This essay examines some of the ways in which the assumption of the essential finitude of the human mind, in contrast to the infinitude of God’s mind, bears on Leibniz’s and Kant’s accounts of our representational capacities. This examination reveals several underappreciated similarities between their views, but also some notable differences that help us pinpoint where and in what ways Kant departs from his celebrated predecessor. The fruits of this examination are a better understanding of Kant’s conception of the discursivity (...) of our understanding, his account of the difference between concepts and intuitions, and the particular flavor of his idealism. -/- . (shrink)

A popular story about Kant's relation to Leibniz presents Kant as a Leibniz-Wolffian by education who, inspired by his encounter with the teachings of Newton and Hume, took on the project of reconciling Leibniz-Wolffian metaphysics with Newtonian science and of responding to epistemological skepticism, a project that led him further and further away from his Leibniz-Wolffian roots and culminated in the total rejection of the Leibniz-Wolffian philosophy in the Critique of Pure Reason. In this essay, four shortcomings of the popular (...) story are identified and several suggestions are made about how to amend and expand the story in order to overcome these shortcomings. Furthermore, some of the most important Leibnizian doctrines that influenced Kant are collected and their role in Kant's philosophy is discussed. (shrink)

Wittgenstein famously criticizes the philosophical practice of analyzing the meaning of words outside their ordinary use in everyday language, whereby often self-made pseudo-problems arise. In order to shed further light on Wittgenstein’s critique, this article makes use of the Remarks on the Philosophy of Psychology. First, starting from the remark in Vol. I, §52, his criticism of the philosophical method of selection and generalization is explained in detail. Next, I give a brief outline of Wittgenstein’s own way of philosophizing by (...) reference to a selection of comments concerning the use of psychological words in everyday language, which will also further elucidate his critique. Finally, I enter into the question which significance everyday language according to Wittgenstein has for philosophy. (shrink)

Response to critical comments by Edward Slowik on my article 'Leibniz on Motion and the Equivalence of Hypotheses' in The Leibniz Review 18 (2008).

Color-vision defects constitute a spectrum of disorders with varying degrees and types of departure from normal human color vision. One form of color-vision defect is dichromacy; by mixing together only two lights, the dichromat can match any light, unlike normal trichromatic humans, who need to mix three. In a philosophical context, our titular question may be taken in two ways. First, it can be taken at face value as a question about visible properties of external objects, and second, it may (...) be interpreted as the more intangible question of “what it’s like” to be color-blind. (shrink)

A survey of color science and color vision.

„Im Grunde gibt es nur die Liebe“, sagte Picasso einst. Kaum ein Mensch durchläuft sein Leben, ohne früher oder später von jenen Gefühlen gebeutelt zu werden, welche als Liebe bezeichnet werden. Aber was ist Liebe überhaupt? Gibt es klare Kriterien, die Liebe notwendigerweise aufweist? Welchen Einfluss haben die kulturellen Vorbedingungen auf unsere Vorstellungen von Liebe?

„Gemeinschaft“ steht in der Philosophie nicht nur im Zusammenhang mit ethischen Überlegungen, sondern auch mit Fragen nach der Natur von sozialen Handlungen und dem charakteristischen Wesen von Kollektiven. Gibt es Bedingungen, die erfüllt sein müssen, damit man von einer Gemeinschaft – in Abgrenzung zu „Gesellschaft“ – sprechen kann?

A survey of arguments for and against the view that colors are physical properties.

Um was geht es beim Bedingungslosen Grundeinkommen? Und weshalb ist es auch ein Thema der Philosophie? Sollte die Gesellschaft grundsätzlich hinterfragen, was unter dem Begriff „Arbeit“ zu verstehen ist? Würde das Grundeinkommen Gerechtigkeit ermöglichen oder verunmöglichen?

Dieses Themendossier zeigt auf, mit welchen Themen sich die Philosophie befasst und wem die Philosophie was nützt. Wo heutzutage Philosophinnen und Philosophen wirken, wird ebenso betrachtet, wie die Frage, weshalb es sich überhaupt lohnt zu philosophieren.

