Full-Blooded Conceptual Realism as a Response to Skeptical Relativism

2021 ◽  
Vol 14 ◽  
pp. 52-64
Author(s):  
Micah Phillips-Gary ◽  
Keyword(s):  
Philosophy Of Mathematics ◽  
Objective Knowledge ◽  
Objective Reality ◽  
Mathematical Objects ◽  
Conceptual Realism ◽  
Skeptical Problem

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.

scholarly journals Full Blooded Conceptual Realism as a Response to Skeptical Relativism

2021 ◽  
Vol 14 (1) ◽  
pp. 53-65
Author(s):  
Micah Phillips-Gary
Keyword(s):  
Philosophy Of Mathematics ◽  
Objective Knowledge ◽  
Objective Reality ◽  
Mathematical Objects ◽  
Conceptual Realism ◽  
Skeptical Problem

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.

Descartes on the Eternal Truths and Essences of Mathematics: An Alternative Reading

Vivarium ◽  
2016 ◽  
Vol 54 (2-3) ◽  
pp. 204-249
Author(s):  
Helen Hattab
Keyword(s):  
Rene Descartes ◽  
Philosophy Of Mathematics ◽  
Ontological Status ◽  
Divine Simplicity ◽  
Historical Evidence ◽  
Eternal Truths ◽  
Mathematical Objects ◽  
Divine Essence ◽  
René Descartes ◽  
Alternative Reading

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.

scholarly journals The Normative Pluriverse

2020 ◽  
Vol 18 (2) ◽  
Author(s):  
Matti Eklund
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Objects ◽  
Normative Realism ◽  
Important Form ◽  
The Way

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.

Naturalism in the Philosophy of Mathematics

Philosophy ◽  
2015 ◽  
Author(s):  
Mary Leng
Keyword(s):  
Natural Science ◽  
Philosophy Of Mathematics ◽  
Sole Source ◽  
Indispensability Argument ◽  
Mathematical Truth ◽  
Methodological Naturalism ◽  
Mathematical Objects ◽  
Ontological Naturalism ◽  
Epistemological Challenge ◽  
Genuine Problem

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.

Hilbertian Structuralism and the Frege-Hilbert Controversy†

2019 ◽  
Vol 27 (3) ◽  
pp. 335-361
Author(s):  
Fiona T Doherty
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Objects ◽  
Structuralist Approach ◽  
Complex Roots ◽  
In Virtue Of

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 $-1$.

Antirealism in the philosophy of mathematics

2018 ◽  
Author(s):  
A.W. Moore
Keyword(s):  
Philosophy Of Mathematics ◽  
Physical Reality ◽  
Mathematical Statement ◽  
Mathematical Objects ◽  
Mathematical Reality

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.

scholarly journals Transformative Experience in Skepticism. The External Standpoint and the Finitude of the Human Condition

Philosophy ◽  
2020 ◽  
Vol 95 (4) ◽  
pp. 395-417
Author(s):  
Rico Gutschmidt
Keyword(s):  
Decision Theory ◽  
Everyday Life ◽  
Human Condition ◽  
Objective Knowledge ◽  
Transformative Experience ◽  
Philosophical Practice ◽  
Transformative Experiences ◽  
Skeptical Problem ◽  
Insight Into ◽  
The Human Condition

AbstractAccording to its quietist readings, skepticism can be dissolved by demonstrating that the notion of ‘absolute objectivity’ is confused. The dissolution of this confusion is supposed to lead us to acquiesce in our finite and plain everyday life without being bothered anymore about the supposed need for objective knowledge. In contrast, I want to propose a transformative reading of skepticism according to which the philosophical practice of skepticism can be ‘epistemically transformative’. To this end, I will transpose L.A. Paul's notion of ‘epistemically transformative experience’ from decision theory to the realm of philosophical practice and argue that the modern skeptical problem of an external standpoint can evoke transformative experiences that lead to a new, albeit non-propositional, insight into the finitude of the human condition.

scholarly journals THE PROBLEM IN THE THEORY OF ARGUMENTATION: IS IT POSSIBLE TO CONVINCE THE INTERLOCUTOR? (A COMMUNICATIVE-PRAGMATIC SOLUTION)

2020 ◽  
pp. 9-40
Author(s):  
Марина Николаевна Вольф ◽  
Игорь Владимирович Берестов
Keyword(s):  
Fundamental Problem ◽  
Philosophy Of Mathematics ◽  
Belief System ◽  
Visual Images ◽  
Pragmatic Approach ◽  
Mathematical Objects ◽  
Free Objects ◽  
Dialectical Approach ◽  
Necessary And Sufficient

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.

INDISPENSABILITY ARGUMENTS AND INSTRUMENTAL NOMINALISM

2012 ◽  
Vol 5 (4) ◽  
pp. 687-709 ◽  
Author(s):  
RICHARD PETTIGREW
Keyword(s):  
Possible Worlds ◽  
Philosophy Of Mathematics ◽  
Scientific Theories ◽  
Mathematical Objects ◽  
Modal Operators ◽  
Indispensability Arguments ◽  
Everyday Reasoning ◽  
Do So

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.

Realism in the philosophy of mathematics

2018 ◽  
Author(s):  
Patricia A. Blanchette
Keyword(s):  
Human Activities ◽  
Philosophy Of Mathematics ◽  
Mathematical Truth ◽  
Mathematical Realism ◽  
Mathematical Objects ◽  
Natural Position ◽  
Potential Problems ◽  
The Subject ◽  
Important Form ◽  
Clear Relation

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.

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