Aus dem vielfältigen Werk von Hermann von Helmholtz versammelt diese Ausgabe die im engeren Sinne philosophischen Abhandlungen, vor allem zur Wissenschaftsphilosophie und Erkenntnistheorie, sowie Vorträge und Reden, bei denen der Autor seine Ausnahmestellung im Wissenschaftsbetrieb nutzte, um die Wissenschaften und ihre Institutionen in der bestehenden Form zu repräsentieren und zu begründen. Ein Philosoph wollte Helmholtz nicht sein, aber er legte der philosophischen Reflexion wissenschaftlicher Erkenntnis und wissenschaftlichen Handelns große Bedeutung bei. Vor allem bezog er, in der Regel ausgehend von seinen (...) fachwissenschaftlichen Forschungen, in den verschiedensten Kontexten zu erkenntnistheoretischen und methodologischen Problemen der Wissenschaften Stellung. Bereits »Ueber die Erhaltung der Kraft« (1847) lässt erkennen, wie verwoben naturwissenschaftliche Grundlagenforschung und philosophische Grundlagenreflexion in seinem Werk sind. Die aus den frühen sinnesphysiologischen Forschungen hervorgegangene empiristische Wahrnehmungslehre trug ihm den Ruf ein, ein maßgeblicher Vertreter des Neukantianismus zu sein. Spätere Arbeiten v.a. zur Geometrie und Arithmetik – das zeigt die vorliegende Ausgabe – stellen jedoch eine radikale Absage an den konstitutiven Kern des Kantianismus (nämlich die Existenz synthetischer Urteile a priori) dar. Helmholtz’ philosophische Beiträge sind bisher in ihrer Vollständigkeit nicht annähernd so gut zugänglich wie sein naturwissenschaftliches Werk. Die Ausgabe enthält außerdem bibliographische Vorberichte zur Einordnung, detaillierte Namens- und Sachregister sowie mit 575 Einträgen für den Zeitraum zwischen 1842 und 2012 die erste umfassende Bibliographie von Helmholtz verfasster Werke überhaupt. »Ich glaube, dass der Philosophie nur wieder aufzuhelfen ist, wenn sie sich mit Ernst und Eifer der Untersuchung der Erkenntnissprocesse und der wissenschaftlichen Methode zuwendet. Da hat sie eine wirkliche und berechtigte Aufgabe.« Helmholtz in einem Brief um 1875. (shrink)

A close look at Frege's proof in "Foundations of Arithmetic" that every number has a successor. The examination reveals a surprising gap in the proof, one that Frege would later fill in "Basic Laws of Arithmetic".

This book is an anthology with the following themes. Non-European Tradition: Bussanich interprets main themes of Hindu ethics, including its roots in ritual sacrifice, its relationship to religious duty, society, individual human well-being, and psychic liberation. To best assess the truth of Hindu ethics, he argues for dialogue with premodern Western thought. Pfister takes up the question of human nature as a case study in Chinese ethics. Is our nature inherently good (as Mengzi argued) or bad (Xunzi’s view)? Pfister ob- (...) serves their underlying agreement, that human beings are capable of becoming good, and makes precise the disagreement: whether we achieve goodness by cultivating autonomous feelings or by accepting external precepts. There are political consequences: whether government should aim to respect and em- power individual choices or to be a controlling authority. Early Greek Thinking: Collobert examines the bases of Homeric ethics in fame, prudence, and shame, and how these guide the deliberations of heroes. She observes how, by depending upon the poet’s words, the hero gains a quasi- immortality, although in truth there is no consolation for each person’s inevi- table death. Plato: Santas examines Socratic Method and ethics in Republic 1. There Socrates examines definitions of justice and tests them by comparison to the arts and sciences. Santas shows the similarities of Socrates’ method to John Rawls’ method of considered judgments in reflective equilibrium. McPherran interprets Plato’s religious dimension as like that of his teacher Socrates. McPherran shows how Plato appropriates, reshapes, and extends the religious conventions of his own time in the service of establishing the new enterprise of philosophy. Ac- cording to Taylor, Socrates believes that humans in general have the task of helping the gods by making their own souls as good as possible, and Socrates’ unique ability to cross-examine imposes on him the special task of helping others to become as good as possible. This conception of Socrates’ mission is Plato’s own, consisting in an extension of the traditional conception of piety as helping the gods. Brickhouse and Smith propose a new understanding of Socratic moral psychology—one that retains the standard view of Socrates as an intellectualist, but also recognizes roles in human agency for appetites and passions. They compare and contrast the Socratic view to the picture of moral psychology we get in other dialogues of Plato. Hardy also proposes a new, non-reductive understanding of Socratic eudaimonism—he argues that Socrates invokes a very rich and complex notion of the “Knowledge of the Good and Bad”, which is associated with the motivating forces of the virtues. Rudebusch defends Socrates’ argument that knowledge can never be impotent in the face of psychic passions. He considers the standard objections: that knowledge cannot weigh incom- mensurable human values, and that brute desire, all by itself, is capable of moving the soul to action. Aristotle: Anagnostopoulos interprets Aristotle on the nature and acquisition of virtue. Though virtue of character, aiming at human happiness, requires a complex awareness of multiple dimensions of one’s experience, it is not properly a cognitive capacity. Thus it requires habituation, not education, according to Aristotle, in order to align the unruly elements of the soul with reason’s knowledge of what promotes happiness. Shields explains Aristotle’s doctrine that goodness is meant in many ways as the doctrine that there are different analyses of goodness for different types of circumstance, just as for being. He finds Aristotle to argue for this conclusion, against Plato’s doctrine of the unity of the Good, by applying the tests for homonymy and as an immediate cons- equence of the doctrine of categories. Shields evaluates the issue as unresolved at present. Russell discusses Aristotle’s account of practical deliberation and its virtue, intelligence (phronesis). He relates the account to contemporary philo- sophical controversies surrounding Aristotle’s view that intelligence is neces- sary for moral virtue, including the objections that in some cases it is unnecessary or even impedes human goodness. Frede examines the advantages and disadvantages of Aristotle’s virtue ethics. She explains the general Greek con- ceptions of happiness and virtue, Aristotle’s conception of phronesis and compares the Aristotle’s ethics with modern accounts. Liske discusses the question of whether the Aristotelian account of virtue entails an ethical-psy- chological determinism. He argues that Aristotle’s understanding of hexis allows for free action and ethical responsibility : By making decisions for good actions we are able to stabilize our character (hexis). Hellenistic and Roman: Annas defends an account of stoic ethics, according to which the three parts of Stoicism—logic, physics, and ethics—are integrated as the parts of an egg, not as the parts of a building. Since by this analogy no one part is a foundation for the rest, pedagogical decisions may govern the choice of numerous, equally valid, presentations of Stoic ethics. Piering interprets the Cynic way of life as a distinctive philosophy. In their ethics, Cynics value neither pleasure nor tradition but personal liberty, which they achieve by self-suffi- ciency and display in speech that is frank to the point of insult. Plotinus and Neoplatonism: Gerson outlines the place of ordinary civic virtue as well as philosophically contemplative excellence in Neoplatonism. In doing so he attempts to show how one and the same good can be both action-guiding in human life and be the absolute simple One that grounds the explanation of everything in the universe. Delcomminette follows Plotinus’s path to the Good as the foundation of free will, first in the freedom of Intellect and then in the “more than freedom” of the One. Plotinus postulates these divinities as not outside but within each self, saving him from the contradiction of an external foundation for a truly free will. General Topics: Halbig discusses the thesis on the unity of virtues. He dis- tinguishes the thesis of the identity of virtues and the thesis of a reciprocity of virtues and argues that the various virtues form a unity (in terms of reciprocity) since virtues cannot bring about any bad action. Detel examines Plato’s and Aristotle’s conceptions of normativity : Plato and Aristotle (i) entertained hybrid theories of normativity by distinguishing functional, semantic and ethical normativity, (ii) located the ultimate source of normativity in standards of a good life, and thus (iii) took semantic normativity to be a derived form of normativity. Detel argues that hybrid theories of normativity are—from a mo- dern point of view—still promising. Ho ̈ffe defends the Ancient conception of an art of living against Modern objections. Whereas many Modern philosophers think that we have to replace Ancient eudaimonism by the idea of moral obligation (Pflicht), Ho ̈ffe argues that Eudaimonism and autonomy-based ethics can be reconciled and integrated into a comprehensive and promising theory of a good life, if we enrich the idea of autonomy by the central elements of Ancient eudaimonism. Some common themes: The topics in Chinese and Hindu ethics are perhaps more familiar to modern western sensibilities than Homeric and even Socratic. Anagnostopoulos, Brickhouse and Smith, Frede, Liske, Rudebusch, and Russell all consider in contrasting ways the role of moral character, apart from intellect, in ethics. Brickhouse / Smith, Hardy, and Rudebusch discuss the Socratic con- ception of moral knowledge. Brickhouse / Smith and Hardy retain the standard view of the so called Socratic Intellectualism. Shields and Gerson both consider the question whether there is a single genus of goodness, or if the term is a homonym. Bussanich, McPherran, Taylor, and Delcomminette all consider the relation between religion and ethics. Pfister, Piering, Delcomminette, and Liske all consider what sort of freedom is appropriate to human well-being. Halbig, Detel, and Ho ̈ffe propose interpretations of main themes of Ancient ethics. (shrink)

