The Human A Priori is a collection of essays by A. W. Moore, one of them previously unpublished and the rest all revised. These essays are all concerned, more or less directly, with something ineliminably anthropocentric in our systematic pursuit of a priori sense-making. Part I deals with the nature, scope, and limits of a priori sense-making in general. Parts II, III, and IV deal with what are often thought to be the three great exemplars of the (...) systematic pursuit of such sense-making: philosophy in the case of Part II, ethics in the case of Part III, and mathematics in the case of Part IV. Much of the attention throughout is devoted to the work of other philosophers: Kant and Wittgenstein feature prominently, and five of the essays take the form of reviews or critical notices of recent work in philosophy. But the interest in never purely exegetical. One of the lessons that emerges from the essays, either in opposition to the views of these other philosophers or by invocation of their views, is that we humans achieve nothing of real significance in philosophy, ethics, or mathematics except from a human point of view, and hence that all three of these pursuits can be said to betoken what may reasonably be called 'the human a priori'. (shrink)

This thesis is concerned with explaining how a subject can acquire a priori knowledge of arithmetic. Every account for arithmetical, and in general mathematical knowledge faces Benacerraf's well-known challenge, i.e. how to reconcile the truths of mathematics with what can be known by ordinary human thinkers. I suggest four requirements that jointly make up this challenge and discuss and reject four distinct solutions to it. This will motivate a broadly Fregean approach to our knowledge of arithmetic and (...) class='Hi'>mathematics in general. Pursuing this strategy appeals to the context principle which, it is proposed, underwrites a form of Platonism and explains how reference to and object-directed thought about abstract entities is, in principle, possible. I discuss this principle and defend it against different criticisms as put forth in recent literature. Moreover, I will offer a general framework for implicit definitions by means of which - without an appeal to a faculty of intuition or purely pragmatic considerations - a priori and non-inferential knowledge of basic mathematical principles can be acquired. In the course of this discussion, I will argue against various types of opposition to this general approach. Also, I will highlight crucial shortcomings in the explanation of how implicit definitions may underwrite a priori knowledge of basic principles in broadly similar conceptions. In the final part, I will offer a general account of how non-inferential mathematical knowledge resulting from implicit definitions is best conceived which avoids these shortcomings. (shrink)

Abstractionism, which is a development of Frege's original Logicism, is a recent and much debated position in the philosophy of mathematics. This volume contains 16 original papers by leading scholars on the philosophical and mathematical aspects of Abstractionism. After an extensive editors' introduction to the topic of abstractionism, the volume is split into 4 sections. The contributions within these sections explore the semantics and meta-ontology of Abstractionism, abstractionist epistemology, the mathematics of Abstractionis, and finally, Frege's application constraint within (...) an abstractionist setting. (shrink)

This article is mainly a critique of Philip Kitcher's book, The Nature of Mathematical Knowledge. Four weaknesses in Kitcher's objection to Kant arise out of Kitcher's failure to recognize the perspectival nature of Kant's position. A proper understanding of Kant's theory of mathematics requires awareness of the perspectival nuances implicit in Kant's theory of pure intuition.

Some modern thinkers have supposed that “cause” is an outworn notion, or at least that it is one of which modern science has no need. This is due mainly to the discovery that, while the scientist can give us general laws as to what in fact happens, he cannot help us to discern the reason for the laws or the inward nature of the forces on which they depend. He can tell us the “that” but not the “why”; he cannot (...) show us in a single case that the effect follows necessarily a priori from the nature of the cause, that any other effect than the one which actually takes place would be logically impossible. He has studied the law of gravitation, but this law does not enable him to see why material bodies should attract each other in this fashion; it is only a generalized statement of the fact that they do. He knows that certain substances, if absorbed by eating, will nourish and others destroy our tissues; but he cannot say why they should do so. He can no doubt analyse them further and discover that, for example, meat is nourishing because it contains a large proportion of nitrogenous matter, but he could not tell a priori whether this nitrogenous matter would be likely to nourish or to poison us. Only where mathematics can be applied do we see necessity in such a way that any alternative becomes inconceivable to us; but mathematics alone can never establish from a quantity present here and now what quantity there will be at a later time or in another part of space. Mathematics can show, e.g. , that, if there is 2 + 2 here and now, there must be 4 here and now, not that, if there is 2 + 2 here and now, there will be 4 in an hour's time or a mile away; and therefore it cannot be made the sole basis of any causal law whatever. (shrink)

