# Mathesis Universalis, Computability and Proof

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Part of the Synthese Library book series (SYLI, volume 412)

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Part of the Synthese Library book series (SYLI, volume 412)

In a fragment entitled *Elementa Nova Matheseos Universalis *(1683?) Leibniz writes “the *mathesis *[…]* *shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the *mathesis *to be “the science of all things that are conceivable.” Leibniz considers all mathematical disciplines as branches of the *mathesis *and conceives the *mathesis *as a general science of forms applicable not only to magnitudes but to every object that exists in our imagination, i.e. that is possible at least in principle. As a general science of forms the *mathesis *investigates possible relations between “arbitrary objects” (“*objets quelconques*”). It is an abstract theory of combinations and relations among objects whatsoever.

In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled *Contributions to a Better-Grounded Presentation of Mathematics. *There is, according to him, a *certain objective connection *among the truths that are germane to a certain homogeneous field of objects: some truths are the “reasons” (“*Gründe*”) of others, and the latter are “consequences” (“*Folgen”*) of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. A *rigorous proof* is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory.

Gottfried Wilhelm Leibniz Mathesis universalis Philosophy of Mathematics Proof Theory logic Ordinal Analysis Characteristica universalis Calculus Ratiocinator mathesis Constructive Mathematics Foundations of Mathematics Craig's interpolation theorem intensional type theory Turing Machine Philosophy Concept of Mathematics and Classification Analytic Philosophy mathematics History of Mathematics philosophy Reverse mathematics Bolzano philosophy Metamathematics Curry–Howard correspondence

- DOI https://doi.org/10.1007/978-3-030-20447-1
- Copyright Information Springer Nature Switzerland AG 2019
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