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

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 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)

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)

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.

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

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

The 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.

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)

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)

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)

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)

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.

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)

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)

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)

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)

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)

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.

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)

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)

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 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)

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)

A discussion of the new interpretation of Adam Smith's system proposed by Ortmann and Walvaerens.

Prawitz conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister. This article resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The paper further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination (...) for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic. (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures (...) in pure mathematics and for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning. (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)

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)

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)

Guessing the outcome of iterations of even most simple arithmetical functions could be an extremely hazardous experience. Not less harder, if at all possible, might be to prove the veracity of even a "sure" guess concerning iterations : this is the case of the famous 3x+1 conjecture. Our purpose here is to study and conceptualize some intuitive insights related to the ultimate (un)solvability of this conjecture.

Religious conservatives in the U.S. have frequently opposed public-health measures designed to combat STDs among minors, such as sex education, condom distribution, and HPV vaccination. Using Rawls’s method of conjecture, I will clear up what I take to be a misunderstanding on the part of religious conservatives: even if we grant their premises regarding the nature and source of sexual norms, the wide-ranging authority of parents to enforce these norms against their minor children, and the potential sexual-disinhibition effects of the (...) above public-health measures, their opposition to at least one of these measures, HPV vaccination, cannot be justified. In fact, their comprehensive doctrines, when properly interpreted, should lead them to back this measure and thereby draw closer to a policy consensus with other citizens regarding teenage sexual health. (shrink)

In this paper I argue that aim-oriented empiricism provides decisive grounds for accepting scientific realism and rejecting instrumentalism. But it goes further than this. Aim-oriented empiricism implies that physicalism is a central part of current (conjectural) scientific knowledge. Furthermore, we can and need, I argue, to interpret fundamental physical theories as attributing necessitating physical properties to fundamental physical entities.

i. a poetic solution of the Goldbach Conjecture; ii. several responses to the Epimenides Paradox; iii. the volitional solution to Russell's Paradox.

A review of Geoffrey Marnell, Mathematical Doodlings: Curiosities, conjectures, and challenges.

I conjecture that the probability amplitudes of a quantum state are fractions of the Planck frequency, stemming from the rich dynamics at the Planck scale. This offers a means to indirectly measure the fundamental properties of quantum spacetime and potentially resolves the measurement problem.

Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Landau's conjecture implies the following unproven statement Φ: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆[2,f(7)]. Let B denote the system of equations: {x_j!=x_k: i,k∈{1,...,9}}∪{x_i⋅x_j=x_k: i,j,k∈{1,...,9}}. The system of equations {x_1!=x_1, x_1 \cdot x_1=x_2, x_2!=x_3, x_3!=x_4, x_4!=x_5, x_5!=x_6, x_6!=x_7, x_7!=x_8, x_8!=x_9} has exactly two solutions in positive integers x_1,...,x_9, namely (1,...,1) and (f(1),...,f(9)). No known system S⊆B with a finite (...) number of solutions in positive integers x_1,...,x_9 has a solution (x_1,...,x_9)∈(N\{0})^9 satisfying max(x_1,...,x_9)>f(9). For every known system S⊆B, if the finiteness/infiniteness of the set {(x_1,...,x_9)∈(N\{0})^9: (x_1,...,x_9) solves S} is unknown, then the statement ∃ x_1,...,x_9∈N\{0} ((x_1,...,x_9) solves S)∧(max(x_1,...,x_9)>f(9)) remains unproven. Let Λ denote the statement: if the system of equations {x_2!=x_3, x_3!=x_4, x_5!=x_6, x_8!=x_9, x_1 \cdot x_1=x_2, x_3 \cdot x_5=x_6, x_4 \cdot x_8=x_9, x_5 \cdot x_7=x_8} has at most finitely many solutions in positive integers x_1,...,x_9, then each such solution (x_1,...,x_9) satisfies x_1,...,x_9≤f(9). The statement Λ is equivalent to the statement Φ. It heuristically justifies the statement Φ . This justification does not yield the finiteness/infiniteness of P(n^2+1). We present a new heuristic argument for the infiniteness of P(n^2+1), which is not based on the statement Φ. Algorithms always terminate. We explain the distinction between existing algorithms (i.e. algorithms whose existence is provable in ZFC) and known algorithms (i.e. algorithms whose definition is constructive and currently known). Assuming that the infiniteness of a set X⊆N is false or unproven, we define which elements of X are classified as known. No known set X⊆N satisfies Conditions (1)-(4) and is widely known in number theory or naturally defined, where this term has only informal meaning. *** (1) A known algorithm with no input returns an integer n satisfying card(X)<ω ⇒ X⊆(-∞,n]. (2) A known algorithm for every k∈N decides whether or not k∈X. (3) No known algorithm with no input returns the logical value of the statement card(X)=ω. (4) There are many elements of X and it is conjectured, though so far unproven, that X is infinite. (5) X is naturally defined. The infiniteness of X is false or unproven. X has the simplest definition among known sets Y⊆N with the same set of known elements. *** Conditions (2)-(5) hold for X=P(n^2+1). The statement Φ implies Condition (1) for X=P(n^2+1). The set X={n∈N: the interval [-1,n] contains more than 29.5+\frac{11!}{3n+1}⋅sin(n) primes of the form k!+1} satisfies Conditions (1)-(5) except the requirement that X is naturally defined. 501893∈X. Condition (1) holds with n=501893. card(X∩[0,501893])=159827. X∩[501894,∞)= {n∈N: the interval [-1,n] contains at least 30 primes of the form k!+1}. We present a table that shows satisfiable conjunctions of the form #(Condition 1) ∧ (Condition 2) ∧ #(Condition 3) ∧ (Condition 4) ∧ #(Condition 5), where # denotes the negation ¬ or the absence of any symbol. No set X⊆N will satisfy Conditions (1)-(4) forever, if for every algorithm with no input, at some future day, a computer will be able to execute this algorithm in 1 second or less. The physical limits of computation disprove this assumption. (shrink)

