The sum of two periodic functions need not be periodic. Consider for example the sum of the functions sin(x) and sin(sqrt(2)x). Fortunately, such a sum is almost periodic and much of the (Fourier) theory of periodic funcitions carries over. Unfortunately, constructively matters are complicated by the fact that the almost periodic functions form an inseparable normed space. As such, it has been a challenge for constructive mathematicians to find a natural treatment of them. In this talk, we will present a simple constructive proof of Bohr's fundamental theorem for almost periodic functions which we then generalise to almost periodic functions as an example to indicate some problems and methods in constructive analysis, i.e. analysis with intuitionistic logic. The lecture will be loosely based on my paper: Almost Periodic Functions, Consturctively
There is now a web of conjectures which all suggest the same thing--that there is a conception of the universe of sets which yields both answers to all of the known unsolvable problems of set theory and which is compatible with all axioms of strong infinity. I shall discuss some of these conjectures and the prospects for proving them.
We define a general concept of a network of analog modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T-> A, and the input-output behaviour of the network with system parameters from A is modelled by a functional F on the set C[T,A] of all continuous streams equipped with the compact-open topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C[T,A] using a contraction principle. We show that if the module functions are continuous then so is the network function F. This is significant in terms of Hadamard's principle. We present a case study involving mechanical systems. We also introduce a compution model on C[T,A], in the sense of recursive analysis, and show that if the module functions are computable then so is F. This is jointwork with John Tucker (Swansea).
The mathematical core of Gödel's philosophical challenge is constituted by the incompleteness theorems when given in their "most satisfactory form", i.e., as Gödel saw it, when formal theories are characterized via Turing computability. As Turing machines codify human mechanical procedures (carried out without appealing to higher cognitive capacities) the question naturally arises, whether the theorems justify the claim that the human mind or intellect has mathematical abilities that are not shared by any machine. Turing admits that steps of "intuition" are needed to transcend particular formal theories; thus, there is a substantive point in contrasting Turing's views with Gödel's assertion, "infinitely surpasses any finite machine". I analyze a seemingly common core that raises cnetral issues in the foundations of mathematics and cognitive science.