I use modal logic and transfinite set-theory to define metaphysical foundations for a general theory of computation. A possible universe is a certain kind of situation; a situation is a set of facts. An algorithm is a certain kind of inductively defined property. A machine is a series of situations that instantiates an algorithm in a certain way. There are finite as well as transfinite algorithms and machines of any degree of complexity (e.g., Turing and super-Turing machines and more). (...) There are physically and metaphysically possible machines. There is an iterative hierarchy of logically possible machines in the iterative hierarchy of sets. Some algorithms are such that machines that instantiate them are minds. So there is an iterative hierarchy of finitely and transfinitely complex minds. (shrink)

The prospects and limitations of defining truth in a finite model in the same language whose truth one is considering are thoroughly examined. It is shown that in contradistinction to Tarski's undefinability theorem for arithmetic, it is in a definite sense possible in this case to define truth in the very language whose truth is in question.

Hegel's Science of Logic makes the just not low claim to be an absolute, ultimate-grounded knowledge. This project, which could not be more ambitious, has no good press in our post-metaphysical age. However: That absolute knowledge absolutely cannot exist, cannot be claimed without self-contradiction. On the other hand, there can be no doubt about the fundamental finiteness of knowledge. But can absolute knowledge be finite knowledge? This leads to the problem of a self-explication of logic (in the sense of (...) Hegel) and further, as will be shown, to a new definition of the dialectical procedure. The stringency of which results from the fact that always exactly that implicit content is explicated that was generated by the preceding explication step itself and is thus concretely comprehensible. At the same time, a new implicit content is generated by this act of explication, which requires a new explication step, and so forth. In the dialectical procedure reinterpreted in this way, dialectical arguments are not beheld, guessed at or even surreptitiously obtained, but are methodically accountable. Thereby dialectics is understood as a self-explication of logic by logical means and thus as a proof of the possibility of ultimate-grounding in the form of absolute and nevertheless finite – and thus also fallible – knowledge. (shrink)

"The ultimate end goal of the finite I and the not-I, i.e., the end goal of the world," writes Schelling in Of the I as the Principle of Philosophy (1795), "is its annihilation as a world, i.e., as the exemplification of finitude." In this paper, I explicate this statement and its theoretical stakes through a comprehensive re-reading of Schelling's 1795 writings: Of the I and Philosophical Letters on Dogmatism and Criticism, written later in the same year, in relation to (...) what Schelling proclaims to be the central problem of all philosophy: the existence of the world (Dasein der Welt). To that end, I analyze Schelling's 1795 conceptions of synthesis and morality, and the structure of the I's striving in the world that they serve to create and uphold – a structure in which the I is torn between a paradisal past and a striven-for future that it constitutively cannot reach as long as the world is there. Schelling's 1795 metaphysics is ultimately caught, I argue, in the tension between two poles: justifying the world, this world of negativity, division, and endless striving – and annihilating or dissolving it in the bliss of absolute identity and freedom, in which there is no world, and no possibility of or need for a world. (shrink)

The problem of approximating a propositional calculus is to find many-valued logics which are sound for the calculus (i.e., all theorems of the calculus are tautologies) with as few tautologies as possible. This has potential applications for representing (computationally complex) logics used in AI by (computationally easy) many-valued logics. It is investigated how far this method can be carried using (1) one or (2) an infinite sequence of many-valued logics. It is shown that the optimal candidate matrices for (1) can (...) be computed from the calculus. (shrink)

The article explores the idea that according to Spinoza finite thought and substantial thought represent reality in different ways. It challenges “acosmic” readings of Spinoza's metaphysics, put forth by readers like Hegel, according to which only an infinite, undifferentiated substance genuinely exists, and all representations of finite things are illusory. Such representations essentially involve negation with respect to a more general kind. The article shows that several common responses to the charge of acosmism fail. It then argues that (...) we must distinguish the well-founded ideality of representations of finite things from mere illusoriness, insofar as for Spinoza we can have true knowledge of what is known only abstractly. Finite things can be seen as well-founded beings of reason. The article also proposes that within Spinoza's framework it is possible to represent a finite thing without drawing on representations of mind-dependent entities. (shrink)

On the rise over the past 20 years has been ‘moderate supernaturalism’, the view that while a meaningful life is possible in a world without God or a soul, a much greater meaning would be possible only in a world with them. William Lane Craig can be read as providing an important argument for a version of this view, according to which only with God and a soul could our lives have an eternal, as opposed to temporally limited, significance, by (...) virtue of our moral choices then making an ultimate difference. I present two major objections to this position. On the one hand, I contend that if God existed and we had souls that lived forever, then, in fact, all our lives would turn out the same. On the other hand, I maintain that, if this objection is wrong, so that our moral choices would indeed make an ultimate difference and thereby confer an eternal significance on our lives (only) in a supernatural realm, then Craig could not capture the view, aptly held by moderate supernaturalists, that a meaningful life is possible in a purely natural world. An eternal significance would be ‘too big’, reducing any meaning possible during an earthly life to nothing by comparison. I conclude that moderate supernaturalists would be wise to avoid appealing to eternal (or infinite) meaning when spelling out the way God and a soul could alone impart a great meaning to our lives; some other notion of greatness should be considered, perhaps one involving a temporary afterlife. (shrink)

