This review investigates the potential of non-orientable topology as a fundamental framework for understanding the Poincaré conjecture and its implications across various scientific disciplines. Integrating insights from Dokuchaev (2020), Rapoport, Christianto, Chandra, Smarandache (under review), and other pioneering works, this article explores the theoretical foundations linking non-orientable spaces to resolving the Poincaré conjecture and its broader implications in theoretical physics, geology, cosmology, and biology.

intro to Part 1 - -/- Most people disliked mathematics when they were at school and they were absolutely correct to do so. This is because maths as we know it is severely incomplete. No matter how elaborated and complicated mathematical equations become, in today's world they're based on 1+1=2. This certainly conforms to the world our physical senses perceive and to the world scientific instruments detect. It has been of immeasurable value to all knowledge throughout history and has elevated (...) science to the lofty status it enjoys. Science is now striving towards Unification - where the subatomic realm, all matter, energy, forces, space and time will be seen as entangled parts of one universe. While 1+1=2 has been vital in getting humanity to this point, it's time to suppress our attachments to the past and realize that whereas 1+1 will always equal 2, it's also capable of equalling the 1 which represents unification. -/- intro to Part 2 -/- b) Division by zero is accepted, in Newtonian maths, to be impossible. But we can regard division by zero as division by nothing i.e. division that has no effect. In this case, 1 divided by 0 is 1. However, to a physicist there is no such thing as nothing (even empty space contains energy). What could the something called 0 actually be? It could be a binary digit. If we use the base of ten (for simplicity) and attach one and zero to it as exponents, we get 10^1 divided by 10^0 = 10^1. If we then cancel 10 from each factor in the expression, we get 1 divided by 0 = 1. At the start of the paragraph, this was referred to as division by nothing. Then 0 was called a binary digit and division by nothing became division by something. The 1 that the division equals is the unified field of space-time. Division by 0 is impossible in Newtonian maths because the result can be infinity. But the word “infinity” can, as the last section of this book shows, apply to the unified field of spacetime. So division by zero is not impossible because it results in the universe, which is obviously possible … a possibility that has always been, and always will be, realized. -/- intro to Part 3 -/- If quantum entanglement has existed in the entire universe forever, everything would be everywhere and everywhen. Space, time and 5th-dimensional hyperspace would not be restricted to certain parts of the Mobius Universe but would exist in every particle. Past, present and future would not exist as the distinct periods which everyday life assumes. All instants of all periods would exist eternally, permitting time travel to any point in the past and to any point in the future. Entanglement may be created by simply zipping along at close to the speed of light - “Quantum entanglement of moving bodies” by Robert M. Gingrich and Christoph Adami in Physical Review Letters 89, 270402 (issue of 30 December 2002) – which might be achieved, according to this book, by warping space so it’s either a fraction of the 90 degrees allowing instantaneous travel or almost at 270 degrees to space as we know it. (shrink)

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert (...) mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture (...) can be generalized and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering and resulting into the “curving of information” (e.g. entanglement). Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism once the philosophical category of the totality is defined formally. The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy and should be related to their shared foundations. (shrink)

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means the (...) invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. (shrink)

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s (...) last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested. (shrink)

Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician (...) Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify a conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed. (shrink)

Poincaré tries to tackle the question "Can science sway us into materialism?". In other words, he analyzes if science - in particular Physics - and its theories are dependent on a materialistic ontological view of the world. His final answer appears to be "no". However, in order to reach this conclusion he resorts to a brief history of Physics, providing an insightful account on the evolution of the concept of atom from Democritus to the, then, recent discoveries on the (...) composition of the atom. In his opinion, science swung and will keep swinging between two opposing approaches over the constitution of nature: the discrete and the continuous approach. (shrink)

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

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry (...) is the study of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them. (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)

In (Weaver 2021), I showed that Boltzmann’s H-theorem does not face a significant threat from the reversibility paradox. I argue that my defense of the H-theorem against that paradox can be used yet again for the purposes of resolving the recurrence paradox without having to endorse heavy-duty statistical assumptions outside of the hypothesis of molecular chaos. As in (Weaver 2021), lessons from the history and foundations of physics reveal precisely how such resolution is achieved.

