There are three elements in this paper. One is what we shall call ‘the Hungarian project’. This is the collected work of Andréka, Madarász, Németi, Székely and others. The second is Molinini’s philosophical work on the nature of mathematical explanations in science. The third is my pluralist approach to mathematics. The theses of this paper are that the Hungarian project gives genuine mathematical explanations for physical phenomena. A pluralist account of mathematical explanation can help us with appreciating the significance of (...) the Hungarian project. The significance consists in the fruitfulness and spread of the project. The spread is wide because the explanations are written in the very familiar language of first-order logic with identity. For this reason, the explanation is understandable to many mathematicians. Because of the methodology adopted in the Hungarian project, the explanations are fruitful in another sense. In the Hungarian project certain questions are asked that would not be asked with a more usual methodology. The Hungarians are distinguished from other scientists in asking logical and mathematical questions, and these both deepen our understanding of the physical theories and induce further spread to mathematics and philosophy. (shrink)

We answer three questions: 1. Can we give a wholly mathematical explanation of a physical phenomenon? 2. Can we give a wholly mathematical explanation for a whole physical theory? 3. What is gained or lost in giving a wholly, or partially, mathematical explanation of a phenomenon or a scientific theory? To answer these questions we look at a project developed by Hajnal Andréka, Judit Madarász, István Németi and Gergely Székely. They, together with collaborators, present special relativity theory in a three-sorted (...) first-order formal language. (shrink)

While there is a longstanding discussion about the interpretation of the extended, general principle of relativity, there seems to be a consensus that the special principle of relativity is absolutely clear and unproblematic. However, a closer look at the literature on relativistic physics reveals a more confusing picture. There is a huge variety of, sometimes metaphoric, formulations of the relativity principle, and there are different, sometimes controversial, views on its actual content. The aim of this paper is to develop a (...) precise language in order to provide a precise formulation of the principle. In view of the fact that the special relativity principle is considered as a universal meta-law, which must be valid for all physical laws in all situations, we try to keep the formalism as general as possible. The benefit of the formal reconstruction is that it makes explicit all the necessary conceptual components of the principle; it brings out many subtle details and the related conceptual problems. (shrink)

We present three natural but distinct formalisations of Einstein’s special principle of relativity, and demonstrate the relationships between them. In particular, we prove that they are logically distinct, but that they can be made equivalent by introducing a small number of additional, intuitively acceptable axioms.

The vision of machines autonomously carrying out substantive conjecture generation, theorem discovery, proof discovery, and proof verification in mathematics and the natural sciences has a long history that reaches back before the development of automatic systems designed for such processes. While there has been considerable progress in proof verification in the formal sciences, for instance the Mizar project’ and the four-color theorem, now machine verified, there has been scant such work carried out in the realm of the natural sciences—until recently. (...) The delay in the case of the natural sciences can be attributed to both a lack of formal analysis of the so-called “theories” in such sciences, and the lack of sufficient progress in automated theorem proving. While the lack of analysis is probably due to an inclination toward informality and empiricism on the part of nearly all of the relevant scientists, the lack of progress is to be expected, given the computational hardness of automated theorem proving; after all, theoremhood in even first-order logic is Turing-undecidable. We give in the present short paper a compressed report on our building upon these formal theories using logic-based AI in order to achieve, in relativity, both machine proof discovery and proof verification, for theorems previously established by humans. Our report is intended to serve as a springboard to machine-produced results in the future that have not been obtained by humans. (shrink)