Notes on the Dialogue between Phenomenology and Mathematics: Husserl and Becker. (original) (raw)
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Mathematical Knowledge and the Origin of Phenomenology: The Question of Symbols in Early Husserl
Studia Phaenomelogica, 2021
The paper is divided into two parts. In the first one, I set forth a hypothesis to explain the failure of Husserl's project presented in the Philosophie der Arithmetik based on the principle that the entire mathematical science is grounded in the concept of cardinal number. It is argued that Husserl's analysis of the nature of the symbols used in the decadal system forces the rejection of this principle. In the second part, I take into account Husserl's explanation of why, albeit independent of natural numbers, the system is nonetheless correct. It is shown that its justification involves, on the one hand, a new conception of symbols and symbolic thinking, and on the other, the recognition of the question of "the formal" and formalization as pivotal to understand "the mathematical" overall.
Intuitionism in the Philosophy of Mathematics: Introducing a Phenomenological Account
The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: What does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics.
History and Philosophy of Logic, 2006
The paper examines the roots of Husserlian phenomenology in Weierstrass's approach to analysis. After elaborating on Weierstrass's programme of arithmetization of analysis, the paper examines Husserl's Philosophy of Arithmetic as an attempt to provide foundations to analysis. The Philosophy of Arithmetic consists of two parts; the first discusses authentic arithmetic and the second symbolic arithmetic. Husserl's novelty is to use Brentanian descriptive analysis to clarify the fundamental concepts of arithmetic in the first part. In the second part, he founds the symbolic extension of the authentically given arithmetic with stepwise symbolic operations. In the process of doing so, Husserl comes close to defining the modern concept of computability. The paper concludes with a brief comparison between Husserl and Frege. While Frege chose to subject arithmetic to logical analysis, Husserl wants to clarify arithmetic as it is given to us. Both engage in a kind of analysis, but while Frege analyses within Begriffsschrift, Husserl analyses our experiences. The difference in their methods of analysis is what ultimately grows into two separate schools in philosophy in the 20th century.
Marmara Üniversitesi İlahiyat Fakültesi Dergisi, 2023
In his later works, the great logician and mathematician Kurt Gödel concentrates his focus on the philosophical problems such as the implications of set theory, the grammar and philosophy of language, objectivity and relativity, the ontological proof of God’s existence, and phenomenology as an exact method. This essay explores how Gödel reads the philosophy (of logic and mathematics) of his time and why he turns his attention to Husserl’s phenomenology for describing the foundations of mathematics. To begin with, Gödel employs Husserl’s significant distinction between Weltanschauung (worldview) philosophy and philosophy as rigorous science: According to the Weltanschauung philosophy, the spirit of time constantly changes so that the ideas discussed and goals attempted are meant to be temporal, and not for the sake of eternal truths, but for that of their own perfection; philosophy as rigorous science, on the other hand, is supratemporal so that its aim is to discover absolute and timeless values. As for the worldview of his time, Gödel sees the development of philosophy and mathematics leaned toward skepticism, pessimism, and positivism. The antinomies of set theory, for instance shaked the grounds on which mathematics and logic are founded. Gödel, too, uses these paradoxes in his incompleteness theorems in order to prove that there are some statements which can neither be proved nor disproved within a system. That also means that arithmetic is not eligible to prove its own consistency. From this, however, Gödel does not come to a conclusion for a nihilism in mathematics and logic: These mere antinomies of set theory do not “necessarily” lead us to logical positivism, and neither to such a materialism, nor to any kind of pessimistic theory of knowledge. The incompleteness theorems assert that there are arithmetical propositions that are true but neither provable nor unprovable within its own calculus, so that arithmetic is intrinsically incomplete. However, instead of Alfred Tarski’s pathological view of examining the detections within the faulty system and then reforming the system all together, Gödel holds that we need to change our methods to find new patterns that describe the antinomies pointing to the unrecoverable reality of the mathematical world. Thus, Gödel does not follow any variation of the Weltanschauung philosophy of his time, either attempting to reduce mathematical realities to mathematical proofs in order to get rid of antinomies, or endeavoring to rescue a complete system of truths by a closed formal system, both Weltanschauung philosophies fail to set forth a realistic method. In this context, Gödel finds the task of phenomenology analogous to what he pursues in terms of a systematic framework for the foundations of mathematics. Husserl’s phenomenology, in Gödel’s account, proliferates the intuition of (mathematical) essences and provides a clarification of meaning of undefinable concepts, such as the antinomies of set theory. Applying the phenomenological reduction to the objective reality of the mathematical world, Gödel believes one obtains a clear experiential reality of the essential characteristics of (mathematical and logical) concepts. Briefly put, what Gödel finds in Husserl’s phenomenology that corresponds to his way of mathematical realism is a thoroughly designated method giving us mathematical essences back again.
One main consequence of the first incompleteness theorem is that the truth of some propositions (given within a mathematical theory, which includes elementary number theory) could not be defined in terms of provability. The question that immediately arises, from an epistemological standpoint, is: in which terms should we, then, define it? When we consider Gödel's philosophical papers (cf. Gödel, Collected works) we can discern two consecutive attitudes as regards the possible response to that question: