It is a striking fact from reverse mathematics that almost all theorems of countable and countably representable mathematics are equivalent to just five subsystems of second order arithmetic. The standard view is that the significance of these equivalences lies in the set existence principles that are necessary and sufficient to prove those theorems. In this article I analyse the role of set existence principles in reverse mathematics, and argue that they are best understood as closure (...) conditions on the powerset of the natural numbers. (shrink)

Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account (...) can be fruitfully applied in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert’s program due to Simpson [1988], and predicativism in the extended form due to Feferman and Schütte. -/- Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only omega-models of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a Pi-1-1 complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than Pi-1-1-CA0. (shrink)

Review of John Stillwell, Reverse Mathematics: Proofs from the Inside Out. Princeton, NJ: Princeton University Press, 2018, pp. 200. ISBN 978-0-69-117717-5 (hbk), 978-0-69-119641-1 (pbk), 978-1-40-088903-7 (e-book).

Mark Wilson argues that the standard categorizations of "Theory T thinking"— logic-centered conceptions of scientific organization (canonized via logical empiricists in the mid-twentieth century)—dampens the understanding and appreciation of those strategic subtleties working within science. By "Theory T thinking," we mean to describe the simplistic methodology in which mathematical science allegedly supplies ‘processes’ that parallel nature's own in a tidily isomorphic fashion, wherein "Theory T’s" feigned rigor and methodological dogmas advance inadequate discrimination that fails to distinguish between explanatory structures that (...) are architecturally distinct. One of Wilson's main goals is to reverse such premature exclusions and, thus, early on Wilson returns to John Locke's original physical concerns regarding material science and the congeries of descriptive concern insofar as capturing varied phenomena (i.e., cohesion, elasticity, fracture, and the transmission of coherent work) encountered amongst ordinary solids like wood and steel are concerned. Of course, Wilson methodologically updates such a purview by appealing to multiscalar techniques of modern computing, drawing from Robert Batterman's work on the greediness of scales and Jim Woodward's insights on causation. (shrink)

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and (...) proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim. (shrink)

Horwich gives a fine analysis of Wittgenstein (W) and is a leading W scholar, but in my view, they all fall short of a full appreciation, as I explain at length in this review and many others. If one does not understand W (and preferably Searle also) then I don't see how one could have more than a superficial understanding of philosophy and of higher order thought and thus of all complex behavior (psychology, sociology, anthropology, history, literature, society). In a (...) nutshell, W demonstrated that when you have shown how a sentence is used in the context of interest, there is nothing more to say. I will start with a few notable quotes and then give what I think are the minimum considerations necessary to understand Wittgenstein, philosophy and human behavior. -/- First one might note that putting “meta” in front of any word should be suspect. W remarked e.g., that metamathematics is mathematics like any other. The notion that we can step outside philosophy (i.e., the descriptive psychology of higher order thought) is itself a profound confusion. Another irritation here (and throughout academic writing for the last 4 decades) is the constant reverse linguistic sexism of “her” and “hers” and “she” or “he/she” etc., where “they” and “theirs” and “them” would do nicely. Likewise, the use of the French word 'repertoire' where the English 'repertory' will do quite well. The major deficiency is the complete failure (though very common) to employ what I see as the hugely powerful and intuitive two systems view of HOT and Searle’s framework which I have outlined above. This is especially poignant in the chapter on meaning p111 et seq. (especially in footnotes 2-7), where we swim in very muddy water without the framework of automated true only S1, propositional dispositional S2, COS etc. One can also get a better view of the inner and the outer by reading e.g., Johnston or Budd (see my reviews). Horwich however makes many incisive comments. I especially liked his summary of the import of W’s anti-theoretical stance on p65. He needs to give more emphasis to ‘On Certainty’, recently the subject of much effort by Daniele Moyal- Sharrock, Coliva and others and summarized in my recent articles. -/- Horwich is first rate and his work well worth the effort. One hopes that he (and everyone) will study Searle and some modern psychology as well as Hutto, Read, Hutchinson, Stern, Moyal-Sharrock, Stroll, Hacker and Baker etc. to attain a broad modern view of behavior. Most of their papers are on academia dot edu and philpapers dot org , but for PMS Hacker see his papers on his Oxford page. -/- He gives one of the most beautiful summaries of where an understanding of Wittgenstein leaves us that I have ever seen. -/- “There must be no attempt to explain our linguistic/conceptual activity (PI 126) as in Frege’s reduction of arithmetic to logic; no attempt to give it epistemological foundations (PI 124) as in meaning based accounts of a priori knowledge; no attempt to characterize idealized forms of it (PI 130) as in sense logics; no attempt to reform it (PI 124, 132) as in Mackie’s error theory or Dummett’s intuitionism; no attempt to streamline it (PI 133) as in Quine’s account of existence; no attempt to make it more consistent (PI 132) as in Tarski’s response to the liar paradoxes; and no attempt to make it more complete (PI 133) as in the settling of questions of personal identity for bizarre hypothetical ‘teleportation’ scenarios.” -/- Finally, let me suggest that with the perspective I have encouraged here, W is at the center of contemporary philosophy and psychology and is not obscure, difficult or irrelevant, but scintillating, profound and crystal clear and that to miss him is to miss one of the greatest intellectual adventures possible. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019) . (shrink)

