William Tait is one of the most distinguished philosophers of mathematics of the last fifty years. This volume collects his most important published philosophical papers from the 1980's to the present. The articles cover a wide range of issues in the foundations and philosophy of mathematics, including some on historical figures ranging from Plato to Gdel.

"The Art of Logical Thinking" is a book written by William Walker Atkinson, an American attorney, merchant, publisher, and author in the late 19th and early 20th centuries. The book was first published in 1909 under the pseudonym Theron Q. Dumont, one of Atkinson's many pen names. The primary focus of "The Art of Logical Thinking" is to provide readers with insights into developing and refining their logical thinking abilities. Atkinson explores various aspects of logical reasoning and critical (...) thinking, aiming to guide readers in making better decisions, solving problems, and approaching issues with a more rational mindset. Key topics covered in the book include: The Nature of Thought: Atkinson delves into the fundamentals of thought processes, examining how ideas are formed and how the mind operates in its thinking capacity. Logical Analysis: The book provides techniques for breaking down complex problems into smaller, more manageable components, facilitating a systematic and logical approach to problem-solving. Deductive and Inductive Reasoning: Atkinson discusses both deductive and inductive reasoning, explaining their principles and applications. Deductive reasoning involves drawing specific conclusions from general principles, while inductive reasoning involves deriving general principles from specific observations. (shrink)

This book is an outgrowth of a series of articles originally published in The Progress Magazine under a pseudonym, in which I sought to account for the prevailing mental unrest regarding subjects of religious and philosophical import. These articles attracted much attention from careful students of the times, and there have been many requests for the republication thereof in book form under my own name. Accordingly, the publishers of the articles requested me to revise the several papers, and to add (...) thereto such further matter as I might think proper. I have gone over the original articles, taking away a little here and adding much there. Several entirely new subjects have been considered and inserted in their proper places in the text. At least one third of the present book is entirely new, having been added to the revised form of the original articles. I have purposely quoted freely from good authorities, that the reader may have the benefit of the thought of a number of competent persons upon the various phases of the subject, in addition to what I have personally thought and written. I have sought merely to tell my story in a plain, simple manner, without any attempt at fine writing, literary excellence, or display of technical philosophical knowledge. My desire has been to place in the hands of the everyday busy man and woman a book from the pages of which they may gain a general knowledge of the causes of the modern mental unrest-an idea of what is going into this great melting pot of thought; what is happening there; and what is likely to be poured forth from it. To many, the numerous new "isms," "ologies" and cults, are entirely new- something sprung full grown from the brains of their founders. The absence of an elementary acquaintance with the history of philosophy, on the part of the average man and woman, has given to many new prophets and founders of cults an unmerited and unwarranted importance and authority. I trust that this book will enable some of those who have been perplexed and worried over these new (?) "isms," and teachings, to learn the true sources of those startling conglomerations of metaphysics, philosophy and theology, and thus once more to regain their mental bearings. It is also my hope that some who have a tendency toward philosophical study may be inspired by this book to go to the proper sources of such knowledge-to the masters of this branch of thought-instead of sitting at the feet of the, too often, freakish, irresponsible, and bizarre founders and teachers of the pseudo schools and cults, who have been revamping the old teachings, dressing them in fantastic garb, and offering them (usually at a high rate of compensation) to those who have felt the hunger for wisdom and the thirst for information, but who lacked the knowledge of the places where these things could be had at first hand, pure and free from the adulterations of these self constituted middle men in the great market of Philosophy. (shrink)

These essays present new analyzes of the central figures of analytic philosophy -- Frege, Russell, Moore, Wittgenstein, and Carnap -- from the beginnings of the analytic movement into the 1930s. The papers do not reflect a single perspective, but rather express divergent interpretations of this controversial intellectual milieu.

There can be no doubt about the value of Frege's contributions to the philosophy of mathematics. First, he invented quantification theory and this was the first step toward making precise the notion of a purely logical deduction. Secondly, he was the first to publish a logical analysis of the ancestral R* of a relation R, which yields a definition of R* in second-order logic.1 Only a narrow and arid conception of philosophy would exclude these two achievements. Thirdly and very importantly, (...) the discussion in §§58-60 of the G r u n d l a g e n defends a conception of mathematical existence, to be found in Cantor (1883) and later in the writings of Dedekind and Hilbert, by basing it upon considerations about meaning which have general application, outside mathematics.2.. (shrink)

The background of these remarks is that in 1967, in ‘’Constructive reasoning” [27], I sketched an argument that finitist arithmetic coincides with primitive recursive arithmetic, P RA; and in 1981, in “Finitism” [28], I expanded on the argument. But some recent discussions and some of the more recent literature on the subject lead me to think that a few further remarks would be useful.

Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will (1) criticize this foundation, (2) propose a quite different one, and (3) note that essentially the latter foundation also underlies the Curry-Howard type theory, and hence Heyting's intuitionistic conception of logic. Thus the Dialectica interpretation (in (...) so far as its aim was to give constructive content to intuitionism) is superfluous. (shrink)

Foundations of a General Theory of Manifolds [Cantor, 1883], which I will refer to as the Grundlagen, is Cantor’s first work on the general theory of sets. It was a separate printing, with a preface and some footnotes added, of the fifth in a series of six papers under the title of “On infinite linear point manifolds”. I want to briefly describe some of the achievements of this great work. But at the same time, I want to discuss its connection (...) with the so-called paradoxes in set theory. There seems to be some agreement now that Cantor’s own understanding of the theory of transfinite numbers in that monograph did not contain an implicit contradiction; but there is less agreement about exactly why this is so and about the content of the theory itself. For various reasons, both historical and internal, the Grundlagen seems not to have been widely read compared to later works of Cantor, and to have been even less well understood. But even some of the more recent discussions of the work, while recognizing to some degree its unique character, misunderstand it on crucial points and fail to convey its true worth. (shrink)

We investigated the properties of the sustained attention to response task . In the SART, participants respond to frequent neutral signals and are required to withhold response to rare critical signals. We examined whether SART performance shows characteristics of speed–accuracy tradeoffs and in addition, we examined whether SART performance is influenced by prior exposure to emotional picture stimuli. Thirty-six participants in this study performed SARTs after being exposed to neutral and negative picture stimuli. Performance in the SART changed rapidly over (...) time and there was a high correlation between participants errors of commission rate and their reaction time to the neutral targets . Regardless of exposure self-reported thoughts significantly correlated with both errors of commission and reaction times. Overall, the results support the view that the SART is a better measure of impulsive responding than sustained attention. (shrink)

The reduction of the lambda calculus to the theory of combinators in [Sch¨ onfinkel, 1924] applies to positive implicational logic, i.e. to the typed lambda calculus, where the types are built up from atomic types by means of the operation A −→ B, to show that the lambda operator can be eliminated in favor of combinators K and S of each type A −→ (B −→ A) and (A −→ (B −→ C)) −→ ((A −→ B) −→ (A −→ C)), (...) respectively.1 I will extend that result to the case in which the types are built up by means of the general function type ∀x : A.B(x) as well as the disjoint union type ∃x : A.B(x)– essentially to the theory of [Howard, 1980]. To extend the treatment of −→ to ∀ we shall need a generalized form of the combinators K and S, and to deal with ∃ we will need to introduce a new form of the combinator S.. (shrink)

We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory does not demand ‘incomplete objects’ in the sense of Frege: bound variables are in principle eliminable. Secondly, the relation of extensional equality for each type is definable (...) in the Curry-Howard theory. (shrink)

The dynamics of populations of self-replicating, hierarchically structured individuals, exposedto accidents which destroy their sub-units, is analyzed mathematically, specifically with regardto the roles of redundancy and sexual repair. The following points emerge from this analysis:0 A population of individuals with redundant sub-structure has no intrinsic steady-statepoint; it tends to either zero or infinity depending on a critical accident rate α c . Increased redundancy renders populations less accident prone initially, but populationdecline is steeper if a is greater than a fixed (...) value α d . Periodic, sexual repair at system-specific intervals prevents continuous decline and stabilizesthe population insofar as it will now oscillate between two fixed population levels. The stabilizing sexual interval increases with increased complexity provided this is accom-panied by appropriate levels of redundancy. The model closely simulates the dynamics of heterosis effects. Repair fitness is a population fitness: the chance of an individual being repaired is a functionof the statistical make-up of the population as a whole at that particular period. Populationsliving at α > α c either engage in sexual repair at the appropriate time or they die out. The mathematical properties of the model illustrate mechanisms which possibly played arole in the evolution of a mortal soma in relation to sexual reproduction. (shrink)

These essays present new analyses of the central figures of analytic philosophy -- Frege, Russell, Moore, Wittgenstein, and Carnap -- from the beginnings of the analytic movement into the 1930s. The papers do not reflect a single perspective, but rather express divergent interpretations of this controversial intellectual milieu.

Of course, I do not mean by the title of this paper to deny the existence of something called.

