_Conjectures and Refutations_ is one of Karl Popper's most wide-ranging and popular works, notable not only for its acute insight into the way scientific knowledge grows, but also for applying those insights to politics and to history. It provides one of the clearest and most accessible statements of the fundamental idea that guided his work: not only our knowledge, but our aims and our standards, grow through an unending process of trial and error.

_Conjectures and Refutations_ is one of Karl Popper's most wide-ranging and popular works, notable not only for its acute insight into the way scientific knowledge grows, but also for applying those insights to politics and to history. It provides one of the clearest and most accessible statements of the fundamental idea that guided his work: not only our knowledge, but our aims and our standards, grow through an unending process of trial and error.

This paper publishes for the first time 132 conjectures by Nicolaus Heinsius on Ovid’s Metamorphoses 1‒4. The value and possible motivations of each proposal are briefly assessed.

In this article, I put forward, as a suspicion only, that Kant never thought Black lives had dignity but only price. I follow Michel-Rolph Trouillot's argument that the Haitian Revolution is unthinkable for Enlightenment philosophers to examine what Kant could have, would have, or should have said about this world-historical event. By making conjectures about Kant's silence on the Haitian Revolution, I also draw from Kant's writings on the American and French Revolutions. If my suspicion is right, then Kantianism cannot (...) accommodate an antiracist program. This ought to change how we talk about racism, slavery, and colonialism in Kant scholarship and in the history of philosophy. (shrink)

The way in which knowledge progresses, and especially our scientific knowledge, is by unjustified anticipations, by guesses, by tentative solutions to our problems, by conjectures. These conjectures are controlled by criticism: that is, by attempted refutations, which include severely critical tests. They may survive these tests; but they can never be positively justified: they can neither be established as certainly true nor even as 'probable'. Criticism of our conjectures is of decisive importance: by bringing out our mistakes it makes us (...) understand the difficulties of the problems which we try to solve. This is how we become better acquainted with our problem, and able to propose more mature solutions: the very refutation of a theory - that is, of a tentative solution to our problem - is always a step forward that takes us nearer the truth. And this is how we can learn from our mistakes. As we learn from our mistakes our knowledge grows, even though we may never know - that is, know for certain. Since our knowledge can grow, there can be no reason here for despair of reason. And since we can never know for certain, the can be no authority here for any claim to authority, for conceit over our knowledge, or for smugness. The essays and lectures of which this book is composed apply this thesis to many topics, ranging from problems of the philosophy and history of the physical and social sciences to historical and political problems. (shrink)

Clayton and Stevens argue that political liberals should engage with the religiously unreasonable by offering religious responses and showing that their religious views are mistaken, instead of refusing to engage with them. Yet they recognize that political liberals will face a dilemma due to such religious responses: either their responses will alienate certain reasonable citizens, or their engagements will appear disingenuous. Thus, there should be a division of justificatory labour. The duty of engagement should be delegated to religious citizens. In (...) this comment, I will argue that the division of justificatory labour is indefensible. This dilemma can be avoided if politicians and political philosophers correctly use conjecture, a form of discourse that involves non-public reason. As a conditional response, conjecture avoids alienating any reasonable citizens. Also, if conjecture is given in a sincere and open-minded manner, then the problem of disingenuousness can be overcome. My comment concludes that while the engagement of politicians and political philosophers does not necessarily jeopardize overlapping consensus, they should be permitted, or perhaps even required, to engage with the religiously unreasonable due to the natural duty of justice. (shrink)

This classic remains one of Karl Popper's most wide-ranging and popular works, notable not only for its acute insight into the way scientific knowledge grows, but also for applying those insights to politics and to history.

We show that both Rado's Conjecture and strong Chang's Conjecture imply that there are no special ℵ2-Aronszajn trees if the Continuum Hypothesis fails. We give similar result for trees of higher heights and we also investigate the influence of Rado's Conjecture on square sequences.

