In this presentation, I'm going to explain Stephen Toulmin's procedural theory of concepts and explanation in order to build on two overlooked ideas from his philosophy of science: methods of representations and inferential techniques. I will argue that developing these notions could be useful for shedding some light on certain specific characteristics of scientific reasoning. In particular, on how scientific inference is related to representational structures, concepts and explanation within scientific practices.

I will explore and illustrate these ideas with by the study of the development of the notion of instantaneous velocity during the passage from Galileo's geometrical physics to analytical mechanics. This episode will illustrate how a change in the method of representation of motion was the key for developing an explicit (algebraic) notion of velocity at each instant. Notion which was a central part on the development of analytical mechanics and, in more general terms, of a new inferential technique in physics.

At the end, I will argue that methods of representations and explanatory schemes are constitutive of scientific inference, I will show how these notions could explain the diversity of inferential practices in science, and I will discuss their relationship with other similar notions from philosophy of science like those of models and model-based reasoning.