In this paper, I examine the relation between Henri Poincaré’s definition of mathematical continuity and Sartre’s discussion of temporality in Being and Nothingness. Poincaré states that a series A, B, and C is continuous when A=B, B=C and A is less than C. I explicate Poincaré’s definition and examine the arguments that he uses to arrive at this definition. I argue that Poincaré’s definition is applicable to temporal series, and I show that this definition of continuity (...) provides a logical basis for Sartre’s psychological explanation of temporality. Specifically, I demonstrate that Poincaré’s definition allows the for-itself to be understood both as connected to a past and future and as distinct from itself. I conclude that the gap between two terms in a temporal series comprises the present and being-for-itself, since it is this gap that occasions the radical freedom to reshape the past into a distinct and different future. (shrink)

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

This article was produced as an introduction to a Portuguese translation of an article by Henri Poincaré titled "The new conceptions of matter". The aim of this introduction was to shortly summarize Poincaré's scientific and philosophical production, to approach the circumstances on which the text was originally presented and, finally, to analyze the relationship - or the lack of it - that Poincaré establishes between science and materialism.

The two greatest mathematicians of the early twentieth century, David Hilbert and Henri Poincaré transformed the mathematics of their time. Their personal interaction was infrequent, until Hilbert invited Poincaré to deliver the first Wolfskehl Lectures in Göttingen in the spring of 1909. A correspondence ensued, which fixed the content and timing of the lecture series. A close reading of the exchange throws light on what Hilbert wanted Poincaré to talk about, and on what Poincaré wanted to (...) present to Hilbert and his colleagues. To answer the latter question, reference is made to the published version of Poincaré’s six talks, with a focus on two of them, concerning the propagation of Hertzian waves, and the theory of relativity. (shrink)

I present the replies that Gottlob Frege, Henri Poincaré, Rudolf Carnap, and Saul Kripke made to the assumption that apriority and necessity are interchangeable synonyms, an assumption that I take, together with the assumptions that there is a split between analytic truths and synthetic truths and that there is a dichotomy between our conceptual schemes and empirical content, as a Kantian dogma.

A recurrent theme in philosophy of science since the early twentieth century is the idea that at least some basic tenets within scientific theories ought to be understood as conventions. Various versions of this idea have come to be grouped together under the label ‘conventionalism’. This chapter presents and discusses some important historical stages in the development of conventionalism. Particular attention is paid to the contributions made by Henri Poincaré, Pierre Duhem and Hans Reichenbach.

Where Consciousness has been described previously, it is only in the context of strictly unconscious means – we leave ourselves under the guise of some manifestation of consciousness from alien means, and lose ourselves in the mysterianism that has formed a true government over conjectural thought on the study of consciousness. This article instead proposes that the information which, to us, occupies a passive basis is formative for consciousness – derivative on a prerogative for formation that is observed in GWT (...) and IIT that rejects any preconceived notion of immediate innateness for consciousness – and that it provides a shape for consciousness that suggests its function to be explicitly limited to its faculties. (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)

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)

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)

The book Poincaré, Philosopher of Science – Problems and Perspectives, edited by María de Paz and Robert DiSalle, is the result of various colloquia and conferences organized by the Portuguese project bearing the same name. The project, initiated by University of Lisbon, brought together scholars of many different countries to speak about the three main philosophical facets of Henri Poincaré: as a philosopher of science in general, as a philosopher of mathematics, and as a philosopher of physics.

