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)

In this chapter we introduce concepts for analyzing proofs, and for analyzing undergraduate and beginning graduate mathematics students’ proving abilities. We discuss how coordination of these two analyses can be used to improve students’ ability to construct proofs. -/- For this purpose, we need a richer framework for keeping track of students’ progress than the everyday one used by mathematicians. We need to know more than that a particular student can, or cannot, prove theorems by induction or contradiction or (...) can, or cannot, prove certain theorems in beginning set theory or analysis. It is more useful to describe a student’s work in terms of a finer-grained framework that includes various smaller abilities that contribute to proving and that can be learned in differing ways and at differing periods of a student’s development. (shrink)

It is quite common to object to an argument by saying that it “proves too much.” In this paper, I argue that the “proving too much” charge can be understood in at least three different ways. I explain these three interpretations of the “proving too much” charge. I urge anyone who is inclined to level the “proving too much” charge against an argument to think about which interpretation of that charge one has in mind.

This paper proves a precisification of Hume’s Law—the thesis that one cannot get an ought from an is—as an instance of a more general theorem which establishes several other philosophically interesting, though less controversial, barriers to logical consequence.

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...) and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

The hard problem of induction is to argue without begging the question that inductive inference, applied properly in the proper circumstances, is conducive to truth. A recent theorem seems to show that the hard problem has a deductive solution. The theorem, provable in ZFC, states that a predictive function M exists with the following property: whatever world we live in, M ncorrectly predicts the world’s present state given its previous states at all times apart from a well-ordered subset. On the (...) usual model of time a well-ordered subset is small relative to the set of all times. M’s existence therefore seems to provide a solution to the hard problem. My paper argues for two conclusions. First, the theorem does not solve the hard problem of induction. More positively though, it solves a version of the problem in which the structure of time is given modulo our choice of set theory. (shrink)

The concept of burden of proof is used in a wide range of discourses, from philosophy to law, science, skepticism, and even in everyday reasoning. This paper provides an analysis of the proper deployment of burden of proof, focusing in particular on skeptical discussions of pseudoscience and the paranormal, where burden of proof assignments are most poignant and relatively clear-cut. We argue that burden of proof is often misapplied or used as a mere rhetorical gambit, with little appreciation of the (...) underlying principles. The paper elaborates on an important distinction between evidential and prudential varieties of burdens of proof, which is cashed out in terms of Bayesian probabilities and error management theory. Finally, we explore the relationship between burden of proof and several (alleged) informal logical fallacies. This allows us to get a firmer grip on the concept and its applications in different domains, and also to clear up some confusions with regard to when exactly some fallacies (ad hominem, ad ignorantiam, and petitio principii) may or may not occur. (shrink)

Justification of theorems plays a vital role in any rational human activity. It is indispensable in science. The deductive method of justifying theorems is used in all sciences and it is the only method of justifying theorems in deductive disciplines. It is based on the notion of proof, thus it is a method of proving theorems. In the Warsaw School of Logic (WSL) – the famous branch of the Lvov-Warsaw School (LWS) – two types of the method: axiomatic deduction (...) method and natural deduction method were developed and practiced. In this paper, both of these methods are briefly discussed with an emphasis on their historical, groundbreaking significance for logic. The axiomatic method by means of rejection (proposed by Jan Łukasiewicz – a co-creator of the WSL), which is the method of the so-called rejection proof (rejection/refutation method) in logical systems and the proving method of generalized natural deduction, which is a hybrid deduction–refutation method of proving theorems, are also outlined in the paper. The author discusses their historical significance. This paper also contains a brief mention of the most significant results which the application of the discussed methods introduced into contemporary scientific research, not only logical one. (shrink)