Horwich gives a fine analysis of Wittgenstein (W) and is a leading W scholar, but in my view they all fall short of a full appreciation, as I explain at length in this review and many others. If one does not understand W (and preferably Searle also) then I don't see how one could have more than a superficial understanding of philosophy and of higher order thought and thus of all complex behavior(psychology, sociology, anthropology, history, literature, society). In a nutshell, (...) W demonstrated that when you have shown how a sentence is used in the context of interest, there is nothing more to say. I will start with a few notable quotes and then give what I think are the minimum considerations necessary to understand Wittgenstein, philosophy and human behavior. -/- First one might note that putting “meta” in front of any word should be suspect. W remarked e.g., that metamathematics is mathematics like any other. The notion that we can step outside philosophy (i.e., the descriptive psychology of higher order thought) is itself a profound confusion. Another irritation here (and throughout academic writing for the last 4 decades) is the constant reverse linguistic sexism of “her” and “hers” and “she” or “he/she” etc., where “they” and “theirs” and “them” would do nicely. Likwise the use of the French word 'repertoire' where the English 'repertory' will do quite well. The major deficiency is the complete failure (though very common) to employ what I see as the hugely powerful and intuitive two systems view of HOT and Searle’s framework which I have outlined above. This is especially poignant in the chapter on meaning p111 et seq.(esp. in footnotes 2-7), where we swim in very muddy water without the framework of automated true only S1, propositional dispositional S2, COS etc. One can also get a better view of the inner and the outer by reading e.g., Johnston or Budd (see my reviews). Horwich however makes many incisive comments. I especially liked his summary of the import of W’s antitheoretical stance on p65. Horwich is first rate and his work well worth the effort. One hopes that he (and everyone) will study Searle and some modern psychology as well as Hutto, Read, Hutchinson, Stern, Moyal-Sharrock, Stroll, Hacker and Baker etc. to attain a broad modern view of behavior. Most of their papers are on academia.edu but for PMS Hacker see http://info.sjc.ox.ac.uk/scr/hacker/DownloadPapers.html. -/- Finally, let me suggest that with the perspective I have encouraged here, W is at the center of contemporary philosophy and psychology and is not obscure, difficult or irrelevant, but scintillating, profound and crystal clear and that to miss him is to miss one of the greatest intellectual adventures possible. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my article The Logical Structure of Philosophy, Psychology, Mind and Language as Revealed in Wittgenstein and Searle 59p(2016). For all my articles on Wittgenstein and Searle see my e-book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Wittgenstein and Searle 367p (2016). Those interested in all my writings in their most recent versions may consult my e-book Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2016’ 662p (2016). -/- All of my papers and books have now been published in revised versions both in ebooks and in printed books. -/- Talking Monkeys: Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B071HVC7YP. -/- The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle--Articles and Reviews 2006-2016 (2017) https://www.amazon.com/dp/B071P1RP1B. -/- Suicidal Utopian Delusions in the 21st century: Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B0711R5LGX . (shrink)

This paper begins by raising a puzzle about what function our use of the word ‘rational’ could serve. To solve the puzzle, I introduce a view I call Epistemic Communism: we use epistemic evaluations to promote coordination among our basic belief-forming rules, and the function of this is to make the acquisition of knowledge by testimony more efficient.

