Traditionally, conceptual thinking is explored via philosophical analysis or psychological experimentation. We seek to complement these mainstream approaches with the perspective of a first person exploration into pure thinking. To begin with, pure thinking is defined as a process and differentiated from its content, the concepts itself. Pure thinking is an active process and not a series of associative thought-events; we participate in it, we immerse ourselves within its active performance. On the other hand, concepts are also of an experiential (...) nature. And yet, little is known about what is it like to have or produce a thought, a concept, or an idea? Is a concept our own construction, a product of our own activity, or is it something we merely discover instead of producing it? We address these issues in a systematic first person enquiry into pure thinking. (shrink)

After a brief outline of the topic of non-language thinking in mathematics the central phenomenological tool in this concern is established, i.e. the eidetic method. The special form of eidetic method in mathematical proving is implicit variation and this procedure entails three rules that are established in a simple geometrical example. Then the difficulties and the merits of analogical thinking in mathematics are discussed in different aspects. On the background of a new phenomenological understanding of the performance of non-language thinking (...) in mathematics the well-known theses of B. L. van der Waerden that mathematical thinking to a great extent proceeds without the use of language is discussed in a new light. (shrink)

This paper argues that the shared intersubjective accessibility of mathematical objects has its roots in a stratum of experience prior to language or any other form of concrete social interaction. On the basis of Husserl’s phenomenology, I demonstrate that intersubjectivity is an essential stratum of the objects of mathematical experience, i.e., an integral part of the peculiar sense of a mathematical object is its common accessibility to any consciousness whatsoever. For Husserl, any experience of an objective nature has as its (...) correlate a “we,” which he terms the “community of monads”. Thus, even before mathematical objects gain expression, formalization, and axiomatization through natural and scientific language, from a phenomenological viewpoint their objectivity has its roots in raw pre-linguistic though intersubjective experience. Accordingly, I demonstrate the different senses in which the experience of mathematical objects is permeated by intersubjectivity, suggesting a picture of mathematical intersubjectivity as pre-linguistic common experience based on Husserl’s idea of a “community of monads”. (shrink)

This dissertation explores and argues for the import of the imagination (Phantasie) in Edmund Husserl's phenomenological method of inquiry. It contends that Husserl's extensive analyses of the imagination influenced how he came to conceive the phenomenological method throughout the main stages of his philosophical career. The work clarifies Husserl's complex method of investigation by considering the role of the imagination in his main methodological apparatuses: the phenomenological, eidetic, and transcendental reductions, and eidetic variation - all of which remained ambiguous despite (...) his extensive programmatic discussions. The work illuminates and clarifies aspects of the Husserlian phenomenological method never before explored. -/- In order to clarify Husserl's eidetic method of inquiry, I propose a new way of thinking about the imagination - as direct intuitive presentation (eigentliche anschauliche Vorstellung) and as horizonal-nexic level of consciousness exhibiting the neutrality, freedom, and possibility as its essential features. Following Husserl's studies of the imagination, I propose a three-level model of consciousness (realizing, imagining, and eidetic) and explore the dynamic flexibility of each level (as horizon within which acts such as judgments or memories can unfold). This model of consciousness allows for a rethinking of the sources and conditions for the possibility of eidetic phenomenological inquiry - topics Husserl was mostly silent about. -/- Through a rethinking of the model of consciousness, I propose a tight and substantial relationship between the natural (everyday) and artificial (methodological, theoretical) attitudes. I argue that the structure and systems of possibilities pertaining to the artificial attitude - i.e., our actual as well as possible methodological tools - are structurally and well as informationally bound to the structure and system of possibilities pertaining to the natural attitude. In order to explore the nature of the relationship between these two attitudes I argue that we must take a closer look at the structure and abilities of imagining consciousness - the sole nexic-horizonal level that can function both naturally and artificially. This insight regarding the nature of consciousness clarifies Husserl's transcendental idealism in its intimate connection to the everyday. Understanding Husserl's philosophical stance is thus purged of all possibility of mistakenly labeling it as entailing immanent detachment, solipsism, or Platonic idealism. (shrink)

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