Der berühmte Ameisenmann E.O. Wilson war schon immer einer meiner Helden - nicht nur ein hervorragender Biologe, sondern eine der winzigen und verschwindenden Minderheit von Intellektuellen, die es zumindest wagt, die Wahrheit über unsere Natur anzudeuten, die andere nicht verstehen oder, soweit sie es verstehen, aus politischen Gründen unermüdlich vermeiden. Leider beendet er seine lange Karriere auf äußerst schäbige Weise als Partei eines ignoranten und arroganten Angriffs auf die Wissenschaft, der zumindest teilweise durch die religiöse Inbrunst seiner Harvard-Kollegenmotiviertist. Es zeigt (...) die abscheulichen Folgen, wenn Universitäten Geld von religiösen Gruppen annehmen, Wissenschaftszeitschriften von großen Namen so bewundert sind, dass sie eine ordnungsgemäße Peer Review vermeiden und wenn Egos außer Kontrolle geraten dürfen. Es führt uns in die Natur der Evolution, die Grundlagen der wissenschaftlichen Methodik, wie Mathematik mit Wissenschaft zusammenhängt, was eine Theorie ausmacht, und sogar, welche Einstellungen zu Religion und Großzügigkeit angemessen sind, wenn wir uns unaufhaltsam dem Zusammenbruch der industriellen Zivilisation nähern. Ich fand Abschnitte in 'Conquest' mit dem üblichen prägnanten Kommentar (obwohl nichts wirklich Neues oder Interessantes, wenn man seine anderen Werke gelesen hat und auf biologie im Allgemeinen ist) in der oftgestylten Prosa, die sein Markenzeichen ist, aber war ziemlich überrascht, dass der Kern des Buches seine Ablehnung inklusiver Fitness (die seit über 50 Jahren ein Standbein der Evolutionsbiologie ist) zugunsten der Gruppenauswahl ist. Man nimmt an, dass von ihm kommen und mit den articles er bezieht sich auf veröffentlicht von sich selbst und Harvard Mathematik Kollege Nowak in großen Peer-Review-Zeitschriften wie Nature, muss es ein wesentlicher Fortschritt sein,, trotz der Tatsache, dass ich wusste, Gruppenauswahl wurde fast überall abgelehnt, da mit jeder großen Rolle in der Evolution. Ich habe zahlreiche Rezensionen im Netz gelesen und viele haben gute Kommentare, aber die, die ich am meisten sehen wollte, war, dass von renommierten Wissenschaftsautor und Evolutionsbiologe Richard Dawkins. Im Gegensatz zu den meisten von Fachleuten, die in Zeitschriften nur für diejenigen mit Zugang zu einer Universität zur Verfügung stehen, ist es leicht im Netz verfügbar, obwohlanscheinend, entschied er sich, es nicht in einer Zeitschrift zu veröffentlichen, da es angemessen abscheulich ist. Leider findet man eine vernichtende Ablehnung des Buches und den acerbic Kommentar über einen wissenschaftlichen Kollegen, den ich je von Dawkins gesehen habe -- über alles in seinem vielen Austausch mit dem verstorbenen und unbeklagten Demagogen und Pseudowissenschaftler Stephan Jay Gould. Obwohl Gould für seine persönlichen Angriffe auf seinen Harvard-Kollegen Wilson berüchtigt war, stellt Dawkins fest, dass ein Großteil von "Conquest" einen unbequem an Goulds häufige Verfehlungen in "bland, unfocussed ecumenicalis" erinnert. Dasselbe gilt mehr oder weniger für Wilsons populäres Schreiben, einschließlich seines jüngsten Buches "The Meaning of Human Existence" – eine weitere schamlose Eigenwerbung seiner diskreditierten Ideen zu Inklusiver Fitness (IF). Dawkins weist darauf hin, dass das berüchtigte 2010-Papier von Nowak, Tarnita und Wilson in Nature von über 140 Biologen, die einen Brief unterzeichnet haben, fast überall abgelehnt wurde und dass es in Wilsons Buch kein Wort darüber gibt. Auch in den folgenden 4 Jahren mit Artikeln, Vorträgen und mehreren Büchern haben sie dies nicht korrigiert. Es gibt keine andere Wahl, als Dawkins trenchant Kommentar zuzustimmen: "Für Wilson nicht zu zugeben, dass er für sich selbst gegen die große Mehrheit seiner professionellen Kollegen spricht - es schmerzt mich, dies von einem lebenslangen Helden zu sagen -- ein Akt mutwilligen Arroganz." Angesichts von Nowaks späterem Verhalten muss man ihn auch einbeziehen. Ich habe das Gefühl, dass einer der fassungslosen Menschen, die man im Fernsehen sieht, interviewt wird, nachdem der nette Mann von nebenan, der seit 30 Jahren alle Kinder babysitten, als Serienmörder entlarvt wird. Dawkins weist auch darauf hin (wie er und andere seit vielen Jahren), dass inklusive Fitness mit dem Neo-Darwinismus (d.h. logischerweise folgt) entsteht und nicht abgelehnt werden kann, ohne die Evolution selbst abzulehnen. Wilson erinnert uns erneut an Gould, der Kreationisten von der einen Seite seines Mundes anprangerte, während er ihnen Trost spendete, indem er endlosen ultraliberalen marxistisch gefärbten Kauderwelsch über Spandrels, unterbrochenes Gleichgewicht und Evolutionspsychologie von der anderen ausspeist. Die Unbestimmtheit und mathematische Opazität (für die meisten von uns) der Mathematik der Gruppen- oder Mehrebenenauswahl ist genau das, was die Sanftmütigen ihnen ermöglichen wollen, dem rationalen Denken in ihren endlosen antiwissenschaftlichen Gerüchten und (in der Wissenschaft) postmodernen Wortsalate zu entkommen. Schlimmer noch, Wilsons "Eroberung" ist ein schlecht durchdachtes und schlampig geschriebenes Durcheinander voller Nonsequiturs, vager Streifzüge, Verwirrungen und Inkohärenz. Eine gute Bewertung, die einige davon im Detail ist, dass von Absolvent Gerry Carter, die Sie im Netz finden können. Wilson hat auch nichts mit unserem gegenwärtigen Verständnis der Evolutionspsychologie (EP) zu tun (siehe z.B. die letzten 300 Seiten von Pinkers 'The Better Angels of our Nature'). Wenn Sie eine seriöse Buchlänge Bericht über die soziale Evolution und einige relevante EP von einem Experten wollen, siehe 'Principles of Social Evolution' von Andrew F.G. Bourke, oder ein nicht ganz so ernster und zugegebenermaßen fehlerhafter und irrender Bericht, aber ein Muss,das Robert Trivers dennoch lesen muss—'The Folly of Fools: The Logic of Deceit and Self-Deception in Human Life' und ältere, aber immer noch aktuelle und durchdringende Werke wie 'The Evolutionof Cooperation':Revised Edition by Robert Axelrod and 'The Biology ofMoral Systems' von Richard Alexander. (shrink)