The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented in USA (...) and EU. It is revealed in the paper that at infinity the snowflake is not unique, i.e., different snowflakes can be distinguished for different infinite numbers of steps executed during the process of their generation. It is then shown that for any given infinite number n of steps it becomes possible to calculate the exact infinite number, Nn, of sides of the snowflake, the exact infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn, of the Koch snowflake as the result of multiplication of the infinite Nn by the infinitesimal Ln. It is established that for different infinite n and k the infinite perimeters Pn and Pk are also different and the difference can be infinite. It is shown that the finite areas An and Ak of the snowflakes can be also calculated exactly (up to infinitesimals) for different infinite n and k and the difference An − Ak results to be infinitesimal. Finally, snowflakes constructed starting from different initial conditions are also studied and their quantitative characteristics at infinity are computed. (shrink)

The Cartesian concept of nature, which has determined modern thinking until the present time, has become obsolete. It shall be shown that Hegel's objective-idealistic conception of nature discloses, in comparison to that of Descartes, new perspectives for the comprehension of nature and that this, in turn, results in possibilities of actualizing Hegel's philosophy of nature. If the argumentation concerning philosophy of nature is intended to catch up with the concrete Being-of-nature and to meet it in its concretion, then this (...) is impossible for the finite spirit in a strictly a priori sense – this is the thesis supported here which is not at all close to Hegel. As the argumentation rather has to consider the conditions of realization concerning the Being-of-nature, too, it is compelled to take up empirical elements – concerning the organism, for instance, system-theoretical aspects, physical and chemical features of the nervous system, etc. With that, on the one hand, empirical-scientific premises are assumed (e.g. the lawlikeness of nature), which on the other hand become (now close to Hegel) possibly able to be founded in the frame of a Hegelian-idealistic conception. In this sense, a double strategy of empirical-scientific concretization and objective-idealistic foundation is followed up, which represents the methodical basic principle of the developed considerations. In the course of the undertaking, the main aspects of the whole Hegelian design concerning the philosophy of nature are considered – space and time, mass and motion, force and law of nature, the organism, the pro-blem of evolution, psychic being – as well as Hegel's basic thesis concerning the philosophy of nature, that therein a tendency toward coherence and idealization manifests itself in the sense of a (categorically) gradually rising succession of nature: from the separateness of space to the ideality of sensation. In the sense of the double strategy of concretization and foundation it is shown that on the one hand possibilities of philosophical penetration concerning actual empirical-scientific results are opened, and on the other hand – in turn – a re-interpretation of Hegel's theorem on the basis of physical, evolution-theoretical and system-theoretical argumentation also becomes possible. In this mutual crossing-over and elucidation of empirical and Hegelian argumentation not only do perspectives of a new comprehension of nature become visible, but also, at the same time – as an essential consequence of this methodical principle – thoughts on the possibilities of actualizing Hegel's philosophy of nature. (shrink)

In this paper it is argued that the conjunction of linguistic ersatzism, the ontologically deflationary view that possible worlds are maximal and consistent sets of sentences, and possible world semantics, the view that the meaning of a sentence is the set of possible worlds at which it is true, implies that no actual speaker can effectively use virtually any language to successfully communicate information. This result is based on complexity issues that relate to our finite computational ability to deal (...) with large bodies of information and a strong, but well motivated, assumption about the cognitive accessibility of meanings of sentences ersatzers seem to be implicitly committed to. It follows that linguistic ersatzism, possible world semantics, or both must be rejected. (shrink)

This paper investigates the view that digital hypercomputing is a good reason for rejection or re-interpretation of the Church-Turing thesis. After suggestion that such re-interpretation is historically problematic and often involves attack on a straw man (the ‘maximality thesis’), it discusses proposals for digital hypercomputing with Zeno-machines , i.e. computing machines that compute an infinite number of computing steps in finite time, thus performing supertasks. It argues that effective computing with Zeno-machines falls into a dilemma: either they are specified (...) such that they do not have output states, or they are specified such that they do have output states, but involve contradiction. Repairs though non-effective methods or special rules for semi-decidable problems are sought, but not found. The paper concludes that hypercomputing supertasks are impossible in the actual world and thus no reason for rejection of the Church-Turing thesis in its traditional interpretation. (shrink)

Validity of physical laws for any aspect of brain activity and strict correlation of mental to physical states of the brain do not imply, with logical necessity, that a complete algorithmic theory of the mind-body relation is possible. A limit of decodability may be imposed by the finite number of possible analytical operations which is rooted in the finiteness of the world. It is considered as a fundamental intrinsic limitation of the scientific approach comparable to quantum indeterminacy and the (...) theorems of logical undecidability. An analysis of these limits, applied to dispositions of future behaviour, suggests that limits of decodability of the psycho-physic relation may actually exist with respect to brain states with self-referential aspects, as they are involved in mental processes. Limits for an algorithmic theory of the mind-body problem suggested by this study are formally similar to other intrinsic limits of the scientific method such as quantum indeterminacy and mathematical undecidability which are also related to self-referential operations. At the metatheoretical level, hard sciences, despite their reliability, universality and objectivity, depend on metatheoretical presuppositions which allow for multiple philosophical interpretations. -/- . (shrink)

Chains of causes appear when the existence of God is discussed. It is claimed by some that these chains must be finite and terminated by God. But these chains seem endless through our knowledge search. This endlessness for the physical reasons for any world event expresses the greatness and complexity of God’s creation and so the transcendence of God. So, only we can put our hands on physical reasons in an endless forage for knowledge. Yet, the endlessness of the (...) physical causes chains confirms the existence of God, and this is what our paper tries using Zorn’s lemma, an equivalent for one of the most celebrated axioms of mathematics, the Axiom of Choice. (shrink)

discussion of the extent to which architects can float about history and the inevitable finitude of architectural possibilities from any historical standpoint.

