Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ (...) problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof. (shrink)

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.

The Four-Colour Theorem (4CT) proof, presented to the mathematical community in a pair of papers by Appel and Haken in the late 1970's, provoked a series of philosophical debates. Many conceptual points of these disputes still require some elucidation. After a brief presentation of the main ideas of Appel and Haken’s procedure for the proof and a reconstruction of Thomas Tymoczko’s argument for the novelty of 4CT’s proof, we shall formulate some questions regarding the connections between the points raised by (...) Tymoczko and some Wittgensteinian topics in the philosophy of mathematics such as the importance of the surveyability as a criterion for distinguishing mathematical proofs from empirical experiments. Our aim is to show that the “characteristic Wittgensteinian invention” (Mühlhölzer 2006) – the strong distinction between proofs and experiments – can shed some light in the conceptual confusions surrounding the Four-Colour Theorem. (shrink)

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried to naturalize (...) mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when one tries to naturalize (...) in this way the traditional view of mathematics, according to which mathematical knowledge is certain and the method of mathematics is the axiomatic method. This paper suggests that, in order to naturalize mathematics through Darwinism, it is better to take the method of mathematics not to be the axiomatic method. (shrink)

According to Phenomenal Conservatism (PC), if it seems to a subject S that P, S thereby has some degree of (defeasible) justification for believing P. But what is it for P to seem true? Answering this question is vital for assessing what role (if any) such states can play. Many have appeared to adopt a kind of non-reductionism that construes seemings as intentional states which cannot be reduced to more familiar mental states like beliefs or sensations. In this paper I (...) aim to show that reductive accounts need to be taken more seriously by illustrating the plausibility of identifying seemings and conscious inclinations to form a belief. I briefly close the paper by considering the implications such an analysis might have for views such as PC. (shrink)

Chase Wrenn argues that there are no epistemic duties. When it appears that we have an epistemic duty to believe, disbelieve or suspend judgement about some proposition P, we are really under a moral obligation to adopt the attitude towards P that our evidence favours. The argument appeals to theoretical parsimony: our conceptual scheme will be simpler without epistemic duties and we should therefore drop them. I argue that Wrenn’s strategy is flawed. There may well be things that we ought (...) to do on epistemic grounds alone. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

Written by experts in the field, this volume presents a comprehensive investigation into the relationship between argumentation theory and the philosophy of mathematical practice. Argumentation theory studies reasoning and argument, and especially those aspects not addressed, or not addressed well, by formal deduction. The philosophy of mathematical practice diverges from mainstream philosophy of mathematics in the emphasis it places on what the majority of working mathematicians actually do, rather than on mathematical foundations. -/- The book begins by first challenging the (...) assumption that there is no role for informal logic in mathematics. Next, it details the usefulness of argumentation theory in the understanding of mathematical practice, offering an impressively diverse set of examples, covering the history of mathematics, mathematics education and, perhaps surprisingly, formal proof verification. From there, the book demonstrates that mathematics also offers a valuable testbed for argumentation theory. Coverage concludes by defending attention to mathematical argumentation as the basis for new perspectives on the philosophy of mathematics. ​. (shrink)

In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There are “verifications” of (...) hypotheses which, while not definitive proofs, provide strong support for those hypotheses, and there are proofs involving an enormous amount of computer hours, which cannot be surveyed by any one mathematician in a lifetime. There have been several attempts to argue that one or another aspect of experimental mathematics shows that mathematics now accepts empirical or inductive methods, and hence shows mathematical apriorism to be false. Assessing this argument is complicated by the fact that there is no agreed definition of what precisely experimental mathematics is. However, I argue that on any plausible account of ’experiment’ these arguments do not succeed. (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' or (...) `pragmatic', but that there must be an element of what it is rational to believe on the evidence, that is, of non-deductive logic. (shrink)

I discuss Smokrović’s work on the normativity of logic (Smokrović 2017, Smokrović 2018). I agree that the classical formal logic is not an adequate model for real-life reasoning. But I present some doubts about his notion of deductive logic and his proposal to model such reasoning in non-monotonic logic. No branch of formal logic by itself is likely to capture real-life inferential links (reasoned-inference). I use the logic of relevance as my case study and extend the pessimistic morals to modern (...) systems of non-classical logic. Finally, I propose a more lax conception of normativty: there is a connection between logical assessment in the broad sense (as sanctioned by the notion of cogency) and the evaluation and criticism of reasoning. (shrink)

