41,995 research outputs found
Intuition, iteration, induction
In Mathematical Thought and Its Objects, Charles Parsons argues that our
knowledge of the iterability of functions on the natural numbers and of the
validity of complete induction is not intuitive knowledge; Brouwer disagrees on
both counts. I will compare Parsons' argument with Brouwer's and defend the
latter. I will not argue that Parsons is wrong once his own conception of
intuition is granted, as I do not think that that is the case. But I will try
to make two points: (1) Using elements from Husserl and from Brouwer, Brouwer's
claims can be justified in more detail than he has done; (2) There are certain
elements in Parsons' discussion that, when developed further, would lead to
Brouwer's notion thus analysed, or at least something relevantly similar to it.
(This version contains a postscript of May, 2015.)Comment: Elaboration of a presentation on December 5, 2013 at `Intuition and
Reason: International Conference on the Work of Charles Parsons', Van Leer
Jerusalem Institute, Jerusale
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
Computer theorem proving in math
We give an overview of issues surrounding computer-verified theorem proving
in the standard pure-mathematical context. This is based on my talk at the PQR
conference (Brussels, June 2003)
Open questions about Ramsey-type statements in reverse mathematics
Ramsey's theorem states that for any coloring of the n-element subsets of N
with finitely many colors, there is an infinite set H such that all n-element
subsets of H have the same color. The strength of consequences of Ramsey's
theorem has been extensively studied in reverse mathematics and under various
reducibilities, namely, computable reducibility and uniform reducibility. Our
understanding of the combinatorics of Ramsey's theorem and its consequences has
been greatly improved over the past decades. In this paper, we state some
questions which naturally arose during this study. The inability to answer
those questions reveals some gaps in our understanding of the combinatorics of
Ramsey's theorem.Comment: 15 page
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
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