The work provides comprehensively definitive, unconditional proofs of Riemann's hypothesis, Goldbach's conjecture, the 'twin primes' conjecture, the Collatz conjecture, the Newcomb-Benford theorem, and the Quine-Putnam Indispensability thesis. The proofs validate holonomic metamathematics, meta-ontology, new number theory, new proof theory, new philosophy of logic, and unconditional disproof of the P/NP problem. The proofs, metatheory, and definitions are also confirmed and verified with graphic proof of intrinsic enabling and sustaining principles of reality.

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)

I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set-theoretic assumptions, to what I will call the Cut-for-Formulas to Cut-for-Sets Theorem: for a set F and a binary relation |- on Power(F), if |- is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order-theoretic variant of the Tukey- Teichmüller Lemma. I then (...) discuss relationships between various cut-conditions in the absence of finitariness or of monotonicity. (shrink)

This paper presents a formalization of the classical proof of completeness in Henkin-style developed by Troelstra and van Dalen for intuitionistic logic with respect to Kripke models. The completeness proof incorporates their insights in a fresh and elegant manner that is better suited for mechanization. We discuss details of our implementation in the Lean theorem prover with emphasis on the prime extension lemma and construction of the canonical model. Our implementation is restricted to a system of intuitionistic propositional logic (...) with implication, conjunction, disjunction, and falsity given in terms of a Hilbert-style axiomatization. As far as we know, our implementation is the first verified Henkin-style proof of completeness for intuitionistic logic following Troelstra and van Dalen's method in the literature. (shrink)

In this paper, I explore how theorists might navigate a course between the twin dangers of piety and excess cynicism when thinking critically about state apologies, by focusing on two government apologies to indigenous peoples: namely, those made by the Australian and Canadian Prime Ministers in 2008. Both apologies are notable for several reasons: they were both issued by heads of government, and spoken on record within the space of government: the national parliaments of both countries. Furthermore, in each (...) case, the object of the apology – that which was apologized for – comes closer to disrupting the idea both countries have of themselves, and their image in the global political community, than any previous apologies made by either government. Perhaps as a result, both apologies were surrounded by celebration and controversy alike, and tracing their consequences – even in the short term – is a difficult business. We avoid excessive piety or cynicism, I argue, when we take several things into account. First, apologies have multiple functions: they narrate particular histories of wrongdoing, they express disavowal of that wrongdoing, and they commit to appropriate forms of repair or renewal. Second, the significance and the success of each function must be assessed contextually. Third, when turning to official political apologies, in particular, appropriate assessment of their capacity to disavow or to commit requires that consider apologies both as performance and as political action. While there remain significant questions regarding the practice of political apology – in particular, its relationship to practices of reparation, forgiveness and reconciliation – this approach can provide a framework with which to best consider them. (shrink)

Smarandache in 2019 has generalized the algebraic structures to NeutroAlgebraic structures and AntiAlgebraic structures. In this paper, authors, for the first time, define the NeutroAlgebra of neutrosophic triplets group under usual+ and x, built using {Zn, x}, n a composite number, 5 < n < oo, which are not partial algebras. As idempotents in Zn alone are neutrals that contribute to neutrosophic triplets groups, we analyze them and build NeutroAlgebra of idempotents under usual + and x, which are not partial (...) algebras. We prove in this paper the existence theorem for NeutroAlgebra of neutrosophic triplet groups. This proves the neutrals assocaited with neutrosophic triplet groups in { Zn, X} under product is a NeutroAlgebra of triplets. We also prove the non-existence theorem of NeutroAlgebra for neutrosophic triplets in case of Zn when n = 2p, 3p and 4p (for some primes p). Several open problems are proposed. Further, the NeutroAlgebras of extended neutrosophic triplet groups have been obtained. (shrink)

Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.

Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...) naïf, il semblerait que la première soit pure et la seconde impure. Des objections à cette vue naïve sont ici considérées et réfutées. Concernant la preuve euclidienne, la question relève de la logique, notamment de la définissabilité arithmétique de l’addition en termes de successeur et de divisibilité telle que démontrée par Julia Robinson, tandis qu’en ce qui concerne la preuve topologique, la question relève de la sémantique, notamment pour tout ce qui touche au problème de savoir ce qui est « inclus » dans le contenu d’énoncés particuliers.A proof is pure, roughly, if it draws only on what is « close » or « intrinsic » to the statement being proved. The infinitude of prime numbers, a classical theorem of arithmetic, is a rich case study for philosophical investigation of purity. Two different proofs of this result are considered, namely the classical Euclidean proof and a more recent « topological » proof by Furstenberg. Naively the former would seem to be pure and the latter to be impure. Objections to these naive views are considered and met. In the case of the former the issue rests on logical matters, specifically the arithmetic definability of addition in terms of successor and divisibility shown by Julia Robinson, while in the case of the latter the issue rests on semantic matters, specifically with respect to what is « contained » in the content of particular statements. (shrink)

‎A universal schema for diagonalization was popularized by N. S‎. ‎Yanofsky (2003)‎, ‎based on a pioneering work of F.W‎. ‎Lawvere (1969)‎, ‎in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function‎. ‎It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema‎. ‎Here‎, ‎we fit more theorems in the universal‎ ‎schema of diagonalization‎, ‎such as Euclid's proof for the infinitude of the primes and new (...) proofs of G. Boolos (1997) for Cantor's theorem on the non-equinumerosity of a set with its powerset‎. ‎Then‎, ‎in Linear Temporal Logic‎, ‎we show the non-existence of a fixed-point in this logic whose proof resembles the argument of Yablo's paradox (1985‎, ‎1993)‎. ‎Thus‎, ‎Yablo's paradox turns for the first time into a genuine mathematico-logical theorem in the framework of Linear Temporal Logic‎. ‎Again the diagonal schema of the paper is used in this proof; and it is also shown that G. Priest's inclosure schema (1997) can fit in our universal diagonal / fixed-point schema‎. ‎We also show the existence of dominating (Ackermann-like) functions (which dominate a given countable set of functions‎, ‎such as primitive recursive functions) in the schema. (shrink)

This paper corrects a mistake I saw students make but I have yet to see in print. The mistake is thinking that logically equivalent propositions have the same counterexamples—always. Of course, it is often the case that logically equivalent propositions have the same counterexamples: “every number that is prime is odd” has the same counterexamples as “every number that is not odd is not prime”. The set of numbers satisfying “prime but not odd” is the same as the set of (...) numbers satisfying “not odd but not not-prime”. The mistake is thinking that every two logically-equivalent false universal propositions have the same counterexamples. Only false universal propositions have counterexamples. A counterexample for “every two logically-equivalent false universal propositions have the same counterexamples” is two logically-equivalent false universal propositions not having the same counterexamples. The following counterexample arose naturally in my sophomore deductive logic course in a discussion of inner and outer converses. “Every even number precedes every odd number” is counterexemplified only by even numbers, whereas its equivalent “Every odd number is preceded by every even number” is counterexemplified only by odd numbers. Please let me know if you see this mistake in print. Also let me know if you have seen these points discussed before. I learned them in my own course: talk about learning by teaching! (shrink)

Horgan and Timmons, with their Moral Twin Earth arguments, argue that the new moral realism falls prey to either objectionable relativism or referential indeterminacy. The Moral Twin Earth thought experiment on which the arguments are based relies in crucial ways on the use of intuitions. First, it builds on Putnam’s well-known Twin Earth example and the conclusions drawn from that about the meaning of kind names. Further, it relies on the intuition that were Earthers and Twin (...) Earthers to meet, they would be able to have genuine moral disagreements. I will argue that the similarities with Putnam’s thought experiment are questionable and so the reliance on Putnam-like intuitions is questionable. I will then further argue that even if we accept the intuitions that Horgan and Timmons rely on, the anti-realist conclusion is not warranted due to there being more to the meaning of kind terms than the argument assumes. Once we develop the meaning of kind terms further we can acknowledge both that Earthers and Twin Earthers refer to different properties with their moral terms, and that in spite of that they can have a substantive disagreement due to a shared meaning component. (shrink)

A brief account of Putnam's Twin Earth thought experiment and its significance, written for a general audience. Sadly, I can't include Helen De Cruz's drawing of it - check out the beautiful book!