We show that both Rado's Conjecture and strong Chang's Conjecture imply that there are no special ℵ2-Aronszajn trees if the Continuum Hypothesis fails. We give similar result for trees of higher heights and we also investigate the influence of Rado's Conjecture on square sequences.

In this short note many conjectures on partitions of integers as summations of prime numbers are presented, which are extension of Goldbach conjecture.

Ce commentaire intégral des Conjectures de Nicolas de Cues manifeste toute la fécondité de l'oeuvre dans sa complexité. Il élucide l'art général des conjectures, en explore et approfondit les sources doctrinales (en particulier lulliennes) et scientifiques : optiques, mathématiques, biologiques et médicales... Il montre ainsi que la théorie cusaine de la connaissance constitue une hénologie des points de vue et met en évidence l'articulation du rapport entre la vérité, objet de toute recherche connaissante, et l'altérité, condition de tout être fini, (...) et notamment de l'homme. La vérité dans l'altérité se révèle alors comme condition de possibilité de tout ce qui est à partir de l'Un - ce qui constitue l'ordre de l'être - et comme condition de possibilité de toute pensée. Il s'agit, dès lors, de saisir la singularité de l'approche cusaine de la réflexion réciproque entre l'ordre de la pensée et l'ordre de l'être. Après avoir analysé le concept central de conjecture, entendu non comme une simple supposition, mais bien comme une connaissance inadéquate du vrai, la première partie déploie les principes de la nouvelle méthode, fondée sur l'analogie fondamentale entre mens humaine et mens divine. La découverte de ces principes nécessite l'exposition d'une théorie du nombre articulée à celle, d'inspiration proclusienne, des quatre unités (divine, intellectuelle, rationnelle et sensible). Cette première partie conduit à la mise en place de figures manuductrices, paradigmatiques pour notre connaissance. La seconde partie du commentaire explicite les exemples d'application de l'ars coniecturalis donnés par le Cusain en mathématiques, en philosophie naturelle et en anthropologie. Le lecteur découvrira ici comment les Conjectures engagent une philosophie de l'esprit et une philosophie critique, fondamentalement liées à une pensée de l'hétéronomie. (shrink)

In this paper I argue that, in at least two cases - his discussions of the temporal precedence o f polytheism over monotheism and of the origins of civil society - we see Hume consigning to historical development certain aspects of reason which, as a comparison with Locke will show, have sometimes been held to be uniform. In the first of these cases Hume has recourse to claims about the general historical development of human thought. In the second case, the (...) origin of the civil institution of justice and government is not linked directly to external circumstances and the principles of human nature, as it is in contractarian theories, but makes a detour through the historical acquisition of certain concepts. Because Hume's position does not conform in any simple sense to Dugald Stewart's 'incontrovertible logical maxim' that the capacities of the human mind have been the same in all ages, Stewart's account of the method of conjectural history is, in any simple sense, inadequate as a description of Hume's practice. (shrink)

We survey the history of Shelah’s conjecture on strongly dependent fields, give an equivalent formulation in terms of a classification of strongly dependent fields and prove that the conjecture implies that every strongly dependent field has finite dp-rank.

This is a conjecture about the conditions and operating structures that are required for the phenomenality of certain mental states. Specifically, full-blown phenomenality is assumed, as contrasted with constrained examples of phenomenal experience such as sensations of color and pain. Propositional attitudes and content, while not phenomenal per se, are standardly concurrent and may condition phenomenal states (e.g., when tied to false beliefs). It is conjectured that full phenomenality natively arises in coherent processes of situated sensory synthesis and representation (...) (with conceptual content) that are looped, mereologically whole and multi-dimensional. And that phenomenal states are typically phase-states within a parameterized conjoint structure of world and experiencer processes that are causally modulated across Markov blankets (which are conditionally independent and may be nested: cf. M. Kirchoff, et. al., 2017, 2018; and T. Burge, 2010, re: anti-individualism). Though they may, it is not accepted that phenomenally conscious states must be targets of higher-order representations (cf. A. Byrne, 2004). (shrink)

The Conjectural Body combines continental philosophy with musicology, popular music studies, and feminist, critical race, and postcolonial theories to offer a unique perspective on issues of gender, race, and the philosophy of music. It is one of the few books in philosophy to take popular music seriously, and is one of the few books in continental feminism to privilege music over the visual.