In this paper I will argue that Boghossian's explanation of how we can acquire a priori knowledge of logical principles through implicit definitions commits a transmission of warrant-failure. To this end, I will briefly outline Boghossian's account, followed by an explanation of what a transmission of warrant-failure consists in. I will also show that this charge is independent of the worry of rule-circularity which has been raised concerning the justification of logical principles and of which Boghossian is fully aware. (...) My argument comes in two steps: firstly, I will argue for the insufficiency of Boghossian's template which is meant to explain how a subject can acquire a warrant for logical principles. I will show however that this insufficiency of his template can be remedied by adopting what I call the Disquotational Step. Secondly, I will argue that incorporating this further step makes his template subject to a transmission of warrant-failure, assuming that certain rather basic and individually motivated principles hold. Thus, Boghossian's account faces a dilemma: either he adopts the Disquotational Step and subjects his account to the charge of a transmission of warrant-failure, or he drops this additional step leaving the account confronted with explaining the gap that has previously been highlighted. I will then suggest various rejoinders that Boghossian might adopt but none of which - I will argue - can resolve the dilemma. Lastly, I will raise and briefly discuss the question whether this worry generalizes to other accounts, such as Hale and Wright's that aim to explain our knowledge of logic and/or mathematics in virtue of implicit definitions. (shrink)

It is proved that the basic framework of the premises and reasoning of Wittgenstein's “Tractatus Logico-philosophicus” corresponds quite well to the transcendental method (as formulated by H. Cohen). Whereas Kant’s philosophy proceeds from the fact of existence of mathematics and mathematised natural science and investigates their conditions of possibility, Wittgenstein proceeds from the fact that propositions of language describe reality and reveals the conditions of possibility of such descriptions. Kant, answering the question about the conditions of possibility of the (...) named sciences, comes to the idea of the transcendental subject and the distinction between the world of phenomena and the thing in itself. Wittgenstein's investigation of the conditions of possibility that the world is described by propositions leads to the assertion that both the world and language are together in logical space. The latter constitutes the a priori and transcendental condition of the possibility that language “reaches out” to reality. For both the theories - in the “Critique of Pure Reason” and in the “Tractatus Logico-Philosophicus” - the idea of the boundary is important. In the “Critique of Pure Reason” it is the boundary of possible experience and cognition, while in the “Tractatus Logico-Philosophicus” it is the boundary of what can be thought and expressed by meaningful propositions. Related to the different definitions of the boundary is the difference in the treatment of mathematised natural science. For “The Critique of Pure Reason” was created in the era of unconditional acceptance of Newtonian mechanics. And the “Tractatus Logico-Philosophicus” was created at the time of the crisis of the Newtonian paradigm and its replacement by other notions of time and space. However, the idea of boundary, which is present in both doctrines, determines closeness in the attitude towards metaphysics between the author of “The Critique of Pure Reason” and the author of “The Tractatus Logico-Philosophicus”. The study also shows that Wittgenstein did not follow logicism in his philosophy of mathematics. For him, both mathematical objects and propositions of logic are constructions. The conviction about the constructive character of mathematical and logical objects shows an affinity with the Kantian tradition in the philosophy of mathematics. (shrink)

Shaffer takes a tour of some perennial questions in this lucid and simply written primer. How do I know I am not dreaming? How does reality differ from a dream? How can we be certain of our knowledge? Varying viewpoints are briefly summarized. The fallibilist view that even a priori mathematical truths and first person reports of feelings and perceptions are subject to error is examined, as is the anti-fallibilist reply that the theoretical possibility of error, without actual evidence, (...) is not sufficient to disturb the certainty of certain kinds of knowledge. Our knowledge of the external world is questioned by Berkeleyan idealism, which treats matter as nothing but a perception of our minds, and by phenomenalism, which treats material objects as nothing but collections of sense data. Opposed to these views are causal theories, which view physical objects as existing independently of observers, and sense data as the result of the causal interaction of material objects and particular bodies. It should be noted that the book's format requires such drastic foreshortening of Descartes' and Berkeley's views that its characterization of the former's system as "one of the great achievements in man's intellectual history" is in no way confirmed by the textual references. Other questions discussed in lively fashion are the self ; identity theory ; epiphenomenalism ; and interactionism. Is the mind best defined as the collection of mental events one has during a lifetime? If these events are all connected with the same body, the mind seemingly cannot survive the death of the body. Finally, theistic and atheistic views about the meaning of life are compared, with the Aristotelian view of teleological causation as the common denominator. The book asks many questions, provides few answers, but serves as an enticement to further exploration.--A. T. (shrink)