In The Matter of Consciousness, in the course of his extended discussion and defense of Frank Jackson’s famous knowledge argument, Torin Alter dismisses some objections on the grounds that they are cases of cheating. Though some opponents of the knowledge argument offer various scenarios in which Mary might come to know what seeing red is like while still in the room, Alter argues that the proposed scenarios are irrelevant. In his view, the Mary case is offered to defend the claim (...) that phenomenal facts cannot be a priori deduced from physical facts. Thus, a proposed scenario constitutes an objection to the knowledge argument only if it presents a case in which Mary’s learning inside the room comes about via a priori deduction from physical facts. Call this the deducibility standard. In what follows, I’ll explore a series of relevant cases in an effort to clarify this standard. Doing so enables us to better understand how cheating should be assessed in this context and thereby also to get clearer on the argumentative dialectic surrounding the Mary case. (shrink)

This dissertation is concerned with two of the largest questions that we can ask about the nature of physical reality: first, whether physical reality begin to exist and, second, what criteria would physical reality have to fulfill in order to have had a beginning? Philosophers of religion and theologians have previously addressed whether physical reality began to exist in the context of defending the Kal{\'a}m Cosmological Argument (KCA) for theism, that is, (P1) everything that begins to exist has a cause (...) for its beginning to exist, (P2) physical reality began to exist, and, therefore, (C) physical reality has a cause for its beginning to exist. While the KCA has traditionally been used to argue for God's existence, the KCA does not mention God, has been rejected by historically significant Christian theologians such as Thomas Aquinas, and raises perennial philosophical questions -- about the nature and history of physical reality, the nature of time, the nature of causation, and so on -- that should be of interest to all philosophers and, perhaps, all humans. While I am not a religious person, I am interested in the questions raised by the KCA. In this dissertation, I articulate three necessary conditions that physical reality would need to fulfill in order to have had a beginning and argue that, given the current state of philosophical and scientific inquiry, we cannot determine whether physical reality began to exist. -/- Friends of the KCA have sought to defend their view that physical reality began to exist in two distinct ways. As I discuss in chapter 2, the first way in which friends of the KCA have sought to defend their view that physical reality began to exist involves a family of a priori arguments meant to show that, as a matter of metaphysical necessity, the past must be finite. If the past is necessarily finite, then the past history of physical reality is necessarily finite. And if having a finite past suffices for having a beginning, then, since the past history of physical reality is necessarily finite, physical reality necessarily began to exist. I show that the arguments which have been offered thus far for the view that the past is necessarily finite do not succeed. Moreover, as I elaborate on in chapter 5, having a finite past does not suffice for having a beginning. -/- As I discuss in chapter 3, the second way in which friends of the KCA have sought to defend their view that physical reality began to exist involves a family of a posteriori arguments meant to show that we have empirical evidence that physical reality has a finite past history. For example, the big bang is sometimes claimed to have been the beginning of physical reality and, since we have excellent empirical evidence for the big bang, we have excellent empirical evidence for the beginning of physical reality. The big bang can be understood in two ways. On the one hand, the big bang can be understood as a theory about the history and development of the observable universe. Understood in that sense, then I agree that the big bang is supported by excellent empirical evidence and by a scientific consensus. On the other hand, some authors (particularly science popularizers, science journalists, and religious apologists) have wrongly interpreted big bang theory as a theory about the beginning of the whole of physical reality. As I argue, while a beginning of physical reality may be consistent with classical big bang theory, classical big bang theory does not provide good reason for thinking that physical reality began to exist. -/- In part II, I turn to discussing three necessary, but not necessarily sufficient, conditions for physical reality to have a beginning. Before discussing the three conditions, in chapter 4, I introduce three metaphysical accounts of the nature of time (A-theory, B-theory, and C-theory) as well as some formal machinery that will subsequently become useful in the dissertation. I introduce the first of the three conditions in chapter 5. According to the Modal Condition, physical reality began to exist only if, at the closest possible worlds without time, physical reality does not exist. I show that this condition helps us to make sense of various views in both theology and philosophy of physics. In chapter 6}, I introduce the second of my three conditions, the Direction Condition, according to which, roughly, physical reality began to exist only if all space-time points agree about the direction of time, so that all space-time points can agree that physical reality's putative beginning took place in their objective past. In chapter 7, I discuss the third condition, the Boundary Condition, according to which physical reality began to exist only if there is a past temporal boundary such that physical reality did not exist before the boundary. I show that there are two senses in which physical reality could be said to have had a past temporal boundary. Lastly, in chapter 8, I show that there is a relationship between my three conditions and classical big bang theory, even though the relationship is not the one usually identified in the literature. -/- In part III, I present four arguments for the view that, at the present stage of philosophical and scientific inquiry, we cannot know whether physical reality satisfies the three necessary conditions to have had a beginning and, consequently, we cannot know whether physical reality had a beginning. As I will prove in chapter 9, no set of observations that we currently have, when conjoined with General Relativity, entails that physical reality satisfies the Direction or Boundary Conditions. As I show in chapter 10, considerations in the philosophical foundations of statistical mechanics entail either that the Cosmos violates the Modal Condition or else that there is a transcendental condition on the possibility of our knowledge of the past that prevents our access to data we would need to gather to determine whether physical reality satisfies the Boundary Condition. In chapter 11, I show that there are a number of live cosmological models according to which physical reality does not satisfy the Boundary Condition. As long as we don't know whether any of those cosmological models are correct, we do not know whether physical reality satisfies the Boundary Condition. Lastly, I turn to confirmation theory and show that, at our present stage of inquiry, ampliative inferences for the conclusion that physical reality satisfies the Modal, Direction, and Boundary Conditions are not successful. (shrink)

