The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the (...) proto-arithmetical ability to arithmetic proper. I argue that enculturation based on neural reuse provides a theoretically sound and fruitful framework for explaining this development. However, I show that a comprehensive explanation must be based on valid theoretical distinctions and involve several stages in the development of arithmetical knowledge. I provide an account that meets these challenges and thus leads to a better understanding of the subject of enculturation. (shrink)

Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially (...) shed light on the development of human numerical abilities, from the proto-arithmetical abilities of subitizing and estimating to counting procedures. Although the current results are far from conclusive and much more work is needed, I argue that AI research should be included in the interdisciplinary toolbox when we try to explain the development and character of numerical cognition and arithmetical intelligence. This makes it relevant also for the epistemology of mathematics. (shrink)

Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an explanation of cognitive phenomena, one that does not include mental representations or mental content in basic minds. Recently, Zahidi and Myin have presented an account of arithmetical cognition that is consistent with the REC view. In this paper, I first evaluate the feasibility of that account by focusing on the evolutionarily developed proto-arithmetical abilities and whether empirical data on them support the radical enactivist (...) view. I argue that although more research is needed, it is at least possible to develop the REC position consistently with the state-of-the-art empirical research on the development of arithmetical cognition. After this, I move the focus to the question whether the radical enactivist account can explain the objectivity of arithmetical knowledge. Against the realist view suggested by Hutto, I argue that objectivity is best explained through analyzing the way universal proto-arithmetical abilities determine the development of arithmetical cognition. (shrink)

One main challenge of non-platonist philosophy of mathematics is to account for the apparent objectivity of mathematical knowledge. Cole and Feferman have proposed accounts that aim to explain objectivity through the intersubjectivity of mathematical knowledge. In this paper, focusing on arithmetic, I will argue that these accounts as such cannot explain the apparent objectivity of mathematical knowledge. However, with support from recent progress in the empirical study of the development of arithmetical cognition, a stronger argument can be provided. I (...) will show that since the development of arithmetic is (partly) determined by biologically evolved proto-arithmetical abilities, arithmetical knowledge can be understood as maximally intersubjective. This maximal intersubjectivity, I argue, can lead to the experience of objectivity, thus providing a solution to the problem of reconciling non-platonist philosophy of mathematics with the (apparent) objectivity of mathematical knowledge. (shrink)

El pensamiento de Agustín de la Riega es, en esencia, revolucionario. No sólo porque a partir de él ha de de-construirse el primer fundamento de la filosofía fenomenológica, sino también porque abre el pensar filosófico a un plano de realidad efectiva (o, siguiendo a Agustín Basave Fernández del Valle, de habencia), que es en verdad una vida… más allá de las dicotomías modernas: objeto/sujeto, inmanencia/trascendencia, ser/ente, etc. La dimensión del haber -con estatuto trans-ontológico- es análogo al plano de fuerzas de (...) Deleuze. Pero lo que en verdad hace a De la Riega martillar la esfera de Pascal, es justamente la tarea liberadora de la filosofía que ha de desencadenar al hombre del destino instrumentalista de la reducción categorial, la diferencia ontológica y el sentido del mundo, construido -desde Jonia hasta Friburgo- por los filósofos de la pura unidad identitaria que someten lo humano al yugo del Ser-Fuerza Oculta y de la deuda ontológica. El pensamiento de De la Riega apela a descentrar la vida del carácter fundacional de la razón como esencializadora o fuente de derecho del ser, invocando al haber en tanto dimensión que acaece por fuera de todo intento de reducción técnico-racional. Descentrar el ser orientándolo hacia la radical diferencia de lo vivo no es un acto propio de la conciencia intencional que especula sobre los aspectos constitutivos del haber, sino que es eminentemente una vivencia de lo fáctico en tanto abierto al estar ahí nomás, ambiguo, plurimórfico y contingente Dicha apertureidad de lo habiente es la que pone a De la Riega en diálogo, y creemos que es esa la misma metodología por medio de la cual descubrir su pensamiento, hasta nuestros días inadvertido, especialmente por las ya obsoletas escuelas académicas de comentaristas fenomenólogos, para quienes es dificultoso salir-se más allá del círculo, y por quienes la filosofía toda ha devenido en una nota al pie de las obras de Heidegger, Merleau-Ponty o Husserl, proscribiendo la posibilidad de un pensar creador y radical. (shrink)