The connection of Hermann Сohen’s “The Logic of Pure Knowledge” with the revolutionary transformations in physics and mathematics at the end of the 19th century is shown. Сohen criticised Kant’s answer to the question “How is mathematics possible”? If Kant refers to a priori forms of pure intuition, Сohen sees in it a restriction of freedom of mathematical thinking by limits of intuition. It has been shown that Cohen's position is in accordance with the main development of (...) mathematics in the last decades of the 19th century, in particular, with K. Weierstrass’ striving to get rid of geometrical or mechanical images and intuitions in the mathematical analysis. Cohen was also well informed about the latest ideas in physics of his time. They are also discussed in “The Logic of Pure Knowledge.” The revolutionary spirit of physics and mathematics was appealing to Cohen, and he felt a corresponding enthusiasm for it. The ongoing scientific revolution is consonant with Cohen’s assertion that the foundations of science are hypotheses. The purity of pure thinking does not guarantee the correctness of any of its constructions. Each step in the development of science requires a critique of existing notions. The development of knowledge from the naive to the critical one shows a movement from a picture of the world as a set of stable things to a picture of continuous movement and change, where change is more important than what changes. Cohen sees such a development in an evolution of the concept of substance in modern physics and he welcomes the replacement of material substance by energy, seeing this movement as a confirmation of critical idealism. Finally, it is discussed whether we can speak of the actuality of Cohen’s logic of pure knowledge. (shrink)

I offer a variant of Putnam’s “permutation argument,” originally an argument against metaphysical realism. This variant is called the “natural permutation argument.” I explain how the natural permutation argument generates a form of referential inscrutability which is not resolvable by consideration of “natural properties” in the sense of Lewis’s response to Putnam. However, unlike the classical permutation argument (which is applicable to nearly all interpretations of all first-order theories), the natural permutation argument only applies to interpretations which have some special (...) symmetries. I give an analysis of the interpretations to which the natural permutation argument does apply, and I explain how, when it fails to apply, the referential inscrutability generated by permutation arguments is resolvable by a Lewisian strategy. In order to demonstrate how these problems of referential inscrutability play out in an a priori setting relevant to philosophy, I discuss the applicability of the natural permutation argument in set-theoretic reasoning. I use the well-known Kunen inconsistency theorem to show that, in Zermelo–Fraenkel set theory, the Axiom of Choice is sufficient to resolve referential inscrutability. I then explain how, as a result of a recent theorem of Daghighi–Golshani–Hamkins–Jeřábek, in certain non-well-founded set theories the natural permutation argument does yield an intractable inscrutability of reference. (shrink)

I argue that we need not accept Quine's holistic conception of mathematics and empirical science. Specifically, I argue that we should reject Quine's holism for two reasons. One, his argument for this position fails to appreciate that the revision of the mathematics employed in scientific theories is often related to an expansion of the possibilities of describing the empirical world, and that this reveals that mathematics serves as a kind of rational framework for empirical theorizing. Two, this (...) holistic conception does not clearly demarcate pure mathematics from applied mathematics. In arguing against Quine, I present a formal account of applied mathematics in which the mathematics employed in an empirical theory plays a role that is analogous to the epistemological role Kant assigned synthetic a priori propositions. According to this account, it is possible to insulate pure mathematics from empirical falsification, and there is a sense in which applied mathematics can also be labeled as a priori. (shrink)

I examine how Kant argues for the transcendental ideality of space. I defend a reading on which Kant accepts the ideality of space because it explains our (actual) knowledge that mathematical judgments are necessarily true. I argue that this reading is preferable over the alternative suggestion that Kant can infer the ideality of space directly from the fact that we have an a priori intuition of space. Moreover, I argue that the reading I propose does not commit Kant to (...) incoherent modal views. If we carefully distinguish between different senses of modality, the fact that our spatial form of intuition is (in some sense) contingent does not undermine the claim that this form can explain how our mathematical judgments are (in some sense) necessary. (shrink)