Bartha (2012) conjectures that, if we meet all of the other objections to Pascal’s wager, then the many-Gods objection is already met. Moreover, he shows that, if all other objections to Pascal’s wager are already met, then, in a choice between a Jealous God, an Indifferent God, a Very Nice God, a Very Perverse God, the full range of Nice Gods, the full range of Perverse Gods, and no God, you should wager on the Jealous God. I argue that (...) his requirement of [strongly] stable equilibrium is not well-motivated. There are other types of Gods, no less worthy of consideration than those that figure in Bartha’s deliberations, which are intuitively no worse wagers than the Jealous God. In particular, I have suggested that one does no worse to wager on a jealous cartel than one does to wager on a Jealous God. Moreover, I argue that there are other types of Gods, no less worthy of consideration than those that figure in Bartha’s deliberations, that make trouble for Pascal’s wager, but not because one would do better to wager on them rather than on a Jealous God. Finally, I argue that, if we suppose that infinitesimal credences are in no worse standing the infinite utilities, then we cannot accept the assumption—built into the relative utilities framework—that there cannot be infinitesimal credences. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for (...) number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

Utterances of simple sentences containing taste predicates (e.g. "delicious", "fun", "frightening") typically imply that the speaker has had a particular sort of first-hand experience with the object of predication. For example, an utterance of "The carrot cake is delicious" would typically imply that the speaker had actually tasted the cake in question, and is not, for example, merely basing her judgment on the testimony of others. According to one approach, this acquaintance inference is essentially an implicature, one generated by the (...) Maxim of Quality together with a certain principle concerning the epistemology of taste (Ninan 2014). We first discuss some problems for this approach that arise in connection with disjunction and generalized quantifiers. Then, after stating a conjecture concerning which operators ‘obviate’ the acquaintance inference and which do not, we build on Anand & Korotkova (2018) and Willer & Kennedy (Forthcoming) by developing a theory that treats the acquaintance requirement as a presupposition, albeit one that can be obviated by certain operators. (shrink)

We suggest that pain processing has a modular architecture. We begin by motivating the (widely assumed but seldom defended) conjecture that pain processing comprises inferential mechanisms. We then note that pain exhibits a characteristic form of judgement independence. On the assumption that pain processing is inferential, we argue that its judgement independence is indicative of modular (encapsulated) mechanisms. Indeed, we go further, suggesting that it renders the modularity of pain mechanisms a default hypothesis to be embraced pending convincing counterevidence. Finally, (...) we consider what a modular pain architecture might look like, and question alleged counterevidence to our proposal. (shrink)

Disagreements that resist rational resolution, often termed “deep disagreements”, have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki’s work on the abc conjecture (...) as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice. (shrink)

Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.

I develop the conjecture that “naturalism” in philosophy names not a thesis but a paradigm in something like Thomas Kuhn’s sense, i.e., a set of commitments, shared by a group of investigators, whose acceptance by the members of the group powerfully influences their day-to-day investigative practice. I take a stab at spelling out the shared commitments that make up naturalism, and the logical and evidential relations among them.

This is a partly provocative essay edited as a humanitarian study in philosophy of science and social philosophy. The starting point is Isaac Asimov’s famous sci-fi novella “Profession” (1957) to be “back” extrapolated to today’s relation between Thomas Kuhn’s “normal science” and “scientific revolutions” (1962). The latter should be accomplished by Asimov’s main personage George Platen’s ilk (called “feeble minded” in the novella) versus the “burned minded” professionals able only to “normal science”. Francis Fukuyama’s “end of history” in post-Hegelian manner (...) is now interpreted to an analogically supposed “end of scientific history” without “scientific revolutions” any more. The relevant dystopia of the prolonged or even “eternal” period of normal science is justified to the contemporary institution of science due to mechanisms such as “peer-review”, “impact-factor rating”, the projects’ competition for funding, etc. Positive feedbacks forcing all scientists needing careers to be more and more orthodox are demonstrated therefore establishing for that dystopia to be the real state of contemporary science. Two counterfactual case studies based correspondingly on Feyerabend’s “Against method” (1975) if Galilei should make his discoveries today and Sokal’s hoax (1996) if he suggested a scientific masterpiece to be really rejected by journals are discussed. Still one case study considering the abundance of Kelvin’s “clouds” on the horizon of today’s physics (dark matter, dark energy, entanglement, quantum gravitation, phenomena refuting the Big Bang, etc.) serves to verify the aforementioned conjecture that science has already entered that dystopia of eternal normal science. The conception of “ontomathematics” implying “creation ex nihilo” being scandalous for the dominating paradigm is sketched as an eventual revolutionary way out. An imaginary and utopic “happy end” reinterpreting the analogical “happy end” of Asimov’s “Profession” finishes the essay “instead of conclusion” relying on the Internet and AI in an increasingly “fluid” and anti-hierarchical society. (shrink)

Around 3 decades ago, Jayant Narlikar & Halton Arp argued on possible variable mass hypothesis cosmology (VMH). In the meantime, the Earth expansion problem has attracted great interest, and recent study gives geodetic evidence that the Earth has been expanding, at least over the recent several decades. Therefore, in the present article discusses some interesting effects related to varying G, but here we argue that instead of varying G we can think of varying mass (M). Among other things we discuss (...) receding planets from the Sun, calculation of Pioneer spacecraft anomaly as proposed by B.G. Sidharth, and also possible slight change of LAGEOS Satellite. Further observation to verify or refute this conjecture is recommended, especially based on geological evidence of expanding earth. (shrink)