This paper connects the hard problem of consciousness to the interpretation of quantum mechanics. It shows that constitutive Russellian pan(proto)psychism (CRP) is compatible with Everett’s relative-state (RS) interpretation. Despite targeting different problems, CRP and RS are related, for they both establish symmetry between micro- and macrosystems, and both call for a deflationary account of Subject. The paper starts from formal arguments that demonstrate the incompatibility of CRP with alternative interpretations of quantum mechanics, followed by showing that RS entails Russellian (...) pan(proto)psychism. Therefore, CRP and RS are mutually supportive. It then provides a unified ontological picture by combining CRP and RS. The challenge faced by CRP, the combination problem, can be resolved by adopting a RS version of quantum mechanics. Technically, this is achieved by a co-consciousness relation capable of explaining the difference between first-person and third-person perspectives. The hierarchical structure of the relation removes any concern on the structural mismatch between the physical and the phenomenal. (shrink)

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In (...) defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

The paper explores the idea that some singular judgements about the natural numbers are immune to error through misidentification by pursuing a comparison between arithmetic judgements and first-person judgements. By doing so, the first part of the paper offers a conciliatory resolution of the Coliva-Pryor dispute about so-called “de re” and “which-object” misidentification. The second part of the paper draws some lessons about what it takes to explain immunity to error through misidentification. The lessons are: First, the so-called Simple (...) Account of which-object immunity to error through misidentification to the effect that a judgement is immune to this kind of error just in case its grounds do not feature any identification component fails. Secondly, wh-immunity can be explained by a Reference-Fixing Account to the effect that a judgement is immune to this kind of error just in case its grounds are constituted by the facts whereby the reference of the concept of the object which the judgement concerns is fixed. Thirdly, a suitable revision of the Simple Account explains the de re immunity of those arithmetic judgements which are not wh-immune. These three lessons point towards the general conclusion that there is no unifying explanation of de re and wh-immunity. (shrink)

As a grand narrative of progress, the utopian project of modernity is primarily concerned with notions of rationalism, universalism, and the development of a metalanguage. The triumph of the Moderate Enlightenment has seen logics of domination, accumulation and individualism incorporated into the project of modernity, with these logics giving rise to globalised capitalism as the metalanguage of modernity and neoliberal economics as the grand narrative of rational progress. The project of modernity is all but complete, requiring only the formality of (...) an end. However, rather than utopia, the foreseeable endpoint of modernity is environmental collapse, with neoliberal economics also serving as a grand narrative of environmental destruction. As an anti-modern response, postmodernism has been a triumphant failure. While there is much to be gained from the postmodern critique of modernity, its incredulity towards metanarratives has left it incapable of forming an adequate response to modernity, especially in regards to action on climate change. Postmodernism is better characterized as a crisis located within modernity itself and it will be argued that rather than the pursuit of the modern or the post-modern, we need to re-imagine ourselves as proto- historical to overcome the impasse of late-capitalism. (shrink)

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers (...) larger than 4 or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic. (shrink)

The revival of the philosophy of mathematics in the 60s following its post-1931 slump left us with two conflicting positions on arithmetic’s ontological relationship to set theory. W.V. Quine’s view, presented in 'Word and Object' (1960), was that numbers are sets. The opposing view was advanced in another milestone of twentieth-century philosophy of mathematics, Paul Benacerraf’s 'What Numbers Could Not Be' (1965): one of the things numbers could not be, it explained, was sets; the other thing numbers could not (...) be, even more dramatically, was objects. Curiously, although Benacerraf’s article appeared in the heyday of Quine’s influence, it declined to engage the Quinean position squarely, even seemed to think it was not its business to do so. Despite that, in my experience, most philosophers believe that Benacerraf’s article put paid to the reductionist view that numbers are sets (though perhaps not the view that numbers are objects). My chapter will attempt to overturn this orthodoxy. (shrink)

