Carl Posy

The Hebrew University of Jerusalem, Philosophy, Faculty Member

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Why Philosophers Should Care about Computational Complexity

One might think that, once we know something is computable, how efficiently it can be computed is... more One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory-the field that studies the resources (such as time, space, and randomness) needed to solve computational problems-leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction, Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.

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Kant’s Mathematical Realism

The Monist, 1984

Though my title speaks of Kant’s mathematical realism, I want in this essay to explore Kant’s rel... more Though my title speaks of Kant’s mathematical realism, I want in this essay to explore Kant’s relation to a famous mathematical anti-realist. Specifically, I want to discuss Kant’s influence on L. E. J. Brouwer, the 20th-century Dutch mathematician who built a contemporary philosophy of mathematics on constructivist themes which were quite explicitly Kantian.1 Brouwer’s theory (called intuitionism) is perhaps most notable for its belief that constructivism (whatever that means) requires us to abandon the traditional (classical) logic of mathematical reasoning in favor of a different canon of reasoning, called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive (or “realistic”) view of mathematics. This means that, according to Brouwer, when we do mathematics we must give up bivalence (the principle that a given sentence either is true or is determinately false), we must no longer use such familiar logical laws as excluded middle, and we must sometimes forebear from the classic method of reductio ad absurdum. All of these are intuitionistically invalid classical principles.

Computability and Constructibility

The MIT Press eBooks, Jun 7, 2013

This chapter contains sections titled: 5.1 Constructive Mathematics, 5.2 Hilbert and Brouwer: Con... more

Inside the Metaphysical Workshop

De Gruyter eBooks, Nov 18, 2019

Of Griffins and Horses

The infinite, the indefinite and the critical turn: Kant via Kripke models

Inquiry: An Interdisciplinary Journal of Philosophy, Oct 15, 2019

This paper aims to show that intuitionistic Kripke models are a powerful tool for interpreting Ka... more This paper aims to show that intuitionistic Kripke models are a powerful tool for interpreting Kant's 'Critical Philosophy'. Part I reviews some old work of mine that applies these models to provide a reading of Kant's second antinomy about the divisibility of matter and to answer several attacks on Kant's antinomies. But it also points out three shortcomings of that original application. First, the reading fails to account for Kant's second antinomy claim that matter is divisible 'ad infinitum' and his first antinomy claim that empirical space extends only 'ad indefinitum'. Secondly, in conformity with the 'assertability' heuristics for Kripke models, it attributes an assertability semantics to Kant. In fact, it attributes a pair of assertability semantics to Kant, but neither of these accords with modern assertabilism. And finally, one must wonder the propriety of using contemporary assertability and contemporary formal logic to save an eighteenth-century text. Part II goes historical to solve the problems about assertability. It outlines the Descartes-Leibniz dialectic about the infinite/indefinite distinction and demonstrates how each position reflects larger philosophical positions. It then demonstrates that Kant's two versions of assertability and his version of the infinite/indefinite dichotomy emerge naturally from his attempts (pre-critical and then critical) to resolve a tension in Leibniz's philosophy. In light of this demonstration, Part III adapts the Kripke model interpretation so that it does reflect that dichotomy. It then refines the Kripke semantics in order to model some of Kant's signature critical doctrines, and it shows how Kant's critical version of the infinite/ indefinite distinction differs from all those of his predecessors and from his own pre-critical version. It also addresses the question of using contemporary logical machinery to interpret Kant. Part IV turns the tables and uses what we have learned about Kant and modern semantics in order to address a pair of issues in the contemporary critical foundations of logic.

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Mathematical Intuitionism

L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its obj... more L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism – from elementary number theory through to Brouwer's uniform continuity theorem – and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth.

