In the past decades, recent paradigm shifts in ethology, psychology, and the social sciences have given rise to various new disciplines like cognitive ethology and evolutionary psychology. These disciplines use concepts and theories of evolutionary biology to understand and explain the design, function and origin of the brain. I shall argue that there are several good reasons why this approach could also apply to human mathematical abilities. I will review evidence from various disciplines (cognitive ethology, cognitive psychology, cognitive archaeology and (...) neuropsychology) that suggests that the human capacity for mathematics is a category-specific domain of knowledge, hard-wired in the brain, which can be explained as the result of natural selection. (shrink)

What are the consequences of evolutionary theory for the epistemic standing of our beliefs? Evolutionary considerations can be used to either justify or debunk a variety of beliefs. This paper argues that evolutionary approaches to human cognition must at least allow for approximately reliable cognitive capacities. Approaches that portray human cognition as so deeply biased and deficient that no knowledge is possible are internally incoherent and self-defeating. As evolutionary theory offers the current best hope for a naturalistic epistemology, evolutionary approaches (...) to epistemic justification seem to be committed to the view that our sensory systems and belief-formation processes are at least approximately accurate. However, for that reason they are vulnerable to the charge of circularity, and their success seems to be limited to commonsense beliefs. This paper offers an extension of evolutionary arguments by considering the use of external media in human cognitive processes: we suggest that the way humans supplement their evolved cognitive capacities with external tools may provide an effective way to increase the reliability of their beliefs and to counter evolved cognitive biases. (shrink)

Our ability for scientific reasoning is a byproduct of cognitive faculties that evolved in response to problems related to survival and reproduction. Does this observation increase the epistemic standing of science, or should we treat scientific knowledge with suspicion? The conclusions one draws from applying evolutionary theory to scientific beliefs depend to an important extent on the validity of evolutionary arguments (EAs) or evolutionary debunking arguments (EDAs). In this paper we show through an analytical model that cultural transmission of scientific (...) knowledge can lead toward representations that are more truth-approximating or more efficient at solving science-related problems under a broad range of circumstances, even under conditions where human cognitive faculties would be further off the mark than they actually are. (shrink)

Our ability for scientific reasoning is a byproduct of cognitive faculties that evolved in response to problems related to survival and reproduction. Does this observation increase the epistemic standing of science, or should we treat scientific knowledge with suspicion? The conclusions one draws from applying evolutionary theory to scientific beliefs depend to an important extent on the validity of evolutionary arguments (EAs) or evolutionary debunking arguments (EDAs). In this paper we show through an analytical model that cultural transmission of scientific (...) knowledge can lead toward representations that are more truth-approximating or more efficient at solving science-related problems under a broad range of circumstances, even under conditions where human cognitive faculties would be further off the mark than they actually are. (shrink)

The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some (...) general unproven assumption. As an alternative, a bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach. (shrink)

The view that the subject matter of epistemology is the concept of knowledge is faced with the problem that all attempts so far to define that concept are subject to counterexamples. As an alternative, this article argues that the subject matter of epistemology is knowledge itself rather than the concept of knowledge. Moreover, knowledge is not merely a state of mind but rather a certain kind of response to the environment that is essential for survival. In this perspective, the article (...) outlines an answer to four basic questions about knowledge: What is the role of knowledge in human life? What is the relation between knowledge and reality? How is knowledge acquired? Is there any a priori knowledge? (shrink)

From antiquity several philosophers have claimed that the goal of natural science is truth. In particular, this is a basic tenet of contemporary scientific realism. However, all concepts of truth that have been put forward are inadequate to modern science because they do not provide a criterion of truth. This means that we will generally be unable to recognize a scientific truth when we reach it. As an alternative, this paper argues that the goal of natural science is plausibility and (...) considers some characters of plausibility. (shrink)

This is a concise, advanced introduction to current philosophical debates about truth. A blend of philosophical and technical material, the book is organized around, but not limited to, the tendency known as deflationism, according to which there is not much to say about the nature of truth. In clear language, Burgess and Burgess cover a wide range of issues, including the nature of truth, the status of truth-value gaps, the relationship between truth and meaning, relativism and pluralism about truth, and (...) semantic paradoxes from Alfred Tarski to Saul Kripke and beyond. Following a brief introduction that reviews the most influential traditional and contemporary theories of truth, short chapters cover Tarski, deflationism, indeterminacy, realism, antirealism, Kripke, and the possible insolubility of semantic paradoxes. The book provides a rich picture of contemporary philosophical theorizing about truth, one that will be essential reading for philosophy students as well as philosophers specializing in other areas. (shrink)

Two concepts of truth as correspondence of ideas with facts are analyzed. One of them is the thought-external fact relation, and the other is the fact-proposition one. The two maps are then composed, and the resulting map is assumed to formalize the concept of truth as adequacy or correspondence of ideas to facts. Besides, some desiderata for a correspondence theory of partial truth are proposed. Finally, the truth criteria employed in science and technology are recalled.