SEMANTIC ARITHMETIC: A PREFACE John Corcoran Abstract Number theory, or pure arithmetic, concerns the natural numbers themselves, not the notation used, and in particular not the numerals. String theory, or pure syntax, concems the numerals as strings of «uninterpreted» characters without regard to the numbe~s they may be used to denote. Number theory is purely arithmetic; string theory is purely syntactical... in so far as the universe of discourse alone is considered. Semantic arithmetic is a broad (...) subject which begins when numerals are mentioned (not just used) and mentioned as names of numbers (not just as syntactic objects). Semantic arithmetic leads to many fascinating and surprising algorithms and decision procedures; it reveals in a vivid way the experiential import of mathematical propositions and the predictive power of mathematical knowledge; it provides an interesting perspective for philosophical, historical, and pedagogical studies of the growth of scientific knowledge and of the role metalinguistic discourse in scientific thought. (shrink)

Orthodoxy holds that there is a determinate fact of the matter about every arithmetical claim. Little argument has been supplied in favour of orthodoxy, and work of Field, Warren and Waxman, and others suggests that the presumption in its favour is unjustified. This paper supports orthodoxy by establishing the determinacy of arithmetic in a well-motivated modal plural logic. Recasting this result in higher-order logic reveals that even the nominalist who thinks that there are only finitely many things should think (...) that there is some sense in which arithmetic is true and determinate. (shrink)

(Goodsell, Journal of Philosophical Logic, 51(1), 127-150 2022) establishes the noncontingency of sentences of first-order arithmetic, in a plausible higher-order modal logic. Here, the same result is derived using significantly weaker assumptions. Most notably, the assumption of rigid comprehension—that every property is coextensive with a modally rigid one—is weakened to the assumption that the Boolean algebra of properties under necessitation is countably complete. The results are generalized to extensions of the language of arithmetic, and are applied to answer (...) a question posed by Bacon and Dorr (2024). (shrink)

An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers.

How do we understand other individuals’ actions? Answers to this question cluster around two extremes: either by ascribing to the observed individual mental states such as intentions, or without ascribing any mental states. Thus, action understanding is either full-blown mindreading, or not mindreading. An intermediate option is lacking, but would be desirable for interpreting some experimental findings. I provide this intermediate option: actions may be understood by ascribing to the observed individual proto-intentions. Unlike intentions, proto-intentions are subject to (...) context-bound normative constraints, therefore being more widely available across development. Action understanding, when it consists in proto-intention ascription, can be a minimal form of mindreading. (shrink)

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two di erent ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers (...) and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier. (shrink)

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

The subject of this article is Modal-Epistemic Arithmetic (MEA), a theory introduced by Horsten to interpret Epistemic Arithmetic (EA), which in turn was introduced by Shapiro to interpret Heyting Arithmetic. I will show how to interpret MEA in EA such that one can prove that the interpretation of EA is MEA is faithful. Moreover, I will show that one can get rid of a particular Platonist assumption. Then I will discuss models for MEA in light of the (...) problems of logical omniscience and logical competence. Awareness models, impossible worlds models and syntactical models have been introduced to deal with the first problem. Certain conditions on the accessibility relations are needed to deal with the second problem. I go on to argue that those models are subject to the problem of quantifying in, for which I will provide a solution. (shrink)

The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis of the world in a Pythagorean manner. Hilbert arithmetic unifies the foundations of mathematics (Peano arithmetic and set theory), foundations of physics (quantum mechanics and information), and philosophical transcendentalism (Husserl’s phenomenology) into a formal theory and mathematical structure literally following Husserl’s tracе of “philosophy as a rigorous science”. In the pathway to that objective, Hilbert arithmetic identifies by itself information related to finite (...) sets and series and quantum information referring to infinite one as both appearing in three “hypostases”: correspondingly, mathematical, physical and ontological, each of which is able to generate a relevant science and area of cognition. Scientific transcendentalism is a falsifiable counterpart of philosophical transcendentalism. The underlying concept of the totality can be interpreted accordingly also mathematically, as consistent completeness, and physically, as the universe defined not empirically or experimentally, but as that ultimate wholeness containing its externality into itself. (shrink)