Ich gebe einen ausführlichen Überblick über 'The Outer Limits of Reason' von Noson Yanofsky aus einer einheitlichen Perspektive von Wittgenstein und Evolutionspsychologie. Ich weise darauf hin, dass die Schwierigkeit bei Themen wie Paradoxon in Sprache und Mathematik, Unvollständigkeit, Unbedenklichkeit, Berechenbarkeit, Gehirn und Universum als Computer usw. allesamt auf das Versäumnis zurückzuführen ist, unseren Sprachgebrauch im geeigneten Kontext sorgfältig zu prüfen, und daher das Versäumnis, Fragen der wissenschaftlichen Tatsache von Fragen der Funktionsweise von Sprache zu trennen. Ich bespreche Wittgensteins Ansichten (...) über Unvollständigkeit, Parakonsistenz und Unentschlossenheit und die Arbeit Wolperts an den Grenzen der Berechnung. Zusammengefasst: The Universe According to Brooklyn---Good Science, Not So Good Philosophy. Wer aus der modernen zweisystems-Sichteinen umfassenden, aktuellen Rahmen für menschliches Verhalten wünscht, kann mein Buch "The Logical Structure of Philosophy, Psychology, Mindand Language in Ludwig Wittgenstein and John Searle' 2nd ed (2019) konsultieren. Diejenigen,die sich für mehr meiner Schriften interessieren, können 'Talking Monkeys--Philosophie, Psychologie, Wissenschaft, Religion und Politik auf einem verdammten Planeten --Artikel und Rezensionen 2006-2019 3rd ed (2019) und Suicidal Utopian Delusions in the 21st Century 4th ed (2019) und andere sehen. (shrink)

Inspired by Bas van Fraassen’s Stance Empiricism, Anjan Chakravartty has developed a pluralistic account of what he calls epistemic stances towards scientific ontology. In this paper, I examine whether Chakravartty’s stance pluralism can exclude epistemic stances that licence pseudo-scientific practices like those found in Scientology. I argue that it cannot. Chakravartty’s stance pluralism is therefore prone to a form of debilitating relativism. I consequently argue that we need (1) some ground or constraint in relation to which epistemic stances can be (...) ranked by degrees, and (2) some way to demarcate science from pseudo-science so that we know what epistemic stances are about. Regarding (1), I argue that empirical detectability can serve as the ground in relation to which epistemic stances are ranked by degrees. Regarding (2), I argue for ranking sciences on a continuum according to established institutional criteria, rather than attempting to draw a strict demarcation. (shrink)

Wilhelm Ostwald’s program of a physical energetics is the attempt at a comprehensive description of nature on the basis of the concept of energy. In his book Energetische Grundlagen der Kulturwissenschaft, first published in 1909, Ostwald applies this conception to the area of culture. His central assumption is that cultural phenomena should be described by the energetic notion of “efficiency relation” (Güteverhältnis). His systematic thesis is that science, when organized according to the Machian “principle of economy,” proves as the (...) highest form of cultural expression, since it instantiates the notion of quality relation most efficiently, that is, “with the lowest energy expenditure.” This view echoes August Comte’s “law of the three stages” and is intended to supply it with a scientific, i.e., energetic foundation. Max Weber regarded Ostwald’s energetic theory of culture as a misguided attempt at an absolutization of the methods of concept formation within the natural sciences. As he wrote in his devastating review essay “’Energetische’ Kulturtheorien” (1909), Ostwald transformed a certain world view (Weltbild) into a scientifically frivolous ideology (Weltanschauung). In particular, Ostwald’s adherence to the Comtean law of three stages and the associated hierarchy of the sciences were criticized by Weber as outdated and completely beside the point. According to Weber, the concepts of the cultural sciences are not at all dependent on natural scientific concepts such as ‘energy.’ In his view, culture cannot be reduced to nature. But exactly this seemed to be the principal aim of Ostwald’s program. In this paper, I will critically investigate Weber’s critique of that program. I shall argue that Ostwald’s assumption of a natural basis of culture can be ‘rescued’ as a methodological device, but that Ostwald’s – thoroughly substantialist – view of energy should be discarded as a metaphysical relict of ancient ‘stuff ontology.’ . (shrink)

This paper surveys Bolzano's Beyträge zu einer begründeteren Darstellung der Mathematik (Contributions to a better-grounded presentation of mathematics) on the 200th anniversary of its publication. The first and only published issue presents a definition of mathematics, a classification of its subdisciplines, and an essay on mathematical method, or logic. Though underdeveloped in some areas (including,somewhat surprisingly, in logic), it is nonetheless a radically innovative work, where Bolzano presents a remarkably modern account of axiomatics and the epistemology of (...) the formal sciences. We also discuss the second, unfinished and unpublished issue, where Bolzano develops his views on universal mathematics. Here we find the beginnings of his theory of collections, for him the most fundamental of the mathematical disciplines. Though not exactly the same as the later Cantorian set theory, Bolzano's theory of collections was used in very similar ways in mathematics, notably in analysis. In retrospect, Bolzano's debut in philosophy was a remarkably successful one, though its fruits would only become generally known much later. (shrink)