According to the conventionalist doctrine of space elaborated by the French philosopher-scientist Henri Poincaré in the 1890s, the geometry of physical space is a matter of definition, not of fact. Poincaré's Hertz-inspired view of the role of hypothesis in science guided his interpretation of the theory of relativity (1905), which he found to be in violation of the axiom of free mobility of invariable solids. In a quixotic effort to save the Euclidean geometry that relied on this axiom, (...) Poincaré extended the purview of his doctrine of space to cover both space and time. The centerpiece of this new doctrine is what he called the "principle of physical relativity," which holds the laws of mechanics to be covariant with respect to a certain group of transformations. For Poincaré, the invariance group of classical mechanics defined physical space and time (Galilei spacetime), but he admitted that one could also define physical space and time in virtue of the invariance group of relativistic mechanics (Minkowski spacetime). Either way, physical space and time are the result of a convention. (shrink)

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)

In this paper, a modified form of Collatz conjecture for neutrosophic numbers n ∈ (Z U I) is defined. We see for any n ∈ (Z U I) the related sequence using the formula (3a + 1) + (3b + 1)I converges to any one of the 55 elements mentioned in this paper. Using the akin formula of Collatz conjecture viz. (3a- 1) + (3b -1)I the neutrosophic numbers converges to any one of the 55 elements mentioned with (...) appropriate modifications. Thus, it is conjectured that every n ∈ (Z U I) has a finite sequence which converges to any one of the 55 elements. (shrink)

The two principal models of design in methodological circles in architecture—analysis/synthesis and conjecture/analysis—have their roots in philosophy of science, in different conceptions of scientific method. This paper explores the philosophical origins of these models and the reasons for rejecting analysis/synthesis in favour of conjecture/analysis, the latter being derived from Karl Popper’s view of scientific method. I discuss a fundamental problem with Popper’s view, however, and indicate a framework for conjecture/analysis to avoid this problem.

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)

The history of western philosophy is split to its core by a long-standing, fundamental dispute. On the one hand there are the so-called empiricists, like Locke, Berkeley, Hume, Mill, Russell, the logical positivists, A. J. Ayer, Karl Popper and most scientists, who hold empirical considerations alone can be appealed to in justifying, or providing a rationale for, claims to factual knowledge, there being no such thing as a priori knowledge – items of factual knowledge that are accepted on grounds other (...) than the empirical. And on the other hand there are the so-called rationalists, like Descartes, Spinoza, Leibniz, Bradley, McTaggart, who hold that a priori knowledge does exist, and even plays a crucial role in science. This long-standing dispute is resolved by the line of thought developed in this essay. The empiricists were right to deny the existence of a priori knowledge where this means factual knowledge of apodictic certainty justified independently of any appeal to experience. They were wrong, however, to deny the existence of a priori knowledge where this means conjectural items of factual knowledge whose acceptance is justified independently of any appeal to experience. Science requires there to be a priori knowledge in this latter sense, and two items of such knowledge will be exhibited. In addition to resolving this empiricist/rationalist dispute, the essay also develops a new conception of science – aim-oriented empiricism – and solves the long-standing problem of what it means to say of a theory that it is unified or explanatory. Furthermore, aim-oriented empiricism does not just change our conception of science; it requires that science itself needs to change. Science needs to become much more like the 17th century conception of natural philosophy, intermingling consideration of testable theories with consideration of untestable metaphysical, epistemological and methodological ideas. The whole relationship between science and the philosophy of science is transformed. The philosophy of science, construed as the exploration and critical assessment of rival ideas about what the aims and methods of science ought to be, ceases to be a meta-discipline, and becomes an integral part of science – or natural philosophy – itself. (shrink)

Let G be an additive subgroup of ℂ, let Wn = {xi = 1, xi + xj = xk: i, j, k ∈ {1, …, n }}, and define En = {xi = 1, xi + xj = xk, xi · xj = xk: i, j, k ∈ {1, …, n }}. We discuss two conjectures. If a system S ⊆ En is consistent over ℝ, then S has a real solution which consists of numbers whose absolute values belong to (...) [0, 22n –2]. If a system S ⊆ Wn is consistent over G, then S has a solution ∈ n in which |xj| ≤ 2n –1 for each j. (shrink)