The endurance of depression, anxiety and suicidal ideation among gay and bisexual men persists despite advances in civil rights and wider social acceptance. While minority stress theory provides a framework for much scholarly debate as to the causes of mental distress among non-heterosexual men, there is a growing interest into the detrimental effects that competitiveness within the gay community itself can have. Past studies have celebrated involvement in gay culture as being associated with better mental health outcomes by tempering the (...) impact of hegemonic heteronormativity. Yet between non-heterosexual men and the general population, there are stark mental health inequalities that require investigation. As a means of proving manhood, men in general are predisposed toward competitiveness and risk, but within a subculture where the attention is exclusively on male sex, the focus is primarily status conscious. This article draws on minority stress theory to consider societal discrimination. It also applies inter-minority gay community stress theory to explore pressures emanating from gay spaces with fixations on masculinity, income and rivalry as major sources of mental health problems in gay and bisexual men. The causes for health disparities illustrated in this article, demonstrate a critical need for public health and social care organizations to respond with innovatory services, based on a firm understanding of stressors arising from interactions between men in gay spaces. (shrink)

This note considers deductive systems for the operator a of unprovability in some particular propositional normal modal logics. We give thus complete syntactic characterization of these logics in the sense of Lukasiewicz: for every formula  either `  or a  (but not both) is derivable. In particular, purely syntactic decision procedure is provided for the logics under considerations.

How can Buddhists prove that non-existent things do not exist? With great difficulty. For the Buddhist, this is not a laughing matter as they are largely global error theorists and, thus, many things are non-existent. The difficulty gets compounded as the Buddhist and their opponent, the non-Buddhist of various kinds, both agree that one cannot prove a thesis whose subject is non-existent. In this paper, I will first present a difficulty that Buddhist philosophers have faced in proving that what (...) they take to be non-existent does not exist. I will then survey two main solutions that they have provided. Those 'solutions' may not solve the problem or solve the problem but creates other problems. I will not survey the Buddhist treatment of the problem of proving about non-existence in order to present a new solution that we have yet to see. Instead, I will articulate a problem about non-existence that is unique to Buddhist philosophers. I will do so in order to present an interesting puzzle about non-existence that has largely escaped attention in the 'Western' literature. (shrink)

A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of n = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. The infinite (...) descent is linked to induction starting from n = 3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for n = 4, one can suggest that the proof for n ≥ 4 was accessible to him. An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995). (shrink)

Kant claims that Aristotles logic as complete, explain the historical and philosophical considerations that commit him to proving the completeness claim and sketch the proof based on materials from his logic corpus. The proof will turn out to be an integral part of Kant’s larger reform of formal logic in response to a foundational crisis facing it.

The Elementary Process Theory (EPT) is a collection of seven elementary process-physical principles that describe the individual processes by which interactions have to take place for repulsive gravity to exist. One of the two main problems of the EPT is that there is no proof that the four fundamental interactions (gravitational, electromagnetic, strong, and weak) as we know them can take place in the elementary processes described by the EPT. This paper sets forth the method by which it can be (...) proven that the EPT agrees with the knowledge that derives from the successful predictions of a modern interaction theory T. This determines a fundamentally new research program in theoretical physics. (shrink)

In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n (...) = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it. (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)

When faced with a rule that they take to be true, and a recalcitrant example, people are apt to say: “The exception proves the rule”. When pressed on what they mean by this though, things are often less than clear. A common response is to dredge up some once-heard etymology: ‘proves’ here, it is often said, means ‘tests’. But this response—its frequent appearance even in some reference works notwithstanding1—makes no sense of the way in which the expression is used. To (...) insist that the exception proves the rule is to insist that whilst this is an exception, the rule still stands; and furthermore, that, rather than undermining the rule, the exception serves to confirm it. This second claim may seem paradoxical, but it should not, once it is realized that what does the confirming is not the exception itself, but rather the fact that we judge it to be an exception; and that what is confirmed is not the rule itself, but rather the fact that we judge it to be a rule. To treat something as an exception is not to treat it as a counterexample that refutes the existence of the rule. Rather it is to treat it as special, and so to concede the rule from which it is excepted. The point comes clearly in the original (probably 17th Century) Latin form: Exceptio probat (figit2) regulam in casibus non exceptis. Exception (i.e. the act of excepting) proves (establishes) the rule in the cases not excepted. Clearly this form of reasoning cannot apply when the rule that we are considering has the form of a simple universal generalization. Here there can be no exceptions, only counterexamples. So what we need, and what will be developed.. (shrink)

A system for the modal logic K furnishes a simple mechanical process for proving theorems.

