Squidgers wrote: ↑Sat Jul 10, 2021 11:20 am
It's a type of mathematical idealism, not a formalist aproach. There are some decent arguments agaijst formalism in there, and he deals with Münchausen trilemma. But obviously, you do you
Obviously I have not read the book, only the short abstract. Based on that it looks like extending point-reductionism of formalism to idealism, which IMO is a very bad idea. I have plenty of respect towards Alain Badiou, for explicating
set theoretical ontology in philosophically lucid way that exposes it to foundational critique and project of constructing
alternative and better ontology based on process philosophical intuitionism with point-free approach, which can be generally characterized as
Whitehead's paradigm.
In that sense, Badiou self-constructs himself and set-theoretical ontology as Worthy Enemy, while his deep critique of math based social alienation and oppression offers the
motivation, and his process philosophical truth theory and notion of 'Event' provide
possibility of liberation and evolution of mathematical onto-dia-logos.
Badious self-identifies as (dialectical) materialist, but his version of mathematical Platonism cannot be considered as anything but idealism.
Intuitionism is ontologically and fundementally an
empirical approach to philosophy of mathematics. So, we start from accepting Zeno's empirical proof by contradiction as empirically valid and true refutation of point reductionism, and that intuitively empirical continuum/continua are not reducible to discrete quantification. Euclid provides intuitively coherent definition of point (definitions 1 and 3): Point has no part, point is end of a line and in that sense a part of a line. Hence, point has no independent substantial existence, but is mereologically intuitied and defined in relation to line and plane and affine parallelism.
On the other hand, if we start from the set theoretical and physicalist point-reductionism, where everly line segment consists of infinity of points, we can't draw a line, as drawing a line from point to point would take infinite time. Hence in ontology of point reductionsim we would lose also Platonic Solids, which were the teleology of Euclids constructive proof strategy. Ethically and aesthetically I like Platonic Solids, and would like to have them around and be able to draw and build e.g. Geodesic Domes.
Of course Euclid is not the final word, e.g. his definition "line has no width" is counter-intuitive and unnecessary, and we can let go of that. But if and when the choise is between Euclid and Platonic solids etc. continuous geometry vs. point-reductionism of Hilbert and Cantor, I choose Zeno, Euclid, Nagarjuna, Bergson, Wittgenstein and Whitehead any day. So I don't think it's just "me", IMHO I'm in a rather good company.