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

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

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

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

Es wird gezeigt, dass Wittgenstein in seiner Frühphilosophie ein nicht-axiomatisches Beweisverständnis entwickelt, für das sich das Problem der Begründung der Axiome nicht stellt. Nach Wittgensteins Beweisverständnis besteht der Beweis einer formalen Eigenschaft einer Formel – z.B. der logischen Wahrheit einer prädikatenlogischen Formel oder der Gleichheit zweier arithmetischer Ausdrücke – in der Transformation der Formel in eine andere Notation, an deren Eigenschaften sich entscheiden lässt, ob die zu beweisende formale Eigenschaft besteht oder nicht besteht. Dieses Verständnis grenzt Wittgenstein gegenüber einem axiomatischen (...) Beweisverständnis ab. Sein Beweisverständnis bedingt ein Programm der Grundlegung der Mathematik, das eine Alternative zu den Ansätzen des Logizismus, Formalismus und Konstruktivismus darstellt. Wittgensteins Ansatz steht im Widerspruch zu den Ergebnissen der Metamathematik, da er die Möglichkeit der Formulierung von Entscheidungsverfahren in der Prädikatenlogik und Arithmetik voraussetzt. Um seinem Ansatz gegenüber der traditionellen Metamathematik Recht zu geben, müsste gezeigt werden, dass sein Beweisverständnis im Bereich der Logik und Arithmetik – der traditionellen Metamathematik zum Trotz – realisierbar ist. (shrink)

In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the (...) concept of ‘denumerability’ as it is presented in set theory as well as his philosophic refutation of Cantor’s Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantor’s Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbert’s problems. (shrink)

The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It (...) is divided into 5 sections. We first show that already in the set theoretical framework, there are different dimensions to the expression foundations of. We then explore these dimensions more thoroughly. After a very short discussion of the links between these dimensions, we move to some of the arguments presented for and against category theory in the foundational landscape. We end up on a more speculative note by examining the relationships between category theory and set theory. (shrink)

In Part I, an attempt is made to survey the original source material on which any detailed assessment of Wittgenstein's remarks on the foundations of mathematics from his middle and later periods ought to be based. This survey is presented within the context of a sketch of Wittgenstein's biography, which also mentions some of the major developments in his thinking. In addition, certain main themes are emphasized; these have to do primarily with the Kantian aspects of Wittgenstein's thought (...) and with his mysticism or the 'religious point of view'. In Part II, Kreisel's critique of Wittgenstein's remarks on the foundations of mathematics, which has been developed since 1958 in a series of published articles, receives close examination, and, in connection with this, different approaches to the philosophical investigation of mathematics are considered which represent genuine alternatives to Wittgenstein's approach. There are separate sections on Lakatos's Proofs and Refutations and Bourbaki's 'L'Architecture des Mathématiques'. Finally, besides a bibliography which surveys the reception of Wittgenstein's views on the foundations of mathematics, there are two substantial appendices, which are supplemental to Part I. The first of these gives the manuscript sources for typescripts 221 and 222-4, and the correspondences in both directions between these typescripts. The second appendix is part of a chronological version of von Wright's catalogue of Wittgenstein's papers, beginning in 1929. (shrink)

In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational (...) models of human cognitive capacities. One interesting area of computational complexity theory is descriptive complexity, which connects the expressive strength of systems of logic with the computational complexity classes. In descriptive complexity theory, it is established that only first-order (classical) systems are connected to P, or one of its subclasses. Consequently, second-order systems of logic are considered to be computationally intractable, and may therefore seem to be unfit to model human cognitive capacities. This would be problematic when we think of the role of logic as the foundations of mathematics. In order to express many important mathematical concepts and systematically prove theorems involving them, we need to have a system of logic stronger than classical first-order logic. But if such a system is considered to be intractable, it means that the logical foundation of mathematics can be prohibitively complex for human cognition. In this paper I will argue, however, that this problem is the result of an unjustified direct use of computational complexity classes in cognitive modelling. Placing my account in the recent literature on the topic, I argue that the problem can be solved by considering computational complexity for humanly relevant problem solving algorithms and input sizes. (shrink)

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

Gentzen’s approach by transfinite induction and that of intuitionist Heyting arithmetic to completeness and the self-foundation of mathematics are compared and opposed to the Gödel incompleteness results as to Peano arithmetic. Quantum mechanics involves infinity by Hilbert space, but it is finitist as any experimental science. The absence of hidden variables in it interpretable as its completeness should resurrect Hilbert’s finitism at the cost of relevant modification of the latter already hinted by intuitionism and Gentzen’s approaches for completeness. This (...) paper investigates both conditions and philosophical background necessary for that modification. The main conclusion is that the concept of infinity as underlying contemporary mathematics cannot be reduced to a single Peano arithmetic, but to at least two ones independent of each other. Intuitionism, quantum mechanics, and Gentzen’s approaches to completeness an even Hilbert’s finitism can be unified from that viewpoint. Mathematics may found itself by a way of finitism complemented by choice. The concept of information as the quantity of choices underlies that viewpoint. Quantum mechanics interpretable in terms of information and quantum information is inseparable from mathematics and its foundation. (shrink)

Da Gintis ein leitender Ökonom ist und ich einige seiner früheren Bücher mit Interesse gelesen habe, erwartete ich einige weitere Einblicke in das Verhalten. Leider, macht er die toten Hände der Gruppenauswahl und Phänomenologie in die Herzstücke seiner Verhaltenstheorien, und das macht die Arbeit weitgehend ungültig. Schlimmer noch, da er hier ein so schlechtes Urteilsvermögen an den Tag stellt, stellt er all seine bisherigen Arbeiten in Frage. Der Versuch, die Gruppenauswahl seiner Freunde in Harvard, Nowak und Wilson wiederzubeleben, vor ein (...) paar Jahren war einer der größten Skandale in der Biologie in den letzten zehn Jahren, und ich habe die traurige Geschichte in meinem Artikel "Altruismus, Jesus und das Ende der Welt – wie die Templeton Foundation kaufte eine Harvard-Professur und angegriffen Evolution, Rationalität und Zivilisation -- Eine Rezension von E.O. Wilson 'The Social Conquest of Earth' (2012) und Nowak und Highfield 'SuperCooperators' (2012). Im Gegensatz zu Nowak scheint Gintis nicht durch religiösen Fanatismus motiviert zu sein, sondern durch den starken Wunsch, eine Alternative zu den düsteren Realitäten der menschlichen Natur zu schaffen, die durch das (nahezu universelle) Unverständnis der grundlegenden menschlichen Biologie und des leeren Schiefers von Verhaltenswissenschaftlern, anderen Akademikern und der breiten Öffentlichkeit leicht gemacht wird. Gintis greift zu Recht (wie schon oft zuvor) Ökonomen, Soziologen und andere Verhaltensforscher an, weil sie keinen kohärenten Rahmen haben, um Verhalten zu beschreiben. Natürlich ist das Notwendige, um Das Verhalten zu verstehen, evolutionär. Leider bietet er selbst keine selbst (nach seinen vielen Kritikern und ich stimme zu), und der Versuch, die faule Leiche der Gruppenauswahl auf alle wirtschaftlichen und psychologischen Theorien zu transplantieren, die er in seiner jahrzehntelangen Arbeit erzeugt hat, macht nur sein gesamtes Projekt ungültig. Obwohl Gintis sich tapfer bemüht, die Genetik zu verstehen und zu erklären, wie Wilson und Nowak, ist er weit davon entfernt, ein Experte zu sein, und wie sie blendet ihn die Mathematik nur für die biologischen Unmöglichkeiten, und das ist natürlich die Norm in der Wissenschaft. Wie Wittgenstein auf der ersten Seite von Kultur und Wert bekannte: "Es gibt keine religiöse Konfession, in der der Missbrauch metaphysischer Ausdrücke für so viel Sünde verantwortlich war wie in der Mathematik." Es war immer glasklar, dass ein Gen, das Verhalten verursacht, das seine eigene Frequenz verringert, nicht bestehen kann, aber dies ist der Kern des Begriffs der Gruppenauswahl. Darüber hinaus ist bekannt und oft gezeigt, dass Gruppenauswahl nur auf inklusive Fitness reduziert (Verwandtise), die, wie Dawkins festgestellt hat, ist nur ein weiterer Name für evolution durch natürliche Selektion. Wie Wilson arbeitet Gintis seit etwa 50 Jahren in dieser Arena und hat sie immer noch nicht begriffen, aber nach dem Skandal hat es nur 3 Tage gedauert, bis ich die relevanteste professionelle Arbeit gefunden, gelesen und verstanden habe, wie in meinem Artikel beschrieben. Es ist ein Irrsinn zu erkennen, dass Gintis und Wilson nicht in der Lage waren, dies in fast einem halben Jahrhundert zu erreichen. Ich diskusere die Fehler der Gruppenauswahl und Phänomenologie, die in der Wissenschaft die Norm sind, als Sonderfälle des nahezu universellen Versagens, die menschliche Natur zu verstehen, die Amerika und die Welt zerstören. Wer aus der modernen zweisystems-Sichteinen umfassenden, aktuellen Rahmen für menschliches Verhalten wünscht, kann mein Buch "The Logical Structure of Philosophy, Psychology, Mindand Language in Ludwig Wittgenstein and John Searle' 2nd ed (2019) konsultieren. Wer sich für mehr meiner Schriften interessiert, kann 'Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019) und Suicidal Utopian Delusions in the 21st Century 4th ed (2019) sehen. (shrink)