In this paper I revisit an important response to the Moral Twin Earth (MTE) challenge: The Common Functional Role strategy (CFR). I argue that CFR is open to a revenge problem. MTE-cases allegedly show that two linguistic communities can be in genuine disagreement even when they are regulated by distinct families of properties. CFR provides a way to reconcile the intuition that the two communities are in genuine disagreement with the claim that the use of moral terms by both (...) communities is causally regulated by different families of properties. This is done by identifying a functional role that those families of properties both fulfill. Still, even if CFR is successful, its proponents need to face a serious revenge problem. Roughly, it could be that the families of properties that regulate each community are equally perfect realizers of the relevant higher-order functional state. I suggest that the proponent of CFR faces a dilemma: either CFR has controversial implications about first-order moral theory, or CFR needs to be coupled with substantive and parochial empirical/metaphysical assumptions to avoid those implications. (shrink)

In a recent paper in this journal, McCall and Lowe (2003) argue that an understanding of Special Relativity reveals that the A theorist’s notion of temporal passage is consistent with the B theory of time. They arrive at this conclusion by considering the twins’ paradox, where one of two twins (T) travels to Alpha Centauri and back and upon her return has aged 30 years, while her earth-bound twin (S) has aged 40 years. This paper argues that their account (...) of temporal passage fails to reconcile the A theoretic notion of temporal passage with the B theory of time since the B theorist is at liberty to adopt this as an account of temporal passage as understood by the B theorist. (shrink)

I argue that the empirical literature on priming effects does not warrant nor suggest the conclusion, drawn by prominent psychologists such as J. A. Bargh, that we have no free will or less free will than we might think. I focus on a particular experiment by Bargh – the ‘elderly’ stereotype case in which subjects that have been primed with words that remind them of the stereotype of the elderly walk on average slower out of the experiment’s room than control (...) subjects – and I show that we cannot say that subjects cannot help walking slower or that they are not free in doing do. I then illustrate how these cases can be reconciled and normalized within a Davidsonian theory of action to show that, in walking slower, subjects are acting intentionally. My argument applies across various experiments, including those of goal priming. In the final section I argue that the only cases in which priming effects are efficacious are so-called Buridan cases. (shrink)

Horgan and Timmons have argued that our intuitions about the semantics of non-moral language and moral language differ, and that while twin-earth semantic intuitions generate one result in Putnam´s twater case, moral twin-earth fails to generate comparable results for moral terms. Horgan and Timmon´s conclude from this that the semantic norms governing the use of natural kind terms differ from the semantic norms governing the use of moral terms. I will argue that Horgan and Timmons’ intuitive moral (...) class='Hi'>twin-earth argument fails to derail the new moral realism. Further, I will discuss Boyd’s semantic theory and raise problems for it that do not rely on the use of moral twin-earth. (shrink)

Fetal reduction is the practice of reducing the number of fetuses in a multiple pregnancy, such as quadruplets, to a twin or singleton pregnancy. Use of assisted reproductive technologies increases the likelihood of multiple pregnancies, and many fetal reductions are done after in vitro fertilisation and embryo transfer, either because of social or health-related reasons. In this paper, I apply Joe Horton’s all or nothing problem to the ethics of fetal reduction in the case of a twin pregnancy. (...) I argue that in the case of a twin pregnancy, there are two intuitively plausible claims: abortion is morally permissible, and it is morally wrong to abort just one of the fetuses. But since we should choose morally permissible acts rather than impermissible ones, the two claims lead to another highly implausible claim: the woman ought to abort both fetuses rather than only one. Yet, this does not seem right. A plausible moral theory cannot advocate such a pro-death view. Or can it? I suggest ways to solve this problem and draw implications for each solution. There are no data in this work. (shrink)

In this article, I examine a new concept of “Twin Memory’ which has emerged in memory classification research of conscious and unconscious memory representations. It is to analyse the presence of twin memory among the various memory systems, and also to provide a platform for the twin memory “anatomy” in the field of cognitive science, neuropsychology and neuroscience.