This collection of newly commissioned essays, edited by NYU philosophers Paul Boghossian and Christopher Peacocke, resumes the current surge of interest in the proper explication of the notion of a priori. The authors discuss the relations of the a priori to the notions of definition, meaning, justification, and ontology, explore how the concept figured historically in the philosophies of Leibniz, Kant, Frege, and Wittgenstein, and address its role in the contemporary philosophies of logic, mathematics, mind, and science. (...) The editors’ Introduction familiarizes the reader with the issues that are to be explored in detail in later parts of the anthology. (shrink)

Robert Hanna works out a unified contemporary Kantian theory of rational human cognition and knowledge. Along the way, he provides accounts of intentionality and its contents, sense perception and perceptual knowledge, the analytic-synthetic distinction, the nature of logic, and a priori truth and knowledge in mathematics, logic, and philosophy. This book is specifically intended to reach out to two very different audiences: contemporary analytic philosophers of mind and knowledge, and contemporary Kantian philosophers or Kant-scholars. At the same time, (...) it rides the crest of a wave of exciting and revolutionary emerging new trends and new work in the philosophy of mind and epistemology, with a special concentration on the philosophy of perception. What is revolutionary in this new wave are its strong emphases on action, on cognitive phenomenology, on disjunctivist direct realism, on embodiment, and on sense perception as a primitive and proto-rational capacity for cognizing the world. Hanna makes a fundamental contribution to this philosophical revolution by giving it a specifically contemporary Kantian twist, and by pushing these new lines of investigation radically further. (shrink)

Kant’s philosophy of mathematics presents two fundamental problems of interpretation: (1) Kant claims that mathematical truths or “judgments” are synthetic a priori; and (2) Kant maintains that intuition is required for generating and/or understanding mathematical statements. Both of these problems arise for us because of developments in mathematics since Kant. In particular, the axiomatization of geometry--Kant’s paradigm of mathematical thinking--has made it seem to some commentators as, for example, Russell, that both (1) and (2) are false (Russell (...) 1919, p. 145).2 If virtually all of mathematics, including geometry, is axiomatizable, it would seem that mathematics results in analytic judgments that are totally independent of sensibility, the source, according to Kant, of intuition. In this paper I will address both of these difficulties. I shall argue that Kant’s understanding of both “synthetic” and “intuition” make his position immune to these criticisms. (shrink)

We examine the sense in which logic is a priori, and explain how mathematical theories can be dichotomized non-trivially into analytic and synthetic portions. We argue that Core Logic contains exactly the a-priori-because-analytically-valid deductive principles. We introduce the reader to Core Logic by explaining its relationship to other logical systems, and stating its rules of inference. Important metatheorems about Core Logic are reported, and its important features noted. Core Logic can serve as the basis for a foundational program (...) that could be called Natural Logicism, an exposition of which will build on the (meta)logical ideas explained here. (shrink)

ABSTRACT There is general agreement that Kant’s thought strongly influenced Heidegger’s. Nevertheless, there is still much work to be done in order to fully appreciate this influence. A central theme to disclose the relation between these authors is the role they give to the transcendental. In this paper I show that Kant’s account of intuition is the focus of Heidegger’s interpretation of Kant in his Die Frage nach dem Ding, since Heidegger interprets Kant’s treatment of intuition as a delimiting of (...) the modern mathematical tendency to determine everything out of pure reason alone. I argue that this interpretation is at the basis of Heidegger’s appropriation of Kant’s synthetic a priori in this work, the aim of which is to take this notion away from the sphere of the objects of mathematical physical science, and place it in the realm of everyday experience. I claim that Heidegger thinks that this move can only be accomplished by removing universality and necessity from Kant’s a priori. Finally, I show that Heidegger’s appropriation of Kant’s synthetic a priori goes together with and illuminates his understanding of transcendence as das Zwischen, ‘the between us and things’. RESUMO É consenso geral que o pensar de Kant influenciou fortemente o pensar de Heidegger. Porém, há ainda muito trabalho a fazer para se conseguir apreciar essa influência plenamente. Um tema crucial para conseguir expor a relação entre esses dois autores é o rol que ambos dão ao transcendental. Neste artigo, vou mostrar que o tratamento que Kant faz da intuição é o foco da interpretação que Heidegger oferece acerca de Kant em sua obra Die Frage nach dem Ding, dado que Heidegger interpreta o jeito como Kant trabalha a intuição como uma delimitação da tendência matemática moderna a determiná- lo tudo pelo só uso da razão pura. Argumento que essa interpretação está na base da apropriação que Heidegger faz do sintético a priori kantiano nesta obra, cujo fim é tirar essa noção fora da esfera dos objetos da ciência físico- matemática e posicioná-la no reino da experiência cotidiana. Sustento que Heidegger pensa que isto só pode ser alcançado ao eliminar a necessidade e a universalidade do a priori kantiano. Finalmente, mostro que a apropriação que Heidegger faz do sintético a priori kantiano vai junto e ilumina sua compreensão da transcendência como o das Zwischen, ‘o entre nós e as coisas’. (shrink)