Originally proposed by John Rawls, the idea of reasoning from conjecture is popular among the proponents of political liberalism in normative political theory. Reasoning from conjecture consists in discussing with fellow citizens who are attracted to illiberal and antidemocratic ideas by focusing on their religious or otherwise comprehensive doctrines, attempting to convince them that such doctrines actually call for loyalty to liberal democracy. Our goal is to criticise reasoning from conjecture as a tool aimed at persuasion and, (...) in turn, at improving the stability of liberal democratic institutions. To pursue this goal, we use as case study real-world efforts to counter-radicalise at-risk Muslim citizens, which, at first glance, reasoning from conjecture seems well-placed to contribute to. This case study helps us to argue that the supporters of reasoning from conjecture over-intellectualise opposition to liberal democracy and what societies can do to counter it. Specifically, they (i) underestimate how few members of society can effectively perform reasoning from conjecture; (ii) overlook that the burdens of judgement, a key notion for political liberals, highlight how dim the prospects of reasoning from conjecture are and (iii) do not pay attention to the causes of religious persons’ opposition to liberal democracy. However, not everything is lost for political liberals, provided that they redirect attention to different and under-researched resources contained in Rawls’s theory. In closing, we briefly explain how such resources are much better placed than reasoning from conjecture to provide guidance relative to counter-radicalisation in societies (i) populated by persons who do not generally hold anything close to a fully worked out and internally consistent comprehensive doctrine and (ii) where political institutions should take responsibility for at least part of the existing alienation from liberal democratic values. (shrink)

To teach the ethics of science to science majors, I follow several teachers in the literature who recommend “persona” writing, or the student construction of dialogues between ethical thinkers of interest. To engage science majors in particular, and especially those new to academic philosophy, I recommend constructing persona dialogues from Henri Poincaré’s essay, “Ethics and Science”, and the non-theological third chapter of Pope Francis’s encyclical on the environment, Laudato si. This pairing of interlocutors offers two advantages. The first is (...) that science students are likely to recognize both names, since Poincaré appears in undergraduate mathematics and physics textbooks, and because Francis is an environmentalist celebrity. Hence students show more interest in these figures than in other philosophers. The second advantage is that the third chapter of Laudato si reads like an implicit rebuttal of Poincaré’s essay in many respects, and so contriving a dialogue between those authors both facilitates classroom discussion, and deserves attention from professional ethicists in its own right. In this paper, I present my own contrived dialogue between Francis and Poincaré, not for assigning to students as a reading, but as a template for an effective assignment product, and as a crib sheet for educators to preview the richly antiparallel themes between the two works. (shrink)

This paper presents folk impressions of the University of Manchester’s difficulties in becoming a great university, but by means of a fiction imitating a distinguished writer from the Indian subcontinent. The impressions concern past efforts and the difficulties they faced.

The aim of this paper is to examine the considerations on stochastic arts in Antiquity and to show how Galen’s analysis concerning the “art of conjecturing” constitutes a preferable alternative to the traditional ways used by philosophers to explain the inherent fallibility in the medical art. By distinguishing the scientific diagnosis from the conjectural one, Galen encompasses all cases relevant to the medical art. The former, because of its general nature, can be theorized. As for the latter, it concerns only (...) the individual and is therefore not likely to enter the theory, except by the method that makes it possible. This paper attempts to outline the modus operandi underlying the technical conjecture in order to illustrate Galen’s deftly developed position on a subject almost exclusively investigated by philosophers until then. (shrink)

W pierwszej połowie XX wieku przyjęło się upatrywać w poglądach H. Poincarégo i P. Duhema przykładów antyrealistycznego stanowiska odnośnie do nauki i jej teorii. Etykietka ta przylgnęła do tych autorów tak mocno, że coraz częstszym dzisiaj głosom tych, którzy sprzeciwiają się takiemu szufladkowaniu ich filozofii, trudno jest przebić się do głównego nurtu dyskusji filozoficznych. W artykule wskazuję, że odczytywanie poglądów obu francuskich autorów jako antyrealistycznych nie znajduje potwierdzenia w ich własnych wypowiedziach. Przeciwnie, ich prace dostarczają mocnych świadectw na rzecz upatrywania (...) w nich prekursorów współczesnych wyrafinowanych stanowisk realistycznych. (shrink)