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and (...) Gödel incompleteness. I hypothesize that only computable and decidable (in Gödel’s sense) structures exist, which alleviates the cosmological measure problem and may help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems. (shrink)

We think of logic as objective. We also think that we are reliable about logic. These views jointly generate a puzzle: How is it that we are reliable about logic? How is it that our logical beliefs match an objective domain of logical fact? This is an instance of a more general challenge to explain our reliability about a priori domains. In this paper, I argue that the nature of this challenge has not been properly understood. I explicate the challenge (...) both in general and for the particular case of logic. I also argue that two seemingly attractive responses – appealing to a faculty of rational insight or to the nature of concept possession – are incapable of answering the challenge. (shrink)

The incredible achievements of modern scientific theories lead most of us to embrace scientific realism: the view that our best theories offer us at least roughly accurate descriptions of otherwise inaccessible parts of the world like genes, atoms, and the big bang. In Exceeding Our Grasp, Stanford argues that careful attention to the history of scientific investigation invites a challenge to this view that is not well represented in contemporary debates about the nature of the scientific enterprise. The historical record (...) of scientific inquiry, Stanford suggests, is characterized by what he calls the problem of unconceived alternatives. Past scientists have routinely failed even to conceive of alternatives to their own theories and lines of theoretical investigation, alternatives that were both well-confirmed by the evidence available at the time and sufficiently serious as to be ultimately accepted by later scientific communities. Stanford supports this claim with a detailed investigation of the mid-to-late 19th century theories of inheritance and generation proposed in turn by Charles Darwin, Francis Galton, and August Weismann. He goes on to argue that this historical pattern strongly suggests that there are equally well-confirmed and scientifically serious alternatives to our own best theories that remain currently unconceived. Moreover, this challenge is more serious than those rooted in either the so-called pessimistic induction or the underdetermination of theories by evidence, in part because existing realist responses to these latter challenges offer no relief from the problem of unconceived alternatives itself. Stanford concludes by investigating what positive account of the spectacularly successful edifice of modern theoretical science remains open to us if we accept that our best scientific theories are powerful conceptual tools for accomplishing our practical goals, but abandon the view that the descriptions of the world around us that they offer are therefore even probably or approximately true. (shrink)

What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical concepts. Numbers and (...) arithmetic, and mathematics in general, have unique features—precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world—that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of non-mathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inference-preserving cross-domain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and “laws” of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus—outside of natural selection—the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without (...) providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

I aim to clarify the relationship between the success of a theory and the truth of that theory. This has been a central issue in the debates between realists and anti-realists. Realists assume that success is a reliable indicator of truth, but the details about the respects in which success is a reliable indicator or test of truth have been largely left to our intuitions. Lewis (Synthese 129:371–380, 2001) provides a clear proposal of how success and truth might be connected, (...) comparing a test of success of our theories to medical tests with low rates of false positives and false negatives. But, contrary to what Lewis claims, I argue that it is not enough for the realist to undercut the claim that success is not a reliable indicator of truth. Rather, the realist must show that our current best theories are likely true. Further, I argue that tests in science are unlike medical tests in a number of important ways. (shrink)

Mathematics poses a difficult problem for methodological naturalists, those who embrace scientific method, and also for ontological naturalists who eschew non-physical entities such as Cartesian souls. Mathematics seems both essential to science but also committed to abstract non-physical entities while methodologically it seems to have no place for experiment or empirical confirmation. The chapter critically reviews a number of responses naturalists have made including logicism, Quinean radical empiricism, and Penelope Maddy’s variant thereof and suggests some further problems both for ontological (...) and for methodological naturalists. (shrink)

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)

We present a logically detailed case-study of Darwinian evolutionary explanation. Special features of Darwin’s explanatory schema made it an unusual theoretical breakthrough, from the point of view of the philosophy of science. The schema employs no theoretical terms, and puts forward no theoretical hypotheses. Instead, it uses three observational generalizations—Variability, Heritability and Differential Reproduction—along with an innocuous assumption of Causal Efficacy, to derive Adaptive Evolution as a necessary consequence. Adaptive Evolution in turn, with one assumption of scale (‘Deep Time’), implies (...) the observational generalization of Adaptation. It is a fascinating methodological task to regiment the premises and make the reasoning both rigorous and clear. Doing so reveals how surprisingly small an amount of mathematics is needed in order to carry out the argument. The investigation also reveals the crucial role played by heritability, and how heritability itself admits of Darwinian explanation. (shrink)

