A transcription and translation of Hermann Hankel's 1867 Vorlesungen über die complexen Zahlen und ihre Functionen, I. Theil: Theorie der Complexen Zahlensysteme, a textbook on complex analysis that played an important role in the transition to modern mathematics in nineteenth century Germany.

This is a bugfix release to correct some errors in.zenodo.json which prevented archiving of 1.0. -/- Version 1.0: This first release of the translation contains: -/- * a full transcription of the Preface and Chapter 1 * translations of select passages in these sections * a full transcription and translation of the title pages and the table of contents through chapter 3 -/- This release was focused on working out (most of) the issues with the markup of a parallel translation (...) and with the build. Subsequent versions will contain more of the text and be more focused on translation! (shrink)

Frege says, at the end of a discussion of formalism in the Foundations of Arithmetic, that his own foundational program “could be called formal” but is “completely different” from the view he has just criticized. This essay examines Frege’s relationship to Hermann Hankel, his main formalist interlocutor in the Foundations, in order to make sense of these claims. The investigation reveals a surprising result: Frege’s foundational program actually has quite a lot in common with Hankel’s. This undercuts (...) Frege’s claim that his own view is completely different from Hankel’s formalism, and motivates a closer examination of where the differences lie. On the interpretation offered here, Frege shares important parts of the formalist perspective, but differs in recognizing a kind of content for arithmetical terms which can only be made available via proof from prior postulates. (shrink)

The paper traces the development and the role of syntactic reduction in Edmund Husserl’s early writings on mathematics and logic, especially on arithmetic. The notion has its origin in Hermann Hankel’s principle of permanence that Husserl set out to clarify. In Husserl’s early texts the emphasis of the reductions was meant to guarantee the consistency of the extended algorithm. Around the turn of the century Husserl uses the same idea in his conception of definiteness of what he calls (...) “mathematical manifolds.” The paper argues that the notion anticipates the notion of reduction in term rewrite theory in computer science. The role of the reduction for Husserl is, however, primarily epistemological: its purpose is to impart clarity to formal mathematics. (shrink)

Husserl’s notion of definiteness, i.e., completeness is crucial to understanding Husserl’s view of logic, and consequently several related philosophical views, such as his argument against psychologism, his notion of ideality, and his view of formal ontology. Initially Husserl developed the notion of definiteness to clarify Hermann Hankel’s ‘principle of permanence’. One of the first attempts at formulating definiteness can be found in the Philosophy of Arithmetic, where definiteness serves the purpose of the modern notion of ‘soundness’ and leads (...) Husserl to a ‘computational’ view of logic. Inspired by Gauss and Grassmann Husserl then undertakes a further investigation of theories of manifolds. When Husserl subsequently renounces psychologism and changes his view of logic, his idea of definiteness also develops. The notion of definiteness is discussed most extensively in the pair of lectures Husserl gave in front of the mathematical society in Göttingen (1901). A detailed analysis of the lectures, together with an elaboration of Husserl’s lectures on logic beginning in 1895, shows that Husserl meant by definiteness what is today called ‘categoricity’. In so doing Husserl was not doing anything particularly original; since Dedekind’s ‘Was sind und sollen die Zahlen’ (1888) the notion was ‘in the air’. It also characterizes Hilbert’s (1900) notion of completeness. In the end, Husserl’s view of definiteness is discussed in light of Gödel’s (1931) incompleteness results. (shrink)

Assuming that there is moral progress, and assuming that the abolition of slavery is an example of it, how does moral progress occur? Is it mainly driven by specific individuals who have gained new moral insights, or by changes in the socio‐economic and epistemic conditions in which agents morally judge the norms and practices of their society, and act upon these judgements? In this paper, I argue that moral progress is a complex process in which changes at the level of (...) belief and changes at the level of institutions and social practices are deeply intertwined, and that changes in the socio‐economic and epistemic conditions of moral agency constitute the main motor of moral progress. I develop my view of moral progress by way of grappling with Michelle Moody‐Adams’ prominent philosophical account of it. My view is less intellectualistic and individualistic than hers, does not presuppose meta‐ethical moral realism, and blurs her distinction between moral progress in beliefs and moral progress in social practices. I point out the limits of humans to progress morally, which are partly grounded in our evolutionary history, and argue that moral progress is always of a ‘local’ nature. (shrink)

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ver since the traditional criteria for medical decision-making capacity (understanding, appreciation, reasoning, evidencing a choice) were formulated, they have been criticized for not taking sufficient account of emotions or values that seem, according to the critics and in line with clinical experiences, essential to decision-making capacity. The aim of this paper is to provide a nuanced and structured overview of the arguments provided in the literature emphasizing the importance of these factors and arguing for their inclusion in competence evaluations. Moreover, (...) a broader reflection on the findings of the literature is provided. Specific difficulties of formulating and measuring emotional and valuational factors are discussed inviting reflection on the possibility of handling relevant factors in a more flexible, case-specific, and context-specific way rather than adhering to a rigid set of operationalized criteria. (shrink)