Mathematicians often intentionally leave gaps in their proofs. Based on interviews with mathematicians about their refereeing practices, this paper examines the character of intentional gaps in published proofs. We observe that mathematicians’ refereeing practices limit the number of certain intentional gaps in published proofs. The results provide some new perspectives on the traditional philosophical questions of the nature of proof and of what grounds mathematical knowledge.

Abduction is considered the most powerful, but also the most controversially discussed type of inference. Based on an analysis of Peirce’s retroduction, Lipton’s Inference to the Best Explanation and other theories, a new theory of abduction is proposed. It considers abduction not as intrinsically explanatory but as intrinsically conditional: for a given fact, abduction allows one to infer a fact that implies it. There are three types of abduction: Selective abduction selects an already known conditional whose consequent is the given (...) fact and infers that its antecedent is true. Conditional-creative abduction creates a new conditional in which the given fact is the consequent and a defined fact is the antecedent that implies the given fact. Propositional-conditional-creative abduction assumes that the given fact is implied by a hitherto undefined fact and thus creates a new conditional with a new proposition as antecedent. The execution of abductive inferences is specified by theory-specific patterns. Each pattern consists of a set of rules for both generating and justifying abductive conclusions and covers the complete inference process. Consequently, abductive inferences can be formalised iff the whole pattern can be formalised. The empirical consistency of the proposed theory is demonstrated by a case study of Semmelweis' research on puerperal fever. (shrink)

Easwaran has given a definition of transferability and argued that, under this definition, randomized arguments are not transferable. I show that certain aspects of his definition are not suitable for addressing the underlying question of whether or not there is an epistemic distinction between randomized and deductive arguments. Furthermore, I demonstrate that for any suitable definition, randomized arguments are in fact transferable.

Midwest Studies In Philosophy, EarlyView.

This paper investigates an important aspect of mathematical practice: that proof is required for a finished piece of mathematics. If follows that non-deductive arguments — however convincing — are never sufficient. I explore four aspects of mathematical research that have facilitated the impressive success of the discipline. These I call the Practical Virtues: Permanence, Reliability, Autonomy, and Consensus. I then argue that permitting results to become established on the basis of non-deductive evidence alone would lead to their deterioration. This furnishes (...) us with a partial rational justification for mathematicians strict insistence on proof. (shrink)

Ralph Johnson argues that mathematical proofs lack a dialectical tier, and thereby do not qualify as arguments. This paper argues that, despite this disavowal, Johnson’s account provides a compelling model of mathematical proof. The illative core of mathematical arguments is held to strict standards of rigour. However, compliance with these standards is itself a matter of argument, and susceptible to challenge. Hence much actual mathematical practice takes place in the dialectical tier.