Online research in philosophy

Ecologists attempt to understand the diversity of life with mathematical models. Often, mathematical models contain simplifying idealizations designed to cope with the blooming, buzzing confusion of the natural world. This strategy frequently issues in models whose predictions are inaccurate. Critics of theoretical ecology argue that only predictively accurate models are successful and contribute to the applied work of conservation biologists. Hence, they think that much of the mathematical work of ecologists is poor science. Against this view, I argue that model building is successful even when models are predictively inaccurate for at least three reasons: models allow scientists to explore the possible behaviors of ecological systems; models give scientists simplified means by which they can investigate more complex systems by determining how the more complex system deviates from the simpler model; and models give scientists conceptual frameworks through which they can conduct experiments and fieldwork. Critics often mistake the purposes of model building, and once we recognize this, we can see their complaints are unjustified. Even though models in ecology are not always accurate in their assumptions and predictions, they still contribute to successful science.

This approach does not define a probability measure by syntactical structures. It reveals a link between modal logic and mathematical probability theory. This is shown (1) by adding an operator (and two further connectives and constants) to a system of lower predicate calculus and (2) regarding the models of that extended system. These models are models of the modal system S₅ (without the Barcan formula), where a usual probability measure is defined on their set of possible worlds. Mathematical probability models can be seen as models of S₅.

To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality (to which there is no direct access), personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about ‘truth’, but the assignment of mathematics to informal reality is not itself formally analysable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed.

Whether a collection of scientific data can be explained only by a unique theory or whether such data can be equally explained by multiple theories is one of the more contested issues in the history and philosophy of science. This paper argues that the case for multiple explanations is strengthened by the widespread failure of Models in mathematical logic to be unique ie categorical. Science is taken to require replicable and explicit public knowledge; this necessitates an unambiguous language for its transmission. Mathematics has been chosen as the vehicle to transmit scientific knowledge, both because of its 'unreasonable effectiveness' and because of its unambiguous nature, hence the vogue of axiomatic systems. But Mathematical Logic tells us that axiomatic systems need not refer to uniquely defined real structures. Hence what is accepted as Science may be only one of several possibilities.

We assess Cartwright's models for probabilistic causality and, in particular, her models for EPR-like experiments of quantum mechanics. Our first objection is that, contrary to econometric linear models, her quasi-linear models do not allow for the unique estimation of parameters. We next argue that although, as Cartwright proves, Reichenbach's screening-off condition has only limited validity, her generalized condition is not empirically applicable. Finally, we show that her models for the EPR are mathematically incorrect and physically implausible.

This book presents a detailed analysis of three ancient models of spatial magnitude, time, and local motion. The Aristotelian model is presented as an application of the ancient, geometrically orthodox conception of extension to the physical world. The other two models, which represent departures from mathematical orthodoxy, are a "quantum" model of spatial magnitude, and a Stoic model, according to which limit entities such as points, edges, and surfaces do not exist in (physical) reality. The book is unique in its discussion of these ancient models within the context of later philosophical, scientific, and mathematical developments.

Some philosophers of mathematics argue that the role of mathematical models in science is merely representational: when scientists use mathematical models they only believe that they are adequate representations of the physical phenomenon under investigation. Others disagree with this view and argue that mathematical models also serve as genuine explanations in science. Consequently, the application of mathematical models in science cannot be treated instrumentally and we ought to be realists about mathematics. I advance two reasons to reject realist conclusion: genuine mathematical explanations are indistinguishable from spurious ones. And, for mathematical models to be explanatory, they have to be “bottom-level”; I deny that we can know which explanations (if any) are bottom level in science. I contend that what plays the explanatory role is the impure function that links physical structures to mathematical structures.

Some philosophers of mathematics argue that the role of mathematical models in science is merely representational: when scientists use mathematical models they only believe that they are adequate representations of the physical phenomenon under investigation. Others disagree with this view and argue that mathematical models also serve as genuine explanations in science. Consequently, the application of mathematical models in science cannot be treated instrumentally and we ought to be realists about mathematics. I advance two reasons to reject realist conclusion: genuine mathematical explanations are indistinguishable from spurious ones. And, for mathematical models to be explanatory, they have to be “bottom-level”; I deny that we can know which explanations (if any) are bottom level in science. I contend that what plays the explanatory role is the impure function that links physical structures to mathematical structures.