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)

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.

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

The way in which knowledge progresses, and especially our scientific knowledge, is by unjustified (and unjustifiable) 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' (in the sense of the probability calculus). 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)

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)

In this paper art history and visual studies, the disciplines that study visual culture, are presented as a field whose conjectural paradigm can be used to understand the epistemic problems associated with abduction. In order to do so, significant statements, concepts and arguments from the work of several specialists in this field have been highlighted. Their analysis shows the fruitfulness and potential for understanding the study of visual culture as a field that is interwoven with the assumptions of abductive cognition.

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)

A provocative collection of technical and popular essays dealing with a variety of scientific and political topics which Popper has treated in his major works. For the most part Popper develops, sharpens, and extends to new areas, themes which he has already explored. The major theme running through the essays is that knowledge grows by unjustified and unjustifiable anticipations, guesses and conjectures. These are controlled by criticisms and refutations. Theories can never be positively justified; they can only prove to be (...) resistant to rational criticism. The boldness of Popper's conjectures demands attempted refutations on the part of the reader.--R. J. B. (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)

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)

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 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.

Rado’s Conjecture is a compactness/reflection principle that says any nonspecial tree of height ω1 has a nonspecial subtree of size ℵ1. Though incompatible with Martin’s Axiom, Rado’s Conjecture turns out to have many interesting consequences that are also implied by certain forcing axioms. In this paper, we obtain consistency results concerning Rado’s Conjecture and its Baire version. In particular, we show that a fragment of PFA, which is the forcing axiom for Baire Indestructibly Proper forcings, is compatible (...) with the Baire Rado’s Conjecture. As a corollary, the Baire Rado’s Conjecture does not imply Rado’s Conjecture. Then we discuss the strength and limitations of the Baire Rado’s Conjecture regarding its interaction with stationary reflection principles and some families of weak square principles. Finally, we investigate the influence of Rado’s Conjecture on some polarized partition relations. (shrink)

A lyrical variation on vv. 27–30, the present passage is, at the same time, a new version of Hec. 631–7. A comparison is instructive in various respects, but Hel. 27 f. will not be quoted in justification of what L gives in vv. 236 f. This wording raises the following objections: the last word, , is outside the metre and not easily attached to the following colon, the beginning of which naturally coincides with the new sentence and new idea If, (...) then, is obelized—and this seems unavoidable—the words. (shrink)

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)

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.

Le De coniecturis de Nicolas de Cues illustre le danger de solipsisme qui menace toute doctrine faisant de la recherche de la « vérité précise » une recherche sans fin. Ce thème n’est pas sans évoquer celui de la traduction. Si la connaissance est individualisée, souligne le Cusain, la compréhension des autres hommes devient en effet problématique : « Personne ne peut jamais concevoir parfaitement la compréhension d’autrui, même si certains s’en approchent plus que d’autres ». Comment pouvons-nous savoir ce (...) que pense un autre homme? Comment pouvons-nous savoir que les signes par lesquels nous communiquons avec les autres ont la même signification pour eux et pour nous? Si les différentes conjectures des différentes personnes sont sans rapport les unes avec les autres, elles sont intraduisibles. Comment donc nous comprendre? Telles sont les questions soulevées par cet article. (shrink)

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)

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.

Réponse aux objections soulevées par Oswald Ducrot à l'encontre de mon approche "gricéenne" de la performativité.

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)

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 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)

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.