I critically examine Nietzsche’s argument in The Will to Power that all the detailed events of the world are repeating infinite times (on account of the merely finite possible arrangements of forces that constitute the world and the inevitability with which any arrangement of force must bring about its successors). Nietzsche celebrated this recurrence because of the power of belief in it to bring about a revaluation of values focused wholly on the value of one’s endlessly repeating life. Belief (...) in recurrence would bring joy to those who had achieved excellence in this life and crush those who had not. I point out, however, that this significance of recurrence must be spoiled by the consideration that within each of the long cycles of recurrence there would have to be, as well, countless variations of this life. And I consider the issue of personal identity within such recurrences. Is it the same person or merely resembling but distinct persons in the recurrences or the variations? If it is the same person in precise recurrences, should the subjectively identical experience be thought of as really additional for that person? (shrink)

The negative zombie argument concludes that physicalism is false from the premises that p ∧¬q is ideally negatively conceivable and that what is ideally negatively conceivable is possible, where p is the conjunction of the fundamental physical truths and laws and q is a phenomenal truth (Chalmers 2002; 2010). A sentence φ is ideally negatively conceivable iff φ is not ruled out a priori on ideal rational reflection. In this paper, I argue that the negative zombie argument is neither a (...) priori nor conclusive. First, I argue that the premises of the argument are true only if there exists an adequate finite ideal reasoner R that believes ◊(p ∧ ¬q) on the basis of not believing p→q on a priori basis. Roughly, a finite reasoner is a reasoner with cognitive limitations (e.g. finite memory). I argue that R is finite only if R reasons nonmonotonically and only approach ideal reflection at the limit of a reasoning sequence. This would render the argument nonconclusive. Finally, I argue that, for some q, R does not believe ◊(p ∧ ¬q) on the basis of not believing p→q on a priori basis (e.g. for q =‘something is conscious’). This would render the choice of an adequate q dependent on empirical information (and the argument a posteriori). I conclude that the negative zombie argument (and, maybe, all zombie arguments) is neither a priori nor conclusive. (shrink)

For Kant, the human cognitive faculty has two sub-faculties: sensibility and the understanding. Each has pure forms which are necessary to us as humans: space and time for sensibility; the categories for the understanding. But Kant is careful to leave open the possibility of there being creatures like us, with both sensibility and understanding, who nevertheless have different pure forms of sensibility. They would be finite rational beings and discursive cognizers. But they would not be human. And this raises (...) a question about the pure forms of the understanding. Does Kant leave open the possibility of discursive cognizers who have different categories? Even if other discursive cognizers might not sense like us, must they at least think like us? We argue that textual and systematic considerations do not determine the answers to these questions and examine whether Kant thinks that the issue cannot be decided. Consideration of his wider views on the nature and limits of our knowledge of mind shows that Kant could indeed remain neutral on the issue but that the exact form his neutrality can take is subject to unexpected constraints. The result would be an important difference between what Kant says about discursive cognizers with other forms of sensibility and what he is in a position to say about discursive cognizers with other forms of understanding. Kantian humility here takes on a distinctive character. (shrink)

There exists a huge number of numerical methods that iteratively construct approximations to the solution y(x) of an ordinary differential equation (ODE) y′(x) = f(x,y) starting from an initial value y_0=y(x_0) and using a finite approximation step h that influences the accuracy of the obtained approximation. In this paper, a new framework for solving ODEs is presented for a new kind of a computer – the Infinity Computer (it has been patented and its working prototype exists). The new computer (...) is able to work numerically with finite, infinite, and infinitesimal numbers giving so the possibility to use different infinitesimals numerically and, in particular, to take advantage of infinitesimal values of h. To show the potential of the new framework a number of results is established. It is proved that the Infinity Computer is able to calculate derivatives of the solution y(x) and to reconstruct its Taylor expansion of a desired order numerically without finding the respective derivatives analytically (or symbolically) by the successive derivation of the ODE as it is usually done when the Taylor method is applied. Methods using approximations of derivatives obtained thanks to infinitesimals are discussed and a technique for an automatic control of rounding errors is introduced. Numerical examples are given. (shrink)

There are possible worlds in which time is circular and finite in duration, forming a loop of, say, 12,000 years. There are also possible worlds in which time is linear and infinite in both directions and in which history is repetitive, consisting of infinitely many 12,000-year epochs, each two of which are exactly alike with respect to all intrinsic, purely qualitative properties. Could one ever have empirical evidence that one inhabits a world of the first kind rather than a (...) world of the second kind? We argue for the affirmative answer, contra Quine, Newton-Smith, and Bergström. Our argument for that conclusion differs from an argument for the same conclusion due to Weir. Weir’s argument is probabilistic and explicitly requires having evidence against determinism. Our argument is a direct appeal to the simplicity of laws, and it involves no probabilistic component. It is modeled on Shoemaker’s argument that one could have evidence of time without change. (shrink)

