Three thinkers of the 19th century revolutionized the science of logic, mathematics, and philosophy. Edmund Husserl (1859-1938), mathematician and a disciple of Karl Weierstrass, made an immense contribution to the theory of human thought. The paper offers a complex analysis of Husserl’s mathematical writings covering calculus of variations, differential geometry, and theory of numbers which laid the ground for his later phenomenological breakthrough. Georg Cantor (1845-1818), the creator of set theory, was a mathematician who changed the mathematical thinking per se. By analyzing the philosophy of set theory this paper shows how was it possible (by introducing into mathematics what philosophers call ‘the subject’). Set theory happened to be the most radical answer to the crisis of foundations. David Hilbert (1862-1943), facing the same foundational crisis, came up with his axiomatic method, indeed a minimalist program whose roots can be traced back to Descartes and Cauchy. Bringing together these three key authors, the paper is the first attempt to analyze how the united efforts of philosophy and mathematics helped to dissolve the epistemological crisis of the 19th century.

We study Saccheri’s three hypotheses on a two right-angled isosceles quadrilateral, with a rectilinear summit side. We claim that in the Hilbert`s foundation of geometry the euclidean parallelism is a theorem and as that it can be used, in the hyperbolic geometry

We write the Euler characteristic X(G) of a four dimensional finite simple geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four dimensional graph satisfying the Gauss-Bonnet relation sum_x K(x) = X(G) can so be rewritten as an average 1-E[K(x,f)]/2 over a collection two dimensional «sectional graph curvatures» K(x,f) through x. Since scalar curvature, the average of all these two dimensional curvatures through a point, is the integrand of the Hilbert action, the integer 2-2 X(G) becomes an integral-geometrically defined Hilbert action functional.

In this note we show that in addition to two integers forming a Pythagorean triple, there also exist two irrational numbers in terms of which this Pythagorean triple can also be obtained. We also put forward a relation between these two pairs and the hypotenuse of a Pythagorean triangle.

We present updates to the problems on Hirzebruch’s 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about contact structures, and about (complementary pairs of) foliations.

In this paper, we give a simple definition of tangents to a curve in elementary geometry. From which, we characterize the existence of the tangent to a curve at a point.

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer’s translation of «Kreisabbildung»), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein’s ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler’s planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann’s Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\»uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem.

In our recent paper [de Lacy Costello et al. 2010] we described the formation of complex tessellations of the plane arising from the various reactions of metal salts with potassium ferricyanide and ferrocyanide loaded gels. In addition to producing colourful tessellations these reactions are naturally computing generalised Voronoi diagrams of the plane. The reactions reported previously were capable of the calculation of three distinct Voronoi diagrams of the plane. As diffusion coupled with a chemical reaction is responsible for the calculation then this is achieved in parallel. Thus an increase in the complexity of the data input does not utilise additional computational resource. Additional benefits of these chemical reactions is that a permanent record of the Voronoi diagram calculation (in the form of precipitate free bisectors) is achieved, so there is no requirement for further processing to extract the calculation results. Previously it was assumed that the permanence of the results was also a potential drawback which limited reusability. This paper presents new data which shows that sequential Voronoi diagram calculations can be performed on the same chemical substrate.

In arXiv:1207.0332 [cs.LO] was proposed a graphic lambda calculus formalism, which has sectors corresponding to untyped lambda calculus and emergent algebras. Here we explore the sector covering knot diagrams, which are constructed as macros over the graphic lambda calculus.

We investigate the most general phase space of configurations, consisting of the collection of all possible ways of assigning elementary attributes, «energies», to elementary positions, «cells». We discuss how this space defines a «universe» with a structure that can be approximately described by a quantum-relativistic physical scenario in three space dimensions. In particular, we discuss how the Heisenberg’s Uncertainty and the bound on the speed of light arise, and what kind of mechanics rules on this space.