The mathematics of McTaggart's paradox

Mc Taggart's celebrated proof of the unreality of time is a chain of implications whose final step asserts that the A-series (i.e. the classification of events as past, present or future) is intrinsically contradictory. This is widely believed to be the heart of the argument, and it is where most attempted refutations have been addressed; yet, it is also the only part of the proof which may be generalised to other contexts, since none of the notions involved in it is specifically temporal. In fact, as I show in the first part of the paper, McTaggart's refutation of the A-series can be easily interpreted in mathematical terms; subsequently, in order to strengthen my claim, I apply the same framework by analogy to the cases of space, modality, and personal identity. Therefore, either McTaggart's proof as a whole may be extended to each of these notions, or it must embed some distinctly temporal element in one of the steps leading up to the contradiction of the A-series. I conclude by suggesting where this element might lay, and by hinting at what I believe to be the true logical fallacy of the proof.

HILBERT BETWEEN THE FORMAL AND THE INFORMAL SIDE OF MATHEMATICS

Abstract: In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution.

Computational modeling for irrigated agriculture planning. Part I: general description and linear programming

Linear programming models are effective tools to support initial or periodic planning of agricultural enterprises, requiring, however, technical coefficients that can be determined using computer simulation models. This paper, presented in two parts, deals with the development, application and tests of a methodology and of a computational modeling tool to support planning of irrigated agriculture activities. Part I aimed at the development and application, including sensitivity analysis, of a multiyear linear programming model to optimize the financial return and water use, at farm level for Jaíba irrigation scheme, Minas Gerais State, Brazil, using data on crop irrigation requirement and yield, obtained from previous simulation with MCID model. The linear programming model outputted a crop pattern to which a maximum total net present value of R$ 372,723.00 for the four years period, was obtained. Constraints on monthly water availability, labor, land and production were critical in the optimal solution. In relation to the water use optimization, it was verified that an expressive reductions on the irrigation requirements may be achieved by small reductions on the maximum total net present value.

Object-oriented Programming Applied to the Development of Structural Analysis and Optimization Software

This paper presents the development of a two-dimensional interactive software environment for structural analysis and optimization based on object-oriented programming using the C++ language. The main feature of the software is the effective integration of several computational tools into graphical user interfaces implemented in the Windows-98 and Windows-NT operating systems. The interfaces simplify data specification in the simulation and optimization of two-dimensional linear elastic problems. NURBS have been used in the software modules to represent geometric and graphical data. Extensions to the analysis of three-dimensional problems have been implemented and are also discussed in this paper.

Nonconventional amide bond formation catalysis: programming enzyme specificity with substrate mimetics

This article reports on the design and characteristics of substrate mimetics in protease-catalyzed reactions. Firstly, the basis of protease-catalyzed peptide synthesis and the general advantages of substrate mimetics over common acyl donor components are described. The binding behavior of these artificial substrates and the mechanism of catalysis are further discussed on the basis of hydrolysis, acyl transfer, protein-ligand docking, and molecular dynamics studies on the trypsin model. The general validity of the substrate mimetic concept is illustrated by the expansion of this strategy to trypsin-like, glutamic acid-specific, and hydrophobic amino acid-specific proteases. Finally, opportunities for the combination of the substrate mimetic strategy with the chemical solid-phase peptide synthesis and the use of substrate mimetics for non-peptide organic amide synthesis are presented.