Authenticity or Autonomy? Leibniz and Kant on Practical Rationality

Springer eBooks, Aug 8, 2008

Review: Desmond Paul Henry, The Truncation of Truth-Functional Calculation

Journal of Symbolic Logic, Mar 1, 1974

Kant’s Philosophy of Mathematics

Kant's Philosophy of Mathematics: Volume 1: The Critical Philosophy and its Roots

The theory of empirical sequences

Journal of Philosophical Logic, 1977

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Immediacy and the Birth of Reference in Kant: The Case for Space

Henry Desmond Paul. The truncation of truth-functional calculation. Notre Dame journal of formal logic, vol. 2 (1961), pp. 193–205

Journal of Symbolic Logic, Mar 1, 1974

Where Have All the Objects Gone?

Southern Journal of Philosophy, Mar 26, 2010

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Intuition and Infinity: A Kantian Theme with Echoes in the Foundations of Mathematics

Royal Institute of Philosophy Supplement, Oct 1, 2008

POSY 'I cannot say, therefore, that the world is infinite in space. .. Any such concept of magnit... more POSY 'I cannot say, therefore, that the world is infinite in space. .. Any such concept of magnitude, as being that of a given infinitude, is empirically impossible, and therefore, in reference to the world as an object of the senses, also absolutely impossible' (A520/B549). Note: Except as indicated, I will use the Kemp Smith (1929) translation of the Critique of Pure Reason (New York: St. Martin's). 2 'Now we have the cosmic whole only in concept, never, as whole in intuition' (A519-520/B546-7). 3 'Consequently, the original representation of space is an a priori intuition, not a concept' (A25/B39-40).

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Intuitionism and Philosophy

Oxford University Press eBooks, Mar 3, 2005

Brouwer's constructivism

Synthese, 1974

In this respect then no one who has even a slight acquaintance with geometry will deny that the n... more In this respect then no one who has even a slight acquaintance with geometry will deny that the nature of this science is in fiat contradiction with the absurd language used by mathematicians, for want of better terms. They constantly talk of 'operations' like 'squaring', 'applying', 'adding' and so on, as if the object were to do something, whereas the true purpose of the whole subject is knowledge -knowledge, moreover, of what eternally exists, not of anything that comes to be this or that at some time and ceases to be. PLATO, Republic, VII, 527

van Dalen Dirk. Intuitionistic logic. Handbook of philosophical logic, Volume III, Alternatives to classical logic, edited by Gabbay D. and Guenthner F., Synthese library, vol. 166, D. Reidel Publishing Company, Dordrecht etc. 1986, pp. 225–339.Felscher Walter. Dialogues as a foundation for intui...

Journal of Symbolic Logic, Jun 1, 1992

Computability: Turing, Gdel, Church, and Beyond

The MIT Press eBooks, Jun 7, 2013

In the 1930s a series of seminal works published by Alan Turing, Kurt Gdel, Alonzo Church, and ot... more In the 1930s a series of seminal works published by Alan Turing, Kurt Gdel, Alonzo Church, and others established the theoretical basis for computability. This work, advancing precise characterizations of effective, algorithmic computability, was the culmination of intensive investigations into the foundations of mathematics. In the decades since, the theory of computability has moved to the center of discussions in philosophy, computer science, and cognitive science. In this volume, distinguished computer scientists, mathematicians, logicians, and philosophers consider the conceptual foundations of computability in light of our modern understanding. Some chapters focus on the pioneering work by Turing, Gdel, and Church, including the Church-Turing thesis and Gdel's response to Church's and Turing's proposals. Other chapters cover more recent technical developments, including computability over the reals, Gdel's influence on mathematical logic and on recursion theory and the impact of work by Turing and Emil Post on our theoretical understanding of online and interactive computing; and others relate computability and complexity to issues in the philosophy of mind, the philosophy of science, and the philosophy of mathematics. Contributors:Scott Aaronson, Dorit Aharonov, B. Jack Copeland, Martin Davis, Solomon Feferman, Saul Kripke, Carl J. Posy, Hilary Putnam, Oron Shagrir, Stewart Shapiro, Wilfried Sieg, Robert I. Soare, Umesh V. Vazirani

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