The problem of the relationship between mathematics and objective reality, which arose in early antiquity, is still a subject of heated discussion. The discussions are mainly about the question that probably was posed most clearly by Immanuel Kant in his Critique of Pure Reason: "How do subjective conditions of thought have objective validity, that is, how do they become conditions of the possibility of all knowledge of objects?" Is it because they are themselves elements of objective reality, or because (...) they are contained in thought a priori, before and independently of experience, or, finally, because they are subjective images of corresponding facets and aspects of objective reality? (shrink)

This is Part I of a two-part study of the foundations of mathematics through the lenses of (i) apriority and analyticity, and (ii) the resources supplied by Core Logic. Here we explain what is meant by apriority, as the notion applies to knowledge and possibly also to truths in general. We distinguish grounds for knowledge from grounds of truth, in light of our recent work on truthmakers. We then examine the role of apriority in the realism/anti-realism debate. We raise (...) a hitherto unnoticed problem for any Orthodox Realist who attempts to explain the a priori. (shrink)

Consider three widely shared claims that have not been discussed vis-à-vis one another. In his Proslogion, Saint Anselm argued that the claim “God exists” is true. If an intuition that a claim c is a useful a-priori justificatory resource, this can only be because such an intuition is a justification that c is true. And if an intuition that c is a justification that c is true, c can stand, not only for mathematical or logical claims, but also for (...) controversial philosophical ones, e.g., “God exists”. This essay addresses to while dialoguing with the literature on Anselm and intuition and articulating an alternative reading of the Proslogion. The alternative reading is that regardless of whether the Proslogion backs up or aims to back up the claim that “God exists” is true, it implicitly articulates the Meaning Argument whose conclusion is that all persons of faith are able to understand that the claim “God exists” is meaningful. This argument, it is argued, is evidence that an intuition that c may be a useful a-priori justificatory resource even if such intuition does not track truth, but merely meaning in being a justification for taking c to be meaningful. It is also supported that an intuition that “God exists” is not a justification that this claim is true. This is an indication that there may be reasons for thinking that the same applies to other controversial philosophical claims. (shrink)

a priori. Since I ended up defending an unpopular answer to this question—"No"—it’s hardly surprising that people have scrutinized the account, or that many have concluded that I stacked the deck in the first place. Of course, this was not my view of the matter. My own judgment was that I’d uncovered the tacit commitments of mathematical apriorists and that the widespread acceptance of mathematical apriorism rested on failure to ask what was needed for knowledge to be a (...) class='Hi'>priori . Nevertheless, my critics have raised important challenges, and have offered rival conceptions that are less demanding. I want to examine their objections to my explication of a priori knowledge, and to explore whether the weaker alternatives succeed in preserving traditional philosophical claims. What follows is a mixture of penitence and intransigence. (shrink)

Wenn die fuhrende "paradigmatische" Philosophie Voraussetzungen wie die der Spaltung der Wirklichkeit in sich schliesst und wenn diese Philosophie empirisch die absolut leere -- ohne jede Moglichkeit der wissenschaftlichen Untersuchung --ist, so ist auch die Behauptung der Unmoglichkeit des Irrefuhrens und der Uberflussigkeit solcher Philosophie sinnvoll. Wenn aber auch die bedeutendste Kritik dieses paradigmatischen Rahmens eine Schwache zeigt, ist auf das Bedurfnis einer "Verweigerung" zu schliessen. Darum bedarf es einer Redefinition des Begriffs des aprioristischen Wissens. "A priori" bezeichnet in (...) diesem Sinne die Beschaftigung mit Wissen, das von den Objekten der empfindlich vernehmbaren Welt absolut unabhangig ist. In Ubereinstimmung mit dieser Definition ist das Verstandnis der Mathematik als der Art des aprioren Wissens moglich, oder epistemologisch gesagt: unter der Aktualitat der erreichten mathematischen Moglichkeit. (shrink)