-/- Shitstorms, Hate Speech oder virale Videos, die zum Klicken, Liken, Teilen bewegen: Die vernetzte Gesellschaft ist von Affekten getrieben und bringt selbst ganz neue Affekte hervor. -/- Die Beiträge des Bandes nehmen die medientechnologischen Entwicklungen unserer Zeit in den Blick und untersuchen sie aus der Perspektive einer kritischen Affekt- und Sozialphilosophie. Sie zeigen: Soziale Medien und digitale Plattformen sind nicht nur Räume des Austauschs, sie erschaffen Affektökonomien – und darin liegt auch ihre Macht. Indem sie neue Formen des sozialen (...) Umgangs stiften und bestimmen, wie wir kommunizieren, verschieben sie auch die politische Topographie. -/- Mit einem Beitrag von Antonio Negri. (shrink)

Muss ein Mensch Selbstachtung haben, um Empfänger der Menschenwürde zu sein? Spielt Menschenwürde auch in der Wirtschaft und im Rechtssystem eine Rolle? Und wie lässt sich „Menschenwürde“ überhaupt definieren? Im Dossier findet man die philosophischen Hintergründe zum Thema „Menschenwürde“.

Dieses Dossier diskutiert die ethischen Aspekte der direkten aktiven Sterbe- hilfe und die Legitimität des Suizids. Hervorgehoben wird dabei die Relevanz der ethischen Perspektive und der Nutzen der philosophischen Diskussion hin- sichtlich dieser brisanten gesellschaftlichen Frage.

On définit souvent la philosophie comme l’« amour de la sagesse ». En grec, le terme « philosophe » qui signifie « ami de la sa- gesse » (philós= ami, sophía= sagesse) se rapporte à une personne qui aspire à une connaissance globale des choses. (1) Mais que sait-on vraiment de la philosophie? Comment procède-t-elle, à quels thèmes s‘intéresse-t-elle?

Welche alltäglichen Probleme regen zu vertieftem Nachdenken an? Was bietet uns die Philosophie an möglichen Antworten an? Können wir von den Philosophinnen und Philosophen tatsächlich etwas lernen? Und: Ist der Begriff „Alltag“ gar selbst ein Untersuchungsgegenstand der Philosophie?

Die Aufgabe der Philosophie besteht nicht darin, den moralischen Zeigefinger zu erheben und vorzugeben, was man tun soll und was nicht. Vielmehr zeigt die Philosophie unterschiedliche Argumente auf, weshalb die eine oder andere Haltung vertretbar oder aus gewissen Gründen gegebenenfalls vertretbarer ist als eine andere. Insofern eine Gesellschaft an Klugheit interessiert ist und sich einer zukünftigen Entwicklung verantwortungsvoll stellt, kann die Philosophie eine wertvolle Aufgabe übernehmen: Die Methode, analytisch das Für und Wider abzuwiegen, bedeutet nicht zuletzt viel Arbeit und Aufmerksamkeit (...) für das Detail. Wie beispielsweise das Kapitel zur Grünen Gentechnik (Seite 6) zeigt, sind die Diskussionen um die zu fällenden moralischen Urteile nicht an ein externes Gremium zu delegieren. Im Gegenteil: Die Diskussion sollte innerhalb der Gesellschaft stattfinden, einerseits um den unterschiedlichen Haltungen Platz zu geben, andererseits aber, weil nur auf diese Weise ein echter Konsens zu Stande kommen kann. Die Rolle der Philosophie zeigt sich ähnlich auch, wenn es um die für die pluralistische Gesellschaft zentrale Toleranz geht. Die wertenden Einstellungen der Bürgerinnen und Bürger verdienen es, ernst genommen zu werden und offen diskutiert zu werden. Nur durch eine tiefgreifende Auseinandersetzung lässt sich verhindern, dass populistische Wortführer mit eigennützigen, aber aufmerksamkeitsstarken Argumenten von den Themen ablenken, die ernsthaft angegangen werden möchten: Die Vision einer friedlichen und freiheitlichen Gesellschaft, das demokratische Ideal, wird nicht mit dem Verweis auf die „Krise“ aufgegeben. Verantwortung interessiert die Bürgerinnen und Bürger schliesslich eben doch mehr, als Donald Trumps neuste Lüge. Das philosophische Themendossier zeigt, dass die akademische Philosophie viel leis-tet, um eine oberflächliche Auseinandersetzung mit gesellschaftlich und zukunftsrelevanten Themen zu verhindern. Auch wenn das auf den ersten Blick nach viel Anstrengung klingt, so zeigt der zweite Blick, dass darin viel Wohlwollen zugunsten der Gesellschaft steckt. Es lohnt sich, den Mut aufzubringen, sich intensiv mit den eigenen Werturteilen auseinanderzusetzen und das Auge vom eigenen Wohl auch auf das Wohle aller schweifen zu lassen. Philosophie kann „ansteckend“ sein und viel Freude hervorrufen: Gerade dann, wenn man neue Ideen mit dazu gewinnt oder andere Argumente für bereits gefasste Entscheidungen kennenlernt. Die Philosophie entwickelt sich fortlaufend weiter und bleibt somit mehr denn je: stets aktuell. (shrink)

Dieses Dossier stellt die Fragen der Debatte des Kommunitarismus: Wie ent- wickeln Menschen ihre Identität? Welche Konsequenzen hat diese Identitäts- bildung auf die Werte einer Gesellschaft? Inwiefern ist der Mensch ein auto- nomes Wesen und inwiefern ist er in Strukturen eingebunden? Und: Ist auch die politische Ebene durch diese Fragen betroffen?

Hat das menschliche Wesen einen klaren Kern, welchen ihn erst zum Menschen macht? Wie wäre dieser zu definieren? Der Mensch verfügt über viele Fähigkeiten, die ihn zu einem vernünftig agierenden Tier machen. Aber ist das der wesentliche Unterschied zu den anderen Tieren? Das vorliegende Dossier geht der Frage nach, was der Mensch ist und was ihn auszeichnet.

Gibt es einen Zusammenhang zwischen Politik und Philosophie? Bei Fragen, wie sich der Staat legitimieren lässt, was Gerechtigkeit ist oder welche Rolle Interkulturalität, Toleranz und Anerkennung in der heutigen Gesellschaft spie- len, bietet die Philosophie Antworten und Denkanstösse.

Dieses Heft befasst sich mit der Bedeutung und der fortlaufenden Entwicklung der Wirtschaftsethik. Es wird gezeigt, wie die Wirtschaftsethik auf die aktuellen Krisen reagiert, warum einem Unternehmen eine moralische Verantwortung zukommt und was man sich unter ethischen Managementprinzipien vorstellen könnte.

Was ist Macht? Hat der Mensch „von Natur aus“ das Bedürfnis, nach Macht zu streben? Ist Macht „schlecht“, weil sie Machtmissbrauch ermöglicht? Wieviele Aspekte das Thema Macht wirklich hat und ob Macht beispielsweise auch etwas mit Gewalt zu tun hat, erfährt man im vorliegenden Dossier.

Ist ein gutes Leben gleichzusetzen mit einem glücklichen Leben? Lässt sich Glück antrainieren? Wie gestaltet sich ein gutes Leben im Alter? Solcherlei Fragen, aber auch die Untersuchung, ob nur eine moralisch gute Lebensführung ein gutes Leben ermöglicht, werden im vorliegenden Dossier bearbeitet.

Dieses Dossier befasst sich mit der Gleichstellung der Geschlechter: Wie muss man sich Gleichberechtigung in der Kindererziehung vorstellen? Können auch Männer Feministen sein? Was ist feministische Philosophie? Welche philoso- phischen Theorien gibt es zu Männern und Frauen?

Wie wird Wissenschaft in der Gesellschaft wahrgenommen? Diese Frage lässt sich nicht leicht beantworten: Ist die gesellschaftliche Wahrnehmung von Wis- senschaft von Misstrauen geprägt? Oder von Unwissen über die Funktionswei- se von wissenschaftlicher Forschung?

Dieses Heft präsentiert philosophische Perspektiven zu Umweltproblemen. Hierbei sollen einerseits die fünf Hauptpositionen der Umweltethik (Anthropozentrismus, Pathozentrismus, Biozentrismus, Holismus & Tiefen-Ökologie) vorgestellt werden. An- dererseits wird anhand der Erhöhung der Grimselstaumauer die Relevanz der Philosophie hinsichtlich des Dilemmas zwischen Nutzen für den Menschen und Schaden an der Natur deutlich gemacht.

Dieses Heft umreisst philosophische Perspektiven zur Zukunft der Alters- und Hin- terbliebenenversicherung (AHV). Welche Pflichten haben die heutigen Generationen gegenüber den zukünftigen? Inwiefern ist die Zukunft der AHV ein philosophisches Thema?

Dieses Dossier stellt die Fragen der Migrationsethikdebatte: Gibt es ein Recht auf Einwanderung in einem liberalen demokratischen Staat wie der Schweiz? Welche Argumente sprechen grundsätzlich für oder gegen offene Grenzen? Was leistet die Philosophie hinsichtlich der Einwanderungsproblematik?

Welche philosophischen Perspektiven gibt es in Bezug auf Rauschmittel? Das Dossier zeigt die Spannbreite: Ob Substanzen, die unsere Leistung erhöhen, sogar moralisch gefordert sein könnten, warum der Staat den Menschen nicht jeden Rausch verbieten sollte, und ob Süchtige Kriminelle sind.

Dieses Heft beinhaltet philosophische Perspektiven der Debatten der personalen Identität und der Liebe. Neben der Darstellung der Kerninhalte der beiden Debatten wird die Frage „Wen liebe ich?“ philosophisch problematisiert. Dabei soll aufgezeigt werden, was unter einer philosophischen Herangehensweise zu verstehen ist und wie sich dies konkret im Fall der angeführten Frage niederschlägt.

Welche Zukunft erwartet die Menschheit? Oder müsste man fragen: Welche möglichen Zukünfte liegen im Gestaltungsbereich des Menschen? Was passiert, wenn wir eine superintelligente Maschine erfinden, die uns sogar ermöglicht, der Vergänglichkeit zu entrinnen? Welchen Einfluss haben die Technologien auf die Arbeit oder auf unsere Wahrnehmung der Zeit?

Dieses Dossier befasst sich mit den philosophischen Perspektiven zum Thema Bewusstsein: Was kennzeichnet „das Geistige“ und hat es ein einheitliches Merkmal? Wie kommt die Welt in den Kopf? Was bedeutet „subjektiv“ und haben auch Tiere und Pflanzen ein Bewusstsein?

Dieses Dossier wirft die Frage auf, welche Wirkung das digitale Zeitalter auf die Menschen hat und zeigt, welcher Zusammenhang zwischen der Philosophie und der Informationstechnologie besteht: Privatsphäre im Internet, künstliche Intelligenz, Computerethik und auch in Bezug auf Datenschutzfragen.

This paper critically examines color relationalism and color relativism, two theories of color that are allegedly supported by variation in normal human color vision. We mostly discuss color relationalism, defended at length in Jonathan Cohen's The Red and the Real, and argue that the theory has insuperable problems.

Albert Einstein ist für seine Arbeiten in der Physik weltberühmt. Nur wenige wissen jedoch, dass Einstein selbst auch philosophische Arbeiten publiziert hat und seine Erkenntnisse weitreichende Folgen für die Philosophie haben. Oder haben „Raum“ und „Zeit“ nichts mit Wissen zu tun?

'Hands Tied' brings together two very different films about hands: Maria Lassnig's Palmistry (1973) and Ayesha Hameed's A Rough History (of the Destruction of Fingerprints) (2016). These works are contextualised and their scope extended further by a roundtable discussion featuring participants Rachel Aumiller, Sam Dolbear, Nadine El-Enany, Amelia Groom, Clio Nicastro, Anja Sunhyun Michaelsen, and M. Ty., who discuss their relation to fate, work, pleasure, touch, and surveillance.

The precarious rights of senior citizens, especially those who are highly educated and who are expected to counsel and guide the younger generations, has stimulated the creation internationally of advocacy associations and opinion leader groups. The strength of these groups, however, varies from country to country. In some countries, they are supported and are the focus of intense interest; in others, they are practically ignored. For this is reason we believe that the creation of a network of all these associations (...) is essential. The proposed network would act as a support for the already-existing policies of the United Nations’ High Commission for Human Rights, of independent experts, and of the Global Alliance for the Rights of Older People. All three have long ago recommended the creation of a recognized instrument for uniting presently scattered efforts. The proposed network, therefore, will seek to promote the international exchange of relevant expertise, and it will reinforce the commitments and actions that single countries are currently taking to meet these objectives. For example, informative public events can be organised to promote particular support initiatives and to provide an opportunity for new members of the network to be presented. The network will promote health for senior citizens, disease prevention, senior mobility, safe free time for seniors, alimentary education, protection against new risks and dangers, as well as equity in the services necessary for seniors to adopt new information and communication technologies. In the case of retired academic members, the network will promote equality with respect to continuing use of digital technologies (particularly email), continuing access to research libraries, and the guaranteed ability for seniors to fund their own research programs and to deliver free seminars. (shrink)

The use of computational tools in the humanities for science 2.0 practices is steadily increasing. This paper examines current research practices of a group of philosophers studying the history of philosophical concepts. We explain the methodology and workflow of these philosophers and provide an overview of tools they currently use in their research. The case study highlights a number of fundamental challenges facing these researchers, including: (i) accessing known relevant research content or resources; (ii) discovering new research content or data; (...) (iii) working collaboratively rather than individually. We propose a mash-up of search, visualization, and awareness tools addressing these challenges and discuss the design of the mash-up, its implementation, and evaluation with the target users. Through our case study, we demonstrate the benefits of a user-centered design approach, as well as the benefits of the concrete mash-up for historians of philosophy, and, importantly, the limitations of these tools for conducting historical and philosophical research. (shrink)

As the title of the article suggests, “The Burqa Ban”: Legal Precursors for Denmark, American Experiences and Experiments, and Philosophical and Critical Examinations, the authors embark on a factually investigative as well as a reflective response. More precisely, they use The 2018 Danish “Burqa Ban”: Joining a European Trend and Sending a National Message (published as a concurrent but separate article in this issue of INTERNATIONAL STUDIES JOURNAL) as a platform for further analysis and discussion of different perspectives. These include (...) case-law at the international level while focusing attention on recent rulings and judicial reasoning by the ECtHR and the ECJ; critical thought-experiments in religion, morality, human rights, and the democratic public space; a contextualized account of burqa-wearing interventions by federal and state governments and, moreover, various courts in the United States; and philosophical commentary and, in some instances, criticism of the Danish and/or European (French, etc.) approach. The different contributions have different aims. The section on case-law at the international level reports on those central judgments that, in effect, helped to pave the path for the Kingdom of Denmark’s burqa ban. Concerning the concurring judges at the ECtHR, the opinions served to uphold a preexisting ban and to grant a wide margin of appreciation to the national authorities, thereby limiting the Court’s own review. -/- As regards to the ECJ, the legality of company rules that contain a policy of neutrality for the workplace was examined, with a similar outcome. The authors who discuss religion, morality, human rights and the democratic public space are endeavoring to, respectively, appeal to ethics as a testing stone for law and to both challenge and address several forms of “expressivist worry” in connection with face veils. In doing so, the authors ask a number of thought-provoking questions that hopefully will inspire public policymakers to careful analysis. While the section that is devoted to American perspectives highlights a comprehensive survey of political and legal responses to, in particular, full-face veils like the burqa, the relevant author also incorporates public perceptions and, in the course of examining these, draws a parallel to “the fate” of the hoodie. The constitutionality of burqa-wearing in America, so it also appears, is partially an open question, but differentiating between religious, political, or personal reasons is a de jure premise. Given that the Danish legislators who drafted law L 219 to ban burqa-wearing in public places rely on a reference to political Islam, they relegate religious and personal reasons to the private domain, thereby also adopting secularism as a premise. This is explored in the last author response of the article, more precisely, in an account of the underlying materialism that, in turn, is applied to Muslim women. If policymakers and legislators engaged in Thinking Things Through exercises, they could, as a minimum, avoid law-making strategies that are not in the spirit of the theory they themselves invoke, albeit tacitly. While the aim of, as it were, arresting culturally self-contradicting legislators is unique for the section in question, all the authors who contribute to the joint research project have one end-goal in common, namely to inform about important perspectives while at the same time opening up for parameters for (more) fruitful, constructive and (if need be) critical debate in the future. With this in mind, four recommendations are presented by the research director for the project. Legally, politically, socially and culturally, conflict-resolution should not translate the relationship between rulers and the ruled into a separation ideology, an instance of controllers versus the controlled. All things being equal, that is the objective limit for a democratic society. (shrink)

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had (...) many partial successes, and generated important advances in logical theory and metatheory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s. (shrink)