Jean-Luc Nancy takes the concept of "essence" in order to indicate its drawbacks on the singularity of being. The concept of essence is not a universal and necessary origin, but contingent and historical meanings for Nancy. This historicity in meaning leads Nancy to question the concept of the individual and the rules of the social/public sphere allocated through individuality. Nancy's argument on the ontological environment of finite beings aims to highlight those beings are mixed singular, not belonging to a (...) universal unit. This allows us to discover that being is singular and also singular-plural to the extent that it is with the other. Thus, essential historical concepts invalidate individual or social organizations at this point. Nancy calls this “finitude” which is the only transcendental concept that makes possible the “being with” (Mitsein). It is possible to think that finitude is the only property to make a community of singularities rather than ready-made concepts of social sciences. I argue that this position is methodological for an alternative socio-ontology. (shrink)

The concept of modes of existence of semiotic entities underlies (post)Greimasian semiotics, yet it seems to have received little attention. Modes of existence can be used in different senses. For Greimas, from the perspective of narrative semiotics, when Michelangelo first receives a block of marble and decides to sculpt the David, his intention is in a virtual mode; as Michelangelo progresses he ends up bringing the David into existence, and his intention comes to the realized mode. In Fontanille’s tensive semiotics, (...) however, modes of existence can have to do with how one can narrow or broaden the scope of our apprehension of the David as our eyes look at it in order to produce a meaningful experience. In this work, the perspectives of narrative and tensive semiotics are contrasted both theoretically and practically applying both to a number of examples. In order to identify all possible modes of existence and all the possibilities of transitioning from one to the other in the examples presented, we resort to the method of finite-state automata from computer science. In the end, we propose a robust narrative account of modes of existence that relies on narrative semiotics for its definition, but into which intent and apprehension from tensive semiotics can be integrated. This work calls for the need of establishing a syntax of modes of existence, since both Greimas and Fontanille construe them as being necessary to account for the production of signification. (shrink)

I want to model a finite, fallible cognitive agent who imagines that p in the sense of mentally representing a scenario—a configuration of objects and properties—correctly described by p. I propose to capture imagination, so understood, via variably strict world quantifiers, in a modal framework including both possible and so-called impossible worlds. The latter secure lack of classical logical closure for the relevant mental states, while the variability of strictness captures how the agent imports information from actuality in the (...) imagined non-actual scenarios. Imagination turns out to be highly hyperintensional, but not logically anarchic. Section 1 sets the stage and impossible worlds are quickly introduced in Sect. 2. Section 3 proposes to model imagination via variably strict world quantifiers. Section 4 introduces the formal semantics. Section 5 argues that imagination has a minimal mereological structure validating some logical inferences. Section 6 deals with how imagination under-determines the represented contents. Section 7 proposes additional constraints on the semantics, validating further inferences. Section 8 describes some welcome invalidities. Section 9 examines the effects of importing false beliefs into the imagined scenarios. Finally, Sect. 10 hints at possible developments of the theory in the direction of two-dimensional semantics. (shrink)

This paper reconsiders Leibniz’s conception of the nature of possible things and offers a novel interpretation of the actualization of possible substances. This requires analyzing a largely neglected notion, the reality of individual essences. Thus far scholars have tended to construe essences as representational items in God’s intellect. We acknowledge that finite essences have being in the divine intellect but insist that they are also grounded in the infinite essence of God, as limitations of it. Indeed, we show that (...) it is critical to understand that this dependence on God’s essence is prior to the dependence on God through divine ideas. Here it is crucial to distinguish questions concerning the ontological status of essences from questions concerning their reality. This yields a fresh view of Leibniz’s theory of creation, which takes seriously his claim that the same thing is first a mere possibility but after creation an actually existent substance. (shrink)

In his classic book “the Foundations of Statistics” Savage developed a formal system of rational decision making. The system is based on (i) a set of possible states of the world, (ii) a set of consequences, (iii) a set of acts, which are functions from states to consequences, and (iv) a preference relation over the acts, which represents the preferences of an idealized rational agent. The goal and the culmination of the enterprise is a representation theorem: Any preference relation that (...) satisfies certain arguably acceptable postulates determines a (finitely additive) probability distribution over the states and a utility assignment to the consequences, such that the preferences among acts are determined by their expected utilities. Additional problematic assumptions are however required in Savage's proofs. First, there is a Boolean algebra of events (sets of states) which determines the richness of the set of acts. The probabilities are assigned to members of this algebra. Savage's proof requires that this be a σ-algebra (i.e., closed under infinite countable unions and intersections), which makes for an extremely rich preference relation. On Savage's view we should not require subjective probabilities to be σ-additive. He therefore finds the insistence on a σ-algebra peculiar and is unhappy with it. But he sees no way of avoiding it. Second, the assignment of utilities requires the constant act assumption: for every consequence there is a constant act, which produces that consequence in every state. This assumption is known to be highly counterintuitive. The present work contains two mathematical results. The first, and the more difficult one, shows that the σ-algebra assumption can be dropped. The second states that, as long as utilities are assigned to finite gambles only, the constant act assumption can be replaced by the more plausible and much weaker assumption that there are at least two non-equivalent constant acts. The second result also employs a novel way of deriving utilities in Savage-style systems -- without appealing to von Neumann-Morgenstern lotteries. The paper discusses the notion of “idealized agent" that underlies Savage's approach, and argues that the simplified system, which is adequate for all the actual purposes for which the system is designed, involves a more realistic notion of an idealized agent. (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers numerically. This can be done on a new kind of a computer – the Infinity Computer – able to work with all these types of numbers. The new computational tools both give (...) possibilities to execute computations of a new type and open new horizons for creating new mathematical models where a computational usage of infinite and/or infinitesimal numbers can be useful. A number of numerical examples showing the potential of the new approach and dealing with divergent series, limits, probability theory, linear algebra, and calculation of volumes of objects consisting of parts of different dimensions are given. (shrink)

