Antirealism in the philosophy of mathematics

Author(s):  
A.W. Moore

Realism in the philosophy of mathematics is the position that takes mathematics at face value. According to realists, mathematics is the science of mathematical objects (numbers, sets, lines and so on); mathematicians, to use the old metaphor, are discoverers, not inventors. Moreover, just as there may be truths about physical reality which we can never know, so too, realists say, there may be truths about mathematical reality which we can never know. It is this claim in particular which antirealists find unacceptable. Equating what can be known in mathematics with what can be proved, they insist that only what can be proved is true. (Only what can be proved: different accounts of what this ‘can’ means, facing different difficulties, generate different positions.) This leads antirealists to recoil not only from realism but also from the practice of mathematicians themselves. For the orthodox assumption that every mathematical statement is either true or false would be invalidated, on the antirealist view, by a statement that was neither provable nor disprovable. Not that antirealists themselves can see it in these terms. For if a statement were neither provable nor disprovable, that would itself be an unprovable truth about mathematical reality. Antirealists must learn how to be circumspect even in defence of their own circumspection.

Vivarium ◽  
2016 ◽  
Vol 54 (2-3) ◽  
pp. 204-249
Author(s):  
Helen Hattab

René Descartes is neither a Conceptualist nor a Platonist when it comes to the ontological status of the eternal truths and essences of mathematics but articulates a view derived from Proclus. There are several advantages to interpreting Descartes’ texts in light of Proclus’ view of universals and philosophy of mathematics. Key passages that, on standard readings, are in conflict are reconciled if we read Descartes as appropriating Proclus’ threefold distinction among universals. Specifically, passages that appear to commit Descartes to a Platonist view of mathematical objects and the truths that follow from them are no longer in tension with the Conceptualist view of universals implied by his treatment of the eternal truths in the Principles of Philosophy. This interpretation also fits the historical evidence and explains why Descartes ends up with seemingly inconsistent commitments to divine simplicity and God’s efficient creation of truths that are not merely conceptually distinct from the divine essence.


2020 ◽  
Vol 18 (2) ◽  
Author(s):  
Matti Eklund

According to a certain pluralist view in philosophy of mathematics, there are as many mathematical objects as there can coherently be. Recently, Justin Clarke-Doane has explored what consequences the analogous view on normative properties would have. What if there is a normative pluriverse? Here I address this same question. The challenge is best seen as a challenge to an important form of normative realism. I criticize the way Clarke-Doane presents the challenge. An improved challenge is presented, and the role of pluralism in this challenge is assessed.


2021 ◽  
Author(s):  
◽  
Scott Waygood

<p>The central claim of this thesis is that geometry is a quasi-empirical science based on the idealisation of the elementary physical operations that we actually perform with pen and paper. This conclusion is arrived at after searching for a theory of geometry that will not only explain the epistemology and ontology of mathematics, but will also fit with the best practices of working mathematicians and, more importantly, explain why geometry gives us knowledge that is relevant to physical reality. We will be considering all the major schools of thought in the philosophy of mathematics. Firstly, from the epistemological side, we will consider apriorism, empiricism and quasi-empiricism, finding a Kitcherian style of quasi-empiricism to be the most attractive. Then, from the ontological side, we will consider Platonism, formalism, Kitcherian ontology, and fictionalism. Our conclusion will be to take a Kitcherian epistemology and a fictionalist ontology. This will give us a kind of quasiempirical-fictionalist approach to mathematics. The key feature of Kitcher's thesis is that he placed importance on the operations rather than the entities of arithmetic. However, because he only dealt with arithmetic, we are left with the task of developing a theory of geometry along Kitcherian lines. I will present a theory of geometry that parallels Kitcher's theory of arithmetic using the drawing of straight lines as the most primitive operation. We will thereby develop a theory of geometry that is founded upon our operations of drawing lines. Because this theory is based on our line drawing operations carried out in physical reality, and is the idealisation of those activities, we will have a connection between mathematical geometry and physical reality that explains the predictive power of geometry in the real world. Where Kitcher uses the Peano postulates to develop his theory of arithmetic, I will use the postulates of projective geometry to form the foundations of operational geometry. The reason for choosing projective geometry is due to the fact that by taking it as the foundation, we may apply Klein's Erlanger programme and build a theory of geometry that encompasses Euclidean, hyperbolic and elliptic geometries. The final question we will consider is the problem of conventionalism. We will discover that investigations into conventionalism give us further reason to accept the Kitcherian quasi-empirical-fictionalist approach as the most appealing philosophy of geometry available.</p>