In the present paper, these authors argue on actual reasons why Hilbert’s axiomatic program to unify gravitation theory and electromagnetism failed completely. An outline of plausible resolution of this problem is given here, based on: a) Gödel’s incompleteness theorem, b) Newton’s aether stream model. And in another paper we will present our calculation of receding Moon from Earth based on such a matter creation hypothesis. More experiments and observations are called to verify this new hypothesis, albeit it is inspired (...) from Newton’s theory himself. (shrink)

This is a pdf of a Mathematica calculation that supplements the paper "Presentist Fragmentalism and Quantum Mechanics" forthcoming in Foundations of Physics. In that paper the Born rule (or at least a progenitor) is derived from experimental conditions on the mutual observations of two fragments. In this pdf the experimental conditions are applied to Hilbert space dimensions 3, 4, and 5. It turns out each of these have a 1-dimensional solution space which, it is hoped, can be interpretated as (...) the phase. (shrink)

Detlefsen (1986) reads Hilbert's program as a sophisticated defense of instrumentalism, but Feferman (1998) has it that Hilbert's program leaves significant ontological questions unanswered. One such question is of the reference of individual number terms. Hilbert's use of admittedly "meaningless" signs for numbers and formulae appears to impair his ability to establish the reference of mathematical terms and the content of mathematical propositions (Weyl (1949); Kitcher (1976)). The paper traces the history and context of Hilbert's reasoning (...) about signs, which illuminates Hilbert's account of mathematical objectivity, axiomatics, idealization, and consistency. (shrink)

After sketching the main lines of Hilbert's program, certain well-known andinfluential interpretations of the program are critically evaluated, and analternative interpretation is presented. Finally, some recent developments inlogic related to Hilbert's program are reviewed.

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished (...) material presented in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the (...) violations of energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (shrink)

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s (...) axiomatic approach was guided primarily by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved. (shrink)

The paper is the final, fifth part of a series of studies introducing the new conceptions of “Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is the proper philosophical sense of both, including philosophy of mathematics and philosophy of physics not less than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization of ontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy (deepening Heidegger’s destruction of it (...) from the pre-Socratics to the Pythagoreans). Husserl’s phenomenology and Heidegger’s derivative “fundamental ontology” as well as his later doctrine after the “turn” are the starting point of the research as established and well known approaches relative to the newly introduced conception of ontomathematics, even more so that Husserl himself started criticizing his “Philosophy of arithmetic” as too naturalistic and psychological turning to “Logical investigations” and the foundations of phenomenology. Heidegger’s “Aletheia” is also interpreted ontomathematically: as a relation of locality and nonlocality, respectively as a motion from nonlocality to locality if both are physically considered. Aristotle’s ontological revision of Plato’s doctrine is “destructed” further from the pre-Socratics' “Logos” or Heideger’s “Language” (after the “turn”) to the Pythagoreans “Numbers” or “Arithmetics” as an inherent and fundamental philosophical doctrine. Then, a leap to contemporary physics elucidates the essence of ontomathematics overcoming the Cartesian abyss inherited from Plato’s opposition of “ideas” versus “things”, and now unifying physics and mathematics, particularly allowing for the “creation from nothing” instead of the quasi-scientific myth of the “Big Bang”. Furthermore, ontomathematics needs another interpretation of arithmetic, propositional logic and set theory in the foundations of mathematics, where the latter two ones are both identified with Boolean algebra, and the former is considered to be a “half of Boolean algebra” in the exact meaning to be equated to it after doubling by a dual anti-isometric counterpart of Peano arithmetic. That unified algebraic realization of the foundations of mathematics is related to Hilbert mathematics in both “narrow and wide senses” where the latter is isomorphic to the qubit Hilbert space, thus underlying all the physical world by the newly introduced substance of quantum information being physically dimensionless and generalizing classical information measured in bits. The substance of information, whether classical or quantum, visualizes the way of the unification of physics and mathematics by merging their foundations in Hilbert arithmetic and Hilbert mathematics: thus how ontomathetics is a “first philosophy”. The relation of ontomathematics to the Socratic “human problematics”, furthermore being fundamental for Western philosophy in Modernity, is discussed. Ontomathematics implies its “substitution” by abstract information (or by “subjectless choice” relevant to it), thus “obliterating the human outline on the ocean beach sand” (by Michel Foucault’s metaphor). A reflection back from the viewpoint of mathematic to Western philosophy as the philosophy of locality ends the study. (shrink)