Ioannis G. Ntagkas

Published in 1891, Edmund Husserl's first book, Philosophie der Arithmetik, aimed to ‘prepare the scientific foundations for a future construction of that discipline’. His goals should seem reasonable to contemporary philosophers of mathematics: "…through patient investigation of details, to seek foundations, and to test noteworthy theories through painstaking criticism, separating the correct from the erroneous, in order, thus informed, to set in their place new ones which are, if possible, more adequately secured. 1"But the ensuing strategy for grounding mathematical knowledge sounds strange to the modern ear. For Husserl cast his work as a sequence of ‘psychological and logical investigations’, providing a psychological analysis "…of the concepts multiplicity, unity, and number, insofar as they are given to use authentically and not through indirect symbolizations. "This emphasis on psychology is a reflection of Husserl's training. As a teenager studying in Leipzig, he attended the lectures of Wilhelm Wundt, a seminal figure in the field of experimental psychology. Wundt held that, via introspection, we can study and classify our inner experiences, in much the same way that scientists study the natural world. 2 People working in his laboratory were therefore trained in procedures for observing and reporting on their own thought processes, as a means of gathering scientific data regarding our cognitive faculties. Bridging the gap between psychology and epistemology, Wundt felt that the results of such inquiry could have normative consequences, since the principles of reasoning employed in the sciences not only have their origins in psychological processes, but, moreover, are justified by the fundamental role they play in thought. His two-volume work, Logik, thus combined empirical considerations with a Kantian emphasis on the way that knowledge depends on our cognitive faculties. 3In The Philosophy of …

We analyze the pointwise convergence of a sequence of computable elements of L1 in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA0, each is equivalent to the assertion that every Gδ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak Königʼs lemma relativized to the Turing jump of any set. It is also equivalent to the conjunction of the statement asserting the existence of a 2-random relative to any given set and the principle of Σ2 collection.

A seminal theorem due to Weyl [14] states that if $(a_n)$ is any sequence of distinct integers, then, for almost every $x \in \mathbb{R}$, the sequence $(a_n x)$ is uniformly distributed modulo one. In particular, for almost every $x$ in the unit interval, the sequence $(a_n x)$ is uniformly distributed modulo one for every computable sequence $(a_n)$ of distinct integers. Call such an $x$ UD random. Here it is shown that every Schnorr random real is UD random, but there are Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random.

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation. In this essay, we describe an approach to the philosophy of mathematics in which it is an important task to understand the roles of our ontological posits and assess the extent to which they enable us to achieve our mathematical goals. We use the history of Dirichlet’s theorem to understand some of the reasons that functions are treated as ordinary objects in contemporary mathematics, as well as some of the reasons one might want to resist such treatment. We also use these considerations to illuminate the formal treatment of functions and objects in Frege’s logical foundation, and we argue that his philosophical and logical decisions were influenced by many of the same factors

Proof theory began in the 1920s as a part of Hilbert’s program, which aimed to secure the foundations of mathematics by modeling infinitary mathematics with formal axiomatic systems and proving those systems consistent using restricted, finitary means. The program thus viewed mathematics as a system of reasoning with precise linguistic norms, governed by rules that can be described and studied in concrete terms. Today such a viewpoint has applications in mathematics, computer science, and the philosophy of mathematics.

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.

As idealized descriptions of mathematical language, there is a sense in which formal systems specify too little, and there is a sense in which they specify too much. On the one hand, formal languages fail to account for a number of features of informal mathematical language that are essential to the communicative and inferential goals of the subject. On the other hand, many of these features are independent of the choice of a formal foundation, so grounding their analysis on a particular choice of a formal system introduces unnecessary specificity. This chapter begins to map out the design features of mathematical language without descending to the level of formal implementation, drawing on examples from the mathematical literature and insights from the design of computational proof assistants and their libraries.

Every branch of mathematics has its subject matter, and one of the distinguishing features of logic is that so many of its fundamental objects of study are rooted in language. The subject deals with terms, expressions, formulas, theorems, and proofs. When we speak about these notions informally, we are talking about things that can be written down and communicated with symbols. One of the goals of mathematical logic is to introduce formal definitions that capture our intuitions about such objects and enable us to reason about them precisely. At the most basic level, syntactic objects can be viewed as strings of symbols. For concreteness, we can identify symbols with particular set-theoretic objects, but for most purposes, it doesn't matter what they are; all that is needed is that they are distinct from one another.

Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. It has been common since the turn of the twentieth century to take correctness to be underwritten by the existence of formal derivations in a suitable axiomatic foundation, but then it is hard to see how this normative standard can be met, given the differences between informal proofs and formal derivations, and given the inherent fragility and complexity of the latter. This essay describes some of the ways that mathematical practice makes it possible to reliably and robustly meet the formal standard, preserving the standard normative account while doing justice to epistemically important features of informal mathematical justification.

The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term εx A denotes some x satisfying A(x), if there is one. In Hilbert's Program, the epsilon terms play the role of ideal elements; the aim of Hilbert's finitistic consistency proofs is to give a procedure which removes such terms from a formal proof. The procedures by which this is to be carried out are based on Hilbert's epsilon substitution method. The epsilon calculus, however, has applications in other contexts as well. The first general application of the epsilon calculus was in Hilbert's epsilon theorems, which in turn provide the basis for the first correct proof of Herbrand's theorem. More recently, variants of the epsilon operator have been applied in linguistics and linguistic philosophy to deal with anaphoric pronouns.

Neurotechnologies are rapidly advancing in the past few years, such that neural prostheses and brain-computer interfaces are no longer things that only appear in science fiction movies. As interactions with neurotechnologies deepen, users have reported feeling that these tools are becoming part of their own selves and minds. The hypothesis of extended cognition can accommodate this intuition, as it maintains that artifacts can become a part of their users’ minds. However, there have also been some stark examples where users have abruptly lost access to their sophisticated tools, demonstrating the sometimes vulnerable and precarious nature of certain advanced technologies. These features seem to challenge the idea that users can stand in the right relation to their tools to meet the criteria of parity that support the extended cognition theorists’ arguments. Notably, these technologies seem to violate a condition of ownership that has been appealed to on various occasions over the past two decades of literature on extended cognition. In addition to arguing that neuroprosthetics can be a part of one’s extended cognitive system, despites apparent challenges to an ownership condition, we will also review the disparate history and evaluate the current status of the ownership criterion in the literature on extended cognition. To proceed, we first review the reasons for the origin of the proposed “ownership” criterion and, second, introduce new challenges arising from emerging technologies such as neuroprosthetics. We argue that the ownership condition has at least three shortcomings: (i) ambiguity of meaning, (ii) ethical pitfalls, and (iii) it fails to capture people’s intuitions about the cognitive status of emerging neurotechnologies. Ultimately, we argue that introducing the concept of co-ownership is a necessary distinction and is better suited to explain how advanced cognitive technologies function.

Recent research has defended the surprising thesis that in many cases the search for truth is better off if the information exchanged between the members of an epistemic community is limited. This is what one may call the limited information thesis. There is, however, the possibility of an even more radical position than this: the thesis that any communication between peers has zero epistemic value and that the search for truth is better off if the truth-inquirer does not take into consideration the truth-claims of her peers. This can be called the solitude thesis. The paper defends the claim that Descartes is a supporter of the solitude thesis with respect to metaphysical inquiry. The defense is facilitated by means of interpreting textual evidence found in Descartes’ essays Discourse on the Method, The Search for Truth and the Meditations on First Philosophy.

They say that argumentative writing skills are best learned through writing argumentative essays. I say that while this is excellent practice for argumentative writing, an important exercise to practice structuring such essays and build critical thinking skills simultaneously is what I call the four-sentence paper. The exercise has the template They say..., I say..., one might object..., I reply... One might object that the assignment oversimplifies argumentative writing, stifles creativity, promotes an adversarial attitude, or that students can’t consider objections well anyway. I reply that a simplified form of argumentative writing is fine for beginners, especially since the template is ubiquitous in philosophy; that any assignment template has room for creativity; that considering objections is consistent with good manners; and that despite some pitfalls of trying to defend a thesis and consider objections, students are capable of considering objections well with proper instruction and practice.

