This post completes the paper “Mathesis Universalis revisited owing to Cantor, Frege, Einstein and Gödel“, offered as a contribution to the Poznań, October 2011, Conference on the Philosophy of Mathematics and Informatics. It is both an abstract and an additional comment involving questions to initiate a discussion.
1. The paper argues that the 17th century program of Mathesis Universalis (MU) gets accomplished in our times though in a fairly different way; the difference is thought-provoking too, and this is the other reason to revisite that famous project.
2. In the new MU, unlike the old one, we have a clear awareness of the enormous complexity of some algorithms, the physical universe, human mind, and human civilization, this complexity resulting in some unsolvable problems; such critical awareness was alien to our ancestors. It was Einstein who with general relativity paved the way to the idea of the evolving universe, and soon people conceived that its evolution tends toward ever more complex structures, up to living cells and further. As far as the complexity reaches the heights involvig infinity (like in Turing’s diagonal argument revealing the existence of unsolvable problems), the way to its treatment has been prepared by Cantor.
3. Leibniz’s project of universal langauage, which due to its precision would enable to solve any well-stated problem whatever, has been carried out by Frege, followed by Hilbert, as far as possible, up to the point in which Gödel (addressing Hilbert’s problems) could have discovered its dramatic limitations (they should have surprised Leibniz enormously, had he got a message about them).
4. However, Gödel’s results do not imply an epistemological pessimism. In his short but enormously seminal communication “Über die Länge von Beweisen” (1936) he reveals the perspective of never ending, but fruitful with each successive stage, process of discovering ever more sophisticated mathematical truths. Those, let mi add, should enable handling ever more complex phenomena, e.g. providing more and more efficient mathematical models and algorithms for natural science, economics, etc.
5. Gödel’s evolutionary vision of the growing mind’s ability to grasp ever more complex mathematical structures, may be seen parallel to the evolutionary vision of the universe as producing ever more complex physical and intellectual structures. Since the latter has been initiated by Einstein’s relativity, the paper starts from the picture of them both merged in a thoughtful talk.
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Questions adressed to expert critics
A. Do you agree that in information age the idea in the focus of our worldview is that of computational complexity? If so, how such a worldview should be named and explained? Do you think, that the reference to the MU project, as made in this paper, is a useful step towards the explanation?
B. As for naming, some authors suggest to use the term informational worlview; see e.g. Hector Zenil’s “Seth Lloyd’s quantum universe view“. The Polish counterpart “światopogląd informatyczny” appeared (presumably first time in Polish literature) in the book by Witold Marciszewski and Paweł Stacewicz “Umysł – komputer – świat. O zagadce umysłu z informatycznego punktu widzenia” (2011). Do you regard these terms, English and Polish, as relevant for the worldview focussed around the concept of computational complexity?
C. Do you share the above (item 4) interpretation of Gödel’s paper on the length of proofs?
D. Do you endorse the opinion (item 5) that Einstein has indirectly contributed to the informational worldview with his idea of evolving universe, provided this evolution’s trend toward growing computational complexity?
Witold Marciszewski