Analytic theologians have proposed numerous “solutions” to the Logical Problem of the Trinity (LPT), mostly versions of Social Trinitarianism (ST) and Relative Identity Trinitarianism (RI). Both types of solution are controversial, but many hold out hope that further “Trinitarian theorizing” may yield some as yet unimagined, and somehow importantly different, solution to the LPT. I first give a precise definition of the LPT and of what would count as a solution to it. I then show how, though there are infinitely (...) many possible solutions, all solutions can be grouped together into a finite, exhaustive taxonomy, based precisely on those features which make them either controversial, heretical, or inconsistent. The taxonomy reveals why ST and RI have been the major proposed solutions, and also proves that there can be no importantly different, new solutions to the LPT. (shrink)

Weak particularism about reasons is the view that the normative valency of some descriptive considerations varies, while others have an invariant normative valency. A defence of this view needs to respond to arguments that a consideration cannot count in favour of any action unless it counts in favour of every action. But it cannot resort to a global holism about reasons, if it claims that there are some examples of invariant valency. This paper argues for weak particularism, and presents a (...) framework for understanding the relationships between practical reasons. A central part of this framework is the idea that there is an important kind of reason-a 'presumptive reason'-which need not be conclusive, but which is neither pro tanto nor prima facie. /// [Richard Holton] Should particularists about ethics claim that moral principles are never true? Or should they rather claim that any finite set of principles will not be sufficient to capture ethics? This paper explores and defends the possibility of embracing the second of these claims whilst rejecting the first, a position termed 'principled particularism'. The main argument that particularists present for their position-the argument that holds that any moral conclusion can be superseded by further considerations-is quite compatible with principled particularism; indeed, it is compatible with the idea that every true moral conclusion can be shown to follow deductively from a finite set of premises. Whilst it is true that these premises must contain implicit ceteris paribus clauses, this does not render the arguments trivial. On the contrary, they can do important work in justifying moral conclusions. Finally the approach is briefly applied to the related field of jurisprudence. (shrink)

Sam Harris (2010) argues that, given our neurology, we can experience well-being, and that seeking to maximize this state lets us distinguish the good from the bad. He takes our ability to compare degrees of well-being as his starting point, but I think that the analysis can be pushed further, since there is a (non-religious) reason why well-being is desirable, namely the finite life of an individual organism. It is because death is a constant possibility that things can be (...) assessed as “for” or “against” one (Champagne 2011a; Smith 2000). Such an account lets us objectively adjudicate moral questions, as Harris desires. However, by anchoring itself in the mortal body as a whole and not just the brain, such an account dampens the claim that neuroscience would have all the answers. Moreover, it pivots on an affirmation of one’s life that can seem mysterious by regular scientific standards. This chapter thus explains why the trade-off is worthwhile. (shrink)

Kant’s Formula of Universal Law (FUL) is generally believed to require you to act only on the basis of maxims that you can will without contradiction to become universal laws. In “Contradiction and Kant’s Formula of Universal Law” (2017), I have proposed to read the FUL instead as requiring that, for any maxim on which you act, you can will two things simultaneously, without volitional self-contradiction: (1) willing the maxim as your own action principle and (2) willing that it become (...) a universal law. In the present essay, I reply to comments by Mark Timmons, Michael Walschots, Paola Romero, and Stefano Lo Re. In response to their comments concerning the application of the FUL, I expand the interpretive framework of the Volitional Self-Contradiction Interpretation. I argue that Kant also constructs the diagnostic volitional self-contradiction as a contradiction between (1) willing a tested maxim to become a universal law and (2) willing what humans, qua finite rational beings, necessarily will, namely the means to their actual and possible future ends. I also clarify how the two ways in which Kant specifies the test are related. Furthermore, I clarify the relation between the Volitional Self-Contradiction interpretation and other interpretations of the FUL, in particular the ‘Practical Contradiction’ and ‘Logical Contradiction’ interpretations, as well as its difference from the Golden Rule. I also address the objection that the volitional self-contradiction is superfluous, clarify the relation between the Formula of Universal Law and the Formula of the Law of Nature, and explain why the will of an immoral agent does not contain a volitional self-contradiction of the type at issue. (shrink)

A number of philosophers have argued that it is hard for finite agents like us to reason and make decisions relying solely on our credences and preferences. They hold that for us to cope with our cognitive limitations, we need binary beliefs as well. For they think that such beliefs, by disposing us to treat certain propositions as true, help us cut down on the number of possibilities we need to consider when we reason. But using Ross and (...) Schroeder as my stalking horse, I argue that such an appeal to binary beliefs does not work. I begin by explaining why there’s supposedly a problem for an account of reasoning that invokes only credences and preferences. I then argue that Ross and Schroeder’s account of belief—as well as other similar accounts—does not help solve the problem. Finally, I consider an alternative approach to solving the problem. This approach, unlike the accounts I criticise, does not hold that having a disposition to treat a proposition as true is necessary for believing it. (shrink)

Recently, it has been proposed to understand a logic as containing not only a validity canon for inferences but also a validity canon for metainferences of any finite level. Then, it has been shown that it is possible to construct infinite hierarchies of "increasingly classical" logics—that is, logics that are classical at the level of inferences and of increasingly higher metainferences—all of which admit a transparent truth predicate. In this paper, we extend this line of investigation by taking a (...) somehow different route. We explore logics that are different from classical logic at the level of inferences, but recover some important aspects of classical logic at every metainferential level. We dub such systems meta-classical non-classical logics. We argue that the systems presented deserve to be regarded as logics in their own right and, moreover, are potentially useful for the non-classical logician. (shrink)