It is often assumed that believing that p for a normative reason consists in nothing more than (i) believing that p for a reason and (ii) that reason’s corresponding to a normative reason to believe that p, where (i) and (ii) are independent factors. This is the Composite View. In this paper, we argue against the Composite View on extensional and theoretical grounds. We advocate an alternative that we call the Prime View. On this view, believing for a normative reason (...) is a distinctive achievement that isn’t exhausted by the mere conjunction of (i) and (ii). Its being an achievement entails that (i) and (ii) are not independent when one believes for a normative reason: minimally, (i) must hold because (ii) holds. Apart from its intrinsic interest, our discussion has important upshots for central issues in epistemology, including the analysis of doxastic justification, the epistemology of perception, and the place of competence in epistemology. (shrink)

In response to recent work on the aggregation of individual judgments on logically connected propositions into collective judgments, it is often asked whether judgment aggregation is a special case of Arrowian preference aggregation. We argue for the converse claim. After proving two impossibility theorems on judgment aggregation (using "systematicity" and "independence" conditions, respectively), we construct an embedding of preference aggregation into judgment aggregation and prove Arrow’s theorem (stated for strict preferences) as a corollary of our second result. Although we (...) thereby provide a new proof of Arrow’s theorem, our main aim is to identify the analogue of Arrow’s theorem in judgment aggregation, to clarify the relation between judgment and preference aggregation, and to illustrate the generality of the judgment aggregation model. JEL Classi…cation: D70, D71.. (shrink)

Representation theorems are often taken to provide the foundations for decision theory. First, they are taken to characterize degrees of belief and utilities. Second, they are taken to justify two fundamental rules of rationality: that we should have probabilistic degrees of belief and that we should act as expected utility maximizers. We argue that representation theorems cannot serve either of these foundational purposes, and that recent attempts to defend the foundational importance of representation theorems are unsuccessful. As a result, we (...) should reject these claims, and lay the foundations of decision theory on firmer ground. (shrink)

Moeser suggested participants default to linear ordering elements but they can be primed to impose either linear or partial ordering. This study seems problematic insofar as ‘greater than’ might be understood to incline participants to favor linear orderings. Recent follow-up studies strongly suggest participants do not default to linear ordering. It seems plausible, moreover, that the observed priming effect is far more pervasive than Moeser countenanced. The present work explores the extent to which priming for linear or partial orders conflicts (...) with the construction of linear or partial ordering lists. Our two experiments suggest friction between priming and task, and also show once the priming tasks are cleared of potentially confounding semantic content, participants perform equally with respect to constructing linear or partial orders under similar priming conditions. (shrink)

In this paper I argue against Twin-Earth externalism. The mistake that Twin Earth arguments rest on is the failure to appreciate the force of the following dilemma. Some features of things around us do matter for the purposes of conceptual classification, and others do not. The most plausible way to draw this distinction is to see whether a certain feature enters the cognitive perspective of the experiencing subject in relation to the kind in question or not. If it (...) does, we can trace conceptual differences to internal differences. If it doesn’t, we do not have a case of conceptual difference. Neither case supports Twin Earth externalism. (shrink)

In this paper I develop an argument against applying a causal theory of mental content to normative concepts. This argument—which I call the Intrapersonal Normative Twin Earth Argument—is inspired by Terry Horgan and Mark Timmons’ Moral Twin Earth Argument. The focus of Horgan and Timmons’ argument is showing that causal theories of mental content conflict with plausible claims about interpersonal normative disagreement. The Intrapersonal Normative Twin Earth Argument, by contrast, is focused on showing that such theories struggle (...) to vindicate plausible claims concerning whether two of an agent’s token normative thoughts have the same or distinct content. (shrink)

Digital Twins are conceptualised in the academic technical discourse as real-time realistic digital representations of physical entities. Originating from product engineering, the Digital Twin quickly advanced into other fields, including the life sciences and earth sciences. Digital Twins are seen by the tech sector as the new promising tool for efficiency and optimisation, while governmental agencies see it as a fruitful means for improving decision-making to meet sustainability goals. A striking example of the latter is the European Commission who (...) wishes to delegate a significant role to Digital Twins in addressing climate change and supporting Green Deal policy. As Digital Twins give rise to high expectations, ambitions, and are being entrusted important societal roles, it is crucial to critically reflect on the nature of Digital Twins. In this article, we therefore philosophically reflect on Digital Twins by critically analysing dominant conceptualisations, the assumptions underlying them, and their normative implications. We dissect the concept and argue that a Digital Twin does not merely fulfil the role of being a representation, but is in fact a steering technique used to direct a physical entity towards certain goals by means of multiple representations. Currently, this steering seems mainly fuelled by a reductionist approach focused on efficiency and optimisation. However, this is not the only direction from which a Digital Twin can be thought and, consequently, designed and deployed. We therefore set an agenda based on a critical understanding of Digital Twins that helps to draw out their beneficial potential, while addressing their potential issues. (shrink)