This article is concerned with what Nietzsche claims about particular kinds of suffering that can emerge in encounters with others. I maintain that, even taking into account statements of Nietzsche’s that contradict or modify his language of solitude, hardness and domination, his acknowledgement of the capacity of witnessing others’ suffering to cause pain does not indicate an intersubjective notion of self-affirmation, but is an instance of a tension he identifies between our inescapable implication in social ways of being, and our need to create ourselves independently to overcome self-alienation. I argue that Nietzsche’s claims about pity are a particular instance of this tension – that is, that while he recognizes that we feel pity, he treats this as an unfortunate affect to be overcome, to be appropriated on an individual basis, rather than as an invitation to be authentically with others – as indicating the possibility of a mutual project of self-affirmation

ABSTRACT This paper adds to efforts to retrieve the long-neglected philosophical contributions of Karoline von Günderrode (1780–1806), and is one of the first to seriously address the political commitments in Günderrode’s work, especially regarding revolution. This idea gains an unusual status in the context of Günderrode’s metaphysics, and is key to understanding the connections between Günderrode’s more obviously philosophical writings and her literary work. I argue that Günderrode’s concept of revolution resembles, in some respects, the ideas of other thinkers of her time, but has ramifications for conceiving of human individuals and their relationships to society and nature that are unique to Günderrode. First, I use a comparison with the work of contemporaries on revolution to justify interpreting Günderrode’s metaphysics in relation to her thought on politics. Günderrode often masks her political thought in the form of plays, and the paper next considers the theme of revolution in her plays Muhammad, the Prophet of Mecca, and Udohla. Lastly, the paper contrasts Günderrode’s position on revolution to claims by Herder and Fichte, arguing that Günderrode’s conception of a cycle of enhancement and decay of natural forces, political power and social cohesion differs from androcentric models that emphasize the development of consciousness.

This chapter considers the philosophical contributions of German writer Karoline von Günderrode (1780–1806). Günderrode is an original, though neglected, thinker engaged with German Idealism and Romanticism, whose writings reflect on the same problems that preoccupied other philosophers working in these traditions. Her work participates in debates regarding the question of free will, the nature of the self, the nature of consciousness, what happens to us after we die, the vocation of humankind, the relationship between the self and nature and between these and the Absolute or the divine, the role of gender in social life, ideals for political arrangements, and the pursuit of virtue and beauty. Günderrode’s writings provide a rich and relatively unexplored perspective on these topics. After a biography and overview of Günderrode’s writings, the chapter summarizes key areas of Günderrode’s thought. These include Günderrode’s metaphysical claims, unique accounts of love and death, models of consciousness and identity, political theory (especially regarding revolution), contributions to early nineteenth century understandings of gender, and nascent ethics and aesthetics. The last section of the chapter considers ways that Günderrode might have influenced several well-known nineteenth century thinkers, notably Clemens Brentano, Georg Friedrich Creuzer, and Bettina Brentano-von Arnim – and, through the latter, the Young Hegelians and American Transcendentalism.

This paper considers the systematic relationship between faith and reason in Kant’s grounding and limiting of moral faith in the Canon of Pure Reason in the Critique of Pure Reason. The first section addresses the interrelationship between Kant’s critique of knowledge and his critique of faith. The second section defines the complex interplay of theoretical and practical issues in Kant’s critical question of what a morally acting agent may hope for regarding the overall outcome of his actions. The third section investigates Kant’s positioning of faith in relation to the epistemic modi of opining and knowing, and defines the specific status of moral faith in Kant. The textually based systematic analyses focus on the complementary relationship in Kant’s work between the way rational critique limits theoretical knowledge in favor of moral faith and the way this critique limits religious faith through moral consciousness.

Johann Heinrich Tieftrunk has a place among the early Kantians in Halle as both a theologian and a philosophical thinker. After situating Tieftrunk within this intellectual history and determining his theological and philosophical position, this paper provides a chronological account of the concept of hope—which lies at the basis of Kant’s moral philosophy—in Tieftrunk’s writings on philosophy of religion. In particular, the discussion centers on the relationship between the foundation of hope in the moral law and the exclusion of a legalistic fulfilment of hope. The “love of the law,” which emerges especially in Tieftrunk’s work, takes a prominent position in the interpretation. The investigation is accompanied by an evaluation of Tieftrunk’s role in the early reception of Kant.