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

This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence (...) which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. (shrink)

Spinoza ’s letter of June 2, 1674 to his friend Jarig Jelles addresses several distinct and important issues in Spinoza ’s philosophy. It explains briefly the core of Spinoza ’s disagreement with Hobbes’ political theory, develops his innovative understanding of numbers, and elaborates on Spinoza ’s refusal to describe God as one or single. Then, toward the end of the letter, Spinoza writes: With regard to the statement that figure is a negation and not anything positive, it is obvious that (...) matter in its totality, considered without limitation [indefinitè consideratam], can have no figure, and that figure applies only to finite and determinate bodies. For he who says that he apprehends a figure, thereby means to indicate simply this, that he apprehends a determinate thing and the manner of its determination. This determination therefore does not pertain to the thing in regard to its being [esse]; on the contrary, it is its non-being [non-esse]. So since figure is nothing but determination, and determination is negation [Quia ergo figura non aliud, quam determinatio, et determinatio negatio est], figure can be nothing other than negation, as has been said. Arguably, what is most notable about this letter is the fate of a single subordinate clause which appears in the last sentence of this passage: et determinatio negatio est. That clause was to be adopted by Hegel and transformed into the slogan of his own dialectical method: Omnis determinatio est negatio. Of further significance is the fact that, while Hegel does credit Spinoza with the discovery of this most fundamental insight, he believes Spinoza failed to appreciate the importance of his discovery. The issue of negation and the possibility of self-negation stand at the very center of the philosophical dialogue between the systems of Spinoza and Hegel, and in this paper I will attempt to provide a preliminary explication of this foundational debate between the two systems. In the first part of the paper I will argue that the “determination is negation” formula has been understood in at least three distinct senses among the German Idealists, and as a result many of the participants in the discussion of this formula were actually talking past each other. The clarification of the three distinct senses of the formula will lead, in the second part of the paper, to a more precise evaluation of the fundamental debate between Spinoza and Hegel regarding the possibility of self-negation. In this part I will evaluate the validity of each interpretation of the determination formula, and motivate the positions of the various participants in the debate. (shrink)

Dynamic Theodicy (DT) is a broad concept we bring up to designate some modern Philosophical Theology attempts to reconcile the necessary and perfect existence of God with the contingent characteristics of human life. In this paper we analyze such approaches and discuss how they have become incomprehensible because the metaphysical assumptions implicit in these explanations have lost their intrinsic relation to the natural human desire for salvation. In the first part we show Charles Hartshorne's DT-model, arising from the modal logic (...) of perfection, and the modern rational problems of this position in making infinite-necessary Being (God) and finite-contingent being (human) compatible. We note that at the heart of the contradictions in this DT account is a dialectical mode of thinking that makes it difficult to find a correct solution to this dichotomy, and to assume a human desire that could be considered related to lifelong goals. In the second part, supported by the proposal of Hans Urs von Balthasar's DT, we develop the concepts of bodily vulnerability, corporeal intentionality, and natural desire for salvation, which come from an Aristotelian-Thomistic thought. This theory is established in order to build an argument, following Alasdair MacIntyre’s ethical framework, on how to make possible the recovery of a metaphysical and anthropological desire that transcends natural aging and goes beyond death. We conclude that both human dependence and the virtues that arise naturally when human beings decide to seek the good of their transcendent condition, make it possible to recover the natural desire for salvation through divine and human love. (shrink)

I want to suggest that it would be bad if compatibilism were true, and that this gives us good reason to think that it isn't. This is, you might think, an outlandish argument, and the considerable burden of this paper is to convince you otherwise. There are two key elements at stake in this argument. The first is that it would be ‐ in a distinctive sense to be explained ‐ bad if compatibilism were true. The thought here is that (...) compatibilism ultimately presents us with a picture on which, in principle, powerful manipulators can effectively guarantee that finite moral agents should become blameworthy. To my mind, this isn't just false ‐ though I think that it is ‐ it is also such that it would be bad (unfortunate, undesirable…) if it were true. The second is that the fact that it would be ‐ in this sense ‐ bad if true gives us reason to think that it isn't. It may be bad that there is no afterlife. But that, in itself, hardly gives us reason to think that there is an afterlife. That is true, but ‐ as others before me have suggested ‐ when the object of the relevant badness is morality itself, the inference seems secure. A more general aim of the paper is to investigate the nature of this very form of argument in itself, and I compare my argument (inter alia) to a recent argument from Sayre–McCord against the possibility of genuine moral dilemmas. (shrink)

This paper proposes a specific approach to understanding the nature of technology that encompasses the entire field of technological praxis, from the making of primitive tools to using the Internet. In that approach, technology is a specific form of human agency that yields to (an imperfect) realization of human control over a technological situation—that is, a situation not governed to an end by natural constraints but by specific human aims. The components of such technological situations are a given collection of (...) natural or artificial beings, humans, human aims, and situation-bound tools. By performing technological situation analysis, the essential form of tool making, the complex system of relationships between science and technology, technological practices with and without machines, the finiteness or imperfectness of any technology, and engineering (i.e., the possibility of the creation of technological situations) can be considered. For a better characterization of the approach to technology, the paper also presents a comparison of other philosophies of technology. Following Feenberg’s comparative analysis, the so-called fundamental question of the philosophy of technology is formulated, its two sides are identified, and it is applied for clarification of our position within philosophy of technology. In our approach, all human praxis can be considered to be technological; more precisely, every human activity has a technological aspect or dimension. (shrink)