In this paper I argue for a priori conjectural scientific knowledge about the world. Physics persistently only accepts unified theories, even though endlessly many empirically more successful disunified rivals are always available. This persistent preference for unified theories, against empirical considerations, means that physics makes a substantial, persistent metaphysical assumption, to the effect that the universe has a (more or less) unified dynamic structure. In order to clarify what this assumption amounts to, I solve the problem of what it means (...) to say of a theory that it is unified. There are, I argue, eight different kinds of unity important in theoretical physics, all varieties of one basic idea. This provides us with a precise way of partially ordering physical theories with respect to their degree of unity. It also leads to a hierarchical view of physics, according to which physics makes a number of increasingly insubstantial metaphysical assumptions concerning the comprehensibility and knowability of the universe. Two of these are identified as constituting a priori conjectures. I conclude by arguing that the view developed in the paper resolves the traditional clash between empiricism and rationalism in the philosophy of science, and has important implications for science, and for academic inquiry more generally. (shrink)

Probability plays a crucial role regarding the understanding of the relationship which exists between mathematics and physics. It will be the point of departure of this brief reflection concerning this subject, as well as about the placement of Poincaré’s thought in the scenario offered by some contemporary perspectives.

I propose and defend the conjecture that what explains why Gettiered subjects fail to know is the fact that their justified true belief depends essentially on unknown propositions. The conjecture follows from the plausible principle about inference in general according to which one knows the conclusion of one’s inference only if one knows all the premises it involves essentially.

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

How were reliable predictions made before Pascal and Fermat's discovery of the mathematics of probability in 1654? What methods in law, science, commerce, philosophy, and logic helped us to get at the truth in cases where certainty was not attainable? The book examines how judges, witch inquisitors, and juries evaluated evidence; how scientists weighed reasons for and against scientific theories; and how merchants counted shipwrecks to determine insurance rates. Also included are the problem of induction before Hume, design arguments for (...) the existence of God, and theories on how to evaluate scientific and historical hypotheses. It is explained how Pascal and Fermat's work on chance arose out of legal thought on aleatory contracts. The book interprets pre-Pascalian unquantified probability in a generally objective Bayesian or logical probabilist sense. (shrink)

In this work we present the function and we determine the nature of conventions and hypotheses for the scientific foundations according with the conventionalist doctrine that arose in France during the turning of the XIX century to the XX. The doctrine was composed by Henri Poincaré, Pierre Duhem and Édouard Le Roy. Moreover, we analyze the relation that conventions and hypotheses can establish with metaphysical thesis through criteria used by scientists in order to determine the preference for certain theories. (...) Thereunto, we promote an immanent interpretation of published works between 1891 and 1905. As result, we reveal that the authors, though being classified as belonging to the same doctrine, don't have only common grounds, but also divergences. Poincaré and Le Roy agree that geometrical conventions are chosen in accordance with convenience criteria. However, they disagree about the value convenience aggregate to scientific knowledge. In regards to natural phenomena, the three authors agree that reality can't be described univocally by the same set of conventions and hypotheses. Yet, Poincaré and Duhem both believe that there are experimental, rational and axiological criteria that justify scientist's satisfaction with certain theories and we indicate how those criteria are related with metaphysics. We conclude that conventionalists, even if warily and implicitly, searched to approach metaphysics in order to justify scientific activity. (shrink)

Until quite recently, it has appeared that eleutheric-conjectural libertarianism (ECL) could not avoid some degree of, very broad, interpersonal utility comparisons (IUCs). And this has been objected to by some of its libertarian critics, notably economists and propertarians. Indeed, this aspect does make the theory less compatible with economics than the rest of the theory and it is thereby a significant problem. This is because one of the main problems that ECL is intended to solve is how an abstract theory (...) of liberty can be compatible with the pro-free-market conclusions of many economists. Being more compatible with economics should make ECL theory more comprehensive, comprehensible, and cogent. (shrink)

I introduce a formalization of probability which takes the concept of 'evidence' as primitive. In parallel to the intuitionistic conception of truth, in which 'proof' is primitive and an assertion A is judged to be true just in case there is a proof witnessing it, here 'evidence' is primitive and A is judged to be probable just in case there is evidence supporting it. I formalize this outlook by representing propositions as types in Martin-Lof type theory (MLTT) and defining a (...) 'probability type' on top of the existing machinery of MLTT, whose inhabitants represent pieces of evidence in favor of a proposition. One upshot of this approach is the potential for a mathematical formalism which treats 'conjectures' as mathematical objects in their own right. Other intuitive properties of evidence occur as theorems in this formalism. (shrink)