Skeptical theism is a popular response to arguments from evil. Many hold that it undermines a key inference often used by such arguments. However, the case for skeptical theism is often kept at an intuitive level: no one has offered an explicit argument for the truth of skeptical theism. In this article, I aim to remedy this situation: I construct an explicit, rigorous argument for the truth of skeptical theism.

In this largely theoretical article, we discuss the relation between a kind of affect, behavioural schemas and aspects of the proving process. We begin with affect as described in the mathematics education literature, but soon narrow our focus to a particular kind of affect – nonemotional cognitive feelings. We then mention the position of feelings in consciousness because that bears on the kind of data about feelings that students can be expected to be able to report. Next we introduce (...) the idea of behavioural schemas as enduring mental structures that link situations to actions, in other words, habits of mind, that appear to drive many mental actions in the proving process. This leads to a discussion of the way feelings can both help cause mental actions and also arise from them. Then we briefly describe a design experiment – a course intended to help advanced undergraduate and beginning graduate mathematics students improve their proving abilities. Finally, drawing on data from the course, along with several interviews, we illustrate how these perspectives on affect and on behavioural schemas appear to explain, and are consistent with, our students’ actions. (shrink)

REVIEW OF: Automated Development of Fundamental Mathematical Theories by Art Quaife. (1992: Kluwer Academic Publishers) 271pp. Using the theorem prover OTTER Art Quaife has proved four hundred theorems of von Neumann-Bernays-Gödel set theory; twelve hundred theorems and definitions of elementary number theory; dozens of Euclidean geometry theorems; and Gödel's incompleteness theorems. It is an impressive achievement. To gauge its significance and to see what prospects it offers this review looks closely at the book and the proofs it presents.

It is argued that the goal of Hilbert's program was to prove the model-theoretical consistency of different axiom systems. This Hilbert proposed to do by proving the deductive consistency of the relevant systems. In the extended independence-friendly logic there is a complete proof method for the contradictory negations of independence-friendly sentences, so the existence of a single proposition that is not disprovable from arithmetic axioms can be shown formally in the extended independence-friendly logic. It can also be proved by (...) means of independence-friendly logic that proof-theoretical consistency of a sentence S implies the existence of a model in which S is not false. Hence the consistency of the axioms of arithmetic in the sense of being not-false in a model can be proved. (shrink)

When we look at 800,000 year ice core data CO2 levels since 1950 have risen at a rate of 123-fold faster than the fastest rate in 800,000 years. When we see that this rise is precisely correlated with global carbon emissions the human link to climate change seems certain and any rebuttal becomes ridiculously implausible. The 800,000 year correlation between CO2 and global temperatures seems to be predicting at least 9 degrees C of more warming based on current CO2 levels.

Saint Thomas Aquinas often argues for the immateriality of the intellect and often employs one argument in particular, which he also attributes to Aristotle. As he explains in his Commentary on the De Anima, "Anything that is in potency with respect to an object, and able to receive it into itself, is, as such, without that object; thus the pupil of the eye, being potential to colors and able to receive them, is itself colorless. ... Since then (the intellect) naturally (...) understands all bodily things, it must be lacking in every bodily nature; just as the sense of sight, being able to know color, lacks all color" (Bk. III, Lect. 7, no. 680). The intellect has no bodily nature since it understands all bodily natures, and it is a condition of knowing a power that it lacks the nature it knows. Thomas Russman, however, in A Prospectus for the Triumph of Realism, argues that what we know about the nature of perception invalidates the assumptions of Aristotle and Aquinas, and because this argument for the immateriality of the intellect depends on an analogy with sensation, its conclusion is undermined. Russman reasons that just as a pink retina can receive the form of green without any hindrance, so a material intellect can receive the forms of all material things without any hindrance. Since Aristotle and Aquinas turned out to be wrong about their understanding of the senses and used this mistake as a basis for their reasoning about the intellect, there is no reason to accept their conclusion that the intellect is immaterial. Although Aquinas and Russman attribute this argument to Aristotle, the version found in De Anima 3. 4 differs from it in one crucial respect. Whereas Thomas claims that the intellect is immaterial because it knows all bodies, Aristotle says that mind is 'unmixed' because it thinks all things (panta noei).(429a18-20) This difference, I contend, makes the argument as Aristotle actually formulated it immune from the criticism of Russman, for Aristotle can draw on his own distinction between being subject to a material alteration (taking on a form in a material way) and engaging in the activity of knowing (taking on the form of an object without matter by becoming one with it). (shrink)