Philosophy ◽  
2015 ◽  
Author(s):  
Mary Leng

In the context of the philosophy of mathematics, the term “naturalism” has a number of uses, covering approaches that look to be fundamentally at odds with one another. In one use, the “natural” in naturalism is contrasted with non-natural, in the sense of supernatural; in this sense, naturalism in the philosophy of mathematics appears in opposition to Platonism (the view that mathematical truths are truths about a body of abstract mathematical objects). Naturalism thus construed takes seriously the epistemological challenge to Platonism presented by Paul Benacerraf in his paper “Mathematical Truth” (cited under Ontological Naturalism). Benacerraf points out that a view of mathematics as a body of truths about a realm of abstract objects appears to rule out any (non-mystical) account of how we, as physically located embodied beings, could come to know such truths. The naturalism that falls out of acceptance of Benacerraf’s challenge as presenting a genuine problem for our claims to be able to know truths about abstract mathematical objects is sometimes referred to as “ontological naturalism,” and suggests a physicalist ontology. In a second use, the “natural” in naturalism is a reference specifically to natural science and its methods. Naturalism here, sometimes called methodological naturalism, is the Quinean doctrine that philosophy is continuous with natural science. Quine and Putnam’s indispensability argument for the existence of mathematical objects places methodological naturalism in conflict with ontological naturalism, since it is argued that the success of our scientific theories confirms the existence of the abstract mathematical objects apparently referred to in formulating those theories, suggesting that methodological naturalism requires Platonism. A final use of “naturalism” in the philosophy of mathematics is distinctive to mathematics, and arises out of consideration of the proper extent of methodological naturalism. According to Quine’s naturalism, the natural sciences provide us with the proper methods of inquiry. But, as Penelope Maddy has noted, mathematics has its own internal methods and standards, which differ from the methods of the empirical sciences, and naturalistic respect for the methodologies of successful fields requires that we should accept those methods and standards. This places Maddy’s methodological naturalism in tension with the original Quinean version of the doctrine, because, Maddy argues, letting natural science be the sole source of confirmation for mathematical theories fails to respect the autonomy of mathematics.


2019 ◽  
Vol 27 (3) ◽  
pp. 335-361
Author(s):  
Fiona T Doherty

ABSTRACT This paper reveals David Hilbert’s position in the philosophy of mathematics, circa 1900, to be a form of non-eliminative structuralism, predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the Frege-Hilbert controversy in virtue of this early structuralist approach. To demonstrate that this historical position deserves contemporary attention I show that Hilbertian structuralism avoids a recent wave of objections against non-eliminative structuralists to the effect that they cannot distinguish between structurally identical but importantly distinct mathematical objects, such as the complex roots of $-1$.