Published in 1903, this book was the first comprehensive treatise on the logical foundations of mathematics written in English. It sets forth, as far as possible without mathematical and logical symbolism, the grounds in favour of the view that mathematics and logic are identical. It proposes simply that what is commonly called mathematics are merely later deductions from logical premises. It provided the thesis for which _Principia Mathematica_ provided the detailed proof, and introduced the work of Frege (...) to a wider audience. In addition to the new introduction by John Slater, this edition contains Russell's introduction to the 1937 edition in which he defends his position against his formalist and intuitionist critics. (shrink)

I discuss whether there are some lessons for philosophical inquiry over the nature of simulation to be learnt from the practical methodology of reengineering. I will argue that reengineering serves a similar purpose as simulations in theoretical science such as computational neuroscience or neurorobotics, and that the procedures and heuristics of reengineering help to develop solutions to outstanding problems of simulation.

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are vulnerable (...) to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

Why did such highly abstract ideas as truth, knowledge, or justice become so important to us? What was the point of coming to think in these terms? This book presents a philosophical method designed to answer such questions: the method of pragmatic genealogy. Pragmatic genealogies are partly fictional, partly historical narratives exploring what might have driven us to develop certain ideas in order to discover what these do for us. The book uncovers an under-appreciated tradition of pragmatic genealogy which cuts (...) across the analytic-continental divide, running from the state-of-nature stories of David Hume and the early genealogies of Friedrich Nietzsche to recent work in analytic philosophy by Edward Craig, Bernard Williams, and Miranda Fricker. However, these genealogies combine fictionalizing and historicizing in ways that even philosophers sympathetic to the use of state-of-nature fictions or real history have found puzzling. To make sense of why both fictionalizing and historicizing are called for, the book offers a systematic account of pragmatic genealogies as dynamic models serving to reverse-engineer the points of ideas in relation not only to near-universal human needs, but also to socio-historically situated needs. This allows the method to offer us explanation without reduction and to help us understand what led our ideas to shed the traces of their practical origins. Far from being normatively inert, moreover, pragmatic genealogy can affect the space of reasons, guiding attempts to improve our conceptual repertoire by helping us determine whether and when our ideas are worth having. (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)

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content (...) to key intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as ex- planatory generality is concerned. (shrink)

Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number (...) of themes in recent philosophy of mathematics (concerning a priority and fictionalism, for example) in revealing new light. (shrink)

Bayesian confirmation theory is rife with confirmation measures. Many of them differ from each other in important respects. It turns out, though, that all the standard confirmation measures in the literature run counter to the so-called “Reverse Matthew Effect” (“RME” for short). Suppose, to illustrate, that H1 and H2 are equally successful in predicting E in that p(E | H1)/p(E) = p(E | H2)/p(E) > 1. Suppose, further, that initially H1 is less probable than H2 in that p(H1) < (...) p(H2). Then by RME it follows that the degree to which E confirms H1 is greater than the degree to which it confirms H2. But by all the standard confirmation measures in the literature, in contrast, it follows that the degree to which E confirms H1 is less than or equal to the degree to which it confirms H2. It might seem, then, that RME should be rejected as implausible. Festa (2012), however, argues that there are scientific contexts in which RME holds. If Festa’s argument is sound, it follows that there are scientific contexts in which none of the standard confirmation measures in the literature is adequate. Festa’s argument is thus interesting, important, and deserving of careful examination. I consider five distinct respects in which E can be related to H, use them to construct five distinct ways of understanding confirmation measures, which I call “Increase in Probability”, “Partial Dependence”, “Partial Entailment”, “Partial Discrimination”, and “Popper Corroboration”, and argue that each such way runs counter to RME. The result is that it is not at all clear that there is a place in Bayesian confirmation theory for RME. (shrink)