The classical theory of types in question is essentially the theory of Martin-Löf [1] but with the law of double negation elimination. I am ultimately interested in the theory of types as a framework for the foundations of mathematics and, for this purpose, we need to consider extensions of the theory obtained by adding ‘well-ordered types,’ for example the type N of the finite ordinals; but the unextended theory will suffice to illustrate the treatment of extensional equality.

To appear in the Proceedings of Logic Colloquium 2006. (32 pages).

To appear in the Proceedings of Logic Colloquium 2006. (28 pages).

1. A Road to Philosophy of Mathematics l became interested in philosophy and mathematics at more or less the same time, rather late in high school; and my interest in the former certainly influenced my attitude towards the latter, leading me to ask what mathematics is really about at a fairly early stage. I don ’t really remember how it was that I got interested in either subject. A very good math teacher came to my school when I was in (...) 9th grade and I got caught up in his course on solid geometry; but he soon left and math then lost its luster again in the hands of teachers who neither liked nor understood it. Calculus wasn’t taught in high school in those days, or at least not in mine: besides geometry we learned some algebra and trigonometry. I doubt that even the word “ philosophy ” passed the lips of any of my teachers. My mother, who worked for a publishing house, brought home for me copies of, among other works, the Jowett translations of Plato’s Dialogues, Will Durant’s Story of Philosophy and Courant and Robbins’ What Is Mathematics?; but I can’t remember why she did that: She wasn’t at all intellectual and, as far as I recall, my interests at the time were mostly confined to sports and girls—in some order. Maybe she just thought it was time for me to develop new interests. After high school, I went in 1948 to Lehigh University, then at least primarily an engineering school, on an athletic scholarship. There I had the good fortune in my first year to have an introduction to philosophy course with Lewis White Beck. He had just moved there from the University of Delaware and shortly thereafter moved on to the University of Rochester, where he became one of the leading lights of American Kant studies. My good luck was compounded when, in my second year, Adolph Gr¨ unbaum arrived at Lehigh, fresh from graduate school at Yale, and stayed at least long enough for me to graduate, before moving to the University of Pittsburgh as Andrew Mellon Professor of Philosophy of Science.. (shrink)

In their article “Three Kinds of Decision-Making Capacity for Refusing Medical Interventions,” Navin et al. (2022) argue that we should recognize two forms of decision-making capacity (DMC) besides...

A simple model, illustrating the transition from a population of free swimming, solitary cells to one consisting of small colonies serves as a basis to discuss the evolution of the cooperative group. The transition is the result of a mutation of the dynamics of cell division, delayed cell separation leads to colonies of four cells. With this mutation cooperative features appear, such as synchronised cell divisions within colonies and coordinated flagellar function which enables the colony to swim in definite directions. (...) The selective advantages under given, environmental conditions are defined and the periods necessary for complete allelic replacement in small populations are calculated for asexual and sexual reproduction. The assumption of a steady-state population during allelic substitution is critically considered, particularly under conditions of competition. It is shown that density-dependent population control must operate in the process of selection. Sexual reproduction slows down the rate of selection even though all cells dre haploid. This phenomenon can be explained in general terms of `organizational dominance', where individual units coordinate the function of their neighbours which may be of a different allelotype.Cooperativity is pointed out as an a priori systemic feature which resides in the sub-units of systems, group formation and coordination appears thus as an almost inevitable event. A particular type of system described as ‘closed cycle of positive fitness interaction’ is discussed in more detail. It has the remarkable feature that its members cannot compete with each other; selection takes place between whole cycles .Gonium has a wide spectrum of `somatic plasticity' which enables it to assume various colonial configurations depending on physiological and environmental conditions. This feature can be explained as the result of dynamic flexibilities on the macro-molecular level. The particular relationship between the vast, molecular complexity and the relative simple dynamics of the cell cycle must lead eventually to the genetic fixation of an environmentally induced phenotype. (shrink)

Given the apocalyptic nature of nuclear weapons, how can states establish an international order that ensures survival while allowing the weapons to be used in controlled ways to discourage great wars, and while allowing nuclear technology to diffuse for civil purposes? How can the possession of nuclear weapons by a few states be reconciled with their renunciation by the majority of states? Which political strategies can best deliver an international nuclear order that is effective, legitimate and durable? These have been (...) central questions in the nuclear age. This article suggests that the effort to construct such an order displayed the characteristics of an enlightenment project, with its emphasis on balance and rationality, the quest for justice and trust among states, the feasibility of instrumental regulation, and the attachment to hope and progress. With the Nuclear Non-Proliferation Treaty at its heart, it necessarily gave precedence to diplomacy and containment over preventive war. The reasons why this conception of nuclear order was discarded by its erstwhile champion, the United States, in favor of one bearing traits of counter-enlightenment, are explored. Its alternative strategy can now be declared a failure. Avoidance of a greater disorder depends on recognition that the problem of nuclear order is more than the problem of proliferation, or of non-compliance, and on recovery whatever the difficulties of the cooperative yet pragmatic sensibility that lay behind the prior approach to order. (shrink)