In this article, I argue that sceptical theists have too narrow a focus: they consider only God’s axiological reasons, ignoring any non-axiological reasons he may have. But this is a mistake: predicting how God will act requires knowing about his reasons in general, and this requires knowing about both God’s axiological and non-axiological reasons. In light of this, I construct and defend a kind of sceptical theism—Deontological Sceptical Theism—that encompasses all of God’s reasons, and briefly illustrate how it renders irrelevant (...) certain charges of excessive sceptical and how it evaporates equiprobability objections. Furthermore, I put forth a simple argument in favor of Deontological Sceptical Theism, which shows that everyone (at least currently) ought to endorse it. (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)

Both of Gettier's examples are not representative of situations in which we would claim knowledge – we do not use language in this way. Therefore, Gettier has not shown that justified true belief is insufficient for knowledge. I am not denying that there is a problem about the definition of knowledge. Several decades earlier, Russell dealt with this problem, using a stopped clock to illustrate it.

Kurt Gödel wrote (1964, p. 272), after he had read Husserl, that the notion of objectivity raises a question: “the question of the objective existence of the objects of mathematical intuition (which, incidentally, is an exact replica of the question of the objective existence of the outer world)”. This “exact replica” brings to mind the close analogy Husserl saw between our intuition of essences in Wesensschau and of physical objects in perception. What is it like to experience a mathematical (...) class='Hi'>proving process? What is the ontological status of a mathematical proof? Can computer assisted provers output a proof? Taking a naturalized world account, I will assess the relationship between mathematics, the physical world and consciousness by introducing a significant conceptual distinction between proving and proof. I will propose that proving is a phenomenological conscious experience. This experience involves a combination of what Kurt Gödel called intuition, and what Husserl called intentionality. In contrast, proof is a function of that process — the mathematical phenomenon — that objectively self-presents a property in the world, and that results from a spatiotemporal unity being subject to the exact laws of nature. In this essay, I apply phenomenology to mathematical proving as a performance of consciousness, that is, a lived experience expressed and formalized in language, in which there is the possibility of formulating intersubjectively shareable meanings. (shrink)

Abstract. The presumptuous philosopher (PP) thought experiment lends more credence to the hypothesis which postulates the existence of a larger number of observers than other hypothesis. The PP was suggested as a purely speculative endeavor. However, there is a class of real world observer-selection effects where it could be applied, and one of them is the possibility of interstellar panspermia (IP). There are two types of anthropic reasoning: SIA and SSA. SIA implies that my existence is an argument that larger (...) total number of observers exists; SSA implies that I should find myself in a region with larger number of observers. However, as S. Armstrong showed, SIA can’t distinguish between different ways how larger number of observers appeared, so it can’t favor IP compared with other ways to get many observers. SSA application here is less controversial as it tells only about relation between regions size: e.g. I am more likely to live in a larger country than in a small one, conditioning that the total number of small countries is also small. Anthropic considerations suggest that the universes in the multiverse with interstellar panspermia will have orders of magnitude more civilizations than universes without IP, and thus we are likely to be in such a universe. This is a strong counterargument against a variant of the Rare Earth hypothesis based on the difficulty of abiogenesis: even if abiogenesis is difficult, IP will disseminate life over billions of planets, meaning we are likely to find ourselves in a universe where IP has happened. This implies that there should be many planets with life in our galaxy, and renders the Fermi paradox even sharper. Either the Great Filter is ahead of us and there are high risks of anthropogenic extinction, or there are many alien civilizations nearby and of the same age as ours—which is itself a global catastrophic risk, as in that case the wave of alien colonization could arrive between 25-500 ky from now. -/- . (shrink)