Scholarly opinions vary on what language is, how it evolved, and from where or what it evolved. Long considered uniquely human, today scholars argue for evolutionary continuity between human language and animal communication systems. But while it is generally recognized that language is an evolving communication system, scholars continue to debate from which species language evolved, and what behavioral and cognitive features are the precursors to human language. To understand the nature of protolanguage, some look for homologs in gene functionality, (...) brain areas, or anatomical structures such as the supralaryngeal vocal tract; others point toward primates, their gestural, vocal, multimodal, and in later evolving hominins also their pantomimic communication systems; and still others draw parallels between the musicality that characterizes language and the pitch found in the numerous sounds produced by animals. Accordingly, protolanguage theories today are multiple and diverse, and protolanguages might have also been diverse. This special issue on Evolving (Proto)Language/s for Lingua bundles several of the protolanguage theories that were put forward at the sixth edition of the Ways to Protolanguage conference series, held at the Calouste Gulbenkian Foundation in Lisbon, in 2019. Not aimed at surveying all the different ways there are to conceptualize, study, and model protolanguage/s, this issue provides interested readers with good overviews on the role played in current theorizing on protolanguage/s by (paleo)anthropology, genetics, physiology, developmental, evolutionary, ecological, and pragmatic research lines. (shrink)

In ‘Forgiveness, an Ordered Pluralism’, Fricker distinguishes two concepts of forgiveness, both of which are deployed in our forgiveness practices: moral justice forgiveness and gifted forgiveness. She then argues that the former is more explanatorily basic than the latter. We think Fricker is right about this. We will argue, however, that contra Fricker, it is a third more minimal concept that is most basic. Like Fricker, we will focus on the function of our practices, but in a way that is (...) informed by research in ethology (research which seeks to explain the function of animal behaviour and determine how these behaviours evolved). (shrink)

The subject of the first section is Carnapian modal logic. One of the things I will do there is to prove that certain description principles, viz. the ''self-predication principles'', i.e. the principles according to which a descriptive term satisfies its own descriptive condition, are theorems and that others are not. The second section will be devoted to Carnapian modal arithmetic. I will prove that, if the arithmetical theory contains the standard weak principle of induction, modal truth collapses to truth. (...) Then I will propose a different formulation of Carnapian modal arithmetic and establish that it is free of collapse. Noteworthy is that one can retain the standard strong principle of induction. I will occupy myself in the third section with Carnapian epistemic logic and arithmetic. Here too it is claimed that the standard weak principle of induction is invalid and that the alternative principle is valid. In the fourth and last section I will get back to the self-predication principles and I will point to some of the consequences if one adds them to Carnapian Epistemic arithmetic. The interaction of self-predication principles and the strong principle of induction results in a collapse of de re knowability. (shrink)

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

A language in which we can express arithmetic and which contains its own satisfaction predicate (in the style of Kripke's theory of truth) can be formulated using just two nonlogical primitives: (the successor function) and Sat (a satisfaction predicate).

This paper is part of a larger project about the relation between mathematics and transcendental philosophy that I think is the most interesting feature of Kant’s philosophy of mathematics. This general view is that in the course of arguing independently of mathematical considerations for conditions of experience, Kant also establishes conditions of the possibility of mathematics. My broad aim in this paper is to clarify the sense in which this is an accurate description of Kant’s view of the relation between (...) mathematics and transcendental philosophy. (shrink)

Galileo proposed what has been called a proto-inertial principle, according to which a body un horizontal motion will conserve its motion. This statement is only true in counterfactual circumstances where no impediments are present. This paper analyzes how Galileo could have been justified in ascribing definite properties to this idealized motion. This analysis is then used to better understand the relation of Galileo’s proto-inertial principle to the classical inertial principle.

