Today I took the stupid O’clock train from Braga to Lisbon, actually managed to tick off some of to so list and went to ArXiv to see if there were any new papers I should be aware of. After “Functional Analysis”, I clicked “GM”. This may have been unwise.
The General Mathematics on ArXiv may actually be more addictive than the TV tropes wiki. Today I found About a virtual subset by Dipl. Math.(FH) Klaus Lange, which contains the following:
The main idea for those virtual subset is to enlarge the set theory.
In general it is neither decidable if V has in minimum one element nor if V have finite
numbers of elements or infinite many.
That leads to the possibility of a virtual set having finite elements but this elements are not
countable. Those finite but not countable virtual subsets will be a new category of the set
He came to this conclusion with reference to Gödel and undecidabilty: you don’t get better mathsiness than that!
I also came across DEGREE OF NEGATION OF EUCLID’S FIFTH POSTULATE (ALLCAPS in the original) by Florentin Smarandache, a paper misfiled as GM by the INTERNATIONAL MAFIA IN SCIENCE. Smarandache is an associate professor of mathematics at the University of New Mexico. Clearly he is rather distinguished, as he has a entry on PlanetMath and a Wikipedia entry with a large number of other articles linking to it all of which where, no doubt, written by completely neutral students of his work and describe highly notable phenomena. Also there is a journal committed solely to discussion of his ideas. Also.
He gives us this.
In this article we present the two classical negations of Euclid’s Fifth Postulate (done by Lobachevski-Bolyai-Gauss, and respectively by Riemann), and in addition of these we propose a partial negation (or a degree of negation) of an axiom in geometry.
The most important contribution of this article is the introduction of the degree of negation (or partial negation) of an axiom and, more general, of a scientific or humanistic proposition (theorem, lemma, etc.) in any field – which works somehow like the negation in fuzzy logic (with a degree of truth, and a degree of falsehood), or like in neutrosophic logic [with a degree of truth, a degree of falsehood, and a degree of neutrality (i.e. neither truth nor falsehood, but ambiguous, unknown, indeterminate)].
It’s good that it works with “a scientific or humanistic proposition…in any field”? Applicability and interdisciplinary work go down well with funding bodies (at least the non-MAFIA controlled ones). Sadly this particular paper is rather light on details
Also notable is that he developed his theory of geometries which are sometime Euclidean and sometimes not because he “observed that
in practice the spaces are not pure, homogeneous, but a mixture of different structures.” I’ve noticed before a tendency amongst less narrow-minded thinkers in mathematics to believe in a very special kind of Platonism, where there exist really real mathematically entities but they aren’t things other mathematicians study. Only the thinker’s own concepts relate to really real things.
And no, I never did say TWFiGM would be every week.