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

Investigation into the sequence structure of the genetic code by means of an informatic approach is a real success story. The features of human language are also the object of investigation within the realm of formal language theories. They focus on the common rules of a universal grammar that lies behind all languages and determine generation of syntactic structures. This universal grammar is a depiction of material reality, i.e., the hidden logical order of things and its relations determined by natural (...) laws. Therefore mathematics is viewed not only as an appropriate tool to investigate human language and genetic code structures through computer sciencebased formal language theory but is itself a depiction of material reality. This confusion between language as a scientific tool to describe observations/experiences within cognitive constructed models and formal language as a direct depiction of material reality occurs not only in current approaches but was the central focus of the philosophy of science debate in the twentieth century, with rather unexpected results. This article recalls these results and their implications for more recent mathematical approaches that also attempt to explain the evolution of human language. (shrink)

This paper is essentially a quantum philosophical challenge: starting from simple assumptions, we argue about an ontological approach to quantum mechanics. In this paper, we will focus only on the assumptions. While these assumptions seems to solve the ontological aspect of theory many others epistemological problems arise. For these reasons, in order to prove these assumptions, we need to find a consistent mathematical context (i.e. time reverse problem, quantum entanglement, implications on quantum fields, Schr¨odinger cat states, the role of (...) observer, the role of mind ). (shrink)

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.

In the last decades two different and apparently unrelated lines of research have increasingly connected mathematics and evolutionism. Indeed, on the one hand different attempts to formalize darwinism have been made, while, on the other hand, different attempts to naturalize logic and mathematics have been put forward. Those researches may appear either to be completely distinct or at least in some way convergent. They may in fact both be seen as supporting a naturalistic stance. Evolutionism is indeed crucial (...) for a naturalistic perspective, and formalizing it seems to be a way to strengthen its scientificity. The paper shows that, on the contrary, those directions of research may be seen as conflicting, since the conception of knowledge on which they rest may be undermined by the consequences of accepting an evolutionary perspective. (shrink)

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the (...) mathematical practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences. (shrink)

Anyone who has read Plato’s Republic knows it has a lot to say about mathematics. But why? I shall not be satisfied with the answer that the future rulers of the ideal city are to be educated in mathematics, so Plato is bound to give some space to the subject. I want to know why the rulers are to be educated in mathematics. More pointedly, why are they required to study so much mathematics, for so long?

Research into ancient physical structures, some having been known as the seven wonders of the ancient world, inspired new developments in the early history of mathematics. At the other end of this spectrum of inquiry the research is concerned with the minimum of observations from physical data as exemplified by Eddington's Principle. Current discussions of the interplay between physics and mathematics revive some of this early history of mathematics and offer insight into the fine-structure constant. Arthur Eddington's (...) work leads to a new calculation of the inverse fine-structure constant giving the same approximate value as ancient geometry combined with the golden ratio structure of the hydrogen atom. The hyperbolic function suggested by Alfred Landé leads to another result, involving the Laplace limit of Kepler's equation, with the same approximate value and related to the aforementioned results. The accuracy of these results are consistent with the standard reference. Relationships between the four fundamental coupling constants are also found. (shrink)

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could (...) mathematics be knowledge about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (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.

How do axioms, or first principles, in ethics compare to those in mathematics? In this companion piece to G.C. Field's 1931 "On the Role of Definition in Ethics", I argue that there are similarities between the cases. However, these are premised on an assumption which can be questioned, and which highlights the peculiarity of normative inquiry.