In this dissertation, I present a novel account of the components that have a peculiar epistemic role in our scientific inquiries, since they contribute to establishing a form of coordination. The issue of coordination is a classic epistemic problem concerning how we justify our use of abstract conceptual tools to represent concrete phenomena. For instance, how could we get to represent universal gravitation as a mathematical formula or temperature by means of a numerical scale? This problem is particularly pressing when (...) justification for using these abstract tools comes, in part or entirely, from knowledge which is not independent from them, thus leading to threats of circularity. Achieving coordination between some abstract conceptual tools and the concrete phenomena that they are supposed to represent is usually a complex process, which involves several epistemic components. Some of these components eventually provide stable conditions for applying those abstract representations to concrete phenomena. It is in this sense of providing certain conditions of applicability that different philosophical traditions, as well as some contemporary reappraisals, view these components as constitutive or a priori. In this work, I present a new gradualist, contextualist, and relational approach to understand these constitutive components of scientific inquiry. It is gradualist inasmuch as the degree to which some component is constitutive depends on three quantifiable features: quasi-axiomaticity, generative potential, and empirical shielding. Since the quantification of these three features impinges on the history and practice of using these components in a scientific context, my approach is a contextualist one. Finally, my approach is relational in a double sense: first, it identifies ordinal relationships among epistemic components with respect to their constitutive character; second, these relationships are relative to a scientific framework of inquiry. After introducing my account and a classic example of constitutively a priori principles, i.e., Friedman’s (2001) analysis of Newtonian mechanics, I turn to my own case studies to demonstrate the advantages of my approach. Firstly, I discuss Okasha’s (2018) view of endogenization as a pervasive theoretical strategy in evolutionary biology and suggest that the constitutive character of the core Darwinian principles progressively increases with endogenization. Secondly, I apply a conceptual distinction between two varieties or scopes of coordination – general coordination and coordination in measurement – to Ohm’s work on electrical conductivity. This distinction allows me to pinpoint to what extent components along different dimensions (e.g., instrumentation, measurement, theorising, etc.) were constitutive of the forms of coordination which Ohm relied on. Thirdly, I discuss the epistemic function of the Hardy-Weinberg principle in the history and practice of population genetics. I assess this principle in terms of my account and identify approximation and stability as two components that are highly constitutive, in that they contribute to justifying its use in population genetics. Finally, applying my account to these case studies enables me to identify at least three qualitatively different types of constitutive components: domain-specific theoretical principles, material components, and domain-independent assumptions underlying reasoning abilities. In the light of my results, I draw some general conclusions on epistemic justification and scientific knowledge. (shrink)

Modality - the question of what is possible and what is necessary - is a fundamental area of philosophy and philosophical research. The Routledge Handbook of Modality is an outstanding reference source to the key topics, problems and debates in this exciting subject and is the first collection of its kind. Comprising thirty-five chapters by a team of international contributors the Handbook is divided into seven clear parts: worlds and modality essentialism, ontological dependence, and modality modal anti-realism epistemology of modality (...) modality in science modality in logic and mathematics modality in the history of philosophy. Within these sections the central issues, debates and problems are examined, including possible worlds, essentialism, counterfactuals, ontological dependence, modal fictionalism, deflationism, the integration challenge, conceivability, a priori knowledge, laws of nature, natural kinds, and logical necessity. The Routledge Handbook of Modality is essential reading for students and researchers in epistemology, metaphysics and philosophy of language. It will also be very useful for those in related fields in philosophy such as philosophy of mathematics, logic and philosophy of science. (shrink)

In this article I investigate a neglected form of radical skepticism that questions whether any of our logical, mathematical and other seemingly self-evident beliefs count as knowledge. ‘A priori skepticism,’ as I will call it, challenges our ability to know any of the following sorts of propositions: (1.1) The sum of two and three is five. (1.2) Whatever is square is rectangular. (1.3) Whatever is red is colored. (1.4) No surface can be uniformly red and uniformly blue at the (...) same time. (1.5) If ‘if p then q’ is true and ‘p’ is true, then ‘q’ is true. (1.6) No statement can be both true and false at the same time and in the same respect. (1.7) If A is taller than B, and B is taller than C, then A is taller than C. (1.8) Everything is identical to itself. (1.9) If the conclusion of an inductive argument is contingent, it is possible for the premises of that argument to be true and its conclusion to be false. (1.10) George W. Bush could have been a plumber. (1.11) George W. Bush could not have been a prime number. (1.12) ‘2 + 3 = 5’ is necessarily true. (shrink)

