The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality.

Some prominent twentieth-century scholars were still opposed to, or uncomfortable with, both irrationals and infinitesimals. Thus, Errett Bishop opposed both the classical development of the real numbers and the use of infinitesimals in teaching calculus [5]. For a discussion, see [9][10][11]31]. 2 Galileo (1564-1642) observed that the natural numbers admit a one-to-one correspondence with their squares. 3 The part-whole principle, which goes back to Euclid, asserts that a (proper) part is smaller than the whole. 4 ''fraction infiniment petite, ou dont le denominateur soit un nombre infini' ' [19, p. 93].

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In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time-dependent symmetries. In particular we describe how that the finite dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time-dependent Hamiltonian function. We also present a very general Lagrangian formulation of the procedure of reduction. As an application we consider the case of the Painlevé equations PI, PII, PIII, PVI and also certain higher order systems appeared in the theory of Frobenius manifolds.Preprint SISSA 11/99/FM