We have reached the peculiar situation where the advance of mainstream science has required us to dismiss as unreal our own existence as free, creative agents, the very condition of there being science at all. Efforts to free science from this dead-end and to give a place to creative becoming in the world have been hampered by unexamined assumptions about what science should be, assumptions which presuppose that if creative becoming is explained, it will be explained away as an illusion. (...) In this paper it is shown that this problem has permeated the whole of European civilization from the Ancient Greeks onwards, leading to a radical disjunction between cosmology which aims at a grasp of the universe through mathematics and history which aims to comprehend human action through stories. By going back to the Ancient Greeks and tracing the evolution of the denial of creative becoming, I trace the layers of assumptions that must in some way be transcended if we are to develop a truly post-Egyptian science consistent with the forms of understanding and explanation that have evolved within history. (shrink)

Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of mathematics are considered (...) and rejected. Constructive empiricism cannot be realist about abstract objects; it must reject even the realism advocated by otherwise ontologically restrained and epistemologically empiricist indispensability theorists. Indispensability arguments rely on the kind of inference to the best explanation the rejection of which is definitive of constructive empiricism. On the other hand, formalist and logicist anti-realist positions are also shown to be untenable. It is argued that a constructive empiricist philosophy of mathematics must be fictionalist. Borrowing and developing elements from both Philip Kitcher's constructive naturalism and Kendall Walton's theory of fiction, the account of mathematics advanced treats mathematics as a collection of stories told about an ideal agent and mathematical objects as fictions. The account explains what true portions of mathematics are about and why mathematics is useful, even while it is a story about an ideal agent operating in an ideal world; it connects theory and practice in mathematics with human experience of the phenomenal world. At the same time, the make-believe and game-playing aspects of the theory show how we can make sense of mathematics as fiction, as stories, without either undermining that explanation or being forced to accept abstract mathematical objects into our ontology. All of this occurs within the framework that constructive empiricism itself provides the epistemological limitations it mandates, the semantic view of theories, and an emphasis on the pragmatic dimension of our theories, our explanations, and of our relation to the language we use. (shrink)

This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure (...) of the physical world. The no-miracles argument is the primary motivation for scientific realism. It is a presupposition of this argument that unobservable entities are explanatory only when they determine the empirical phenomena they explain. I argue that mathematical entities should also be seen as explanatory only when they determine the empirical facts they explain, namely, the modal structure of the physical world. Thus, scientific realism commits us to a metaphysical determination relation between mathematics and physical modality that has not been previously recognized. The requirement to account for the metaphysical dependence of modal physical structure on mathematics limits the class of acceptable solutions to the applicability problem that are available to the scientific realist. (shrink)

I discuss ways in which the linguistic form of mathimatics helps us think mathematically.

The imperviousness of mathematical truth to anti-objectivist attacks has always heartened those who defend objectivism in other areas, such as ethics. It is argued that the parallel between mathematics and ethics is close and does support objectivist theories of ethics. The parallel depends on the foundational role of equality in both disciplines. Despite obvious differences in their subject matter, mathematics and ethics share a status as pure forms of knowledge, distinct from empirical sciences. A pure understanding of principles (...) is possible because of the simplicity of the notion of equality, despite the different origins of our understanding of equality of objects in general and of the equality of the ethical worth of persons. (shrink)

This paper argues that new light may be shed on mathematical reasoning in its non-pathological forms by careful observation of its pathologies. The first section explores the application to mathematics of recent work on fallacy theory, specifically the concept of an ‘argumentation scheme’: a characteristic pattern under which many similar inferential steps may be subsumed. Fallacies may then be understood as argumentation schemes used inappropriately. The next section demonstrates how some specific mathematical fallacies may be characterized in terms of (...) argumentation schemes. The third section considers the phenomenon of correct answers which result from incorrect methods. This turns out to pose some deep questions concerning the nature of mathematical knowledge. In particular, it is argued that a satisfactory epistemology for mathematical practice must address the role of luck. (shrink)

The literature on the indispensability argument for mathematical realism often refers to the ‘indispensable explanatory role’ of mathematics. I argue that we should examine the notion of explanatory indispensability from the point of view of specific conceptions of scientific explanation. The reason is that explanatory indispensability in and of itself turns out to be insufficient for justifying the ontological conclusions at stake. To show this I introduce a distinction between different kinds of explanatory roles—some ‘thick’ and ontologically committing, others (...) ‘thin’ and ontologically peripheral—and examine this distinction in relation to some notable ‘ontic’ accounts of explanation. I also discuss the issue in the broader context of other ‘explanationist’ realist arguments. (shrink)

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)

Fefermans argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.