Aristotle, in his Metaphysics, not only tries to establish a relationship that is direct, coherent, inter-operational and "precise" between this science, its name as a science, and its object of study, but also begins an indignation that tries to set a science — materially adequate and formally correct — to study τὸ ὂν ᾗ ὂν. In order to complete this task, Aristotle does an in-focus strategy that consists on the diffusion of τὸ ὄν in its categories, that allows Aristotle the (...) possibility of formulate a science ἐπιστήμη to study τὸ ὂν ᾗ ὂν at the same time that supports the diffusion of τὸ ὄν in the "different categories or supreme genres" of τὸ ὄν. Due to this argumentation, Aristotle distinguishes the ἐπιστήμη σοφία of any other type of knowledge: the former is fundamentally an enquiry "about the first causes (πρῶτα αἴτια) and the principles (ἀρχὰς)". With this conception in mind, Aristotle was able to characterize several related forms of the ἐπιστήμη σοφία which, as a whole, eventually allow him to establish a science of the principles and causes "maximally universal and referred to…everything real", and in consequence, to "that that is, as long as something is". In this work, I will try to reconstruct such arguments, and offer some observations and conjectures referred to these notions that plausibly incline the scale in favor of an ontological reading of Aristotle’s Metaphysics, in order to contribute to clarify the confusion that exists between the aim of the discipline with the justification to establish the existence of its own object. (shrink)

Krisen, heißt es manchmal, erfordern klare Ansagen: Bei Behauptungen wissen wir, woran wir sind. Vermutungen hingegen sind unklar und stehen der Übernahme von Ver­ant­wor­tung entgegen. In diesem Essay wird mit den Mitteln der Sprachphilosophie ge­zeigt, dass vermutende Sprechakte für die Krisenkommunikation in der Corona-Pandemie richtig und wichtig sind. Weder sind Vermutungen anfälliger für Unklarheit als andere Sprechakte noch sind sie besser dazu geeignet, Verantwortung abzuweisen. Im Gegen­teil: In einer Situation, die durch Unsicherheit geprägt ist, sind Vermutungen besonders wertvoll für das (...) Kommunizieren ungesicherter Hypothesen. (shrink)

In this paper, I propose to look closely at certain crucial aspects of the logic of Rawls' argument in Political Liberalism and related subsequent writings. Rawls' argument builds on the notion of comprehensiveness, whereby a doctrine encompasses the full spectrum of the life of its adherents. In order to show the mutual conflict and irreconcilability of comprehensive doctrines, Rawls needs to emphasise the comprehensiveness of doctrines, as their irreconcilability to a large extent emanates from that comprehensiveness. On the other hand, (...) in order to show the possibility and plausibility of the political liberal solution he needs to emphasise that most of these doctrines are reasonable: i.e., they are willing to cede a portion of their authority to political liberalism for the right reasons. Yet, if they are willing to cede a portion of their authority to a political conception they cannot be as comprehensive as we initially thought they were. All these elements highlight the tension in the argument itself. I suggest that many of these tensions can be removed by making Rawls' account more flexible. In this context I propose certain amendments to Rawls' account, which may overcome some of the tensions mentioned above. (shrink)

In 1980 F. Wattenberg constructed the Dedekind completion

d of the Robinson non-archimedean field

and established basic algebraic properties of

d [6]. In 1985 H. Gonshor established further fundamental properties of

d [7].In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion

d in transcendental number theory were considered. We dealing using set theory ZFC
(
-model of ZFC).Given (...) an class of analytic functions of one complex variable f  

z, we investigate the arithmetic nature of the values of fz at transcendental points en, n  .Main results are: (i) the both numbers e
 and e   are irrational, (ii) number ee is transcendental. Nontrivial generalization of the Lindemann- Weierstrass theorem is obtained. (shrink)