This essay concerns the question of how we make genuine epistemic progress through conceptual analysis. Our way into this issue will be through consideration of the paradox of analysis. The paradox challenges us to explain how a given statement can make a substantive contribution to our knowledge, even while it purports merely to make explicit what one’s grasp of the concept under scrutiny consists in. The paradox is often treated primarily as a semantic puzzle. However, in “Sect. 1” I argue (...) that the paradox raises a more fundamental epistemic problem, and in “Sects.1 and 2” I argue that semantic proposals—even ones designed to capture the Fregean link between meaning and epistemic significance—fail to resolve that problem. Seeing our way towards a real solution to the paradox requires more than semantics; we also need to understand how the process of analysis can yield justification for accepting a candidate conceptual analysis. I present an account of this process, and explain how it resolves the paradox, in “Sect. 3”. I conclude in “Sect. 4” by considering the implications for the present account concerning the goal of conceptual analysis, and by arguing that the apparent scarcity of short and finite illuminating analyses in philosophically interesting cases provides no grounds for pessimism concerning the possibility of philosophical progress through conceptual analysis. (shrink)

Leibniz and Kant were heirs of a biblical theistic tradition which viewed miraculous activity in the world as both possible and actual. But both were also deep explanatory rationalists about the natural world: more committed than your average philosophical theologian to its thoroughgoing intelligibility. These dual sympathies—supernaturalist religion and empirical rationalism—generate a powerful tension across both philosophers’ systems, one that is most palpable in their accounts of empirical miracles—that is, events in nature that violate one or more of the natural (...) laws. The primary goal of this chapter is to exhibit their respective efforts to resolve this tension, and to show how Kant’s noumenal ignorance doctrine allows him to set aside some key difficulties that Leibniz’s model leaves unresolved. The chapter concludes with the unorthodox suggestion that both systems leave conceptual space for the idea that finite free beings, rather than just God, can produce empirical miracles. (shrink)

As the ongoing literature on the paradoxes of the Lottery and the Preface reminds us, the nature of the relation between probability and rational acceptability remains far from settled. This article provides a novel perspective on the matter by exploiting a recently noted structural parallel with the problem of judgment aggregation. After offering a number of general desiderata on the relation between finite probability models and sets of accepted sentences in a Boolean sentential language, it is noted that a (...) number of these constraints will be satisfied if and only if acceptable sentences are true under all valuations in a distinguished non-empty set W. Drawing inspiration from distance-based aggregation procedures, various scoring rule based membership conditions for W are discussed and a possible point of contact with ranking theory is considered. The paper closes with various suggestions for further research. (shrink)

Causal finitism, the view that the causal history of any event must be finite, has garnered much philosophical interest recently—especially because of its applicability to the Kalām cosmological argument. The most prominent argument for causal finitism is the Grim Reaper argument, which attempts to show that, if infinite causal histories are possible, then other paradoxical states of affairs must also be possible. However, this style of argument has been criticized on the grounds of (i) relying on controversial modal principles, (...) and (ii) providing a false diagnosis of the paradoxes involved. In this paper, I develop a new kind of Grim Reaper argument immune to these criticisms. I show that, by using insights from the literature on time travel, causal finitists should instead argue that infinite causal histories are problematically inexplicable, as they entail the possibility of unexplained foiling mechanisms. The fruits of this paper are that (i) a novel supporting argument for the Kalām is developed, and (ii) along the way of building this argument, it is shown that the literatures on time travel and causal finitism are deeply and intimately connected. (shrink)

In his Wittgenstein on Rules and Private Language, Saul Kripke famously raised two sorts of problems for responses to the meaning skeptic that appealed to how we were disposed to use our words in the past. The first related to the fact that our “dispositions extend to only finitely many cases” while the second related to the fact that most of us have “dispositions to make mistakes.” The second of these problems has produced an enormous, and still growing, literature on (...) the purported ‘normativity’ of meaning, but the first has received (at least comparatively) little attention. It will be argued here, however, that (1) the fact that we can be disposed to make mistakes doesn’t present a serious problem for many disposition-based responses to the skeptic, and (2) considerations of the ‘finiteness’ of our dispositions point, on their own, to an important way that the relation between meaning and use must be understood as ‘normative’. In particular, the assumption that words have determinate meanings, more so than the assumption that they have ‘correctness’ conditions, rules out the possibility of understanding the relation between meaning and use in the more descriptive fashion commonly associated with dispositionalism. Nevertheless, if the relation of meaning and use is understood in a more normative key, responding to the meaning-skeptic with facts about one’s dispositions will turn out to be perfectly legitimate, though it will prevent one from being the sort of ‘dispositionalist’ that Kripke gives, without any real argument, a monopoly on disposition-based responses. (shrink)

The aim of this paper is to introduce a system of dynamic deontic logic in which the main problems related to the de finition of deontic concepts, especially those emerging from a standard analysis of permission in terms of possibility of doing an action without incurring in a violation of the law, are solved. The basic idea is to introduce two crucial distinctions allowing us to differentiate (i) what is ideal with respect to a given code, which fixes the types (...) of action that are abstractly prescribed, and what is ideal with respect to the specific situation in which the agent acts, and (ii) the transitions associated with actions and the results of actions, which can obtain even without the action being performed. (shrink)