It is true of many truths that I do not believe them. It is equally true, however, that I cannot rationally assert of any such truth both that it is true and that I do not believe it. To explain why this is so, I will distinguish absence of belief from disbelief and argue that an assertion of “p, but I do not believe that p” is paradoxical because it is indefensible, i.e. for reasons internal to it unable to convince. (...) A closer examination of the irrationality involved will show that such is the skeptic’s predicament, trying to convince us to bracket knowledge claims we have good grounds to take ourselves to be entitled to. Even if the sceptic cannot be proven wrong, his challenge still demands an answer, if not a treatment. In this paper, I argue that the cure lies in epidemiology rather than epistemology: instead of attacking the sceptic head-long, I commend guerilla tactics, vaccinating our fellow non-sceptics against the sceptical virus. I will not argue that the sceptic is wrong, necessarily wrong or that he cannot be believed, but that he cannot convince. Scepticism requires a leap of faith: something we may justifiably refrain from even on the sceptic’s own standards. (shrink)

The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations used (...) in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology. (shrink)

Most mathematicians and many physicists believe that Zeno's paradox and Kant's antinomy were solved by Cantor and others, but this is a misunderstanding. This paper expose such misunderstanding. And once we accept the validity of Zeno's and Kant's arguments, we suggest that their paradoxes are resolved by structural realism and eternalism.

The author has established a mathematical theory about the system of freedom in which components of freedom are ruled by the largest freedom principle, explaining how one invariant reality can be equated with the dynamical universe. Freedom as a whole is the reality, and components of freedom show variable phenomena and become a dynamic system. In freedom, component equality leads to sequence equality; therefore, various sequences coexist in the system. Because there are incompatible sequences for any sequence, the interior of (...) freedom cannot be a static sequence. In order for the system to be a whole, there must be some connecting sequences between any two sequences. Then, at every part of freedom, it is always possible to find a group of three independent sequences that, for most components, is located inside. For the sequence group, there is a sequence through which most components flow in and out. The most abundant three - sequence group and most abundant connecting sequence correspond to the space - time structure. Other incompatible sequences correspond to particles, and interactions between these sequences correspond to interactions between particles. The interactions have some symmetries similar to those in physics, such as SU (3) AND SU (2)×U (1), thus proving the feasibility of the hypothesis: the universe is equivalent with the system of freedom. (shrink)

Mümkün varlıkları aracı kılmadan Vâcibü’l-Vücûd’un varlığını ispatlama çabalarının bir sonucu olan sıddıkîn burhanı ilk defa Müslüman filozoflar tarafından dillendirildi. İbn Sînâ (ö. 428/1037) da Fârâbî’nin etkisiyle yeni bir burhan açıkladı ve buna sıddıkîn adını verdi. Molla Sadrâ (ö. 1050/1641) varlığın asaleti ilkesini mutasavvıflardan, teşkîk ilkesini de Sühreverdî’den iktibas ederek yeni bir sıddıkîn burhanı dillendirdi. Bu burhanın, varlığın asaleti, basîtliği/yalınlığı, teşkîkî ve ma’lûlün illete ihtiyacı olmak üzere bazı öncülleri vardır. O, bu öncülleri açıkladıktan sonra teselsüle ihtiyaç duymadan Vâcibü’l-Vücûd’un varlığını ispatlar. Onun (...) burhanı diğer sıddıkîn burhanlarına nispetle bazı üstün özelliklere sahiptir. Öncelikle bu burhan varlığın asaleti ilkesi üzerine bina edildiğinden dayanağı varlık kavramı değil, varlığın nesnel hakikatidir. İkincisi bu burhan Vâcibü’l-Vücûd’un zâtını ispatlamakla birlikte Vâcibü’l-Vücûd’un birliğini ve kemâlî sıfatlarını ispatlar. Üçüncüsü bu burhana göre mümkün varlıklar hiçbir aracı olmadan Vâcibü’l-Vücûd’a bağımlı olduklarından burhanın açıklanmasında teselsül ve kısır döngüden yararlanma ihtiyacı yoktur. Molla Sadrâ’nın teselsül ve kısır döngüden de yararlanarak Vâcibü’l-Vücûd’un varlığını ispatladığı burhanlar da vardır. Molla Sadrâ bu burhanıyla hem Vâcibü’l-Vücûd’un varlığını ve hem de onun kemâlî sıfatlarını ispatladığını iddia eder. Bu çalışmada bu iddia incelecek ve ispat-ı vacip meselesinde Molla Sadrâ’nın ne gibi yenilikler getirdiği ele alınacaktır. ABSTRACT A result of the attempts to prove the existence of The Necessary Existence without the mediation of possible beings, proof of the truthful was first expressed by Muslim philosophers called it sıddıkin. Mullā Ṣadrā’s “proof” that he explains has certain premises such as the fundamental of existence, indivisibility of the existence, existential gradation, effect’s need for cause. After explanining these premises, Mullā Ṣadrā then proves the Necessary Existence without needing succession. Mullā Ṣadrā’s proof of the truthful has certain superior features tan the proofs of truthful expressed before him. First of all, as this proof is built upon the principle of the fundamental of existence, its foundation is the external truth of existence, not the concept of existence itself. Secondly, this proof proves the unity and perfection attributes of The Necessary Existence as well as proving the existence of the Necessary Existence. Thirdly, according to this proof, as all possible existences are dependent on The Necessary Existence without any mediation, there is no need to benefit from succession and vicious circle in explaining the proof. However, Mullā Ṣadrā proves the Necessary Existence by benefitting from succession and vicious circle in other proofs. Mullā Ṣadrā asserts that this proof demonstrates the presence of The Necessary Existence and its perfection attributions. This study examines this assertion and addresses what kind of innovations was brought by Mullā Ṣadrā in terms of proving The Necessary Existence. (shrink)