Im Artikel wird der Fortschritt von Fichtes Methodenverständnis gegenüber den enzyklopädischen Wissenssystemen – wie beispielsweise bei Wolff, Zeller oder Lambert – herausgearbeitet. Anfang des 19. Jahrhunderts hatte sich die Erkenntnis durchgesetzt, dass weder die Philosophie noch die aufkeimenden Sozialwissenschaften und auch nicht die Naturforschung, wo sie es mit dem Leben zu tun hat, als bloßer Formalismus nach dem Modell der Mathematik oder Physik aufzubauen waren. Fichte gelang es demgegenüber mit seiner Ich-Konstruktion nicht nur das innere Verhältnis der differenzierten Vermögen (...) des Subjekts dadurch abzuleiten, dass er die Form der Erkenntnis zugleich als Form des Wollens behauptete; mehr noch: Die in der synthetischen Methode erscheinende Vernunftgesetzlichkeit ist in einen alle anderen Erkenntnismethoden übergreifenden Rang erhoben. Das Prinzip des kantschen "Naturzwecks" der Einheit von Ursache und Wirkung, ist darin zum grundlegenden Tätigkeitsprinzip der Subjekt-Objekt-Auffassung gesteigert. The article highlights the progress made by Fichte's understanding of methods compared to the encyclopedic knowledge systems - such as those of Wolff, Zeller or Lambert. At the beginning of the 19th century, it had become clear that neither philosophy nor the emerging social sciences nor natural science, where it deals with life, could be constructed as a mere formalism based on the model of mathematics or physics. In contrast, Fichte's construction of the ego not only succeeded in deriving the inner relationship between the differentiated faculties of the subject by asserting that the form of knowledge was also a form of will; moreover, the laws of reason that appear in the synthetic method were elevated to a level that overshadowed all other methods of knowledge. The principle of Kant's "natural purpose" of the unity of cause and effect is elevated to the fundamental principle of activity of the subject-object conception. (shrink)

The foundational ideas of David Hilbert have been generally misunderstood. In this dissertation prospectus, different aims of Hilbert are summarized and a new interpretation of Hilbert's work in the foundations of mathematics is roughly sketched out. Hilbert's view of the axiomatic method, his response to criticisms of set theory and intuitionist criticisms of the classical foundations of mathematics, and his view of the role of logical inference in mathematical reasoning are briefly outlined.

Peirce and Dewey were generally more concerned with the process of scientific activity than purely mathematical work. However, their accounts of knowledge production afford some insights into the epistemology of mathematical postulates, especially definition and axioms. Their rejection of rationalist metaphysics and their emphasis on continuity in inquiry provides the pretext for the pragmatic a priori – hypothetical and operational assumptions whose justification relies on their fruitfulness in the long run. This paper focuses on the application of this idea to (...) the epistemology of definitions and an account of progress in mathematics, although it has broader implications for the study of conceptual change and the function and basis of presuppositions in the sciences. (shrink)

MethodologyA new hypothesis on the basic features characterizing the Foundations of Mathematics is suggested.Application of the methodBy means of it, the several proposals, launched around the year 1900, for discovering the FoM are characterized. It is well known that the historical evolution of these proposals was marked by some notorious failures and conflicts. Particular attention is given to Cantor's programme and its improvements. Its merits and insufficiencies are characterized in the light of the new conception of the FoM. (...) After the failures of Frege's and Cantor's programmes owing to the discoveries of an antinomy and internal contradictions, respectively, the two remaining, more radical programmes, i.e. Hilbert's and Brouwer's, generated a great debate; the explanation given here is their mutual incommensurability, defined by means of the differences in their foundational features.ResultsThe ignorance of this phenomenon explains the inconclusiveness of a century-long debate between the advocates of these two proposals. Which however have been so greatly improved as to closely approach or even recognize some basic features of the FoM.Discussion on the resultsYet, no proposal has recognized the alternative basic feature to Hilbert's main one, the deductive organization of a theory, although already half a century before the births of all the programmes this alternative was substantially instantiated by Lobachevsky's theory on parallel lines. Some conclusive considerations of a historical and philosophical nature are offered. In particular, the conclusive birth of a pluralism in the FoM is stressed. (shrink)

We study a new formal logic LD introduced by Prof. Grzegorczyk. The logic is based on so-called descriptive equivalence, corresponding to the idea of shared meaning rather than shared truth value. We construct a semantics for LD based on a new type of algebras and prove its soundness and completeness. We further show several examples of classical laws that hold for LD as well as laws that fail. Finally, we list a number of open problems. -/- .

Grete Hermann’s essay “Die naturphilosophischen Grundlagen der Quantenmechanik” has received much deserved scholarly attention in recent years. In this paper, I follow the lead of Elise Crull who sees in Hermann’s work the general outlines of a neo-Kantian interpretation of quantum theory. In full support of this view, I focus on Hermann’s central claim that limited spatio-temporal, and even analogically causal, representations of events exist within an overall relational structure of entangled quantum mechanical states that defy any unified spatio-temporal (...) description. In my view, Hermann also advances an important transcendental argument that perspectival spatio-temporal representations of nature have their foundations in general relational networks that are not spatio-temporal. The key point is that the adoption of a perspectival system within the general network induces the representation but only for that context. These ideas are consistent with a perspectival subject–object principle in Kant and also with Weyl’s work on Lie groups and their representations. (shrink)

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the (...) mathematical practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences. (shrink)