A speculative investigation of how Frege's logical views change between Begriffsschrift and Grundgesetze and how this might have affected the formal development of logicism.

The Turing machine halting problem can be explained by several factors, including arithmetic logic irreversibility and memory erasure, which contribute to computational uncertainty due to information loss during computation. Essentially, this means that an algorithm can only preserve information about an input, rather than generate new information. This uncertainty arises from characteristics such as arithmetic logical irreversibility, Landauer's principle, and memory erasure, which ultimately lead to a loss of information and an increase in entropy. To measure this uncertainty (...) and loss of information, the concept of arithmetic logical entropy can be used. The Turing machine and its equivalent, general recursive functions can be understood through the λ calculus and the Turing/Church thesis. However, there are certain recursive functions that cannot be fully understood or predicted by other algorithms due to the loss of information during logical-arithmetic operations. In other words, the behaviour of these functions cannot be completely determined at every stage of the computation due to a lack of information in their definition. While there are some cases where the behaviour of recursive functions is highly predictable, the lack of information in most cases makes it impossible for algorithms to determine if a program will halt or not. This inability to predict the outcome of the computation is the essence of the halting problem of the Turing machine. Even in cases where more information is available about the program, it is still difficult to determine with certainty if the program will halt or not. This also highlights the importance of the Turing oracle machine, which introduces information from outside the computation to compensate for the lack of information and ultimately decide the result of the computation. (shrink)

In the first section of this paper we show that i Π1 ≡ W⌝⌝lΠ1 and that a Kripke model which decides bounded formulas forces iΠ1 if and only if the union of the worlds in any path in it satisflies IΠ1. In particular, the union of the worlds in any path of a Kripke model of HA models IΠ1. In the second section of the paper, we show that for equivalence of forcing and satisfaction of Πm-formulas in a linear Kripke (...) model deciding Δ0-formulas it is necessary and sufficient that the model be Σm-elementary. This implies that if a linear Kripke model forces PEMprenex, then it forces PEM. We also show that, for each n ≥ 1, iΦn does not prove ℋ(IΠn's are Burr's fragments of HA. (shrink)

In this essay I argue that Berkeley is proto-phenomenologist. The term phenomenology will chiefly be understood in terms of the approach of Edmund Husserl. Berkeley is attentive to the correct use of significations in philosophical exposition, the subjective character of experience, the motility of the perceiver and the transcendence of things. Like the phenomenologists he rejects materialism, naturalism and scepticism. He seeks to preserve the evidences of ordinary perception, setting out an account of scientific theory that can cohere with (...) them. In the first part of this essay I go through some of the more clear-cut ways in which Berkeley anticipates Husserl and some of the latter’s successors. In the second part I lay out their criticisms of Berkeley, also indicating what they hold in common. In the third and final part I consider some of the ways in which Berkeley can meet these criticisms, either readily or through certain qualifications, and I bring out his most significant contributions as a proto-phenomenologist. My aim is to show there are more than traces of the phenomenological attitude in Berkeley. In his efforts to save the appearances, he goes back to that world in which we live and breathe, if not quite have our being. (shrink)

We argue that, under the usual assumptions for sufficiently strong arithmetical theories that are subject to Gödel’s First Incompleteness Theorem, one cannot, without impropriety, talk about *the* Gödel sentence of the theory. The reason is that, without violating the requirements of Gödel’s theorem, there could be a true sentence and a false one each of which is provably equivalent to its own unprovability in the theory if the theory is unsound.

Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that (...) arithmetical knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology. (shrink)

The theoretical biology movement originating in Britain in the early 1930’s and the biosemiotics movement which took off in Europe in the 1980’s have much in common. They are both committed to replacing the neo-Darwinian synthesis, and they have both invoked theories of signs to this end. Yet, while there has been some mutual appreciation and influence, particularly in the cases of Howard Pattee, René Thom, Kalevi Kull, Anton Markoš and Stuart Kauffman, for the most part, these movements have developed (...) independently of each other. Focussing on morphogenesis understood as vegetative semiosis, in this paper I will argue that the ideas of these movements are commensurate. Furthermore, synthesising them would enable us to see life processes as proto-narratives. Doing so will involve synthesising biohermeneutics, Peircian biosemiotics with Waddington’s theoretical biology and Piagetian genetic structuralism, and this, I claim, would strengthen the challenge of these traditions to mainstream biology. At the same time, this should contribute to overcoming the opposition between the sciences and the humanities, developing a broader tradition of Schellingian thought which involves developing the humanities and then demanding of the physical and biological sciences that they are consistent with and can make intelligible the emergence of humans as conceived by the humanities. (shrink)

In Plato’s Statesman, the Eleatic Stranger applies a specialized method of inquiry—the “method of collection and division”, or “method of division”—in order to discover the nature of statecraft. This paper articulates some consequences of the fact that the method is both a tool for identifying natural kinds—that is, a tool for carving the world by its joints (Phaedrus 265b-d)—and social kinds—that is, the kinds depending on human beings for their existence and explanation. A central goal of the paper is to (...) illuminate the extent to which this use of the method of division allows us to identify Plato as an early historical forerunner of racialism, which is an ideology according to which humanity divides into races differentiated by heritable physiological, cultural, and intellectual traits, as a way of vindicating oppressive and exploitative social, political, and economic systems. -/- I defend an interpretation of the Stranger’s claim, much discussed in the literature, that the division of humankind into Greek and barbarian is unnatural (Politicus 262c-263a). I argue that, in the Stranger’s view, this division reflects subjective illusion and prejudice, rather than the fundamental, and teleological, structure of human social organization, which concerns how human beings rationally cooperate to self-produce as a species. Nonetheless, the Stranger’s alternative theory of the natural structure of human society, I suggest, is proto-racial in another way. Through a brief consideration of the Stranger’s affirmative and complex division of kinds in the city, I argue that he re-introduces naturalistic foundations for unjust human hierarchies through his alternative theory of natural kinds and human social teleology. (shrink)

We take an argument of Gödel's from his ground‐breaking 1931 paper, generalize it, and examine its validity. The argument in question is this: "the sentence G says about itself that it is not provable, and G is indeed not provable; therefore, G is true".

This paper presents the main aspects of the proto-Wenzi’s philosophy, with a focus on its intricate relationship with the Laozi. They show that the proto-Wenzi advocates a philosophy of quietude, not only in terms of its content, but also through the rhetoric it uses to create a harmonious synthesis of diverse, and at times even incompatible, ideas.

I argue that the reception of Hegel in the sub-field of history and philosophy of science has been in part impeded by a misunderstanding of his mature metaphilosophical views. I take Alan Richardson’s influential account of the rise of scientific philosophy as an illustration of such misunderstanding, I argue that the mature Hegel’s metaphilosophical views place him much closer to the philosophers who are commonly taken as paradigms of scientific philosophy than it is commonly thought. Hegel is commonly presented as (...) someone who conceived of philosophy as a science that relied on the solitary genius of the individual thinker, and as a science whose propositions could not and should not be made accessible to “the common people”. Against this view, I argue that Hegel in fact thought that philosophy was a thoroughly anti-individualistic activity, and that he emphasized the importance of the intersubjective accessibility of philosophical discourse. I argue that when we carefully reconstruct Hegel’s reasons for his break with Schelling, and if we pay close attention to his explicit metaphilosophical pronouncements, we can see that he in fact adhered to what I call a “proto-modernist” conception of philosophy as a science. I conclude by pointing out how the mischaracterization of Hegel has served to obscure the existence of a strand of scientific philosophy that emerged by way of an immanent critique of Hegel, namely Marxist philosophy. (shrink)

In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain (...) the ... (shrink)