I begin by arguing that a consistent general naturalism must be understood in terms of methodological maxims rather than metaphysical doctrines. Some specific maxims are proposed. I then defend a generalized naturalism from the common objection that it is incapable of accounting for the normative aspects of human life, including those of scientific practice itself. Evolutionary naturalism, however, is criticized as being incapable of providing a sufficient explanation of categorical moral norms. Turning to the epistemological norms of science itself, particularly (...) those governing the empirical testing of specific models, I argue that these should be regarded as conditional rather than categorical and that, as such, can be given a naturalistic justification. The justification, however, is more cognitive than evolutionary. The historical development of science is found to be a better place for applying evolutionary ideas. After briefly considering the possibility of a naturalistic understanding of mathematics and logic, I turn to the problem of reconciling scientific realism with an evolutionary picture of scientific development. The solution, I suggest, is to understand scientific knowledge as being “perspectival” rather than absolutely objective. I first argue that scientific observation, whether by humans or instruments, is perspectival. This argument is extended to scientific theorizing which is regarded not as the formulation of universal laws of nature but as the construction of principles to be used in the construction of models to be applied to specific natural systems. The application of models, however, is argued to be not merely opportunistic but constrained by the methodological presumption that we live in a world with a definite causal structure even though we can understand it only from various perspectives. (shrink)

Psychologism in logic is the doctrine that the semantic content of logical terms is in some way a feature of human psychology. We consider the historically influential version of the doctrine, Psychological Individualism, and the many counter-arguments to it. We then propose and assess various modifications to the doctrine that might allow it to avoid the classical objections. We call these Psychological Descriptivism, Teleological Cognitive Architecture, and Ideal Cognizers. These characterizations give some order to the wide range of modern views (...) that are seen as psychologistic because of one or another feature. Although these can avoid some of the classic objections to psychologism, some still hold. (shrink)

Excerpts from Robert Nozick's "Invariances" Necessary truths are invariant across all possible worlds, contingent ones across only some.

Throughout, the book combines daring speculations with detailed investigations to portray the nature and status of rationality and the essential role that...

Hilary Kornblith argues for a naturalistic approach to investigating knowledge. Knowledge, he explains, is a feature of the natural world, and so should be investigated using scientific methods. He offers an account of knowledge derived from the science of animal behavior, and defends this against its philosophical rivals. This controversial and refreshingly original book offers philosophers a new way to do epistemology.

This article reviews the transition between post-Fregean anti-naturalistic epistemology and contemporary naturalistic epistemologies. It traces the revival of naturalism to Quine’s critique of the "a priori", and Kuhn’s defense of historicism, and use the arguments of Quine and Kuhn to identify a position, "traditional naturalism", that combines naturalistic themes with the claim that epistemology is a normative enterprise. Pleas for more radical versions of naturalism are articulated, and briefly confronted.

According to the semantic view of scientific theories, theories are classes of models. I show that this view -- if taken seriously as a formal explication -- leads to absurdities. In particular, this view equates theories that are truly distinct, and it distinguishes theories that are truly equivalent. Furthermore, the semantic view lacks the resources to explicate interesting theoretical relations, such as embeddability of one theory into another. The untenability of the semantic view -- as currently formulated -- threatens to (...) undermine scientific structuralism. (shrink)

Exact sciences are described as sciences whose theories are formalized. These are contrasted to inexact sciences, whose theories are not formalized. Formalization is described as a broader category than mathematization, involving any form/content distinction allowing forms, e.g., as represented in theoretical models, to be studied independently of the empirical content of a subject-matter domain. Exactness is a practice depending on the use of theories to control subject-matter domains and to align theoretical with empirical models and not merely a state of (...) a science. Inexact biological sciences tolerate a degree of “mismatch” between theoretical and empirical models and concepts. Three illustrations from biological sciences are discussed in which formalization is achieved by various means: Mendelism, Weismannism, and Darwinism. Frege’s idea of a “conceptual notation” is used to further characterize the notion of a form/content distinction. (shrink)

The broader context for the formal darwinism project established by two of the commentators, in terms of reconciling the Modern Synthesis with Darwinian arguments over design and in terms of links to other types of selection and design, is discussed and welcomed. Some overselling of the project is admitted, in particular of whether it claims to consider all organic design. One important fundamental question raised in two commentaries is flagged but not answered of whether design is rightly represented by an (...) optimisation program, and another from one commentary of whether the coreplicon dissolves in the face of multi-generational imprinting. Calls for the project to be extended to design at levels above and below the individual are considered sympathetically, but judged impractical at the high level of abstraction of the project. All claims of substantive technical error are emphatically rejected. Close technical readings are welcomed that, among other things, represent the project as 'axiomatizing fitness'. The prospects for the project are set out in the light of this highly varied set of commentaries. (shrink)