An axiological analysis of the basis of the 1997 Constitution of the Republic of Poland, determined mainly in the Preamble, makes it possible to put forward a thesis that this axiology is not, at least in reference to the principle, eclectic. In respect of the meta-axiological settlements, this is a tradition of natural-law type, recognizing the objective grounding of values and law. The accepted solutions are also convergent with the axiology typical of the international protection of human rights. -/- Résumé (...) L’analyse des fondements axiologiques de la Constitution polonaise de 1997, inscrits avant tout dans le préambule, permet de formuler la thèse selon laquelle cette axiologie n’est pas, en principe, éclectique. La réflexion sur les valeurs s’inscrit dans la vision classique des fondements du droit, remontant à l’antiquité. Partant de là, se dégage une tradition juridique appuyée sur le droit naturel, reconnaissant la force objective des valeurs et du droit. Ces conclusions rejoignent l’axiologie classique de la protection internationale des droits de l’Homme. -/- Zusammenfassung Die Analyse der vor allem durch die Präambel definierten axiologischen Grundlagen der Verfassung der Republik Polen von 1997 ermöglicht die Aufstellung der These, dass diese Axiologie nicht eklektisch ist, zumindest nicht prinzipiell. Hier enthüllt sich ein Denken von den Werten, das sich in die klassische, bis in die Antike zurückreichende Reflexion über die Grundlagen des Rechts einfügt. Wegen der metaaxiologischen Entscheidungen ist das eine Tradition naturrechtlichen Typs, die die objektive Begründung der Werte und des Rechts anerkennt. Die angenommenen Lösungen stimmen auch mit der für den internationalen Schutz der Menschenrechte typischen Axiologie überein. (shrink)

The current definition of Constructive mathematics as “mathematics within intuitionist logic” ignores two fundamental issues. First, the kind of organization of the theory at issue. I show that intuitionist logic governs a problem-based organization, whose model is alternative to that of the deductive-axiomatic organization, governed by classical logic. Moreover, this dichotomy is independent of that of the kind of infinity, either potential or actual, to which respectively correspond constructive mathematical and classical mathematical tools. According to this view a (...) mathematical theory is based on the choices regarding these two dichotomies. As an example of this kind of foundation, arithmetic is rationally re-founded on constructive mathematical tools and the model of the problem-based organization. In conclusion, constructive mathematics is not only mathematics making use of constructive tools in intuitionist logic but also organized according to around a basic problem, solved by a method discovered using intuitionist logic. (shrink)

The attempts of theoretically solving the famous puzzle-dictum of physicist Eugene Wigner regarding the “unreasonable” effectiveness of mathematics as a problem of analytical philosophy, started at the end of the 19th century, are yet far from coming out with an acceptable theoretical solution. The theories developed for explaining the empirical “miracle” of applied mathematics vary in nature, foundation and solution, from denying the existence of a genuine problem to structural theories with an advanced level of mathematical formalism. Despite (...) this variation, methodologically fundamental questions like “Which is the adequate theoretical framework for solving Wigner’s conjecture?” and “Can the logico-mathematical formalism solve it and is it entitled to do it?” did not receive answers yet. The problem of the applicability of mathematics in the physical reality has been treated unitarily in some sense, with respect to the semantic-conceptual use of the constitutive terms, within both the structural and non-structural theories. This unity (of consistency) applied to both the referred objects and concepts per se and the aims of the investigations. For being able to make an objective study of the possible alternatives of the existent theories, a foundational approach of them is needed, including through semantic-conceptual distinctions which to weaken the unity of consistency. (shrink)

Abstract (German) Dieses Buch faßt meine Forschungen zur Semiotik Giordano Brunos und zu deren Grundlagen in der Kosmologie und in der Gedächtnistheorie zusammen, wobei sowohl die Heterogenität als auch der Universalität des Denkers den Charakter dieses Buches geprägt haben. Es wird kein glattes Bild seiner Person oder seines Werkes angeboten, kein bejahender Jubel über eine epochale Leistung; wichtiger war mir die bis ins Detail gehende, manchmal mühselige Interpretation seiner Systementwürfe, wobei ich versucht habe, auch deren formales und technisches Niveau (...) deutlich zu machen. Nicht seine Weltanschauung soll dargestellt werden, sondern sein Versuch, eine Vielfalt von Detaillösungen in eine wohlproportionierte, wenn auch ins Unendliche reichende Wissensarchitektur zu integrieren. Als Startsignal mag folgende Frage dienen: Inwiefern ist Giordano Bruno Semiotiker? Der Philosoph Bruno wird von Jacobi als Beispiel spinozistischer Tendenzen wieder ins Gespräch gebracht; später wird er für die spekulative Naturphilosophie, die aus Kants Schatten heraustritt, z.B. für Schelling und Hegel, interessant, und schließlich wird er als Märtyrer der Freiheit von den Antiklerikalen im Risorgimento hochgehoben. Der Heroenkult um die „Kopernikanische Wende" weiß mit Bruno nichts anzufangen, er ist kein Naturwissenschaftler, wie man ihn sich im 20 . Jh . wünscht. Von Yates ( 1964 ) wird er als Hermetiker und Ahnherr der Rosenkreuzer entdeckt, und plötzlich soll er auch noch ein Semiotiker sein ? Die Etikette „Semiotiker" ist allerdings weniger riskant als jene früher an Bruno herangetragenen. Bruno ist Philosoph, in einem allgemeinen Sinn , insofern er alle wesentlichen Aspekte: metaphysische, natur philosophische, ethische, semiotische, poetologische zum Entwurf eines möglichst kohärenten Bildes vereint, und er ist ein Renaissance-Philosoph, d .h. das antike Erbe, besonders jenes in den orientalischen Tiefen der Kulturgeschichte (Ägypten und Chaldäer) ist sein Ausgangspunkt. Selbst den Entwurf des Copernicus versteht Bruno in erster Linie als eine Wiederentdeckung antiker Einsichten. Es gibt jedoch in Brunos Schaffen einen roten Faden: die Architektur unseres Wissens, das Gedächtnis der Kultur- und Wissensgemeinschaft und natürlich desjenigen, der wie Bruno dieses Wissen zu einer Synthese vereinigen will . Seine Philosophie versucht, wie jede große Philosophie, alles, was wir wissen und was wir daraus an weiterem impliziten Wissen erschließen können, in einer Einheit zu denken, und diese Einheit ist notwendigerweise nur als Zeichensystem (begrifflich, argumentativ, bildhaft, als imaginierte Welt) zu realisieren. -/- Abstract (English) This book summarizes my research on Giordano Bruno's semiotics and its foundations in cosmology and memory theory, whereby both the heterogeneity and the universality of the thinker have shaped the character of this book. No smooth picture of his person or his work is offered, no affirmative jubilation over an epochal achievement; What was more important to me was the sometimes laborious interpretation of his system designs, which went down to the last detail, and I also tried to make their formal and technical level clear. It is not his worldview that should be presented but his attempt to integrate various detailed solutions into a well-proportioned, albeit infinitely reaching, architecture of knowledge. The following question may serve as a starting signal: To what extent is Giordano Bruno a semiotician? Jacobi brings up the philosopher Bruno again as an example of Spinozist tendencies; later, he becomes important for the speculative natural philosophy emerging from Kant's shadow, e.g., for Schelling and Hegel. He is finally extolled as a martyr of freedom by the anticlericals in the Risorgimento. The hero cult around the "Copernican turn" does not know what to do with Bruno; he is not a natural scientist as one would wish in the 20th century. Yates (1964) discovered him as a hermetic, even an ancestor of the Rosicrucians. The etiquette "semiotic" is certainly less risky than the associations that Bruno used to have. Bruno is a philosopher, in a general sense, in that he combines all the essential aspects: metaphysical, natural philosophical, ethical, semiotic, poetological to project a picture that is as coherent as possible, and he is a Renaissance philosopher, i .e . the ancient heritage, especially that in the oriental depths of cultural history (Egypt and Chaldeans) is his starting point. Bruno sees even the draft of Copernicus primarily as a rediscovery of ancient insights. However, there is a common thread in Bruno's work: the architecture of our knowledge, the memory of the cultural and knowledge community and, of course, of those who, like Bruno, want to unite this knowledge into a synthesis. His philosophy, like every great philosophy, tries to think of everything that we know and what we can deduce from it as further implicit knowledge as a unit, and this unit is necessarily only available as a system of signs (conceptual, argumentative, pictorial, as an imagined world). to realize. 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English title: The Fundations of Normativity by Kant and Spener. In one of the chapters in Conflict of Faculties, Kant presents so-called “Spener’s problem”. Kant refers to the conception of the 17th century theologian Philipp Jakob Spener, concerning the problem of radical moral transformation of individuals. Spener became famous as one of the founders and main exponents of pietistic theology, which in the 18th century became one of the most significant intellectual trends at the University of Königsberg and about which (...) one may assume, that it could influence the evolution of Kantian philosophy. The goal of the article is to compare the main assumptions of Spener’s theology with Kantian theory of moral motivation. The analysis focuses on the possibility of describing the phenomenon of normativity in both conceptions. (shrink)