THE ART OF CAUSAL CONJECTURE Glenn Shafer Table of Contents Chapter 1. Introduction........................................................................................ ...........1 1.1. Probability Trees..........................................................................................3 1.2. Many Observers, Many Stances, Many Natures..........................................8 1.3. Causal Relations as Relations in Nature’s Tree...........................................9 1.4. Evidence............................................................................................ ...........13 1.5. Measuring the Average Effect of a Cause....................................................17 1.6. Causal Diagrams..........................................................................................20 1.7. Humean Events............................................................................................23 1.8. Three Levels of Causal Language................................................................27 1.9. An Outline of the Book................................................................................27 Chapter 2. Event Trees............................................................................................... .....31 2.1. Situations and Events...................................................................................32 2.2. The Ordering of Situations and Moivrean Events.......................................35 2.3. Cuts................................................................................................ ..............39 2.4. Humean Events............................................................................................43 (...) 2.5. Moivrean Variables......................................................................................49 2.6. Humean Variables........................................................................................53 2.7. Event Trees for Stochastic Processes...........................................................54 2.8. Timing in Event Trees.................................................................................56 2.9. Intersecting Event Trees...............................................................................60 2.10. Notes on the Literature...............................................................................61 Chapter 3. Probability Trees...........................................................................................63 3.1. Some Types of Probability Trees.................................................................64 ii 3.2. Axioms for the Probabilities of Moivrean Events.......................................68 3.3. Zero Probabilities....................................................................................... ..70 3.4. A Sample-Space Analysis of the Event-Tree Axioms.................................72 3.5. Probabilities and Expected Values for Variables.........................................74 3.6. Martingales......................................................................................... .........79 3.7. The Expectation of a Variable in a Cut........................................................83 3.8. Conditional Expected Value and Conditional Expectation.........................87 Chapter 4. The Meaning of Probability...........................................................................91 4.1. The Interpretation of Expected Value..........................................................92 4.2. The Interpretation of Expectation................................................................95 4.3. The Long Run..............................................................................................98 4.4. Changes in Belief.........................................................................................101 4.5. The Empirical Validation of Probability......................................................106 4.6. The Diversity of Uses of Probability...........................................................108 4.7. Notes on the Literature.................................................................................110 Chapter 5. Independent Events.......................................................................................113 5.1. Independence........................................................................................ .......114 5.2. Weak Independence.....................................................................................118 5.3. The Principle of the Common Cause...........................................................121 5.4. Conditional Independence............................................................................128 5.5. Notes on the Literature.................................................................................133 Chapter 6. Events Tracking Events.................................................................................135 6.1. Tracking............................................................................................ ...........137 6.2. Tracking and Conditional Independence.....................................................142 6.3. Stochastic Subsequence...............................................................................143 6.4. Singular Diagrams for Stochastic Subsequence...........................................147 6.5. Conjunctive and Interactive Forks...............................................................149 Chapter 7. Events as Signs of Events..............................................................................153 iii 7.1. Sign................................................................................................ ..............154 7.2. Weak Sign................................................................................................ ....159 7.3. The Ethics of Causal Talk............................................................................160 7.4. Screening Off...............................................................................................16 2 Chapter 8. Independent Variables...................................................................................167 8.1. Unconditional Independence........................................................................170 8.2. Conditional Independence............................................................................175 8.3. Independence for Partitions.........................................................................177 8.4. Independence for Families of Variables......................................................182 8.5. Individual Properties of the Independence Relations...................................186 Chapter 9. Variables Tracking Variables........................................................................189 9.1. Tracking and Conditional Independence: A Summary...............................190 9.2. Strong Tracking............................................................................................ 192 9.3. Strong Tracking and Conditional Independence..........................................198 9.4. Stochastic Subsequence...............................................................................201 9.5. Functional Dependence................................................................................203 9.6. Tracking in Mean.........................................................................................204 9.7. Linear Tracking............................................................................................ 207 9.8. Tracking by Partitions..................................................................................210 9.9. Tracking by Families of Variables...............................................................212 Chapter 10. Variables as Signs of Variables...................................................................215 10.1. Sign................................................................................................ ............219 10.2. Linear Sign................................................................................................ .222 10.3. Scored Sign................................................................................................ 225 10.4. Families of Variables.................................................................................227 Chapter 11. AnTheory of Event Trees.............................................................229 11.1. Event Trees as Sets of Sets........................................................................230 11.2. Event Trees as Partially Ordered Sets........................................................232 iv 11.3. Regular Event Trees...................................................................................240 11.4. The Resolution of Moivrean Variables......................................................244 11.5. Humean Events and Variables...................................................................246 Chapter 12. Martingale Trees..........................................................................................247 12.1. Examples of Decision Trees......................................................................249 12.2. The Meaning of Probability in a Decision Tree.........................................253 12.3. Martingales......................................................................................... .......257 12.4. The Structure of Martingale Trees.............................................................261 12.5. Probability and Causality...........................................................................265 12.6. Lower and Upper Probability.....................................................................269 12.7. The Law of Large Numbers.......................................................................272 12.8. Notes on the Literature...............................................................................274 Chapter 13. Refining............................................................................................ ...........275 13.1. Examples of Refinement............................................................................277 13.2. A Constructive Definition of Finite Refinement.......................................281 13.3. Axioms for Refinement..............................................................................282 13.4. Lifting Moivrean Events and Variables.....................................................288 13.5. Refining Martingale Trees.........................................................................288 13.6. Grounding........................................................................................... .......294 Chapter 14. Principles of Causal Conjecture..................................................................299 14.1. The Diversity of Causal Explanation.........................................................302 14.2. The Mean Effect of the Happening of a Moivrean Event..........................305 14.3. The Effect of a Humean Variable..............................................................311 14.4. Attribution and Generality.........................................................................316 14.5. The Statistical Measurement of the Effect of a Cause...............................319 14.6. Measurement by Experiment.....................................................................320 14.7. Using Our Knowledge of How Things Work............................................322 v 14.8. Sampling Error...........................................................................................329 14.9. The Sampling Frame..................................................................................329 14.10. Notes on the Literature.............................................................................330 Chapter 15. Causal Models.............................................................................................3 31 15.1. The Causal Interpretation of Statistical Prediction....................................333 15.2. Generalizing to a Family of Exogenous Variables....................................337 15.3 Some Joint Causal Diagrams......................................................................339 15.4. Causal Path Diagrams................................................................................342 15.5. Causal Relevance Diagrams.......................................................................346 15.6. The Meaning of Latent Variables..............................................................352 15.7. Notes on the Literature...............................................................................357 Chapter 16. Representing Probability Trees...................................................................359 16.1. Three Graphical Representations...............................................................361 16.2. Skeletal Simplifications.............................................................................368 16.3. Martingale Trees in Type Theory...............................................................371 Appendix A. Huygens’s Probability Trees.....................................................................379 Huygens’s Manuscript in Translation..................................................................380 Appendix B. Some Elements of Graph Theory..............................................................385 B1. Undirected Graphs........................................................................................385 B2. Directed Graphs............................................................................................38 6 Appendix C. Some Elements of Order Theory...............................................................393 C1. Partial and Quasi Orderings.........................................................................393 C2. Singular and Joint Diagrams for Binary Relations.......................................394 C3. Lattices............................................................................................ .............395 C4. The Lattice of Partitions of a Set..................................................................396 Appendix D. The Sample-Space Framework for Probability.........................................399 D1. Probability Measures....................................................................................399 D2. Variables........................................................................................... ...........400 vi D3. Families of Variables...................................................................................401 D4. Expected Value............................................................................................402 D5. The Law of Large Numbers.........................................................................405 D6. Conditional Probability................................................................................406 D7. Conditional Expected Value........................................................................407 Appendix E. Prediction in Probability Spaces................................................................409 E1. Conditional Distribution...............................................................................411 E2. Regression on a Single Variable...................................................................412 E3. Regression on a Partition or a Family of Variables......................................415 E4. Linear Regression on a Single Variable.......................................................418 E5. Linear Regression on a Family of Variables................................................422 Appendix F. Sample-Space Concepts of Independence.................................................425 F1. Overview............................................................................................ ...........426 F2. Independence Proper.....................................................................................432 F3. Unpredictability in Mean..............................................................................434 F4. Simple Uncorrelatedness..............................................................................437 F5. Mixed Uncorrelatedness...............................................................................438 F6. Partial Uncorrelatedness...............................................................................440 F7. Independence for Partitions..........................................................................442 F8. Independence for Families of Variables.......................................................445 F9. The Basic Role of Uncorrelatedness.............................................................448 F10. Dawid’s Axioms.........................................................................................449 Appendix G. Prediction Diagrams..................................................................................453 G1. Path Diagrams............................................................................................ ..454 G2. Generalized Path Diagrams..........................................................................462 G3. Relevance Diagrams.....................................................................................466 G4. Bubbled Relevance Diagrams......................................................................475 Appendix H. Abstract Stochastic Processes...................................................................479 vii H1. Probability Conditionals and Probability Distributions...............................477 H2. Abstract Stochastic Processes......................................................................479 H3. Embedding Variables and Processes in a Sample Space.............................480 References.......................................................................................... ..............................491. (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)