This dissertation is a study of the relationship between dunamis and energeia in Aristotle’s ontology. Throughout his writings, Aristotle employs these terms to uncover what I call a proto-phenomenological description of the different ways of being. While contemporary scholarship has suggested the significance of dunamis and energeia for Aristotle’s understanding of being, the relationship between these terms has often been interpreted as mutually exclusive. Accordingly, dunamis would be understood as subordinate to energeia, which would function as the sole primary (...) term of Aristotle’s ontology. I argue that it is a mistake to consider dunamis and energeia as non-reciprocal and subordinate terms. Furthermore, I suggest that this mistake often leads to an underestimation of the dynamic character of Aristotle’s proto-phenomenological account of being. To recover this dimension of Aristotle’s thinking, I claim that dunamis and energeia ought to be understood as reciprocal and co-constitutive terms characterized by a constant and dynamic interrelation. To defend this interpretation, I turn to Aristotle’s conceptions of nature, movement, and soul as discussed in the Metaphysics, Physics, and De Anima. I argue that each of these key terms provide a concrete illustration of how dunamis and energeia function as co-constitutive principles for the manifestation of the being of natural beings. I propose that this proto-phenomenological approach to being remains Aristotle’s most significant and enduring contribution to Western ontology. (shrink)

This paper demonstrates that Edmund Husserl’s frequently overlooked 1890 manuscript, “On the Logic of Signs,” when closely investigated, reveals itself to be the hermeneutical touchstone for his seminal 1891 Philosophy of Arithmetic. As the former comprises Husserl’s earliest attempt to account for all of the different kinds of signitive experience, his conclusions there can be directly applied to the latter, which is focused on one particular type of sign; namely, number signs. Husserl’s 1890 descriptions of motivating and replacing signs (...) will be respectively employed to clarify his 1891 understanding of the authentic and inauthentic presentations of numbers via number signs. Moreover, his schematic classification of replacement-signs in Semiotic will illuminate the reasons why he believed the number system to be necessary for the operation of replacing number signs. (shrink)

Elementary patterns of resemblance notate ordinals up to the ordinal of Pi^1_1-CA_0. We provide ordinal multiplication and exponentiation algorithms using these notations.

Second-order Peano Arithmetic minus the Successor Axiom is developed from first principles through Quadratic Reciprocity and a proof of self-consistency. This paper combines 4 other papers of the author in a self-contained exposition.

Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it (...) isn’t. In fact, 2FA is not conservative over n-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic. (shrink)

Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.

The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results.

This paper presents proof that Buss's S22 can prove the consistency of a fragment of Cook and Urquhart's PV from which induction has been removed but substitution has been retained. This result improves Beckmann's result, which proves the consistency of such a system without substitution in bounded arithmetic S12. Our proof relies on the notion of "computation" of the terms of PV. In our work, we first prove that, in the system under consideration, if an equation is proved and (...) either its left- or right-hand side is computed, then there is a corresponding computation for its right- or left-hand side, respectively. By carefully computing the bound of the size of the computation, the proof of this theorem inside a bounded arithmetic is obtained, from which the consistency of the system is readily proven. This result apparently implies the separation of bounded arithmetic because Buss and Ignjatović stated that it is not possible to prove the consistency of a fragment of PV without induction but with substitution in Buss's S12. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system that we have considered is strictly equational, which is a property on which our proof relies. (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)

Proof uses forcing on perfect trees for 2-quantifier sentences in the language of arithmetic. The result extends to exact pairs for the hyperarithmetic degrees.

This paper describes both an exegetical puzzle that lies at the heart of Frege’s writings—how to reconcile his logicism with his definitions and claims about his definitions—and two interpretations that try to resolve that puzzle, what I call the “explicative interpretation” and the “analysis interpretation.” This paper defends the explicative interpretation primarily by criticizing the most careful and sophisticated defenses of the analysis interpretation, those given my Michael Dummett and Patricia Blanchette. Specifically, I argue that Frege’s text either are inconsistent (...) with the analysis interpretation or do not support it. I also defend the explicative interpretation from the recent charge that it cannot make sense of Frege’s logicism. While I do not provide the explicative interpretation’s full solution to the puzzle, I show that its main competitor is seriously problematic. (shrink)