This paper contains a close analysis of Frege's proofs of the axioms of arithmetic §§70-83 of Die Grundlagen, with special attention to the proof of the existence of successors in §§82-83. Reluctantly and hesitantly, we come to the conclusion that Frege was at least somewhat confused in those two sections and that he cannot be said to have outlined, or even to have intended, any correct proof there. The proof he sketches is in many ways similar to that given (...) in Grundgesetze der Arithmetik, but fidelity to what Frege wrote in Die Grundlagen and in Grundgesetze requires us to reject the charitable suggestion that it was this (beautiful) proof that he had in mind in §§82-83. (shrink)

Ein Beispiel für die Rezeption und Fortführung der schopenhauerschen Logik findet man in den Mathematiklehrbüchern Friedrich Adolph Wilhelm Diesterwegs (1790–1866), In diesem Aufsatz werden die historische und systematische Dimension dieser Anwendung von Logikdiagramme auf die Mathematik skizziert. In Kapitel 2 wird zunächst die frühe Rezeption der schopenhauerschen Logik und Philosophie der Mathematik vorgestellt. Dabei werden einige oftmals tradierte Vorurteile, die das Werk Schopenhauers betreffen, in Frage gestellt oder sogar ausgeräumt. In Kapitel 3 wird dann die Philosophie der (...) class='Hi'>Mathematik und der Logik Schopenhauers vorgestellt. Es wird gezeigt, dass Schopenhauers Logikdiagramme auf Ideen der Mathematiker Leonhard Euler (1707–1783), Bernhard Friedrich Thibaut (1775–1832) und Franz Ferdinand Schweins (1780–1856) zurückgehen und dass Schopenhauer diese Logikdiagramme auch bereits zur Analyse mathematischer Begriffe verwendet. Kapitel 4 beschäftigt sich mit der Schopenhauer-Rezeption Diesterwegs. Es wird die Frage behandelt, inwieweit Diesterweg in der Mathematik von Schopenhauer beeinflusst war und welche Rolle dieser Einfluss auf sein Verständnis von Logik und Mathematik hatte. In Kapitel 5 wird schließlich gezeigt, dass Diesterweg die schopenhauersche Begriffsanalyse der Mathematik mit Hilfe von Logikdiagrammen aufgreift und weiterführt. Da Diesterweg einer der einflussreichsten Pädagogen und Mathematikdidaktiker des 20. Jahrhunderts war, ist anzunehmen, dass sich Spuren dieser Methode noch heute finden lassen. Inwiefern dies wahrscheinlich ist, wird in der Zusammenfassung (Kapitel 6) thematisiert. (shrink)

Epstein and Carnielli's fine textbook on logic and computability is now in its second edition. The readers of this journal might be particularly interested in the timeline `Computability and Undecidability' added in this edition, and the included wall-poster of the same title. The text itself, however, has some aspects which are worth commenting on.

It is undeniable Poincaré was a very famous and influential scientist. So, possibly because of it, it was relatively easy for him to participate in the heated discussions of the foundations of mathematics in the early 20th century. We can say it was “easy” because he didn't get involved in this subject by writing great treatises, or entire books about his own philosophy of mathematics (as other authors from the same period did). Poincaré contributed to the philosophy (...) of mathematics by writing short essays and letters. (shrink)

Moral reasoning traditionally distinguishes two types of evil:moral (ME) and natural (NE). The standard view is that ME is the product of human agency and so includes phenomena such as war,torture and psychological cruelty; that NE is the product of nonhuman agency, and so includes natural disasters such as earthquakes, floods, disease and famine; and finally, that more complex cases are appropriately analysed as a combination of ME and NE. Recently, as a result of developments in autonomous agents in cyberspace, (...) a new class of interesting and important examples of hybrid evil has come to light. In this paper, it is called artificial evil (AE) and a case is made for considering it to complement ME and NE to produce a more adequate taxonomy. By isolating the features that have led to the appearance of AE, cyberspace is characterised as a self-contained environment that forms the essential component in any foundation of the emerging field of Computer Ethics (CE). It is argued that this goes someway towards providing a methodological explanation of why cyberspace is central to so many of CE's concerns; and it is shown how notions of good and evil can be formulated in cyberspace. Of considerable interest is how the propensity for an agent's action to be morally good or evil can be determined even in the absence of biologically sentient participants and thus allows artificial agents not only to perpetrate evil (and fort that matter good) but conversely to `receive' or `suffer from' it. The thesis defended is that the notion of entropy structure, which encapsulates human value judgement concerning cyberspace in a formal mathematical definition, is sufficient to achieve this purpose and, moreover, that the concept of AE can be determined formally, by mathematical methods. A consequence of this approach is that the debate on whether CE should be considered unique, and hence developed as a Macroethics, may be viewed, constructively,in an alternative manner. The case is made that whilst CE issues are not uncontroversially unique, they are sufficiently novel to render inadequate the approach of standard Macroethics such as Utilitarianism and Deontologism and hence to prompt the search for a robust ethical theory that can deal with them successfully. The name Information Ethics (IE) is proposed for that theory. Itis argued that the uniqueness of IE is justified by its being non-biologically biased and patient-oriented: IE is an Environmental Macroethics based on the concept of data entity rather than life. It follows that the novelty of CE issues such as AE can be appreciated properly because IE provides a new perspective (though not vice versa). In light of the discussion provided in this paper, it is concluded that Computer Ethics is worthy of independent study because it requires its own application-specific knowledge and is capable of supporting a methodological foundation, Information Ethics. (shrink)

