This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed (...) by Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (shrink)

We argue that Wittgenstein’s philosophical perspective on Gödel’s most famous theorem is even more radical than has commonly been assumed. Wittgenstein shows in detail that there is no way that the Gödelian construct of a string of signs could be assigned a useful function within (ordinary) mathematics. — The focus is on Appendix III to Part I of Remarks on the Foundations of Mathematics. The present reading highlights the exceptional importance of this particular set of remarks and, (...) more specifically, emphasises its refined composition and rigorous internal structure. (shrink)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.

This book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

This book presents and survey of the foundations of mathematics. The emphasis is on a mathematical comparison of systems rather than on any exhaustive development of analysis within a single system. Nevertheless, for most systems considered, enough details are given for the development of arithmetic, and the method of constructing the other notions of analysis is indicated. The elements of the general theory of cardinal and ordinal numbers are also furnished in the course of this work.

Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.

The physical foundations of mathematics in the theory of emergent space-time-matter were considered. It is shown that mathematics, including logic, is a consequence of equation which describes the fundamental field. If the most fundamental level were described not by mathematics, but something else, then instead of mathematics there would be consequences of this something else.

Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.

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

A new hypothesis on the basic features characterising the Foundations of Mathematics is suggested. By means of them the entire historical development of Mathematics before the 20th Century is summarised through a table. Also the several programs, launched around the year 1900, on the Foundations of Mathematics are characterised by a corresponding table. The major difficulty that these programs met was to recognize an alternative to the basic feature of the deductive organization of a theory (...) - more precisely, to Hilbert’s main tenet. Ironically, already half a century before the births of these programs the alternative organization has been substantially represented by Lobachevsky's theory on parallel lines. Moreover, although each program’s founder recognised the basic features in a partial way only, all together these programs represented just the four possible foundational approaches. (shrink)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27–29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set (...) theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career. Other topics covered are forcing axioms, inner model theory, the partition calculus, and the theory of ultrafilters. This volume should make a suitable introduction to Woodin's work and the concerns which motivate it. The papers should be of interest to graduate students and researchers in both mathematics and philosophy of mathematics, particularly in set theory, foundations and related areas. (shrink)

Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is (...) to provide an arena for exploring relations and interactions between mathematical fields, their relative strengths, etc. Given the different goals, there is little point to determining a single foundation for all of mathematics. (shrink)

The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books.

This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The first two sections focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current (...) research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The further sections then build on this and are centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics. (shrink)

The Dutch mathematician and philosopher L. E. J. Brouwer (1881-1966) developed a foundation for mathematics called 'intuitionism'. Intuitionism considers mathematics to consist in acts of mental construction based on internal time awareness. According to Brouwer, that awareness provides the fundamental structure to all exact thinking. In this note, it will be shown how this strand of thought leads to an intuitionistic function that is effectively computable yet non-recursive.

A Non-Technical Introduction to Ethnomethodological Investigations of the Foundations of Mathematics through the Use of a Theorem of Euclidean Geometry* I ...

Book Information The Foundations of Mathematics in the Theory of Sets. The Foundations of Mathematics in the Theory of Sets J. P. Mayberry Cambridge Cambridge University Press 2000 xx + 424 Hardback US$80.00 By J. P. Mayberry. Cambridge University Press. Cambridge. Pp. xx + 424. Hardback:US$80.00.

In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this (...) use of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H(ℵ1) and H(ℵ2). Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics. (shrink)

In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this (...) use of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H(ℵ1) and H(ℵ2). Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics. (shrink)

Wittgenstein’s philosophy of mathematics is often devalued due to its peculiar features, especially its radical departure from any of standard positions in foundations of mathematics, such as logicism, intuitionism, and formalism. We first contrast Wittgenstein’s finitism with Hilbert’s finitism, arguing that Wittgenstein’s is perspicuous or surveyable finitism whereas Hilbert’s is transcendental finitism. We then further elucidate Wittgenstein’s philosophy by explicating his natural history view of logic and mathematics, which is tightly linked with the so-called rule-following problem (...) and Kripkenstein’s paradox, yielding vital implications to the nature of mathematical understanding, and to the nature of the certainty and objectivity of mathematical truth. Since the incompleteness theorems, foundations of mathematics have mostly lost their philosophical driving force, and anti-foundationalism has become prevalent and pervasive. Yet new foundations have nevertheless emerged and come into the scene, namely categorical foundations. We articulate the foundational significance of category theory by explicating three forms of foundations, i.e., global foundations, local foundations, and conceptual foundations. And we explore the possibility of categorical unified science qua pluralistic unified science, arguing for the categorical unity of science on both mathematical and philosophical grounds. We then turn to an issue in conceptual foundations of mathematics, namely the nature of the concept of space. We elucidate Wittgenstein’s intensional conception of space in relation to Brouwer’s theory of space continua and to the modern conception of space as point-free structure in category theory and algebraic geometry. We finally give a bird’s-eye view of mathematical philosophy from a Wittgensteinian perspective, and further sheds new light on Wittgenstein’s constructive structuralism and his view of incompleteness and contradictions in mathematics. (shrink)