The two purposes of this essay. The general philosophical problem with most versions of social libertarianism and how this essay will proceed. The specific problem with liberty explained by a thought-experiment. The positive and abstract theory of interpersonal liberty-in-itself as ‘the absence of interpersonal initiated constraints on want-satisfaction’, for short ‘no initiated impositions’. The individualistic liberty-maximisation theory solves the problems of clashes, defences, and rectifications without entailing interpersonal utility comparisons or libertarian consequentialism. The practical implications of instantiating liberty: three rules (...) of liberty-in-practice, 1) initial ultimate control of one’s body, 2) initial ultimate control of one’s used resources, 3) consensual interpersonal interactions and resource transfers. These rules are economically efficient and maximise general want-satisfaction. Private property and legal remedies are additional practical institutional aspects, but to which ‘initiated impositions’ then apply prima facie. Libertarian law is often mistaken for complete libertarianism. Moral explanations are a separate issue. The three main moral theories imply libertarianism, but it can be morally posited independently of them. Critical rationalism and its application. No empirical or argumentative support for theories. An important ambiguity with ‘justification’. How the epistemology applies to the theory of liberty and its application but remains separable in principle. Conclusion: there are further published explanations but this should be enough to generate useful criticism. Appendix replying to some typical comments. (shrink)

“There could be no fairer destiny for any. . . theory than that it should point the way to a more comprehensive theory in which it lives on, as a limiting case.” ALBERT EINSTEIN..

Sir Karl Popper is well known for explicating science in falsificationist terms, for which his degree of corroboration formalism, C(h,e,b), has become little more than a symbol. For example, de Queiroz and Poe in this issue argue that C(h,e,b) reduces to a single relative (conditional) probability, p(e,hb), the likelihood of evidence e, given both hypothesis h and background knowledge b, and in reaching that conclusion, without stating or expressing it, they render Popper a verificationist. The contradiction they impose is easily (...) explained--de Queiroz and Poe fail to take account of the fact that Popper derived C(h,e,b) from absolute (logical) probability and severity of test, S(e,h,b), where critical evidence, p(e,b), is fundamental. Thus, de Queiroz and Poe's conjecture that p(e,hb) = C(h,e,b) is refuted. Falsificationism, not verificationism, remains a fair description of the parsimony method of inference used in phylogenetic systematics, not withstanding de Queiroz and Poe's mistaken understanding that "statistical" probability justifies that method. Although de Queiroz and Poe assert that maximum likelihood has the power "to explain data", they do not successfully demonstrate how causal explanation is achieved or what it is that is being explained. This is not surprising, bearing in mind that what is assumed about character evolution in the accompanying likelihood model M cannot then be explained by the results of a maximum likelihood analysis. (shrink)

Through close examination of the texts, the author shows how 15th-century philosopher Nicholas of Cusa developed an understanding of uncertainty that opened the way for human intelligence, despite its inherent weaknesses, to find out more about ourselves, the world, and what lies beyond.

This article discusses textually problematic passages in Catullus 63, a particularly corrupt poem from a particularly corrupt manuscript tradition. It proposes new conjectures and revives several old ones. Throughout there are notes on punctuation, conjecture attribution and an analysis of the structure of Attis’ lament.

