865 research outputs found
Finitary and Infinitary Mathematics, the Possibility of Possibilities and the Definition of Probabilities
Some relations between physics and finitary and infinitary mathematics are
explored in the context of a many-minds interpretation of quantum theory. The
analogy between mathematical ``existence'' and physical ``existence'' is
considered from the point of view of philosophical idealism. Some of the ways
in which infinitary mathematics arises in modern mathematical physics are
discussed. Empirical science has led to the mathematics of quantum theory. This
in turn can be taken to suggest a picture of reality involving possible minds
and the physical laws which determine their probabilities. In this picture,
finitary and infinitary mathematics play separate roles. It is argued that
mind, language, and finitary mathematics have similar prerequisites, in that
each depends on the possibility of possibilities. The infinite, on the other
hand, can be described but never experienced, and yet it seems that sets of
possibilities and the physical laws which define their probabilities can be
described most simply in terms of infinitary mathematics.Comment: 21 pages, plain TeX, related papers from
http://www.poco.phy.cam.ac.uk/~mjd101
Logics of Finite Hankel Rank
We discuss the Feferman-Vaught Theorem in the setting of abstract model
theory for finite structures. We look at sum-like and product-like binary
operations on finite structures and their Hankel matrices. We show the
connection between Hankel matrices and the Feferman-Vaught Theorem. The largest
logic known to satisfy a Feferman-Vaught Theorem for product-like operations is
CFOL, first order logic with modular counting quantifiers. For sum-like
operations it is CMSOL, the corresponding monadic second order logic. We
discuss whether there are maximal logics satisfying Feferman-Vaught Theorems
for finite structures.Comment: Appeared in YuriFest 2015, held in honor of Yuri Gurevich's 75th
birthday. The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-23534-9_1
Takeuti's proof theory in the context of the Kyoto School
Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project
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