Der Autor analysiert die methodologische Grundlegung der Transzendentalphilosophie durch Fichte. Als entscheidenden Punkt arbeitet er den durch Fichte vollzogenen Paradigmenwechsel heraus Danach kommt der Systembegründung nicht mehr ein vorwissenschaftlicher Status zu, sondern diese wird Moment der Systemkonstitution und Systemkonstruktion. Das erlaubt Fichte, die philosophische Methode in ihrer Strenge von der Methode der Mathematik abzuheben. -/- The author analyzes the methodological basis of Fichte's transcendental philosophy. As a decisive point, he works out the paradigm shift that Fichte carried out. According (...) to this, the justification of the system no longer has a pre-scientific status, but becomes a moment of the system constitution and system construction. This allows Fichte to distinguish the philosophical method in its rigor from the method of mathematics. (shrink)

Foundations of Relational Realism presents an intuitive interpretation of quantum mechanics, based on a revised decoherent histories interpretation, structured within a category theoretic topological formalism. -/- If there is a central conceptual framework that has reliably borne the weight of modern physics as it ascends into the twenty-first century, it is the framework of quantum mechanics. Because of its enduring stability in experimental application, physics has today reached heights that not only inspire wonder, but arguably exceed the limits of (...) intuitive vision, if not intuitive comprehension. For many physicists and philosophers, however, the currently fashionable tendency toward exotic interpretation of the theoretical formalism is recognized not as a mark of ascent for the tower of physics, but rather an indicator of sway—one that must be dampened rather than encouraged if practical progress is to continue. -/- In this unique two-part volume, designed to be comprehensible to both specialists and non-specialists, the authors chart out a pathway forward by identifying the central deficiency in most interpretations of quantum mechanics: That in its conventional, metrical depiction of extension, inherited from the Enlightenment, objects are characterized as fundamental to relations—i.e., such that relations presuppose objects but objects do not presuppose relations. The authors, by contrast, argue that quantum mechanics exemplifies the fact that physical extensiveness is fundamentally topological rather than metrical, with its proper logico-mathematical framework being category theoretic rather than set theoretic. -/- By this thesis, extensiveness fundamentally entails not only relations of objects, but also relations of relations. Thus, the fundamental quanta of quantum physics are properly defined as units of logico-physical relation rather than merely units of physical relata as is the current convention. Objects are always understood as relata, and likewise relations are always understood objectively. In this way, objects and relations are coherently defined as mutually implicative. The conventional notion of a history as “a story about fundamental objects” is thereby reversed, such that the classical “objects” become the story by which we understand physical systems that are fundamentally histories of quantum events. -/- These are just a few of the novel critical claims explored in this volume—claims whose exemplification in quantum mechanics will, the authors argue, serve more broadly as foundational principles for the philosophy of nature as it evolves through the twenty-first century and beyond. (shrink)

I prove that invoking the univalence axiom is equivalent to arguing 'without loss of generality' (WLOG) within Propositional Univalent Foundations (PropUF), the fragment of Univalent Foundations (UF) in which all homotopy types are mere propositions. As a consequence, I argue that practicing mathematicians, in accepting WLOG as a valid form of argument, implicitly accept the univalence axiom and that UF rightly serves as a Foundation for Mathematical Practice. By contrast, ZFC is inconsistent with WLOG as it is applied, (...) and therefore cannot serve as a foundation for practice. (shrink)

Standard histories of mathematics and of analytic philosophy contend that work on the foundations of mathematics was motivated by a crisis such as the discovery of paradoxes in set theory or the discovery of non-Euclidean geometries. Recent scholarship, however, casts doubt on the standard histories, opening the way for consideration of an alternative motive for the study of the foundations of mathematics—unification. Work on foundations has shown that diverse mathematical practices could be integrated into (...) a single framework of axiomatic systems and that much of mathematics could be expressed in a single language. The new framework was the product of an interdisciplinary coalition whose ideas resemble those later adopted by the Vienna Circle and logical empiricists. (shrink)

K. Marx’s 200th jubilee coincides with the celebration of the 85 years from the first publication of his “Mathematical Manuscripts” in 1933. Its editor, Sofia Alexandrovna Yanovskaya (1896–1966), was a renowned Soviet mathematician, whose significant studies on the foundations of mathematics and mathematical logic, as well as on the history and philosophy of mathematics are unduly neglected nowadays. Yanovskaya, as a militant Marxist, was actively engaged in the ideological confrontation with idealism and its influence on modern (...) class='Hi'>mathematics and their interpretation. Concomitantly, she was one of the pioneers of mathematical logic in the Soviet Union, in an era of fierce disputes on its compatibility with Marxist philosophy. Yanovskaya managed to embrace in an originally Marxist spirit the contemporary level of logico-philosophical research of her time. Due to her highly esteemed status within Soviet academia, she became one of the most significant pillars for the culmination of modern mathematics in the Soviet Union. In this paper, I attempt to trace the influence of the complex socio-cultural context of the first decades of the Soviet Union on Yanovskaya’s work. Among the several issues I discuss, her encounter with L. Wittgenstein is striking. (shrink)

This article articulates and assesses an imperatival approach to the foundations of mathematics. The core idea for the program is that mathematical domains of interest can fruitfully be viewed as the outputs of construction procedures. We apply this idea to provide a novel formalisation of arithmetic and set theory in terms of such procedures, and discuss the significance of this perspective for the philosophy of mathematics.

This paper introduces a novel object that has less structure than, and is ontologically prior to the natural numbers. As such it is a candidate model of the foundation that lies beneath the natural numbers. The implications for the construction of mathematical objects built upon that foundation are discussed.

I identify two versions of the scientific anti-realist’s selectionist explanation for the success of science: Bas van Fraassen’s original and K. Brad Wray’s newer interpretation. In Wray’s version, psycho-social factors internal to the scientific community – viz. scientists’ interests, goals, and preferences – explain the theory-selection practices that explain theory-success. I argue that, if Wray’s version were correct, then science should resemble art. In art, the artwork-selection practices that explain artwork-success appear faddish. They are prone to radical change over time. (...) Theory-selection practices that explain theory-success in science are however not faddish. They are mostly stable; that is, long-lived and consistent over time. This is because scientists (explicitly or implicitly) subscribe to what I will call the testability norm: scientific theories must make falsifiable claims about the external physical world. The testability norm and not psycho-sociology explains the theory-selection practices that explain theory-success in science. Contra Wray, scientific anti-realists can then maintain that the external physical world (as expressed in the testability norm) explains theory-success. (shrink)

In this paper I challenge Paolo Palmieri’s reading of the Mach-Vailati debate on Archimedes’s proof of the law of the lever. I argue that the actual import of the debate concerns the possible epistemic (as opposed to merely pragmatic) role of mathematical arguments in empirical physics, and that construed in this light Vailati carries the upper hand. This claim is defended by showing that Archimedes’s proof of the law of the lever is not a way of appealing to a non-empirical (...) source of information, but a way of explicating the mathematical structure that can represent the empirical information at our disposal in the most general way. (shrink)

The paper justifies the following theses: The totality can found time if the latter is axiomatically represented by its “arrow” as a well-ordering. Time can found choice and thus information in turn. Quantum information and its units, the quantum bits, can be interpreted as their generalization as to infinity and underlying the physical world as well as the ultimate substance of the world both subjective and objective. Thus a pathway of interpretation between the totality via time, order, choice, and information (...) to the substance of the world is constructed. The article is based only on the well-known facts and definitions and is with no premises in this sense. Nevertheless it is naturally situated among works and ideas of Husserl and Heidegger, linked to the foundation of mathematics by the axiom of choice, to the philosophy of quantum mechanics and information. (shrink)