Internet users fill in interactive forms with multiple fields, check/uncheck checkboxes, select options and agree to submit. People give their consents without keeping track of them. Dominance of the machine producing human consent is ubiquitous. Humanless bureaucratic procedures become embedded into routine usage of digital products and services automating human behavior. This bureaucracy does not make individuals wait in conveyor-like lines (which sometimes can cause a collective action), it patiently waits or suddenly pops up in an annoying message requiring immediate (...) action and consuming time. It follows pre-installed procedures so cannot be distracted or appealed to. It lacks what Sofia Ranchordás calls ‘administrative empathy’. Users can hardly reach through bureaucratic procedures and communicate with another human. An individual involved in these sequences of actions contributes to increasing the quantity of bureaucratic procedures. The algorithmic bureaucracy power resides in the limited ability of human beings to make conscious and informed decisions when dealing with high complexity levels. This ubiquity makes it virtually impossible for individuals to track and verify statuses and control binding power of all legal actions on the platform. The authors apply critical analysis to explore social implications, threats and possible alternatives for corporate and governmental algorithmic bureaucracy in the emerging digital society. (shrink)

In this paper, I argue that the first and the third premises of the zombie argument cannot be jointly true: zombies are either inconceivable beings or the possible existence of them does not threaten the physicalist standpoint. The tenability of the premises in question depends on how we understand the concept of a zombie. In the paper, I examine three popular candidates to this concept, namely zombies are creatures who lack consciousness, but are identical to us in their (a) functional (...) organization, (b) entire physical makeup, and (c) microphysical structure. The main aim of the paper is to argue that none of these conceptions conveys a consistent zombie-concept to us, which, at the same time, would be dangerous for physicalism. In the conclusion, I argue that the source of this failure can be found in the logical fallaciousness of the argument, namely the premises simply presuppose the truth of the conclusion. (shrink)

The credibility of digital computer simulations has always been a problem. Today, through the debate on verification and validation, it has become a key issue. I will review the existing theses on that question. I will show that, due to the role of epistemological beliefs in science, no general agreement can be found on this matter. Hence, the complexity of the construction of sciences must be acknowledged. I illustrate these claims with a recent historical example. Finally I temperate this diversity (...) by insisting on recent trends in environmental sciences and in industrial sciences. (shrink)

We report on an exploratory study of the way eight mid-level undergraduate mathematics majors read and reflected on four student-generated arguments purported to be proofs of a single theorem. The results suggest that mid-level undergraduates tend to focus on surface features of such arguments and that their ability to determine whether arguments are proofs is very limited -- perhaps more so than either they or their instructors recognize. We begin by discussing arguments (purported proofs) regarded as texts and validations of (...) those arguments, i.e., reflections of individuals checking whether such arguments really are proofs of theorems. We relate the way the mathematics research community views proofs and their validations to ideas from reading comprehension and literary theory. We then give a detailed analysis of the four student-generated arguments and finally analyze the eight students' validations of them. (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)

We explain and explore class-theoretic potentialism—the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the $\mathsf {.2}$ and $\mathsf {.3}$ axioms). We then discuss the significance of these results for the different kinds of class-theoretic potentialists.