Whether some condition is equivalent to a conjunction of some conditions has been a major issue in analytic philosophy. Examples include: knowledge, acting freely, causation, and justice. Philosophers have striven to offer analyses of these, and other concepts, by showing them equivalent to such a conjunction. Timothy Williamson offers a number of arguments for the idea that knowledge is ‘prime’, hence not equivalent to or composed by some such conjunction. I focus on one of his arguments: the requirement that such (...) conjuncts must be freely recombinable. Although there has been a great deal of discussion of Williamson's arguments, the flaw I describe has gone unnoticed. Williamson's argument is expressed in terms of conditions, and cases of the condition. Does the condition include specific information, or is the specific information only part of the case? His argument equivocates between more and less general specifications of the conditions. Once this distinction is clarified, his argument can be seen to be vitiated by this conflation. Neither option yields a sound argument for Williamson's desired conclusion. (shrink)

Michael Scriven’s (1959) example of identical twins (who are said to be equal in fitness but unequal in their reproductive success) has been used by many philosophers of biology to discuss how fitness should be defined, how selection should be distinguished from drift, and how the environment in which a selection process occurs should be conceptualized. Here it is argued that evolutionary theory has no commitment, one way or the other, as to whether the twins are equally fit. This is (...) because the theory of natural selection is fundamentally about the fitnesses of traits, not the fitnesses of token individuals. A plausible philosophical thesis about supervenience entails that the twins are equally fit if they live in identical environments, but evolutionary biology is not committed to the thesis that the twins live in identical environments. Evolutionary theory is right to focus on traits, rather than on token individuals, because the fitnesses of token organisms (as opposed to their actual survivorship and degree of reproductive success) are almost always unknowable. This point has ramifications for the question of how Darwin’s theory of evolution and R. A. Fisher’s are conceptually different. (shrink)

In special cases of partial twinning, two heads, each supporting a more-orless normal human mental life, emerge from a single torso. It is often argued that there must be two people in such a case, even if there is only one biological organism. That would pose a problem for ‘animalism’, the view that people are organisms. The paper argues that it is very hard to say what sort of non-organisms the people in such cases would be. Reflection on partial twinning (...) is no more comfortable for those who think we’re not organisms than for those who think we are. We may have to accept that a single person could have two separate mental lives. (shrink)

We give a review and critique of jury theorems from a social-epistemology perspective, covering Condorcet’s (1785) classic theorem and several later refinements and departures. We assess the plausibility of the conclusions and premises featuring in jury theorems and evaluate the potential of such theorems to serve as formal arguments for the ‘wisdom of crowds’. In particular, we argue (i) that there is a fundamental tension between voters’ independence and voters’ competence, hence between the two premises of most jury theorems; (...) (ii) that the (asymptotic) conclusion that ‘huge groups are infallible’, reached by many jury theorems, is an artifact of unjustified premises; and (iii) that the (nonasymptotic) conclusion that ‘larger groups are more reliable’, also reached by many jury theorems, is not an artifact and should be regarded as the more adequate formal rendition of the ‘wisdom of crowds’. (shrink)

Time contraction is not a property of the moving body. Permanent contraction is against the constancy of the speed of light. Einstein uses wrong units of measurement and comes to wrong conclusions. We use the translation of the article "On the electrodynamics of moving bodies".