A strongly independent preorder on a possibly in finite dimensional convex set that satisfi es two of the following conditions must satisfy the third: (i) the Archimedean continuity condition; (ii) mixture continuity; and (iii) comparability under the preorder is an equivalence relation. In addition, if the preorder is nontrivial (has nonempty asymmetric part) and satisfi es two of the following conditions, it must satisfy the third: (i') a modest strengthening of the Archimedean condition; (ii') mixture continuity; and (iii') completeness. (...) Applications to decision making under conditions of risk and uncertainty are provided. (shrink)

This paper focuses on Newman’s approach to what we might call “the problem of the partiality and unity of the sciences.” The problem can be expressed in the form of a question: “If all human knowing is finite and partial, then on what grounds can one know of the unity and wholeness of all the sciences?” Newman’s solution to the problem is openly theistic, since it appeals to one’s knowledge of God. For Newman, even if I exclusively pursue my (...) own partial science as a physicist, or psychologist, or historian, and even if I do not understand much about the content of the other sciences, nevertheless it is still possible for me to grasp the comprehensive ground of the unity of all the sciences, by virtue of my knowledge of God. The problem, however, is that this solution seems to rely on the sort of intellectual imperialism that Newman criticizes throughout much of his work. For this solution seems to assert the unity of the sciences only by placing one science—theology—above all the others as a supervening Über-science. The aim of this paper is to defend Newman against this charge of imperialism, and to show that his thought is not only more plausible, but also more nuanced, than might appear at first sight. (shrink)

Developing a suggestion of Wittgenstein, I provide an account of truth tables as formulas of a formal language. I define the syntax and semantics of TPL (the language of Tabular Propositional Logic), and develop its proof theory. Single formulas of TPL, and finite groups of formulas with the same top row and TF matrix (depiction of possible valuations), are able to serve as their own proofs with respect to metalogical properties of interest. The situation is different, however, for groups (...) of formulas whose top rows differ. For them I provide (i) a tree-style system of ‘row tree proofs’, which is shown to be sound and complete, and (ii) an alternative, re-writing strategy. (shrink)

The electronic and muonic hydrogen energy levels are calculated very accurately [1] in Quantum Electrodynamics (QED) by coupling the Dirac Equation four vector (c ,mc2) current covariantly with the external electromagnetic (EM) field four vector in QED’s Interactive Representation (IR). The c -Non Exclusion Principle(c -NEP) states that, if one accepts c as the electron/muon velocity operator because of the very accurate hydrogen energy levels calculated, the one must also accept the resulting electron/muon internal spatial and time coordinate operators (ISaTCO) (...) derived directly from c without any assumptions. This paper does not change any of the accurate QED calculations of hydrogen’s energy levels, given the simplistic model of the proton used in these calculations [1]. The Proton Radius Puzzle [2, 3] may indicate that new physics is necessary beyond the Standard Model (SM), and this paper describes the bizarre, and very different, situation when the electron and muon are located “inside” the spatially extended proton with their Centers of Charge (CoCs) orbiting the proton at the speed of light in S energy states. The electron/muon center of charge (CoC) is a structureless point that vibrates rapidly in its inseparable, non-random vacuum whose geometry and time structure are defined by the electron/muon ISaTCO discrete geometry. The electron/muon self mass becomes finite in a natural way due to the ISaTCOs cutting off high virtual photon energies in the photon propagator. The Dirac-Maxwell-Wilson (DMW) Equations are derived from the ISaTCO for the EM fields of an electron/muon, and the electron/muon “look” like point particles in far field scattering experiments in the same way the electric field from a sphere with evenly distributed charge “e” “looks” like a point with the same charge in the far field (Gauss Law). The electron’s/muon’s three fluctuating CoC internal spatial coordinate operators have eight possible eigenvalues [4, 5, 6] that fall on a spherical shell centered on the electron’s CoM with radius in the rest frame. The electron/muon internal time operator is discrete, describes the rapid virtual electron/positron pair production and annihilation. The ISaTCO together create a current that produce spin and magnetic moment operators, and the electron and muon no longer have “intrinsic” properties since the ISaTCO kinematics define spin and magnetic moment properties. (shrink)

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the violations (...) of energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (shrink)

In this paper, building on recent and longstanding work (Warren 2001, Friedman 2013, Glezer 2018), I investigate how the account of the essences or natures of material substances in the Metaphysical Foundations is related to Kant’s demand for the completeness of the system of nature. We must ascribe causal powers to material substances for the properties of those substances to be observable and knowable. But defining those causal powers requires admitting laws of nature, taken as axioms or principles of natural (...) science, which govern anything that can be constructed mathematically or “given as an object of experience” (MAN, 4:474–75). Presenting a complete system of nature requires adumbrating the properties of material substances that can be objects of experience. But it is not possible to show how those properties are involved in explanations of the phenomena without involving the laws that govern interactions between material substances. For Kant, or so I will argue, it is not possible to account for the natures of things, or for the features of laws of nature, independently of each other. The paper will substantiate the Finitist Account (FA) of Kant on laws. The Finitist Account has it that modal judgments about the possible proofs that can be made, or interactions that can be explained, are (1) based on concrete intuitive reasoning and motivated by the desire to avoid appeal to the actual infinite, (2) grounded in finite decision procedures, which are (3) based on reliable systems of axioms or rules of inference. (shrink)