# The Relation Between Physics and Mathematics in Thomas Aquinas's Division and Methods of the Sciences

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This thesis addresses the question â How can mathematics provide knowledge of physical objects?â which was provoked by Thomas Aquinasâ s inclusion of â intermediateâ (i.e., physicomathematical) sciences in his Division and Methods of the Sciences. In examining Aquinasâ s process of division, I paid special attention to the way he distinguishes the sciences according to the formal ratio of their objects, an important development upon Boethiusâ s framework. This led me to discuss the modes of abstraction proper to each science and, in turn, how their distinct epistemic foundations seem to prevent one science from being meaningfully applied to the study of another. However, in the case of physics, the accident quantity is implicitly included in the definition of its objects, suggesting that mathematics can, in some way, inform their study (even though mathematical propositions themselves are neither true nor false from the standpoint of extramental reality). I concluded that the knowledge obtained through physico-mathematical sciences is conditional in an ontological sense, for the mathematical systems that these sciences employ cannot be more than hypothetical depictions of observed phenomena. Nevertheless, insofar as the conclusions of a given mathematical model are corroborated by physical data, the hypothesis of the model is validated. In fact, mathematicsâ indifference to the material world is of remarkable value to the physicist. As an ordered system of the imagination, mathematics enables the physicist to reinterpret the material world according to its quantitative aspects in an idealized setting. In this way, mathematics can become an indispensable tool in the physicistâ s quest to locate and abstract the universal natures of physical bodies.