To counter a general belief that all the paradoxes stem from a kind of circularity (or involve some self--reference, or use a diagonal argument) Stephen Yablo designed a paradox in 1993 that seemingly avoided self--reference. We turn Yablo's paradox, the most challenging paradox in the recent years, into a genuine mathematical theorem in Linear Temporal Logic (LTL). Indeed, Yablo's paradox comes in several varieties; and he showed in 2004 that there are other versions that are equally paradoxical. Formalizing these (...) versions of Yablo's paradox, we prove some theorems in LTL. This is the first time that Yablo's paradox(es) become new(ly discovered) theorems in mathematics and logic. (shrink)

It is widely believed that Twin-Earth-style thought experiments show that the contents of a person's thoughts fail to supervene on her intrinsic properties. Several recent philosophers have made the further claim that Twin-Earth-style thought experiments produce metaphysically necessary conditions for the possession of certain concepts. I argue that the latter view is false, and produce counterexamples to several proposed conditions. My thesis is of particular interest because it undermines some attempts to show that externalism is incompatible with privileged (...) access. (shrink)

This paper presents two cases demonstrating that theories allowing the environment to partially determine the content of moral concepts provide incorrect truth-conditions for moral terms. While typical Moral Twin Earth cases seek to establish that these theories fail to account formoral disagreement, neither case here essentially involves interpersonal disagreement. Both involve a single person retaining moral beliefs despite recognizing actual or potential mismatches with the purportedly content-determining facts. This lets opponents of such theories grant objections that standard Moral (...) class='Hi'>Twin Earth cases fail to demonstrate disagreement, and argue more straightforwardly that they generate implausible truth-conditions for moral claims. (shrink)

In the present paper, we prove the normalization theorem and the consistency of the first-order classical logic with disjunctive syllogism. First, we propose the natural deduction system SCD for classical propositional logic having rules for conjunction, implication, negation, and disjunction. The rules for disjunctive syllogism are regarded as the rules for disjunction. After we prove the normalization theorem and the consistency of SCD, we extend SCD to the system SPCD for the first-order classical logic with disjunctive syllogism. It (...) can be shown that SPCD is conservative extension to SCD. Then, the normalization theorem and the consistency of SPCD are given. (shrink)

Mark Timmons and Terry Horgan have argued that the new moral realism, which rests on the causal theory of reference, is untenable. While I do agree that the new moral realism is untenable, I do not think that Timmons and Horgan have succeeded in showing that it is. I will lay out the case for new moral realism and Horgan and Timmons’ argument against it, and then argue that their argument fails. Further, I will discuss Boyd’s semantic theory as well (...) as attempts to improve upon it, raise serious problems for these semantic accounts, and suggest an alternative view that accounts for our use of moral terms. (shrink)

Sri Guru Granth Sahib, the holy scripture of the Sikhs, contains numerous references to the worship of the divine in Nature. The Sikh scripture declares that human beings' purpose is to achieve a blissful state and be in harmony with the Earth and all creation. Millions of Sikhs recite Gurbani daily wherein the divine is remembered using the symbolism from Nature, esp. air, water, sun, moon, trees, animals, and the Earth. The human mind loses communion with Nature and ultimately with (...) God by being self-conceited. It causes misery all around, is a repeated assertion of Sikhism. The contemporary environmental crisis is an outcome of the actions of such a self-conceited human mind. By affirming God's immanence and His presence in the creation, the Sikh religion imparts the spirit of self-righteousness to the entire subject of Nature. Sikhism is a remarkable religious and cultural phenomenon; several important themes emerge within its universe of beliefs. On the ecological front, the theology of Sikhism suggests that humans must live in harmony with Nature. The Sikh Gurus exemplified many of these teachings, and their examples continue to inspire contemporary social, religious, and environmental leaders in their efforts to protect the planet. In this presentation, the prime environmental teachings of Sikhism are shared with the community. (shrink)

Any intermediate propositional logic can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert’s $\varepsilon $ -calculus. The first and second $\varepsilon $ -theorems for classical logic establish conservativity of the $\varepsilon $ -calculus over its classical base logic. It is well known that the second $\varepsilon $ -theorem fails for the intuitionistic $\varepsilon $ -calculus, as prenexation is impossible. The paper investigates the effect of adding critical $\varepsilon $ (...) - and $\tau $ -formulas and using the translation of quantifiers into $\varepsilon $ - and $\tau $ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ${\varepsilon \tau }$ -calculi. The “extended” first $\varepsilon $ -theorem holds if the base logic is finite-valued Gödel–Dummett logic, and fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second $\varepsilon $ -theorem also holds for finite-valued first-order Gödel logics. The methods used to prove the extended first $\varepsilon $ -theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic. (shrink)