William Walker's original analysis of John Locke's An Essay Concerning Human Understanding offers a challenging and provocative assessment of Locke's importance as a thinker, bridging the gap between philosophical and literary-critical discussion of his work. He presents Locke as a foundational figure who defines the epistemological and ontological ground on which eighteenth-century and Romantic literature operate and eventually diverge. He is revealed as a crucial figure for emerging modernity, less the familiar empiricist innovator and more the proto-Nietzschean thinker (...) whose text fosters hitherto unsuspected instabilities and promotes a new kind of rhetorical force to counterbalance them. Walker's reading of Locke is at once finely attentive to the text and engagingly resourceful in placing the Essay in its broadest philosophical and historical context. (shrink)

A commonly used experimental procedure for the study of granulocyte kinetics involves the labelling and subsequent tracing of granulocyte DNA. Following the introduction of a label into the system, observations are made periodically on the concentration of label in the DNA of granulocytes taken from the circulating blood. A mathematical model for the expected value of this concentration has been derived, studied, and related to experimental observations from studies using P32 as a label. Insofar as the derivation of the model (...) accurately incorporates the relevant aspects of granulocyte kinetics, the model will be useful in the interpretation of the experimental observations in terms of these kinetics.Among other things, study of the model indicates that the assumption of “flash” labelling made with respect to some labels is quite crucial and needs to be examined critically. It has also been necessary to make adjustments to allow for the observed emergence of labelled cells from the marrow soon after label introduction. In addition, despite a high degree of confounding of parameter effects it has been possible to suggest bounds on some of the parameters of the model for some species. The bounds will be refined as the model is improved and more data become available. Study and development of the model continues with particular interest in generalizations to include the diseased state chronic granulocytic leukemia. (shrink)

Some of the founding documents of our modern human rights culture assert that, by virtue of natural law, the will of God, the will of a Supreme Being, or some kind of natural world order, all humans have a right to civil liberties. In Areopagitica, Milton rejects this way of grounding the claim to civil liberties. Instead, he argues for civil liberties on pragmatic grounds, but also on the premise that members of political societies are entitled to civil liberties from (...) their governors only insofar as those members are rational and virtuous. His argument for civil liberties is also grounded in the view that the proper function of government includes propagating virtue in those it governs, assessing their rationality and moral virtue, and extending civil liberties to them in accordance with this assessment. Arguing in this way, Milton opposes the notion that, simply by virtue of being human, all members of political societies have a specific set of rights which their governments, and indeed all other people on earth, are bound to respect. He thus has more in common with Isocrates and Renaissance humanists than he does with the defenders of our modern human rights culture. (shrink)

In the major political prose works which he published from 1649 to 1654, Milton argues that it was not the parliamentarians but Charles Stuart and his supporters who were the real rebels during the wars of the 1640s. He claims that during this period, the parliamentarians did not fight to overturn law, church, and government, but to preserve peace, to maintain the old, orthodox form of Christianity which had only partially been re-established in England, and to defend English law and (...) the civil liberties it safeguarded. He disavows any hostility to true monarchy and asserts the right of all peoples to choose for themselves whatever form of government they wish. He argues that, since by 1649 Charles Stuart had long been deposed, there was no regicide but merely a legal execution, one which was also consistent with the ancient constitution of England and the political thought of the champion of ancient Roman laws and customs, Cicero. All of this supports several recent accounts of Interregnum political thought and rhetoric and challenges much of the work, from Christopher Hill on, which makes Milton out to be a radical. (shrink)

This article provides an account of what may reasonably be inferred from Sallust’s historical writing about how he understands civil liberty, what he feels is necessary for it to exist in any given political society, why he feels it is important, and the extent to which he feels it is properly enjoyed by the plebeian citizens of Rome. On the basis of this account, the article revises recent arguments presented by Quentin Skinner, Philip Pettit and others concerning Sallust’s political thought (...) and how it is related to Aristotelian, republican, ‘neo-Roman’, and liberal traditions of political thought. (shrink)