The answer is an emphatic yes, as I explain in this article. I argue that non-deductive justification is in fact pervasive in mathematics, and (...) that it is in good epistemic standing. (shrink)

The scientific study of consciousness, also referred to as consciousness science, is a young scientific field devoted to understanding how conscious experiences and the brain relate. It comprises a host of theories, experiments, and analyses that aim to investigate the problem of consciousness empirically, theoretically, and conceptually. This thesis addresses some of the questions that arise in these investigations from a formal and mathematical perspective. These questions concern theories of consciousness, experimental paradigms, methodology, and artificial consciousness. -/- Regarding theories of (...) consciousness, the thesis contributes to the understanding of the mathematical structure that some of the formal theories in the field propose. The work presented here targets the theory of consciousness known as Integrated Information Theory (IIT) and the neuroscientific theory known as Predictive Processing Theory or Free Energy Principle in its Active Inference form (AI-PP). The thesis provides axiomatic definitions of the mathematical structures that constitute these theories, and uses these definitions to address some of the open questions surrounding the theories. For AI-PP, this includes a rigorous derivation of the formula for Active Inference via Free Energy minimisation and a proof of compositionality of Free Energy. For IIT, this includes resolutions of some of the criticisms of IIT's formal scope and applications, but also the identification of new issues that concern the formalism and its derivation. When possible, the definitions are provided in the mathematical framework of category theory. -/- Regarding experiments, the thesis addresses the main paradigm for testing and falsifying theories of consciousness currently applied in the field. This paradigm consists of comparing the conscious experience that a theory predicts with the conscious experience that is inferred from behavioural data or report by use of measures of consciousness. The thesis provides a formal model of this paradigm and shows that under a certain condition—if inference and prediction are independent—, any minimally informative theory of consciousness can always be falsified. This is deeply problematic since the field’s reliance on report or behaviour to infer conscious experiences, in conjunction with the general structure of most contemporary theories of consciousness, implies such independence. This observation provides the exact formal underpinning of the well-known unfolding argument. The thesis analyses the origin of the problem and identifies precisely which changes are required to avoid this problem in future research. The thesis furthermore shows that the problem of falsifying theories of consciousness, and of empirical comparisons of theories of consciousness more generally, follows from a pervasive closure paradigm in consciousness science, that consists of taking a neuroscientific account of the brain as input to a theory of consciousness, so as to explain what consciousness is, without allowing for modifications or adaptations of the neuroscientific account that would accommodate consciousness as part of the brain's functioning. As is shown in the thesis, this paradigm has implications that point to a fundamental need of revision. -/- Regarding methodological and conceptual questions, the thesis contributes to the foundations of structural research in consciousness science. Structural research aims to use mathematical structures or mathematical spaces, instead of verbal descriptions or simple categorisations, to represent conscious experiences scientifically, for example when building theories of consciousness, or when exploring new empirical avenues to measure consciousness. Despite considerable advances in this realm, there was, prior to this thesis, no explicit definition of what a mathematical structure of conscious experience should be; that is, how the attribution of mathematical structure to conscious experiences should be systematically understood. Perhaps the most important contribution of this thesis to the field is to propose such a definition. The definition, a structural concept, extends existing approaches wherever available, and provides a basis for developing a common formal language to study consciousness, bridging developments as far apart as psychophysics and phenomenology. In addition, and independently of this proposal, the thesis offers a critical analysis of which metaphysical premises need to be presumed in structural research, whether the use of particular formal tools (such as structure-preserving mappings or homomorphisms) is justified, and how structural theories of consciousness could otherwise be built in the first place. An attempt to expand the results from consciousness to more general problems in philosophy of science is made in the context of the well-known Newman problem. -/- Regarding the question of artificial consciousness—can AI feel?—, the thesis contributes two results that take the form of no-go theorems, as known from physics. The first no-go theorem shows that if consciousness is relevant for the temporal evolution of a system's states—if it is dynamically relevant—, then contemporary AI systems cannot be conscious. That is because AI systems run on CPUs, GPUs, TPUs or other processors which have been designed and verified to adhere to computational dynamics that systematically preclude or suppress deviations. The second no-go theorem is situated in the context of computational functionalism, a view which posits that consciousness is a computation. The theorem shows that if computational functionalism holds true, consciousness cannot be a Turing computation. Rather, it must be a novel type of computation that has recently been proposed by Geoffrey Hinton, called mortal computation. -/- This thesis is part of a global effort to pioneer a mathematical perspective in consciousness science, now called Mathematical Consciousness Science. The hope behind the research carried out in this PhD is to illustrate the power and usefulness of mathematical approaches in different areas of consciousness science, and in doing so, to lay the foundations for future mathematical work that complements and supports empirical and theoretical work in the further development of this exciting field. (shrink)

