This lecture is intended as an introduction to some of the metaphysical and epistemological issues that are usually raised in connection with mathematical thought. These issues include, among other things, the nature of abstract objects like numbers and sets, some of the well-known arguments for the existence of such objects as well as the epistemological problems that the existence of those objects incur. There will also be some brief discussions of certain anti-realist approaches to mathematical ontology.

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Let A=[a_₀, a_₁, ..., a_{n-1}] ∈ ℂ^ℓ be a finite complex sequence of length ℓ periodically continued with period ℓ. ‎The periodic autocorrelation function PAF(A,s) of A at lag s is‎ PAF(A, s) =∑ {j=0}^{ℓ-1} a_joverline{a_{j+s}}, s=0,1, ..., ℓ-1. ‎Two sequences‎ ‎A and B of the same length ℓ. ‎form a Legendre pair (A,B) if‎ ‎PAF(A,s)+PAF(B,s)=-2, s=1,2, ?, ⌊n/2⌋. ‎Binary Legendre pairs of length ℓ are pertinent to the construction of Hadamard matrices of order ℓ +2 which can exist only if ℓ is odd‎. ‎It is conjectured that there is a binary Legendre pair of every odd length ℓ which is stronger than the Hadamard-matrix conjecture‎. ‎Several infinite classes of Legendre pairs are known‎. ‎The most prominent constructions are defined via characters of finite fields‎. ‎In particular‎, ‎there are Legendre pairs of length ℓ for every prime ℓ =p and every Mersenne number ℓ =2^s-1‎, ‎respectively‎. ‎The smallest undecided case is ℓ=11. ‎In the first part of the talk we summarize the known results on binary Legendre pairs‎. ‎In the second part we discuss quaternary Legendre pairs of length ℓ‎. ‎In contrast to binary Legendre pairs they can exist for even ℓ as well‎. ‎Very recently‎, ‎the first infinite construction was found by Jedwab and Pender completing our earlier semi-construction‎. ‎We give also constructions of quaternary Legendre pairs of length ℓ for all remaining even ℓ < 42‎. ‎The smallest open case is ℓ =42 and the case ℓ =46 is particularly interesting since the existence of a quaternary Legendre pair of this length would imply the existence of a quaternary Hadamard matrix of order 94. ‎Finally‎, ‎we mention generalizations to k-ary sequences‎.

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We address Krasner's problem (1956) for Polish ultrametric spaces and provide a (hopefully) satisfactory solution. In particular, we discuss a correspondence between the isometry groups of Polish ultrametric spaces belonging to some natural subclasses and various collections of generalized wreath products proposed in the literature. This is joint work with R. Camerlo and A. Marcone.

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