Two fundamental errors led Einstein to reject generally covariant gravitational field equations for over two years as he was developing his general theory of relativity. The first is well known in the literature. It was the presumption that weak, static gravitational fields must be spatially flat and a corresponding assumption about his weak field equations. I conjecture that a second hitherto unrecognized error also defeated Einstein's efforts. The same error, months later, allowed the hole argument to convince Einstein that (...) all generally covariant gravitational field equations would be physically uninteresting. (shrink)

Determinism is established in quantum mechanics by tracing the probabilities in the Born rules back to the absolute (overall) phase constants of the wave functions and recognizing these phase constants as pseudorandom numbers. The reduction process (collapse) is independent of measurement. It occurs when two wavepackets overlap in ordinary space and satisfy a certain criterion, which depends on the phase constants of both wavepackets. Reduction means contraction of the wavepackets to the place of overlap. The measurement apparatus fans out the (...) incoming wavepacket into spatially separated eigenpackets of the chosen observable. When one of these eigenpackets together with a wavepacket located in the apparatus satisfy the criterion, the reduction associates the place of contraction with an eigenvalue of the observable. The theory is nonlocal and contextual. Keywords:. (shrink)

A conjectural inference is proposed, aimed at producing conjectural theorems from formal conjectures assumed as axioms, as well as admitting contradictory statements as conjectural theorems. To this end, we employ Paraconsistent Informational Logic, which provides a formal setting where the notion of conjecture formulated by an epistemic agent can be defined. The paraconsistent systems on which conjectural deduction is based are sequent formulations of the C-systems presented in Carnielli-Marcos [CAR 02b]. Thus, conjectural deduction may also be considered to be (...) a tool for investigating the properties of paraconsistency in general. (shrink)

We ordinarily assume that we have reliable knowledge of our immediate surroundings, so much so that almost all the time we entrust our lives to the truth of what we take ourselves to know, without a moment’s thought. But if, as Karl Popper and others have maintained, all our knowledge is conjectural, then this habitual assumption that our common sense knowledge of our environment is secure and trustworthy would seem to be an illusion. Popper’s philosophy of science, in particular, fails (...) to do justice to the distinction we ordinarily draw between secure knowledge and mere conjecture. But Popper’s philosophy of science, in particular his attempted solution to the problem of induction, is defective. It fails to take into account that physics, in only accepting unified theories, even though endlessly empirically more successful disunified rivals are always available, makes the persistent metaphysical assumption that all disunified theories are false. Once this point is acknowledged, it becomes clear that a new conception of scientific method is required which sees science as making a hierarchy of metaphysical assumptions concerning the comprehensibility and knowability of the universe. This provides a framework of relatively unproblematic assumptions and associated methods of science within which much more problematic assumptions and associated methods can be critically assessed and improved. This hierarchical view seems at first to intensify the problem of distinguishing certainty from conjecture, in that it emphasizes that scientific knowledge, and even humble common sense knowledge, contain usually unacknowledged cosmological conjectures. But actually it explicates the basis we have for drawing the distinction between trustworthy knowledge and mere conjecture, and even goes some way towards providing a rationale for this distinction, in so far as one exists. (shrink)

A collective understanding that traces a debate between ‘what is science?’ and ‘what is a science about?’ has an extraction to the notion of scientific knowledge. The debate undertakes the pursuit of science that hardly extravagance the dogma of pseudo-science. Scientific conjectures invoke science as an intellectual activity poured by experiences and repetition of the objects that look independent of any idealist views (believes in the consensus of mind-dependence reality). The realistic machinery employs in an empiricist exposition of the objective (...) phenomenon by synchronizing the general method to make observational predictions that cover all the phenomena of the particular entity without any exception. The formation of science encloses several epistemological purviews and a succession of conjectures cum refutation that a newer theorem could reinstate. My attempt is to advocate a holistic plea of scientific conjectures that outruns the restricted regulation of experience or testable hypothesis to render the validity of a chain of logical reasoning (deductive or inductive) of basic scientific statements. The milieu of scientific intensification integrates speculation that loads efficiency towards a new experimental dimension where the reality is not itself objective or observers relative; in fact the observed phenomenon divulges in the constructive progression of preferred methods of falsifiability and uncertainty. (shrink)

The text of Fronto is in a very corrupt state, and the startling discrepancies which exist between different collations, as well as the unintelligibility of many of the readings deciphered, seem to justify a good deal of conjectural emendation. I append some attempts to complete or restore the sense in some passages of the Greek letters which seem hitherto to have been left in an unsatisfactory state. Not having had access to the MS, I have relied on the collations of (...) Naber and others for a statement of what is to be found in the palimpsest. (shrink)

