This paper serves as a kind of field guide to certain passages in the literature which bear upon the foundational theory of abstract objects. The foundational theory assimilates ideas from key philosophers in both the analytical and phenomenological traditions. I explain how my foundational theory of objects serves as a common ground where analytic and phenomenological concerns meet. I try to establish how the theory offers a logic that systematizes a well-known phenomenological kind of entity, and I try to show (...) the various ways the theory systematizes the ideas of many analytic philosophers. The ideas of Plato, Leibniz, Frege, Russell, Goedel and even Kripke become connected through those of Brentano, Meinong, Husserl, and Mally. -/- The field guide will not only document the passages in which the distinction between two kinds of predication originates, but also document the other surprising, and often unrelated, contexts where the distinction reappears in the work of others. It will also document ways in which the theory can be used to represent precisely the ideas of the philosophers mentioned above. The resulting guide will bring together the works of many different authors, including some clearly within the analytic tradition, some clearly within the phenomenological tradition, and some who straddle the divide. (shrink)

In this paper I argue that it is more difficult to see how Godel's incompleteness theorems and related consistency proofs for formal systems are consistent with the views of formalists, mechanists and traditional intuitionists than it is to see how they are consistent with a particular form of mathematical realism. If the incompleteness theorems and consistency proofs are better explained by this form of realism then we can also see how there is room for skepticism about Church's Thesis and the (...) claim that minds are machines. (shrink)

In 1928 Edmund Husserl wrote that “The ideal of the future is essentially that of phenomenologically based (“philosophical”) sciences, in unitary relation to an absolute theory of monads” (“Phenomenology”, Encyclopedia Britannica draft) There are references to phenomenological monadology in various writings of Husserl. Kurt Gödel began to study Husserl’s work in 1959. On the basis of his later discussions with Gödel, Hao Wang tells us that “Gödel’s own main aim in philosophy was to develop metaphysics—specifically, something like the monadology of (...) Leibniz transformed into exact theory—with the help of phenomenology.” (A Logical Journey: From Gödel to Philosophy, p. 166) In the Cartesian Meditations and other works Husserl identifies ‘monads’ (in his sense) with ‘transcendental egos in their full concreteness’. In this paper I explore some prospects for a Gödelian monadology that result from this identification, with reference to texts of Gödel and to aspects of Leibniz’s original monadology. (shrink)

In a lecture manuscript written around 1961, Gödel describes a philosophical path from the incompleteness theorems to Husserl's phenomenology. It is known that Gödel began to study Husserl's work in 1959 and that he continued to do so for many years. During the 1960s, for example, he recommended the sixth investigation of Husserl's Logical Investigations to several logicians for its treatment of categorial intuition. While Gödel may not have been satisfied with what he was able to obtain from philosophy and (...) Husserl's phenomenology, he nonetheless continued to recommend Husserl's work to logicians as late as the 1970s. In this paper I present and discuss the kinds of arguments that led Gödel to the work of Husserl. Among other things, this should help to shed additional light on Gödel's philosophical and scientific ideas and to show to what extent these ideas can be viewed as part of a unified philosophical outlook. Some of the arguments that led Gödel to Husserl's work are only hinted at in Gödel's 1961 paper, but they are developed in much more detail in Gödel's earlier philosophical papers. In particular, I focus on arguments concerning Hilbert's program and an early version of Carnap's program.§1. Some ideas from phenomenology. Since Husserl's work is not generally known to mathematical logicians, it may be helpful to mention briefly a few details about his background. (shrink)

Gödel has argued that we can cultivate the intuition or perception of abstractconcepts in mathematics and logic. Gödel's ideas about the intuition of conceptsare not incidental to his later philosophical thinking but are related to many otherthemes in his work, and especially to his reflections on the incompleteness theorems.I describe how some of Gödel's claims about the intuition of abstract concepts are related to other themes in his philosophy of mathematics. In most of this paper, however,I focus on a central (...) question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that uses some ideas of a philosopher to whom Gödel also turned in this connection: Edmund Husserl. The answer depends on how we understand the conscious directedness toward objects and the meaning of the term abstract in the context of a theory of the intentionality of cognition. (shrink)

Gödel has argued that we can cultivate the intuition or ‘perception’ of abstractconcepts in mathematics and logic. Gödel's ideas about the intuition of conceptsare not incidental to his later philosophical thinking but are related to many otherthemes in his work, and especially to his reflections on the incompleteness theorems.I describe how some of Gödel's claims about the intuition of abstract concepts are related to other themes in his philosophy of mathematics. In most of this paper, however,I focus on a central (...) question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that uses some ideas of a philosopher to whom Gödel also turned in this connection: Edmund Husserl. The answer depends on how we understand the conscious directedness toward ‘objects’ and the meaning of the term ‘abstract’ in the context of a theory of the intentionality of cognition. (shrink)

There is a significant convergence between some contemporary panpsychist theories and idealism. Traditional phenomenal, macro idealism has been widely criticised for its implausible account of physical facts. This paper examines three idealism friendly contemporary panpsychist philosophies and suggests that with some revisions a plausible realist and physical idealism can be articulated. Taking from information-theoretic models of consciousness and metaphysical arguments for the ontological primacy of mind, it is suggested via a Schelling inspired perspective that aspects of consciousness??? intrinsic nature provide (...) a promising approach to the decomposition problem which is currently a significant challenge to pan/cosmopsychist theories. (shrink)

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the “unreasonable effectiveness of mathematics in (...) philosophy”—to adapt Wigner’s phrase—is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate. (shrink)

Gödel's philosophical views were to a significant extent influenced by the study not only of Leibniz or Husserl, but also of Kant. Both Gödel and Kant aimed at the secure foundation of philosophy, the certainty of knowledge and the solvability of all meaningful problems in philosophy. In this paper, parallelisms between the foundational crisis of metaphysics in Kant's view and the foundational crisis of mathematics in Gödel's view are elaborated, especially regarding the problem of finding the “secure path of a (...) science” for both mathematics and philosophy. Gödel's temporal subjectivism and metaphysical conceptual objectivism are presented as positively or negatively motivated by Kant's viewpoints. A remark on Gödel's collapse of modalities (in accordance with the collapse of objective time) is added. (shrink)

The view that mathematics deals with ideal objects to which we have epistemic access by a kind of perception has troubled many thinkers. Using ideas from Husserl’s phenomenology, I will take a different look at these matters. The upshot of this approach is that there are non-material objects and that they can be recognized in a process very closely related to sense perception. In fact, the perception of physical objects may be regarded as a special case of this more universal (...) way of recognizing objects of any kind. (shrink)

I examine various claims to the effect that Cantor's Continuum Hypothesis and other problems of higher set theory are ill-posed questions. The analysis takes into account the viability of the underlying philosophical views and recent mathematical developments.

Both Carnap and Quine made significant contributions to the philosophy of mathematics despite their diversed views. Carnap endorsed the dichotomy between analytic and synthetic knowledge and classified certain mathematical questions as internal questions appealing to logic and convention. On the contrary, Quine was opposed to the analytic-synthetic distinction and promoted a holistic view of scientific inquiry. The purpose of this paper is to argue that in light of the recent advancement of experimental mathematics such as Monte Carlo simulations, limiting mathematical (...) inquiry to the domain of logic is unjustified. Robustness studies implemented in Monte Carlo Studies demonstrate that mathematics is on par with other experimental-based sciences. (shrink)