Recent advances in neuroscience lead to a wider realm for philosophy to include the science of the Darwinian-evolved computational brain, our inner world producing organ, a non-recursive super- Turing machine combining 100B synapsing-neuron DNA-computers based on the genetic code. The whole system is a logos machine offering a world map for global context, essential for our intentional grasp of opportunities. We start from the observable contrast between the chaotic universe vs. our orderly inner world, the noumenal cosmos. So far, philosophy (...) has been rehearsing our thoughts, our human-internal world, a grand painting of the outer world, how we comprehend subjectively our experience, worked up by the logos machine, but now we seek a wider horizon, how humans understand the world thanks to Darwinian evolution to adapt in response to the metaphysical gap, the chasm between the human animal and its environment, shaping the organism so it can deal with its variable world. This new horizon embraces global context coded in neural structures that support the noumenal cosmos, our inner mental world, for us as denizens of the outer environment. Kant’s inner and outer senses are fundamental ingredients of scientific philosophy. Several sections devoted to Heidegger, his lizard debunked, but his version of the metaphysical gap & his doctrine of the logos praised. Rorty and others of the behaviorist school discussed also. (shrink)

Evolutionary accounts of the origin of human morality may be speculative to some extent, but they contain some very plausible claims, such as the claim that ethics evolved as a response to the demands of group living. Regarding the phenomenon of moral progress, it has been argued both that it is ruled out by an evolutionary approach, and that it can be explained by it. It has even been claimed that an evolutionary account has the potential to advance progress in (...) the moral domain. This paper explores the complex relationship between evolutionary explanations of morality and the possibility of moral progress. It seeks to answer the question as to what these explanations are able to tell us about the possibility of moral progress and the ways in which such progress can be achieved. It is argued that evolutionary explanations can inform moral education and other forms of moral enhancement, and that increased evolutionary knowledge figures among the changes in the circumstances of morality that can lead to moral progress. Evolutionary explanations can show us certain limits to the possibility for humans of progressing morally as well as certain enabling conditions. It is argued that both aspects – enhancement and changes in the circumstances – are equally important for the achievement of moral progress. This is illustrated by means of two examples of areas in which moral progress seems possible: our relationship towards the distant poor and our relationship towards non-human animals. (shrink)

t. 1. Die Grundlegung der systematischen Ästhetik.--t. 2. Die Bestätigung der systematischen Ästhetik in den Arten der Künste und an ihrer Geschichte; 26.

Excerpt from Die Systematischen Begriffe in Kants Vorkritischen Schriften: Nach Ihrem Verhaltniss zum Kritischen Idealismus Schriften aufgenommen hatte, heranzieht und in Rucksicht auf die Schrift vom Jahre 1763 der einzig mogliche Beweisgrund zu einer Demonstration des Daseins Gottes beleuchtet. Ueber haupt hat er die Frage von der Beweisbarkeit Gottes durch Anfuhrung der vielen Beweis und Widerlegungsversuche ein gehend erortert. Zu einer literarischen Wichtigkeit hat aber erst Kuno Fischer die Frage uber den Werth der vorkritischen Schriften erhoben. Seine Untersuchungen werden uns (...) in der Folge viel fach beschaftigen. Doch erscheint es zur Orientirung uber die Methode, nach welcher wir selbst die Frage behandeln, zweck massig, uber das von K. Fischer geubte Verfahren, das spaterhin sehr im Einzelnen gepruft werden soll, vorlaufig nur, soweit es die Methode angeht, ein Urtheil zu fassen. Kuno Fischer sucht und macht Unterschiede. Dies muss vor Allem anerkannt werden. Auch gluckt es ihm, die Gedanken, welche er aus den Schriften vor dem Jahre 1770 heraushebt, in eine die spatere Entdeckung praedestinirende Beziehung zu einander zu setzen. Diese uberraschenden Zusammenhange werden oftmals und immer in neu erscheinenden Wendungen vorgefuhrt, so dass ein Bild von dem Entwickelurigsgange des Denkers entworfen wird. Ob dieses Bild treu ist, ob die Gedanken, aus denen es zusammengesetzt wird, genau aufgenommen und richtig geordnet sind, das wird an seinem Orte gepruft werden mussen. Hier fragen wir nur: Kann es den Lernenden, ge schweige den Nachforschenden fordern, wenn Einzelnes aus dem Systeme, wie der Ausdruck wiederholentlich gebraucht wird. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

Science-fiction (SF) has become a reference point in the discourse on the ethics and risks surrounding artificial intelligence (AI). Thus, AI in SF—science-fictional AI—is considered part of a larger corpus of ‘AI narratives’ that are analysed as shaping the fears and hopes of the technology. SF, however, is not a foresight or technology assessment, but tells dramas for a human audience. To make the drama work, AI is often portrayed as human-like or autonomous, regardless of the actual technological limitations. Taking (...) science-fictional AI too literally, and even applying it to science communication, paints a distorted image of the technology's current potential and distracts from the real-world implications and risks of AI. These risks are not about humanoid robots or conscious machines, but about the scoring, nudging, discrimination, exploitation, and surveillance of humans by AI technologies through governments and corporations. AI in SF, on the other hand, is a trope as part of a genre-specific mega-text that is better understood as a dramatic means and metaphor to reflect on the human condition and socio-political issues beyond technology. (shrink)

