The proof-theoretic investigation of Mathematics has been one of the major achievements of XX century Science: it gave us mathematical rigour and modern computing. The latter, its major fall-out, is changing our live. Yet, we need to go beyond its techniques and philosophy. First because, in view also of its successful paradigms, Mathematics has now reached a remarkable level of rigour and we are no longer scared of the novel geometric intelligibility of physical space, which originated Frege's "royal way out" from the "delirium" in Geometry (see [Tappenden95] for more on Frege's view). Second, because we may take advantage, also in computing, by an enlargement of our foundational paradigms, beyond the traditional linguistic-finitistic certitudes. In a sense, we should try to bring together the two "foundational ways" that split at the end of XIX century, as I tried to summarise in the introduction. In short, the foundation of Mathematics lies in: 1. Logic; 2. Formalisms; 3. Regularities of phenomenal space.
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