Truth, Paradoxes, Freedom, Applications, and Knowledge

This chapter begins by showing that with the problems of mathematical existence and determinate truth solved, a sophisticated inferentialist theory of mathematics leads to mathematical conventionalism. A philosophical worry harkening back to the Carnap/Quine debate is addressed before a number of issues in the philosophy of mathematics are given conventionalist treatments. The chapter discusses how conventionalists can handle the set-theoretic paradoxes, the freedom of mathematics, the many applications of mathematics to the physical world, and then provides a naturalistic epistemology of mathematics, even addressing the epistemology of consistency. In almost all of these cases, the discussion in the chapter shows that a conventionalist theory deals with these issues in a more satisfying way than other approaches to the philosophy of mathematics.

On a Computational Approach to Micro- and Macro-modelling of Damage in Brittle and Quasi-brittle Materials

Computational modelling of damage in brittle and quasi-brittle materials needs some coupling between micro- and macroscopic crack initiation and evolution, up to their non-negligible softening behaviour. Most such approaches contain ad hoc evaluations, with some physical and engineering motivations, namely those connected with massive application of steel fibre-reinforced concrete and similar composites in building projects, but without any proper mathematical existence and convergence analysis for the time development of damage. This paper presents a possibility of such deterministic analysis on a selected model problem of structural dynamics, supplied by comments to useful directions of generalization. Several application examples document the feasibility of such approach, up to its software implementation and real data validation.

Monotonicity, Convexity, Continuity, and Isomorphism: The Psychological Scaffolding of Arithmetic

With the natural numbers as our starting point, we obtain the arithmetic structure of real (as in R) addition and multiplication without relying on any algebraic tools; in particular, we leverage monotonicity, convexity, continuity, and isomorphism. Natural addition arises by minimizing against monotonicity. Rational addition arises from natural addition by minimizing against convexity. Real addition arises from rational addition via any one of three methods; unique convex extension, unique continuous extension, and unique monotonic extension. Real multiplication arises from real addition via isomorphism. Following these mathematical developments, we argue that each of the leveraged mathematical concepts ---monotonicity, convexity, continuity, and isomorphism --- enjoys, prior to its formal mathematical existence, an intuitive psychological existence. Taken together, these lines of argument suggest a way for psychological representation of algebraic structure to emerge from non-algebraic --- and psychologically plausible --- ingredients.

7. Counting infinity

‘Counting infinity’ returns to the mathematics of infinity, discussing Cantor’s remarkable theory of how to count infinite sets, and the discovery that there are different sizes of infinity. For example, the set of all integers is infinite, and the set of all real numbers (infinite decimals) is infinite, but these infinities are fundamentally different, and there are more real numbers than integers. The ‘numbers’ here are called transfinite cardinals. For comparison, another way to assign numbers to infinite sets is mentioned, by placing them in order, leading to transfinite ordinals. It ends by asking whether the old philosophical distinction between actual and potential infinity is still relevant to modern mathematics, and examining the meaning of mathematical existence.