My understanding is that proximate explanations concern adaptive mechanism and that ultimate explanations concern adaptive rationale. Viewed in this light, the two kinds of explanation are quite distinct, but they interact in a complementary way to give a full understanding of biological adaptations. In contrast, Laland et al. (2013)—following a literal reading of Mayr (Science 134:1501–1506, 1961)—have characterized ultimate explanations as concerning any and all mechanisms that have operated over the course of an organism’s evolutionary history. This has unfortunate consequences, (...) such as allowing random drift to form the basis for ultimate explanations, and allowing proximate and ultimate explanations to bleed into each other until their distinction is meaningless. Here, I suggest Laland et al’s explanatory framework of “reciprocal causation” is not conducive to successful biological science, and that they have misunderstood key elements of the theory of Darwinian adaptation. (shrink)

In this paper, we develop an account of the justification thinkers have for employing certain basic belief-forming methods. The guiding idea is inspired by Reichenbach's work on induction. There are certain projects in which thinkers are rationally required to engage. Thinkers are epistemically justified in employing any belief-forming method such that "if it doesn't work, nothing will" for successfully engaging in such a project. We present a detailed account based on this intuitive thought and address objections to it. We conclude (...) by commenting on the implications that our account may have for other important epistemological issues and debates. (shrink)

In this paper I examine various ways in whichphilosophers have made connections between truth andnatural selection. I introduce several versions ofthe view that mechanisms of true belief generationarise as a result of natural selection and argue thatthey fail to establish a connection between truth andnatural selection. I then turn to scientific truthsand argue that evolutionary accounts of the origin ofscientific truth generation mechanisms also fail. Iintroduce David Hull's selectionist model ofscientific development and argue that his account ofscientific success does not (...) rely on connecting truthand natural selection. I argue that Hull's model,which severs the connection between truth andselection, can account for some aspects of scientificchange, but it still leaves us plenty of questionsabout what aspects of our individual cognitive make-upcontribute to scientific change and how they do so. I introduce an evolutionary approach to scientificcognition that shows how some of these questions canbe answered without making an explanatory appeal toselection for true belief generating mechanisms. (shrink)

Formal languages are widely regarded as being above all mathematical objects and as producing a greater level of precision and technical complexity in logical investigations because of this. Yet defining formal languages exclusively in this way offers only a partial and limited explanation of the impact which their use actually has. In this book, Catarina Dutilh Novaes adopts a much wider conception of formal languages so as to investigate more broadly what exactly is going on when theorists put these tools (...) to use. She looks at the history and philosophy of formal languages and focuses on the cognitive impact of formal languages on human reasoning, drawing on their historical development, psychology, cognitive science and philosophy. Her wide-ranging study will be valuable for both students and researchers in philosophy, logic, psychology and cognitive and computer science. (shrink)

Groundbreaking mathematician Gregory Chaitin gives us the first book to posit that we can prove how Darwin’s theory of evolution works on a mathematical level.

This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only does (...) this engaging book present the Platonist-Naturalist debate over mathematics in a comprehensive fashion, but it also sheds considerable light on non-mathematical aspects of a dispute that is central to contemporary philosophy. (shrink)

Scientific realism is the position that the aim of science is to advance on truth and increase knowledge about observable and unobservable aspects of the mind-independent world which we inhabit. This book articulates and defends that position. In presenting a clear formulation and addressing the major arguments for scientific realism Sankey appeals to philosophers beyond the community of, typically Anglo-American, analytic philosophers of science to appreciate and understand the doctrine. The book emphasizes the epistemological aspects of scientific realism and contains (...) an original solution to the problem of induction that rests on an appeal to the principle of uniformity of nature. (shrink)

This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis--the success of mathematical physics ...

Bringing together evolutionary biology, psychology, and philosophy, Henry Plotkin presents a new science of knowledge, one that traces an unbreakable link between instinct and our ability to know.

A complete philosophy of mathematics must address Paul Benacerraf’s dilemma. The requirements of a general semantics for the truth of mathematical theorems that coheres also with the meaning and truth conditions for non-mathematical sentences, according to Benacerraf, should ideally be coupled with an adequate epistemology for the discovery of mathematical knowledge. Standard approaches to the philosophy of mathematics are criticized against their own merits and against the background of Benacerraf’s dilemma, particularly with respect to the problem of understanding the distinction (...) between pure and applied mathematics and the effectiveness of applied mathematics in the natural sciences and engineering. The evaluation of these alternatives provides the basis for articulating a philosophically advantageous Aristotelian inherence concept of mathematical entities. An inherence account solves Benacerraf’s dilemma by interpreting mathematical entities as nominalizations of structural spatiotemporal properties inhering in existent spatiotemporal entities. (shrink)