Online research in philosophy

Many people think that mathematical models are built using well-known “mathematical things” such as numbers and geometry. But since the 19th century, mathematicians have investigated various non-numerical and non-geometrical structures: groups, fields, sets, graphs, algorithms, categories etc. What could be the most general distinguishing feature that would separate mathematical models from non-mathematical ones? I would describe this feature by using such terms as autonomous, isolated, stable, self-contained, and – as a summary – formal. Autonomous and isolated – because mathematical models can be investigated “on their own” in isolation from the modeled objects. And one can do this for many years without any external information flow. Stable – because any modification of a mathematical model is qualified explicitly as defining a new model. No implicit modifications are allowed. Self- contained – because all properties of a mathematical model must be formulated explicitly. The term “formal model” can be used to summarize all these features.

The translation of a mathematical model into a numerical one employs various modifications in order to make the model accessible for computation. Such modifications include discretizations, approximations, heuristic assumptions, and other methods. The paper investigates the divergent styles of mathematical and numerical models in the case of a specific piece of code in a current atmospheric model. Cognizance of these modifications means that the question of the role and function of scientific models has to be reworked. Neither are numerical models pure intermediaries between theory and data, nor are they autonomous tools of inquiry. Instead, theory and data are transformed into a new symbolic form of research due to the fact that computation has become an essential requirement for every scientific practice. Therefore the question is posed: What do numerical (climate) models really represent?

Baker (2005) claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given without postulating mathematical objects and thus cannot be used by the mathematical realist. I also show that, despite this, mathematics keeps playing an important methodological role in the explanation and does not reduce to a merely computational or descriptive framework.

The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without caution, as the use of anachronistic notation tends to impede, rather than enhance, our ability to recognize the emergent nature of mathematical objects.

Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers numerically. This can be done on a new kind of a computer – the Infinity Computer – able to work with all these types of numbers. The new computational tools both give possibilities to execute computations of a new type and open new horizons for creating new mathematical models where a computational usage of infinite and/or infinitesimal numbers can be useful. A number of numerical examples showing the potential of the new approach and dealing with divergent series, limits, probability theory, linear algebra, and calculation of volumes of objects consisting of parts of different dimensions are given.

This paper attempts to motivate skepticism about the reality of mathematical objects. The aim of the paper is not to provide a general critique of mathematical realism, but to demonstrate the insufficiency of the arguments advanced by Michael Resnik. I argue that Resnik’s use of the concept of immanent truth is inconsistent with the treatment of mathematical objects as ontologically and epistemically continuous with the objects posited by the natural sciences. In addition, Resnik’s structuralist program, and his denial of relational properties, is incompatible with a realist metaphysics about mathematical objects.

In this paper I propose a position in the ontology of mathematics which is inspired mainly by a case study in the mathematical discipline if-theory. The main theses of this position are that mathematical objects are introduced by mathematicians and that after mathematical objects have been introduced, they exist as objectively accessible abstract objects.

In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a “nature” than that confers on them.

Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical economics could mean. Method: An algebraic simple model system is subjected to a deeper structure of underlying variables. With an algorithm simulating the steps in taking a limit of second order difference quotients the error terms are studied at the background of their algebraic expression. Results: With the algorithm that was applied to a simple quadratic polynomial system we found unstably amplified round-off errors. The possibility of numerical chaos is already known but not in such a simple system as used in our paper. The amplification of the errors implies that it is not possible with computer means to constructively show that the algebra and numerical analysis will ‘on the long run’ converge to each other and the error term will vanish. The algebraic vanishing of the error term cannot be demonstrated with the use of the computer because the round-off errors are amplified. In philosophical terms, the amplification of the round-off error is equivalent to the continuum hypothesis. This means that the requirement of (numerical) construction of mathematical objects is no safeguard against inference-only conclusions of qualities of (numerical) mathematical objects. Unstably amplified round-off errors are a same type of problem as the ordering in size of transfinite cardinal numbers. The difference is that the former problem is created within the requirements of constructive mathematics. This can be seen as the reward for working numerically constructive.

This article examines one aspect of Thomas Aquinas' understanding of abstraction. It shows in which way, according to Aquinas, universal material objects and individual material objects are the starting point for mathematical objects. It comes to the conclusion that for Aquinas there are not only universal mathematical objects (circle, line), but also individual mathematical objects (this circle, that line). Universal mathematical objects are properties of universal material objects and individual mathematical objects are properties of individual material objects. One type of abstractio formae leads from individual material objects to universal mathematical objects, a second type from universal material objects to universal mathematical objects, and a third type from individual material objects to individual mathematical objects. Therefore, the concept of abstractio formae is ambiguous.