This text is designed to teach students how to read and write proofs in mathematics and to acquaint them with how mathematicians investigate problems and formulate conjecture.

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}.

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)

ABSTRACT Between the first two Critiques, Kant wrote what he called a “conjectural history” of the development of human freedom through a reading of Genesis. In the essay, reason itself is conceived of in terms of its “genesis,” and Kant primarily reads “Genesis” as an account of reason’s ascension or becoming. Just as humankind becomes itself through the Fall, so too does reason simultaneously come into its own. Adam indeed acts as a template for the conception of moral agency that (...) Kant will go on to develop in much more detail in the second critique. More significantly, however, as I argue below, the “genesis” and ascension of reason is only made possible through Eve’s covering up of herself. I argue that the act naturalizes both shame and clothing onto a feminine body conceived as un-reasonable. It is against this foil of unreason that reason is able to reach its own heights. I argue that Kant’s reading of Genesis in the “Conjectures” naturalizes women and clothing as superfluous: to humankind and to the ascent of reason alike. However, their function as the condition of possibility for Kant’s own system will show that they are anything but superfluous. Rather, they are essential. (shrink)

We prove that a combinatorial consequence of the negation of the PCF conjecture for intervals, involving free subsets relative to set mappings, is not implied by even the strongest known large cardinal axiom.

This paper analyzes the relationship between the meanings of the particular aesthetic judgments (I) pronounced by any person on artwork and the artwork itself. The purpose is to state the following hypothesis: it is false that a consumer is stupid and unable to recognize the quality work of art and meaning of, it is more, all the phrases enunciated in natural language is not a conjecture, but that recognize and reflect the quality of the work. A work of art (...) like any other human product, is likely to be measured and therefore valued, fact, despite the opinion of the artists. Preliminary results obtained lead us to that decision there are common patterns for each of the elements of the art world, which also requires the formulation of a much more simple and elegant. (shrink)

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)

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.

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)