Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of "invariance" and "conceptual stability" that single out the mathematical conceptualization from any other form of knowledge, and give unity to it. Yet, this very conceptualization is deeply rooted in our "acts of experience", as Weyl says, beginning with our presence in the world, first in space and time as living beings, up to the most complex attempts we make by language to give an account of it. I will try to sketch the origin of some key steps in organizing perception and knowledge by "mathematical tools", as mathematics is one of the many practical and conceptual instruments by which we categorize, organise and "give a structure" to the world. It is conceived on the "interface" between us and the world, or, to put it in husserlian terminology, it is "de

Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s

It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl’s role, and in particular on Brouwer’s reaction to Weyl’s allegiance to the cause of intuitionism. This short episode certainly raises a number of questions: what made Weyl give up his own program, spelled out in “Das Kontinuum”, how come Weyl was so well-informed about Brouwer’s new intuitionism, in what respect did Weyl’s intuitionism differ from Brouwer’s intuitionism, what did Brouwer think of Weyl’s views,........? To some of these questions at least partial answers can be put forward on the basis of some of the available correspondence and notes. The present paper will concentrate mostly on the historical issues of the intuitionistic episode in Weyl’s career. Weyl entered the foundational controversy with a bang in 1920 with his sensational paper “On the new foundational crisis in mathematics ” 1. He had already made a name for himself in the foundations of mathematics in 1918 with his monograph “The Continuum” [Weyl 1918] ; this contained in addition to a technical logical – mathematical construction of the continuum, a fairly extensive discussion of the shortcomings of the traditional construction of the continuum on the basis of arbitrary — and hence also impredicative — Dedekind cuts. This book did not cause much of a stir in mathematics, that is to say, it was ritually quoted in the literature but, probably, little understood. It had to wait for a proper appreciation until the phenomenon of impredicativity was better understood 2. The paper “On the new foundational crisis in mathematics ” had a totally different effect, it was the proverbial stone thrown into the quiet pond of mathematics. Weyl characterised it in retrospect with the somewhat apologetic words: Only with some hesitation I acknowledge these lectures, which reflect in their style, which was here and there really bombastic, the mood of excited times — the times immediately following the First World War. 3 Indeed, Weyl’s “New crisis ” reads as a manifesto to the mathematical community, it uses an evocative language with a good many explicit references to the political

When the microequations of a dynamical system generate complex macrobehaviour, there can be an explanatory gap between the small-scale and large-scale descriptions of the same system. The microdynamics may be simple, but its relationship to the macrobehaviour may seem impenetrable. This phenomenon, known as emergence, poses problems for the nature of scientific understanding. How do we reconcile two radically different modes of description? Emergence is formulated using the powerful tools of algorithmic information and computational theory. This provides the ground for an extension and generalisation of the phenomenon. Mathematics itself is analysed as an emergent system, linking formalist notions of mathematics as a string manipulation game with the more abstract ideas and proofs that occupy mathematicians. A philosophical problem that has plagued emergence is whether the whole can be more than the sum of its parts. This possibility, known as strong emergence, manifests when emergent macrostructures introduce brand new causal dynamics into a system. A new perspective on this