While the consequences of mathematically-based software, algorithms and strategies have become ever wider and better appreciated, ethical reflection on mathematics has remained primitive. We review the somewhat disconnected suggestions of commentators in recent decades with a view to piecing together a coherent approach to ethics in mathematics. Calls for a Hippocratic Oath for mathematicians are examined and it is concluded that while lessons can be learned from the medical profession, the relation of mathematicians to those affected by their (...) work is significantly different. There is something to be learned also from the codes of conduct of cognate but professionalised quantitative disciplines such as engineering and accountancy, as well as from legal principles bearing on professional work. We conclude with recommendations that professional societies in mathematics should sponsor an (international) code of ethics, institutional mission statements for mathematicians and syllabuses of ethics courses for incorporation into mathematics degrees. (shrink)

This study determined the cognitive skills achievement in mathematics of elementary pre-service teachers as a basis for improving problem-solving and critical thinking which was analyzed using Piaget's seven logical operations namely: classification, seriation, logical multiplication, compensation, ratio and proportional thinking, probability thinking, and correlational thinking. This study utilized an adopted Test on Logical Operations (TLO) and descriptive research design to describe the cognitive skills achievement and to determine the affecting factors. Overall, elementary pre-service teachers performed with sufficient understanding in (...) dealing with the TLO. However, in each logical operation, students were found to have insufficient understanding of probability thinking and correlational thinking. These insufficiencies in other logical operations were attributed to their misconceptions on some mathematical terms, misinterpretation of the problems, poor comprehension skills, poor problem-solving skills, and poor performance in mathematics in general. (shrink)

A belief one has is safe if either (i) it could not easily be false or (ii) in any nearby world in which it is false, it is not formed using the method one uses to form one’s actual belief. It seems our mathematical beliefs are safe if mathematical pluralism is true: if, loosely put, almost any consistent mathematical theory is true. It seems, after all, that in any nearby world where one’s mathematical beliefs differ from one’s actual beliefs, one (...) would believe some other true, consistent theory. Focusing on Justin Clarke-Doane’s recent discussion, I argue the thesis that mathematical beliefs are safe given pluralism faces some obstacles. I argue (i) is true of mathematical belief given pluralism only if we deny plausible claims about the interpretation of non-pluralists who many of us could easily be. Unless strong metasemantic theses are true, it is plausible many of us could easily deny or refuse to believe a consistent and true mathematical theory we actually believe. Since philosophical arguments and controversies permeate the methodology of foundational mathematics, I argue we cannot confidently distinguish the methods we use in mathematics between worlds, thus raising doubts about (ii). (shrink)

Some courses achieve existence, some have to create Professional Issues and Ethics in existence thrust upon them. It is normally Mathematics; but if you don’t do it, we will a struggle to create a course on the ethical be.” I accepted. or social aspects of science or mathematics. The gift of a greenfield site and a bull- This is the story of one that was forced to dozer is a happy occasion, undoubtedly. But exist by an unusual confluence (...) of outside cirwhat to do next? It seemed to me I should cumstances. ensure the course satisfied these require- In the mid 1990s, the University of New ments: South Wales instituted a policy that all its • It should look good to students, to staff. (shrink)