A lean hylomorphism stands as a metaphysical holy grail. An embarrassing feature of traditional hylomorphic ontologies is prime matter. Prime matter is both so basic that it cannot be examined (in principle) and its engagement with the other hylomorphic elements is far from clear. One particular problem posed by prime matter is how it is to be understood both as a principle of individuation for material substances and as pure potency. I present Thomas Aquinas’s way of squeezing some intelligibility out (...) of prime matter by modeling it on the idea of logical genus. Such a modeling provides insight into understanding prime matter as substratum, as maximally indeterminate, and as ontologically vague. One of the unusual but exciting things that fall out from this analysis of prime matter is the Entirety Thesis: “For any substance x, if x has prime matter then the prime matter of x is the same* as x,” where ‘same*’ is understood as “indeterminately identical.”. (shrink)

The “four-color” theorem seems to be generalizable as follows. The four-letter alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA plan of any alive being. Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters. That admits to be formulated as a “four-letter theorem”, and thus one can (...) search for a properly mathematical proof of the statement. It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one. Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions. The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letter theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted. (shrink)

The previous two parts of the paper demonstrate that the interpretation of Fermat’s last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen - Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason’s theorem and partly similar to that in Part II. The concept of (probabilistic) measure (...) of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason’s theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason’s theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about “quantum number theory”. It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the “nonstandard bijection” and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart. (shrink)

The paper is a continuation of another paper published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum (...) contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense, in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem. (shrink)

Peer review is often taken to be the main form of quality control on academic research. Usually journals carry this out. However, parts of maths and physics appear to have a parallel, crowd-sourced model of peer review, where papers are posted on the arXiv to be publicly discussed. In this paper we argue that crowd-sourced peer review is likely to do better than journal-solicited peer review at sorting papers by quality. Our argument rests on two key claims. First, crowd-sourced peer (...) review will lead on average to more reviewers per paper than journal-solicited peer review. Second, due to the wisdom of the crowds, more reviewers will tend to make better judgments than fewer. We make the second claim precise by looking at the Condorcet Jury Theorem as well as two related jury theorems developed specifically to apply to peer review. (shrink)

Strategies for effectively communicating scientific findings to the public are an important and growing area of study. Recognizing that some complex subjects require recipients of information to take a more active role in constructing an understanding, we sought to determine whether it was possible to increase subjects’ intellectual effort via “priming” methodologies. In particular, we asked whether subconsciously priming “intellectual virtues”, such as curiosity, perseverance, patience, and diligence might improve participants’ effort and performance on various cognitive tasks. In the first (...) experiment, we found no significant differences in either effort or understanding between IV-primed and neutrally-primed individuals across two different priming techniques. The second experiment measured the effect of IV-priming on intellectual effort in simpler, shorter-duration puzzles and exploration activities; here, we observed an effect, but given its low strength and short duration, we do not believe that priming of IVs is a promising strategy for science communication. (shrink)

If the concept of “free will” is reduced to that of “choice” all physical world share the latter quality. Anyway the “free will” can be distinguished from the “choice”: The “free will” involves implicitly certain preliminary goal, and the choice is only the mean, by which it can be achieved or not by the one who determines the goal. Thus, for example, an electron has always a choice but not free will unlike a human possessing both. Consequently, and paradoxically, the (...) determinism of classical physics is more subjective and more anthropomorphic than the indeterminism of quantum mechanics for the former presupposes certain deterministic goal implicitly following the model of human freewill behavior. The choice is usually linked to very complicated systems such as human brain or society and even often associated with consciousness. In its background, the material world is deterministic and absolutely devoid of choice. However, quantum mechanics introduces the choice in the fundament of physical world, in the only way, in which it can exist: All exists in the “phase transition” of the present between the uncertain future and the well-ordered past. Thus the present is forced to choose in order to be able to transform the coherent state of future into the well-ordering of past. The concept of choice as if suggests that there is one who chooses. However quantum mechanics involves a generalized case of choice, which can be called “subjectless”: There is certain choice, which originates from the transition of the future into the past. Thus that kind of choice is shared of all existing and does not need any subject: It can be considered as a low of nature. There are a few theorems in quantum mechanics directly relevant to the topic: two of them are called “free will theorems” by their authors, Conway and Kochen, and according to them: “Do we really have free will, or, as a few determined folk maintain, is it all an illusion? We don’t know, but will prove in this paper that if indeed there exist any experimenters with a modicum of free will, then elementary particles must have their own share of this valuable commodity” “The import of the free will theorem is that it is not only current quantum theory, but the world itself that is non-deterministic, so that no future theory can return us to a clockwork universe”. Those theorems can be considered as a continuation of the so-called theorems about the absence of “hidden variables” in quantum mechanics. (shrink)