The topic of a priori knowledge is approached through the theory of evidence. A shortcoming in traditional formulations of moderate rationalism and moderate empiricism is that they fail to explain why rational intuition and phenomenal experience count as basic sources of evidence. This explanatory gap is filled by modal reliabilism -- the theory that there is a qualified modal tie between basic sources of evidence and the truth. This tie to the truth is then explained by the theory of (...) concept possession: this tie is a consequence of what, by definition, it is to possess (i.e., to understand) one’s concepts. A corollary of the overall account is that the a priori disciplines (logic, mathematics, philosophy) can be largely autonomous from the empirical sciences. (shrink)

In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There (...) are “verifications” of hypotheses which, while not definitive proofs, provide strong support for those hypotheses, and there are proofs involving an enormous amount of computer hours, which cannot be surveyed by any one mathematician in a lifetime. There have been several attempts to argue that one or another aspect of experimental mathematics shows that mathematics now accepts empirical or inductive methods, and hence shows mathematical apriorism to be false. Assessing this argument is complicated by the fact that there is no agreed definition of what precisely experimental mathematics is. However, I argue that on any plausible account of ’experiment’ these arguments do not succeed. (shrink)

To evaluate Hume's thesis that causal claims are always empirical, I consider three kinds of causal statement: ?e1 caused e2 ?, ?e1 promoted e2 ?, and ?e1 would promote e2 ?. Restricting my attention to cases in which ?e1 occurred? and ?e2 occurred? are both empirical, I argue that Hume was right about the first two, but wrong about the third. Standard causal models of natural selection that have this third form are a priori mathematical truths. Some are obvious, (...) others less so. Empirical work on natural selection takes the form of defending causal claims of the first two types. I provide biological examples that illustrate differences among these three kinds of causal claim. (shrink)

A physical and mathematical framework for the analysis of probabilities in quantum theory is proposed and developed. One purpose is to surmount the problem, crucial to any reconciliation between quantum theory and space-time physics, of requiring instantaneous “wave-packet collapse” across the entire universe. The physical starting point is the idea of an observer as an entity, localized in space-time, for whom any physical system can be described at any moment, by a set of (not necessarily pure) quantum states compatible with (...) his observations of the system at that moment. The mathematical starting point is the theory of local algebras from axiomatic relativistic quantum field theory. A function defining thea priori probability of mistaking one local state for another is analysed. This function is shown to possess a broad range of appropriate properties and to be uniquely defined by a selection of them. Through a general model for observations, it is argued that the probabilities defined here are as compatible with experiment as the probabilities of conventional interpretations of quantum mechanics but are more likely to be compatible, not only with modern developments in mathematical physics, but also with a complete and consistent theory of measurement. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

In this article, I discuss Hawthorne'€™s contextualist solution to Benacerraf'€™s dilemma. He wants to find a satisfactory epistemology to go with realist ontology, namely with causally inaccessible mathematical and modal entities. I claim that he is unsuccessful. The contextualist theories of knowledge attributions were primarily developed as a response to the skeptical argument based on the deductive closure principle. Hawthorne uses the same strategy in his attempt to solve the epistemologist puzzle facing the proponents of mathematical and modal realism, but (...) this problem is of a different nature than the skeptical one. The contextualist theory of knowledge attributions cannot help us with the question about the nature of mathematical and modal reality and how they can be known. I further argue that Hawthorne'€™s account does not say anything about a priori status of mathematical and modal knowledge. Later, Hawthorne adds to his account an implausible claim that in some contexts a gettierized belief counts as knowledge. (shrink)

The present paper deals with the ontological status of numbers and considers Frege ́s proposal in Grundlagen upon the background of the Post-Kantian semantic turn in analytical philosophy. Through a more systematic study of his philosophical premises, it comes to unearth a first level paradox that would unset earlier still than it was exposed by Russell. It then studies an alternative path, that departin1g from Frege’s initial premises, drives to a conception of numbers as synthetic a priori in a (...) more Kantian sense. On this basis, it tentatively explores a possible derivation of basic logical rules on their behalf, suggesting a more rudimentary basis to inferential thinking, which supports reconsidering the difference between logical thinking and AI. Finally, it reflects upon the contributions of this approach to the problem of the a priori. (shrink)

Why believe in the a priori? The answer is clear: there are many examples, drawn from mathematics, logic and philosophy, of knowledge that does not seem to be empirical. It does not seem possible that this knowledge could be justified or revised “by experience.” It must be justified in some other way, justified a priori.

