Mathematics is traditionally considered being an apriori discipline consisting of purely analytic propositions. The aim of the present paper is to offer arguments against this entrenched view and to draw attention to the experiential dimension of mathematical knowledge. Following Husserl’s interpretation of physical knowledge as knowledge constituted by the use of instruments, I am trying to interpret mathematical knowledge also as acknowledge based on instrumental experience. This interpretation opens a new view on the role of the logicist program, both in philosophy of mathematics and in philosophy of science., Matematika je tradičně považována za apriori disciplínu tvořenou čistě analytickými výroky. Cílem této práce je nabídnout argumenty proti tomuto zakořeněnému pohledu a upozornit na zkušenostní rozměr matematických znalostí. V návaznosti na Husserlovu interpretaci fyzických znalostí jako poznání vytvořených použitím nástrojů se snažím interpretovat matematické znalosti také jako uznání na základě instrumentální zkušenosti. Tato interpretace otevírá nový pohled na roli logického programu, a to jak ve filozofii matematiky, tak ve filozofii vědy., and Ladislav Kvasz
The aim of the present paper is to offer a new analysis of the multifarious relations between mathematics and reality. We believe that the relation of mathematics to reality is, just like in the case of the natural sciences, mediated by instruments (such as algebraic symbolism, or ruler and compass). Therefore the kind of realism we aim to develop for mathematics can be called instrumental realism. It is a kind of realism, because it is based on the thesis, that mathematics describes certain patterns of reality. And it is instrumental realism, because it pays atten-tion to the role of instruments by means of which mathematics identifies these patterns. The article concludes by offering solutions to some famous semantic paradoxes based on the diagonal construction as corroboration for this claim., Cílem příspěvku je nabídnout novou analýzu rozmanitých vztahů mezi matematikou a realitou. Věříme, že vztah matematiky k realitě je, stejně jako v případě přírodních věd, zprostředkován nástroji (např. Algebraickou symbolikou, pravítkem a kompasem). Proto se druh realismu, který chceme rozvíjet pro matematiku, nazývá instrumentální realismus. Je to druh realismu, protože je založen na tezi, že matematika popisuje určité vzorce reality. A je to instrumentální realismus, protože věnuje pozornost roli nástrojů, pomocí kterých matematika tyto vzorce identifikuje. Článek uzavírá nabídku některých slavných sémantických paradoxů založených na diagonální konstrukci jako důkaz tohoto tvrzení., and Ladislav Kvasz
In this study, analytical models for predicting groundwater contamination in isotropic and homogeneous porous formations are derived. The impact of dispersion and diffusion coefficients is included in the solution of the advection-dispersion equation (ADE), subjected to transient (time-dependent) boundary conditions at the origin. A retardation factor and zero-order production terms are included in the ADE. Analytical solutions are obtained using the Laplace Integral Transform Technique (LITT) and the concept of linear isotherm. For illustration, analytical solutions for linearly space- and time-dependent hydrodynamic dispersion coefficients along with molecular diffusion coefficients are presented. Analytical solutions are explored for the Peclet number. Numerical solutions are obtained by explicit finite difference methods and are compared with analytical solutions. Numerical results are analysed for different types of geological porous formations i.e., aquifer and aquitard. The accuracy of results is evaluated by the root mean square error (RMSE).
A 2D hydrodynamic (labeled as CAR) model has been proposed in a rectangular Cartesian coordinate system with two axes within the horizontal plane and one axis along the vertical direction (global coordinates), considering the effects of bed slope on both pressure distribution and bed shear stresses. The CAR model satisfactorily reproduces the analytical solutions of dam-break flow over a steep slope, while the traditional Saint-Venant Equations (labeled as SVE) significantly overestimate the flow velocity. For flood events with long duration and large mean slope, the CAR and the SVE models present distinguishable discrepancies. Therefore, the proposed CAR model is recommended for applications to real floods for its facility of extending from 1D to 2D version and ability to model shallow-water flows on steep slopes.