By Francis Bailly, Giuseppe Longo
This publication identifies the organizing options of actual and organic phenomena through an research of the rules of arithmetic and physics. Our objective is to suggest a conversation among varied conceptual universes and hence to supply a unification of phenomena. The function of 'order" and symmetries within the foundations of arithmetic is associated with the most invariants and ideas, between them the geodesic precept (a final result of symmetries), which govern and confer cohesion to varied actual theories. additionally, an test is made to appreciate causal buildings, a principal section of actual intelligibility, when it comes to either symmetries and symmetry breakings. A contrast among the rules of (conceptual) development and of proofs, either in physics and in arithmetic, courses lots of the paintings.
the significance of mathematical instruments can be highlighted to elucidate changes within the types for physics and biology which are proposed via non-stop and discrete arithmetic, reminiscent of computational simulations.
on account that biology is very advanced and never to boot understood at a theoretical point, we recommend a "unification via concepts" which at least should still precede mathematization. This constitutes an summary for unification additionally according to highlighting conceptual transformations, complicated issues of passage and technical irreducibilities of 1 box to a different. certainly, we believe the following a really universal monist perspective, specifically the view that residing items are "big luggage of molecules". the most query notwithstanding is to appreciate which "theory" may also help higher comprehend those luggage of molecules. they're, certainly, quite "singular", from the actual perspective. Technically, we convey this singularity during the notion of "extended criticality", which supplies a logical extension of the serious transitions which are identified in physics. The presentation is usually saved at a casual and conceptual point.
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Additional info for Mathematics and the Natural Sciences: The Physical Singularity of Life (Advances in Computer Science and Engineering: Texts)
Then” (ﬁrst dimension), of completely formal, even mechanic calculus (second dimension), but additionally in a 3rd conceptual size, those buildings of (and in) time and house, which mingle it, much more so than the 2 others, with the diﬀerent types of wisdom. And the epistemological challenge then poses itself as an research of the structure of the invariants of language and of evidence, those invariants which we name “logic” and “formal systems,” in addition to the invariants of time, and of area, upon which we build our geometries, those “human constructs . . . in our areas of humanity” as Husserl says within the “Origin of Geometry” (see below). the matter is hence posed from the research of this very strange type of wisdom that is arithmetic, from its cognitive roots, be they pre-human, to its communicable show, with its millions of mediating degrees. Axiomatic conventions and logico-formal evidence are literally however the final result of a structure of which means, universal notations of innovations rooted in “our residing practices,” to place it as Wittgenstein may do, in our “acts of expertise” (Weyl): logico-formal research is an important accompaniment to this latter a part of the epistemological method, the research of facts, of yes proofs, however it is insuﬃcient (it is basically “incomplete,” a few theorems inform us). The foundational analyses of arithmetic needs to therefore be prolonged from the learn of deduction and of axiomatics to that four Hilbert, as a superb mathematician, will deal with rather good another way. due to the Beltrami-Klein interpretation of non-Euclidean geometries inside of Euclidean geometry, he'll provide an accurate immersion (interpretation) of his axioms for geometry inside mathematics through research (Hilbert, 1899). yet, for the latter, he'll now not search for a “logical meaning,” not like Frege. certainly, as soon as geometry (ies) is (are) interpreted inside of mathematics, a finitary evidence of its (thus, their) coherence (of non-contradiction) could suffice for its foundational research, solely and solely founded round its challenge of coherence; its a pity it doesn’t paintings, simply because mathematics doesn't have, itself, any mathematics (finitary) facts of coherence. on the contrary, we deal with with an enormous piling of infinities, or by way of evidence based upon geometrical judgments, see bankruptcy 2. arithmetic January 12, 2011 9:34 international medical e-book - 9in x 6in Mathematical ideas and actual items arithmetic 23 of the structure of recommendations and of constructions; yet this is often very unlikely with no parallel research of the structure of the actual item and of conception. 1. 2. 2 The “transcendent” in physics and in arithmetic there is not any doubt that there exists a truth past ourselves, which enters into “friction” with our activities upon it and which, furthermore, “canalizes” them. Husserl makes use of a observe from the idealist culture to designate this truth: he considers the idea of transcendence. In a really universal interpretation of this notice, and particularly independently from Husserl, the subsequent deduction is mostly made, ﬁrst in physics, then in arithmetic: the “properties” of the realm (physical, numerical, mathematical, .