A collective understanding that traces a debate between 'what is science?’ and ‘what is a science about?’ has an extraction to the notion of scientific knowledge. The debate undertakes the pursuit of science that hardly extravagance the dogma of pseudo-science. Scientific conjectures invoke science as an intellectual activity poured by experiences and repetition of the objects that look independent of any idealist views (believes in the consensus of mind-dependence reality). The realistic machinery employs in an empiricist exposition of the objective (...) phenomenon by synchronizing the general method to make observational predictions that cover all the phenomena of the particular entity without any exception. The formation of science encloses several epistemological purviews and a succession of conjectures cum refutation that a newer theorem could reinstate. My attempt is to advocate a holistic plea of scientific conjectures that outruns the restricted regulation of experience or testable hypothesis to render the validity of a chain of logical reasoning (deductive or inductive) of basic scientific statements. The milieu of scientific intensification integrates speculation that loads efficiency towards a new experimental dimension where the reality is not itself objective or observers relative; in fact the observed phenomenon divulges in the constructive progression of preferred methods of falsifiability and uncertainty. (shrink)

This article delineates the notion of conjectural artworks—that is, ways of thinking and explaining formal and relational phenomena by visual means—and presents an appraisal and review of the use of such visual ways in the work of Chilean biologists and philosophers Humberto Maturana and Francisco Varela. Particularly, the article focuses on their recurrent uses of Cellular Automaton, that is, discrete, locally interacting, rule-based mathematical models, as conjectural artworks for understanding the concepts of autopoiesis, structural coupling, cognition and enaction: (i.e. Protobio (...) and Bittorio). Additionally, the article proposes a new model of conjectural artwork based on an extension of cellular automaton: random Boolean networks namely, binary systems with variable local connections. Such model, as it is argued, is useful to connect the theoretical frameworks by Maturana and Varela, especially structural coupling and enaction, with other relevant fields such as biosemiotics’ Umwelt-research and cognitive landscapes in neurodynamics, and to advance and explore the concepts of structurally coupled categorization and generalization. (shrink)

In this article we present a conjecture regarding the biology of subjectivity and feeling, based on biophysical and phenomenological considerations. We propose that feeling, as a subjective phenomenon, would come to life as a process of resistance to variance hypothesized to occur during the unfolding of cognition and behaviours in the wakeful and emoting individual. After showing how the notion of affect, when considered from a biological standpoint, suggests an underlying process of resistance to variance, we discuss how vigilance, (...) emotional arousal and attentional behaviours reflect a dynamics of controlled over-excitation related to cognitive integration and control. This can be described as a form of resistance to variance. We discuss how such a dynamics objectively creates an internal state of tension and affectedness in the system that could be associated with subjective states. Such a dynamics is shaped by the system's need to cope with its own inertia, to engage in intentional behaviours, attend, preserve coherence, grapple with divergent cognitive, emotional and motivational tendencies, and delayed auto-perturbations of the brain-body system. More generally, it is related to the need to respect the hierarchy of the various influences which affect its internal dynamics and organization. (shrink)

Elaborating on an old conjecture by Blackett, we formulate a new conjecture about the neutrality of matter according to which any physical system possesses an active electric charge proportional to its mass. We discuss limits on the conjecture coming from existing laboratory experiments on the neutrality of matter and from the observation of the global surface electric field of the Earth. In a cosmological setting, we show that a cosmic rotation of the Universe is inevitable if our (...) conjecture is true and if the magnetic fields observed today in large-scale gravitationally bounded objects and cosmic voids have a primordial origin. Finally, we discuss the possibility that the angular momentum of spiral galaxies could be a signature of a rotating Universe. (shrink)

We investigate how weak square principles are denied by Chang’s Conjecture and its generalizations. Among other things we prove that Chang’s Conjecture does not imply the failure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\omega_1, 2}}$$\end{document}, i.e. Chang’s Conjecture is consistent with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_{\omega_1, 2}}$$\end{document}.