In this book, David Stump traces alternative conceptions of the a priori in the philosophy of science and defends a unique position in the current debates over conceptual change and the constitutive elements in science. Stump emphasizes the unique epistemological status of the constitutive elements of scientific theories, constitutive elements being the necessary preconditions that must be assumed in order to conduct a particular scientific inquiry. These constitutive elements, such as logic, mathematics, and even some fundamental laws of (...) nature, were once taken to be a priori knowledge but can change, thus leading to a dynamic or relative a priori. Stump critically examines developments in thinking about constitutive elements in science as a priori knowledge, from Kant’s fixed and absolute a priori to Quine’s holistic empiricism. By examining the relationship between conceptual change and the epistemological status of constitutive elements in science, Stump puts forward an argument that scientific revolutions can be explained and relativism can be avoided without resorting to universals or absolutes. (shrink)

Question order effects are commonly observed in self-report measures of judgment and attitude. This article develops a quantum question order model (the QQ model) to account for four types of question order effects observed in literature. First, the postulates of the QQ model are presented. Second, an a priori, parameter-free, and precise prediction, called the QQ equality, is derived from these mathematical principles, and six empirical data sets are used to test the prediction. Third, a new index is derived (...) from the model to measure similarity between questions. Fourth, we show that in contrast to the QQ model, Bayesian and Markov models do not generally satisfy the QQ equality and thus cannot account for the reported empirical data that support this equality. Finally, we describe the conditions under which order effects are predicted to occur, and we review a broader range of findings that are encompassed by these very same quantum principles. We conclude that quantum probability theory, initially invented to explain order effects on measurements in physics, appears to be a powerful natural explanation for order effects of self-report measures in social and behavioral sciences, too. (shrink)

There is a widely held view that the expressions ‘necessary truth’, ‘a priori truth’ and ‘analytic truth’ either express the same concept or, at least, refer to all and only the same items. Philosophers who hold this view, and who are sometimes described as ‘empiricists’, often draw the conclusion that the truths of logic and mathematics, if necessary, are also a priori and are, in some important sense, empty or not about the world. The subject matter of (...) these disciplines, then, is said to differ in a philosophically important way from that of the empirical sciences, such as physics or chemistry. Rationalists, in contrast, have traditionally held that some a priori truths, either of logic or mathematics, are synthetic and, hence, non-analytic: i.e., there are synthetic a priori truths. (shrink)

Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual (...) thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking. (shrink)

I argue for the claim that there are instances of a priori justified belief – in particular, justified belief in moral principles – that are not analytic, i.e., that cannot be explained solely by the understanding we have of their propositions. §1–2 provides the background necessary for understanding this claim: in particular, it distinguishes between two ways a proposition can be analytic, Basis and Constitutive, and provides the general form of a moral principle. §§3–5 consider whether Hume's Law, properly (...) interpreted, can be established by Moore's Open Question Argument, and concludes that it cannot: while Moore's argument – appropriately modified – is effective against the idea that moral judgments are either reductively analyzable or Constitutive-analytic, a different argument is needed to show that they are not Basis-analytic. Such an argument is supplied in §6. §§7–8 conclude by considering how these considerations bear on recent discussions of “alternative normative concepts”, on the epistemology of intuitions, and on the differences between disagreement in moral domains and in other a priori domains such as logic and mathematics. (shrink)

In the article, an argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modelling, not the world, but our activities in the world.

This volume contains ten new essays on a priori knowledge by authors from Canada, the United States, Australia, & Europe Topics addressed include the nature, explanation, & indispensability of a priori knowledge, its connection with analytic truth, its place in mathematics, in logic, & in empirical theory, & the contribution of Kant & Quine to these topics. The focus is on twentieth-century contributions to these issues, but most essays also address earlier discussions at some length, & the (...) essays that focus on Kant also relate his views to more recent discussions. Contents: Introduction. A Rationalist Manifesto. Analyticity, Undeniability, & Truth. Analyticity & the A Priori. The Doubtful A Priori. Some Remarks on Indiscernibility. The Constitutive A Priori. Kant's A Priori Methods for Recognising Necessary Truths. EPR as A Priori Science. Gaps, Gluts, & Paradox. An Essay on Material Necessity. (shrink)