The focus of this chapter is on efforts to create a new mathematics, with my prime interest being the role of mathematics in comprehending a world consisting first and foremost of processes, and examining what developments in mathematics are required for this. I am particularly interested in developments in mathematics able to do justice to the reality of life. Such mathematics could provide the basis for advancing ecology, human ecology and ecological economics and thereby assist (...) in the transformation of society and civilization so that we augment life rather than undermining the conditions for our existence. It was in the process of grappling with these problems that I was drawn to investigate the tradition of intuitionism in mathematics and the role of intuition in mathematics, science and philosophy, and then to consider Whitehead’s work on mathematics and its philosophy in relation to these. (shrink)

Nietzsche has a surprisingly significant and strikingly positive assessment of mathematics. I discuss Nietzsche's theory of the origin of mathematical practice in the division of the continuum of force, his theory of numbers, his conception of the finite and the infinite, and the relations between Nietzschean mathematics and formalism and intuitionism. I talk about the relations between math, illusion, life, and the will to truth. I distinguish life and world affirming mathematical practice from its ascetic perversion. For Nietzsche, (...) math is an artistic and moral activity that has an essential role to play in the joyful wisdom. (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)

_ Source: _Volume 6, Issue 2-3, pp 120 - 142 This paper aims to contribute to the debate over epistemic versus non-epistemic readings of the ‘hinges’ in Wittgenstein’s _On Certainty_. I follow Marie McGinn’s and Daniele Moyal-Sharrock’s lead in developing an analogy between mathematical sentences and certainties, and using the former as a model for the latter. However, I disagree with McGinn’s and Moyal-Sharrock’s interpretations concerning Wittgenstein’s views of both relata. I argue that mathematical sentences as well as certainties are (...) true and are propositions; that some of them can be epistemically justified; that in some senses they are not prior to empirical knowledge; that they are not ineffable; and that their primary function is epistemic as much as it is semantic. (shrink)

At the beginning of the present century, a series of paradoxes were discovered within mathematics which suggested a fundamental unclarity in traditional mathemati­cal methods. These methods rested on the assumption of a realm of mathematical idealities existing independently of our thinking activity, and in order to arrive at a firmly grounded mathematics different attempts were made to formulate a conception of mathematical objects as purely human constructions. It was, however, realised that such formulations necessarily result in a (...) class='Hi'>mathematics which lacks the richness and power of the old ‘platonistic’ methods, and the latter are still defended, in various modified forms, as embodying truths about self-existent mathematical entities. Thus there is an idealism-realism dispute in the philosophy of mathematics in some respects parallel to the controversy over the existence of the experiential world to the settle­ment of which lngarden devoted his life. The present paper is an attempt to apply Ingarden’s methods to the sphere of mathematical existence. This exercise will reveal new modes of being applicable to non-real objects, and we shall put forward arguments to suggest that these modes of being have an importance outside mathematics, especially in the areas of value theory and the ontology of art. (shrink)

In this paper, I study how mathematicians are presented in western popular culture. I identify five stereotypes that I test on the best-known modern movies and television shows containing a significant amount of mathematics or important mathematician characters: (1) Mathematics is highly valued as an intellectual pursuit. (2) Little attention is given to the mathematical content. (3) Mathematical practice is portrayed in an unrealistic way. (4) Mathematicians are asocial and unable to enjoy normal life. (5) Higher mathematics (...) is ... (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)

This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal (...) science, mathematical ontology: what does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

Otávio Bueno* * and Steven French.** ** Applying Mathematics: Immersion, Inference, Interpretation. Oxford University Press, 2018. ISBN: 978-0-19-881504-4 978-0-19-185286-2. doi:10.1093/oso/9780198815044. 001.0001. Pp. xvii + 257.

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)

The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last (...) hundred years. This article explains the distinction and why it has proved to be one of the great organizing themes of mathematics. (shrink)