At Aeschylus Agamemnon 985 the manuscript reading ψαμμίας ἀκάτα is corrupt, giving neither meter nor sense. Wilamowitz’ conjecture ψάμμος ἄμπτα has met with some editorial approval, but its sense is dubious and should be rejected. I propose instead ψάλλον ἀκταῖς, “they were plucking on the shore”, referring to the performance of a paean on the lyre by the Greek fleet departing for, or, less likely, arriving at, Troy. The fleet’s departure would be an appropriate time for the soldiers to (...) perform a paean, for which the lyre was a common accompaniment. Plucking a lyre without strumming, however, as ψάλλον implies, seems to have been a less common performance technique, sometimes culturally marked as ‘Asian’. While there is evidence that a paean with plucked accompaniment might have seemed unexceptional to a Greek audience, if Aeschylus did intend an Asian connotation, I suggest that the Chorus, having just witnessed Agamemnon tread on the purple cloths, would thereby portray the Greek army as already tainted by tryphe on their departure for Troy. Whether ‘Greek’ or ‘Trojan’, the conjectured army’s paean would form an effective counterpoint to the lyre-less threnos in the Chorus’ heart that dominates the ode. (shrink)

This article proposes two emendations to the text of Apollonius Citiensis and Galen of Pergamum’s De Praecognitione. The problematic transmissions are isolated and a conjectural emendation suggested.

In this article, I propose a conjecture concerning the transmission of Spinoza’s Korte Verhandeling in the 1670s involving Leibniz. On the basis of a report about Spinoza’s philosophy written down by Leibniz after some conversations with Tschirnhaus in early 1676, I suggest that Tschirnhaus may have had in his possession a manuscript copy of KV and that his account of Spinoza’s doctrine to Leibniz was colored by this text. I support the hypothesis partly by means of external evidence, but (...) mainly through a comparative analysis of Leibniz’s report and the doctrine contained in KV, showing that the report in important respects corresponds better to this text than to Ethics. I finally point to the importance that this hypothesis, if true, would have for our knowledge of Tschirnhaus’ role in the first diffusion of Spinoza’s philosophy outside Holland and for our understanding Leibniz’s reception of Spinoza in the mid-1670s. (shrink)

In this article, I propose a conjecture concerning the transmission of Spinoza’s Korte Verhandeling (KV) in the 1670s involving Leibniz. On the basis of a report about Spinoza’s philosophy written down by Leibniz after some conversations with Tschirnhaus in early 1676, I suggest that Tschirnhaus may have had in his possession a manuscript copy of KV and that his account of Spinoza’s doctrine to Leibniz was colored by this text. I support the hypothesis partly by means of external evidence, (...) but mainly through a comparative analysis of Leibniz’s report and the doctrine contained in KV, showing that the report in important respects corresponds better to this text than to Ethics. I finally point to the importance that this hypothesis, if true, would have for our knowledge of Tschirnhaus’ role in the first diffusion of Spinoza’s philosophy outside Holland and for our understanding Leibniz’s reception of Spinoza in the mid-1670s. (shrink)

In this article, I propose a conjecture concerning the transmission of Spinoza’s Korte Verhandeling in the 1670s involving Leibniz. On the basis of a report about Spinoza’s philosophy written down by Leibniz after some conversations with Tschirnhaus in early 1676, I suggest that Tschirnhaus may have had in his possession a manuscript copy of KV and that his account of Spinoza’s doctrine to Leibniz was colored by this text. I support the hypothesis partly by means of external evidence, but (...) mainly through a comparative analysis of Leibniz’s report and the doctrine contained in KV, showing that the report in important respects corresponds better to this text than to Ethics. I finally point to the importance that this hypothesis, if true, would have for our knowledge of Tschirnhaus’ role in the first diffusion of Spinoza’s philosophy outside Holland and for our understanding Leibniz’s reception of Spinoza in the mid-1670s. (shrink)