The implicit dimension of enthymemes is investigated from a pragmatic perspective to show why a premise can be left unexpressed, and how it can be used strategically. The relationship between the implicit act of taking for granted and the pattern of presumptive reasoning is shown to be the cornerstone of kairos and the fallacy of straw man. By taking a proposition for granted, the speaker shifts the burden of proving its un-acceptability onto the hearer. The resemblance of the tacit (...) premise with what is commonly acceptable or has been actually stated can be used as a rhetorical strategy. (shrink)

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings (...) of modal operators in terms of rules of inference. (shrink)

Conservativeness has been proposed as an important requirement for deflationary truth theories. This in turn gave rise to the so-called ‘conservativeness argument’ against deflationism: a theory of truth which is conservative over its base theory S cannot be adequate, because it cannot prove that all theorems of S are true. In this paper we show that the problems confronting the deflationist are in fact more basic: even the observation that logic is true is beyond his reach. This seems to conflict (...) with the deflationary characterization of the role of the truth predicate in proving generalizations. However, in the final section we propose a way out for the deflationist — a solution that permits him to accept a strong theory, having important truth-theoretical generalizations as its theorems. (shrink)

In this paper we motivate the ‘principles of trust’, chance-credence principles that are strictly stronger than the New Principle yet strictly weaker than the Principal Principle, and argue, by proving some limitative results, that the principles of trust conflict with Humean Supervenience.

In a companion paper (Byrne and Green 2023) we disentangled the main characterizations of naïve realism and argued that illusions provide the best proving ground for naïve realism and its main rival, representationalism. According to naïve realism, illusions never involve perceptual error. We assessed two leading attempts to explain apparent perceptual error away, from William Fish and Bill Brewer, and concluded that they fail. This paper considers another prominent attempt, from Craig French and Ian Phillips, and also sketches the (...) case for representationalism. (shrink)

Different authors offer subtly different characterizations of naïve realism. We disentangle the main ones and argue that illusions provide the best proving ground for naïve realism and its main rival, representationalism. According to naïve realism, illusions never involve per- ceptual error. We assess two leading attempts to explain apparent perceptual error away, from William Fish and Bill Brewer, and conclude that they fail. Another lead- ing attempt is assessed in a companion paper, which also sketches an alternative representational account.

I present a version of Kit Fine's stratified semantics for the logic RWQ and define a natural family of related structures called RW hyperdoctrines. After proving that RWQ is sound with respect to RW hyperdoctrines, we show how to construct, for each stratified model, a hyperdoctrine that verifies precisely the same sentences. Completeness of RWQ for hyperdoctrinal semantics then follows from completeness for stratified semantics, which is proved in an appendix. By examining the base category of RW hyperdoctrines, we (...) find reason to be worried about the ontology of stratified models. (shrink)

Presupposition has been described in the literature as closely related to the listener’s knowledge and the speaker’s beliefs regarding the other’s mind. However, how is it possible to know or believe our interlocutor’s knowledge? The purpose of this paper is to find an answer to this question by showing the relationship between reasoning, presumption and language. Presupposition is analyzed as twofold reasoning process: on the one hand, the speaker by presupposing a proposition presumes that his interlocutor knows it; on the (...) other hand, the listener reconstructs the propositions taken for granted and assesses them against the shared presumptions. The possibility of reconstructing a presupposition is distinguished from its assessment, where the consistency of the presupposition with the shared or common ground is evaluated, and its reasonableness established. The analysis of presuppositions from an argumentative perspective provides an instrument for evaluating the reasonableness of a presupposition and understanding its dialogical effect. On this view, the dialogical force of a presupposition lies in its presumptive nature, which sets and shifts the burden of proving its unacceptability or unreasonableness. (shrink)