Collaborative invitation to spin-off research

[Federica]

...

Yes, I noticed it as well! I haven't watched it either. I wanted to review this post first:

viewtopic.php?p=24770#p24770

[Federica]

I have watched it now. My impression is, there's a frank intention to bring into concrete mathematical fruition the idea of a psychic space (collective unconscious) shared by thinking humans. In other words, no dissociative boundaries, as far as mathematical archetypes (concepts) are concerned. I like that. But then, as it seems, the intention falls short of its own ambition, and the reasoning becomes somewhat contradictory. On the one hand, there is the explicit purpose to stop considering mathematical archetypes as separate from the thinker(s): they are said to reside in a shared space in which they are accessible as is, to thinking human beings. But in fact, this ideal connection is “inner” and “integrated” only in this propositional, definitional sense. Beyond the initial statement, and to all intents and purposes, the connection is treated as outer, determined by statically conceived archetypes.

His conception is that with the ongoing development of mathematics, we open new vistas. But these vistas amount to an expansion of the set of thinkable mental pictures of archetypes. These archetypes are static, externalized objects, rather than intimate experiences. And so evolution means getting to know (integrate) new archetypes in objective consciousness.

As a consequence, mathematics is seen as a portal towards an extended sense-perceptible reality, rather than a portal towards spiritual reality. There’s the idea that more perceptual dimensions are to become conscious, once new mathematical archetypes are integrated in awareness (like the flat 3D space is about to become, as he says).

It's still difficult to resist the conception of centrality, predominance, of the material world. At the end of the day, it irresistibly attracts trains of thought. Like, if we integrate new mathematical archetypes, the final benefit has to be an augmented perception of the world, a new vision of it. But more mathematics applied to external nature, means adding more layers of perceptual constraints, adding grids to grids, whilst the mystery of nature - which speaks of inner worlds other and higher than man’s growing (but still groping) mathematical self-discipline - remains hidden.


PS: he says he's open to exploring the question further with the audience, and that he reads all the comments...

[Federica]

Incidentally, and interestingly, the reference to curved versus flat space in Frenkels last video is clarified (for the non mathematician) in one of this week's Aeon essays: Beyond Causality, by Gordon Gillespie.

Beyond this detail, in this essay there's an insightful take on mathematics as the only discipline able to offer a structural but not causal understanding of reality, thereby becoming the bridge across the traditional separation between humanities and natural sciences. The author offers a thought-provoking overview on the state of our current understanding of reality, that touches on naive realism, naive idealism, Newton, Kant, Einstein, Wittgenstein, Riemann and more, which has been worth reading for me.

Ludwig Wittgenstein once said: ‘I want to show the colourfulness of mathematics.’ In that spirit, I placed mathematics at the centre of my project because, in my view, mathematics searches along more of these many paths than any other intellectual discipline. It is connected on a deep level both with the natural sciences and the humanities. It bridges the gulf between them, and it does so by putting certain metaphysical and epistemological dogmas into question

The best mediator of a conciliatory view that avoids the mistake of the naive realist and the naive idealist is mathematics. Mathematics gives us shining proof that understanding some aspect of the world does not always come down to uncovering some intricate causal web, not even in principle. Determination is not explanation. And mathematics, rightly understood, demonstrates this in a manner that lets us clearly see the mutual dependency of mind and nature.
For mathematical explanations are structural, not causal.

Kant’s transcendental idealism doesn’t only suffer from the fact that Euclidean geometry turned out to be not quite as constitutive as he thought. More severely, his conception of empirical knowledge, as an act of understanding through which conceptually formed ‘judgments’ miraculously emerge from mere ‘sensations’, remains completely obscure, as even well-meaning readers of the Critique must admit. But we can attribute at least one fundamental insight to Kant: mind and world are no separate spheres that must first be connected, so that the question arises as to how exactly this might be achieved.

Mathematics highlights the limits of natural scientific explanation. This becomes even clearer when we consider how the idea of an all-explanatory physical theory or ‘world formula’ came about in the first place. In other words, how did scientists come to believe, or at least hope, that a mathematical description of nature on the most fundamental level exists, with which every worldly phenomenon is explainable in the sense that its entire causal history can be derived from basic laws, at least in principle?