Author(s):  
Марина Николаевна Вольф ◽  
Игорь Владимирович Берестов

В настоящей статье мы предлагаем вариант коммуникативно-прагматического подхода к определению содержания предложений, высказываемых агентами в дискуссии. Мы намерены: 1) предложить такую трактовку процесса убеждения, в соответствии с которой он может быть признан успешным – даже если отдаётся должное фундаментальной проблеме знаковой коммуникации и признаётся, что агенты в процессе коммуникации передают друг другу только знаки, а их значения оказываются недоступными собеседнику; 2) показать, что популярные современные подходы (прагма-диалектический подход, AGM-подход, DBR-подход и др.) к процессу убеждения не предлагают такой трактовки процесса убеждения; 3) установить точные и проверяемые условия, необходимые и достаточные для признания процесса убеждения в нашей трактовке успешным. Наш подход использует современные подходы к трактовке содержания предложения (прежде всего ментальный холизм – М. Харрелл, Н. Блок и др.), а также экстраполирует современные структуралистские подходы к пониманию математических объектов в философии математики (Д. Гильберт, М. Резник, С. Шапиро) на знаковую коммуникацию в ситуации убеждающей аргументации. Мы отклоняем решение проблемы знаковой коммуникации, состоящее в том, что вполне возможен трансфер визуальных образов вне зависимости от их содержания или значения, из чего следует, что коммуникация агентов всё-таки возможна посредством передачи таких образов. Это решение, фактически, предлагает обсуждать не знаковую, а беззнаковую коммуникацию, но, во-первых, передача убеждений как лингвистически выражаемых объектов в таком случае невозможна, и, во-вторых, есть сильные аргументы в пользу того, что даже не имеющие содержания объекты различных агентов различны, что также делает трансфер невозможным. Мы определяем риторическое значение предложения, высказываемого агентом в дискуссии. Этим значением является также предложение, антецедентом которого является полное описание системы убеждений агента, а консеквентом – исходное предложение. Также мы признаём процесс убеждения риторически успешным, если по завершении этого процесса аудитория не может возразить тезису убеждающего при условии, что убеждающий придаёт своим предложениям и предложениям аудитории некоторое (хотя бы риторическое) значение. То, что риторическое значение не зависит от того, какое значение приписывают своим словам сами собеседники, позволяет показать, что, несмотря на проблему знаковой коммуникации, убеждающий может иметь риторический успех. Поскольку наш подход ориентирован исключительно на риторический успех в процессе убеждения посредством знаковой коммуникации, наша трактовка процесса убеждения имеет коммуникативно-прагматический характер. Как и подобает прагматическому подходу к процессу убеждения, наш подход не требует от убеждающего высказывать только те предложения, которые он сам понимает и считает истинными, и те аргументы, которые он сам понимает, считает корректными и приемлемыми. The paper propose a variant of a communicative-pragmatic approach to determining the meaning of proposition, expressed by agents in the discussion. We intend to: 1) to offer an interpretation of the process of persuasion, according to which it can be recognized as successful – even if it pays tribute to the fundamental problem of sign communication and admits that agents in the process of communication transmit only signs to each other, and the meanings of these are inaccessible to the interlocutor; 2) to show that popular contemporary approaches (Pragma-dialectical approach, AGM-approach, DBR-approach, etc.) to the process of persuasion do not offer this kind of interpretation the process of persuasion; 3) to establish precise and verifiable conditions which are necessary and sufficient to admit the process of persuasion in our interpretation as successful. Our approach uses contemporary approaches to the interpretation of the meaning of the proposition (first of all, the mental holism – M. Harrell, N. Blok, etc.), and also extrapolates actual structuralist approaches to the understanding of mathematical objects in the philosophy of mathematics (D. Hilbert, M. Resnik, S. Shapiro) on sign communication in a situation of persuasive argumentation. We reject the solution to the problem of sign communication, which in fact consists that it is quite possible to transfer visual images regardless of their content or meaning, which implies that the communication of agents is still possible through the transmission of such images. This solution, in fact, suggests discussing unsigned rather than signed communication, but, first, the transfer of beliefs as linguistically expressed objects is not possible in this case, and, second, there are strong arguments in favor of the fact that even the content-free objects of different agents are different, which also makes transfer impossible. We determine the rhetorical meaning of the proposition expressed by agents in the discussion. Such a meaning is also a proposition, the antecedent of which is a complete description of the agent's belief system, and the consequent is the original proposition. We also admit the persuasion process as rhetorically successful if, at the end of this process, the audience cannot object to the persuader's thesis, on the assumption that the persuader attaches some (at least rhetorical) significance to his or her suggestions and those of the audience. The fact that the rhetorical meaning does not depend on the meaning attributed to their words by the interlocutors themselves, allows us to show that, despite of the problem of sign communication the persuader can have a rhetorical success. Since our approach is focused solely on rhetorical success in the process of persuasion through sign communication, our interpretation of the process of persuasion has a communicative and pragmatic character. As befits a pragmatic approach to the persuasion process, our approach does not require the persuader to express only those proposition that he himself understands and considers to be true, and those arguments that he himself understands, considers its as correct and acceptable.