Origin of the Yellowstone hotspot & Columbia River Basalts has remained uncertain until now. Here, we present evidence of meteorite impact origin. The hotspot is shallow, only 200 km deep, invalidating a theory of mantle plume origin. The hotspot track runs from the Yellowstone National Park in NW Wyoming to the volcanic Modoc Plateau in NE California. We present evidence of apparent remnants of an impact crater existing in the Modoc, a large multi-ring structure at least 160 km diameter. Much (...) of the complex crater has become obliterated by later Cascadia and Sierra orogenies. The crater has a tall 4,100 foot central uplift cone, locally known as Chalk Mountain, consisting of diatomaceous earth, presumably the rebound cone of a meteorite impact. This falsifies a theoretical prohibition of cosmic impact volcanism. Based on recent insights into explosive volcanism a plausible mechanism is proposed for how meteorite impacts can lead to resurgent calderas of the Yellowstone type, & of apparent LIP origin from thin crust ET impacts, invalidating theoretical constructs of mantle plumes. We mention chaotic terrains, including plutons, arising antipodal to impacts. (shrink)

Guided by key insights of the four great philosophers mentioned in the title, here, in review of and expanding on our earlier work (Burchard, 2005, 2011), we present an exposition of the role played by language, & in the broader sense, λογοζ, the Logos, in how the CNS, the brain, is running the human being. Evolution by neural Darwinism has been forcing the linguistic nature of mind, enabling it to overcome & exploit the cognitive gap between an animal and its (...) world by recognizing environmental structures. Our work was greatly influenced by Heidegger’s lecture notes on metaphysics (Heidegger, 1935). We found agreement with recent progress in neuroscience, but also mathematical foundations of language theory, equating Logos with the mathematical concept of structure. The mystery of perception across the gap is analyzed as radiation and molecules impinging on sensory neurons that carry linguistic information about gross environmental structures, and only remotely about the physical reality of elementary particles. The most important logical brain function is Ego or Self, guiding the workings of the brain as a logos machine. Ego or Self operates from neurons in frontopolar cortex with global receptive fields. The logos machine can function only by availing itself of global context, its internally stored noumenal cosmos NK, and the categorical-conceptual apparatus CCA, updated continually through the neural default mode network (Raichle, 2005). In the Transcendental Deduction, Immanuel Kant discovered that Ego or Self is responsible for conscious control in perception relying on concepts & categories for a fitting percept to be incorporated intoNK. The entire CNS runs as a “movie-in-the-brain” (Parvizi & Damasio, 2001), at peak speed processing simultaneously in a series of cortical centers a stack of up to twelve frames in gamma rhythm of 25 ms intervals. We equate global context, or NK, with our human world, Heidegger’s Dasein being-in-the-world, and are able to demonstrate that the great philosopher in EM parallels neuro-science concerning the human mind. (shrink)

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add Zermelo's choice function. (...) This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (shrink)

Original, or 'old', institutional economics (OIE) – also known as 'institutionalism' – played a key role in its early stages; it could be said that it was once the 'mainstream economics' of the time. This period ran approximately from the first important contributions of Thorstein Veblen in 1898 to the implementation of the New Deal in the early 1930s, where many institutionalists played a significant role. However, notwithstanding its promising scientific and institutional affirmation, institutional economics underwent a period of marked (...) decline that spanned from the mid-1930s to the late 1980s, when a new season for institutional economics was set in motion. In order to cast some light on this complex issue – without any claim of completeness – we have organised the work as follows: in the first section we consider the main interpretations of this phenomenon. In the subsequent sections we analyse a number of 'endogenous' aspects which might have played a significant role in the period of decline: (i) the relations of institutional economics with Keynes's macroeconomic theory; (ii) the links between theoretical and empirical analysis and the supposed lack of a clear theory; (iii) the interdisciplinary orientation. (shrink)

Elementary textbook introducing several topics from numerical mathematics, with many exercises and solutions. For students and practitioners from engineering and natural sciences.

In our previous work we proved the conjecture NP = PSPACE by advanced proof theoretic methods that combined Hudelmaier’s cut-free sequent calculus for minimal logic with the horizontal compressing in the corresponding minimal Prawitz-style natural deduction. In this Addendum we show how to prove a weaker result NP = coNP without referring to HSC. The underlying idea is to omit full minimal logic and compress only “naive” normal tree-like ND refutations of the existence of Hamiltonian cycles in given non-Hamiltonian graphs, (...) since the Hamiltonian graph problem in NPcomplete. Thus, loosely speaking, the proof of NP = coNP can be obtained by HSC-elimination from our proof of NP = PSPACE. (shrink)