Wittgenstein’s philosophy of mathematics is often devalued due to its peculiar features, especially its radical departure from any of standard positions in foundations of mathematics, such as logicism, intuitionism, and formalism. We first contrast Wittgenstein’s finitism with Hilbert’s finitism, arguing that Wittgenstein’s is perspicuous or surveyable finitism whereas Hilbert’s is transcendental finitism. We then further elucidate Wittgenstein’s philosophy by explicating his natural history view of logic and mathematics, which is tightly linked with the so-called rule-following problem (...) and Kripkenstein’s paradox, yielding vital implications to the nature of mathematical understanding, and to the nature of the certainty and objectivity of mathematical truth. Since the incompleteness theorems, foundations of mathematics have mostly lost their philosophical driving force, and anti-foundationalism has become prevalent and pervasive. Yet new foundations have nevertheless emerged and come into the scene, namely categorical foundations. We articulate the foundational significance of category theory by explicating three forms of foundations, i.e., global foundations, local foundations, and conceptual foundations. And we explore the possibility of categorical unified science qua pluralistic unified science, arguing for the categorical unity of science on both mathematical and philosophical grounds. We then turn to an issue in conceptual foundations of mathematics, namely the nature of the concept of space. We elucidate Wittgenstein’s intensional conception of space in relation to Brouwer’s theory of space continua and to the modern conception of space as point-free structure in category theory and algebraic geometry. We finally give a bird’s-eye view of mathematical philosophy from a Wittgensteinian perspective, and further sheds new light on Wittgenstein’s constructive structuralism and his view of incompleteness and contradictions in mathematics. (shrink)

A presupposition of this paper is that "mathematical" entities exhibit referential problems not affecting other sorts of entities. This view places constraints both on semantics for mathematical theories and on formal semantics generally. A main goal of the paper is to illustrate how "sets" can be avoided in semantics by utilizing "properties". This method is then exploited in the case of mathematics to obtain interpretations involving no "mathematical entities" but nevertheless producing "platonistic" truth-Value distributions.

Mathematical methods in biology occupy a somewhat peculiar position, and the attitude of many biologists toward them is similar to that of many practical engineers toward what is called pure scientific research. The modern progressive engineer recognizes the value of pure science, which seeks for truth regardless of any possibility of practical applications; yet he still frequently shows a definite dislike towards such investigations. The whole history of civilization demonstrates that discoveries which, at the time they were made, did not (...) show any signs of the possibility of being turned into practical use, now lie at the basis of many useful engineering devices. As Professor R. A. Millikan has said, we owe our present-day civilization to Galileo. Every educated person, be he pure scientist or engineer, recognizes this fact. And yet the frame of mind of the “practical” man is not too receptive to investigations which do not show at least some remote prospects of practical applications. Unless he can see at least hypothetically some practical results of such research, he will not give it his whole-hearted enthusiasm. The ultimate criterion in the evaluation of any research still remains its practical use. Those pure scientists who have had close contact with what is called “industrial research” know only too well the truth of the above statement. (shrink)

It turns out, time and time again, in order to make serious progress in f.o.m., we need to take actual reasoning and actual development into account at precisely the proper level. If we take these into account too much, then we are faced with information that is just too difficult to create an exact science around - at least at a given state of development of f.o.m. And if we take these into account too little, our findings will not have (...) the relevance to mathematical practice that could be achieved. (shrink)

This pioneering book demonstrates the crucial importance of Wittgenstein's philosophy of mathematics to his philosophy as a whole. Marion traces the development of Wittgenstein's thinking in the context of the mathematical and philosophical work of the times, to make coherent sense of ideas that have too often been misunderstood because they have been presented in a disjointed and incomplete way. In particular, he illuminates the work of the neglected 'transitional period' between the Tractatus and the Investigations.