2021 ◽  
Vol 14 ◽  
pp. 52-64
Author(s):  
Micah Phillips-Gary ◽  

In this paper, I discuss full-blooded Platonism (the claim that all possible mathematical objects exist) as a response to the skeptical problem in the philosophy of mathematics as to how empirical beings can cognize non-empirical mathematical objects. I then attempt to develop an analogous position regarding the applicability of concepts to reality in response to the skeptical problem regarding how we can cognize an objective reality through human-constructed concepts. If all concepts meeting certain minimal conditions structure reality under some aspect, then objective knowledge is possible, regardless of how these concepts arose historically.


2012 ◽  
Vol 5 (4) ◽  
pp. 687-709 ◽  
Author(s):  
RICHARD PETTIGREW

In the philosophy of mathematics, indispensability arguments aim to show that we are justified in believing that mathematical objects exist on the grounds that we make indispensable reference to such objects in our best scientific theories (Quine, 1981a; Putnam, 1979a) and in our everyday reasoning (Ketland, 2005). I wish to defend a particular objection to such arguments called instrumental nominalism. Existing formulations of this objection are either insufficiently precise or themselves make reference to mathematical objects or possible worlds. I show how to formulate the position precisely without making any such reference. To do so, it is necessary to supplement the standard modal operators with two new operators that allow us to shift the locus of evaluation for a subformula. I motivate this move and give a semantics for the new operators.


Author(s):  
Patricia A. Blanchette

Mathematical realism is the view that the truths of mathematics are objective, which is to say that they are true independently of any human activities, beliefs or capacities. As the realist sees it, mathematics is the study of a body of necessary and unchanging facts, which it is the mathematician’s task to discover, not to create. These form the subject matter of mathematical discourse: a mathematical statement is true just in case it accurately describes the mathematical facts. An important form of mathematical realism is mathematical Platonism, the view that mathematics is about a collection of independently existing mathematical objects. Platonism is to be distinguished from the more general thesis of realism, since the objectivity of mathematical truth does not, at least not obviously, require the existence of distinctively mathematical objects. Realism is in a fairly clear sense the ‘natural’ position in the philosophy of mathematics, since ordinary mathematical statements make no explicit reference to human activities, beliefs or capacities. Because of the naturalness of mathematical realism, reasons for embracing antirealism typically stem from perceived problems with realism. These potential problems concern our knowledge of mathematical truth, and the connection between mathematical truth and practice. The antirealist argues that the kinds of objective facts posited by the realist would be inaccessible to us, and would bear no clear relation to the procedures we have for determining the truth of mathematical statements. If this is right, then realism implies that mathematical knowledge is inexplicable. The challenge to the realist is to show that the objectivity of mathematical facts does not conflict with our knowledge of them, and to show in particular how our ordinary proof-procedures can inform us about these facts.


Author(s):  
Stewart Shapiro

This article examines a number of issues and problems that motivate at least much of the literature in the philosophy of mathematics. It first considers how the philosophy of mathematics is related to metaphysics, epistemology, and semantics. In particular, it reviews several views that account for the metaphysical nature of mathematical objects and how they compare to other sorts of objects, including realism in ontology and nominalism. It then discusses a common claim, attributed to Georg Kreisel that the important issues in the philosophy of mathematics do not concern the nature of mathematical objects, but rather the objectivity of mathematical discourse. It also explores irrealism in truth-value, the dilemma posed by Paul Benacerraf, epistemological issues in ontological realism, ontological irrealism, and the connection between naturalism and mathematics.


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