Various cases of conjoined twinning have been presented as problems for the animalist view that we are animals. In some actual and possible cases of human dicephalus that have been discussed in the literature, it is arguable that there are two persons but only one human animal. It is also tempting to believe that there are two persons and one animal in possible instances of craniopagus parasiticus that have been described. Here it is argued that the animalist can admit that (...) these are cases in which human persons are not animals, without forfeiting the title “animalist.” It is also shown that this is not only an option but also a well-motivated and plausible option for the animalist. Seeing this requires getting clear on what the word “we” should be thought to include in the animalist's claim that we are animals. Here animalism is defended against twinning objections by figuring out how to view the scope of the animalist's identity claim. (shrink)

This paper offers a simple response to the Moral Twin Earth (MTE) objection to Naturalist Moral Realism (NMR). NMR typically relies on an externalist metasemantics such as a causal theory of reference. The MTE objection is that such a theory predicts that terms like ‘good’ and ‘right’ have a different reference in certain twin communities where it’s intuitively clear that the twins are talking about the same thing when using ‘good’. I argue that Boyd’s causal regulation theory, the (...) original target of the MTE objection, was never vulnerable to this objection. The theory contains an epistemic constraint on reference which implies that either the property that causally regulates uses of ‘good’ isn’t different for the twin communities or, in scenarios where the reference is different, the communities diverge in ways where it’s not intuitively clear that ‘good’ has the same reference for them. (shrink)

Computer assisted theorem proving is an increasingly important part of mathematical methodology, as well as a long-standing topic in artificial intelligence (AI) research. However, the current generation of theorem proving software have limited functioning in terms of providing new proofs. Importantly, they are not able to discriminate interesting theorems and proofs from trivial ones. In order for computers to develop further in theorem proving, there would need to be a radical change in how the software functions. Recently, (...) machine learning results in solving mathematical tasks have shown early promise that deep artificial neural networks could learn symbolic mathematical processing. In this paper, I analyze the theoretical prospects of such neural networks in proving mathematical theorems. In particular, I focus on the question how such AI systems could be incorporated in practice to theorem proving and what consequences that could have. In the most optimistic scenario, this includes the possibility of autonomous automated theorem provers (AATP). Here I discuss whether such AI systems could, or should, become accepted as active agents in mathematical communities. (shrink)

An excerpt of "The SignalGlyph Project and Prime Numbers" (a chapter of the book THE IMAGE OF LANGUAGE) that attempts to illustrate how dimensional limitations of mathematical language have obscured recognition of the system of patterning in the distribution of prime numbers.

The standard representation theorem for expected utility theory tells us that if a subject’s preferences conform to certain axioms, then she can be represented as maximising her expected utility given a particular set of credences and utilities—and, moreover, that having those credences and utilities is the only way that she could be maximising her expected utility. However, the kinds of agents these theorems seem apt to tell us anything about are highly idealised, being always probabilistically coherent with infinitely precise (...) degrees of belief and full knowledge of all a priori truths. Ordinary subjects do not look very rational when compared to the kinds of agents usually talked about in decision theory. In this paper, I will develop an expected utility representation theorem aimed at the representation of those who are neither probabilistically coherent, logically omniscient, nor expected utility maximisers across the board—that is, agents who are frequently irrational. The agents in question may be deductively fallible, have incoherent credences, limited representational capacities, and fail to maximise expected utility for all but a limited class of gambles. (shrink)