The Weirdest Math Video You'll See Today

This is great! 😂❤️💓💕 See FOM Rejection, ...

I think it's about what happens when you're missing the middle term in all your deductions, like this:

The machine becomes your worst nightmare, ... Look at this video on instability in Newton's Method:

Notice this: (apologies, my blackboard is rather small!)

Newton's method fails to find the root at 1 - phi, because the magnitude of the derivative is greater than one at that point so the associated recurrence relation diverges. But geometrically the two solutions are symmetrical. So the question is "How can we design numerical methods which use these geometric symmetries (i.e. geometric fixpoints) to find all possible solutions, not just the stable ones?" See Wildberger at 10 minutes 42 seconds, on how to take photographs of a parabola at different scales. In a projective plane, a parabola looks like an ellipse! See Aristotle on The Continuum for some motivation behind the idea of modelling the process of computing numerical solutions to differential equations. See also this theorem of Appolonius at 39 minutes 42 seconds, and this theorem at 27 minutes 43 seconds. For the connection, see Quaternions Visualised.

So the idea is to develop numerical algorithms where we solve for both roots at the same time, using whichever is the most numerically stable and translating that approximation to the least numerically stable. This is the approach I used here, in Surfing the Cloud, on page 17:

This can all be done with a rational paramaterization of the conic. See this lecture, from 10 minutes 50 seconds:

Back to projective conics: see from 27 minutes 33 seconds, which shows the study of an hyperbola in homogeneous coordinate space. Note in particular the duality at 31 minutes 33 seconds. More on this here:

I think projective representations of conics will turn out to be what we need for a sound basis of tensor calculus, more amenable to numerical analysis and with fewer black holes! See All About e, and Tensor Calculus, and this abstract POINT class (line 322 or thereabouts) in Standard ML: https://github.com/IanANGrant/red-october/blob/master/src/dynlibs/ffi/MagnificentPairs.sml. So what I would do, if I had a place to work, and a decent computer, and maybe some interested students, would be to write Standard ML programs implementing these abstract POINT types for projective linear algebra, as described here at 6 minutes 56 seconds on a basis of rational paramaterization, using compiled primitives described in Writing Assembler using Standard ML Functors, and within a Cartesian Closed Category defined in the framework described here: On Tarski's Definition of Truth "Convention-T". The fundamental elements of the spaces would be Markov random fields, and the aim would be to recover the exact representations from these stochastic approximations, as limit objects of some kind, looking a lot like the Real closed field.

On Markov fields rather than Bayes nets, see Julia Galef on the Penrose Triangle and belief:

See my comment on that video:

6:33 If you accepted the reporter's criticism initially, then it must've been because you had another reason to believe that hand-sanitizing being over-hygeinic was a valid complaint. That is why you kept the belief. And the same applies to the religious example. The religious belief that sex outside of matrimony was wrong must have made sense at the time for reasons other than merely that it was written down in that particular religious text, and those reasons wouldn't necessarily go away when you no longer accept the authority of that particular text. One of those other reasons could be the society in which you live, which may still be composed largely of adherents to that religion. What I am trying to get at here is that cyclical, non-linear networks are inevitable in social contexts, and then "orphaned beliefs" are going to be pretty rare things, on the whole.

 Just about every single page of this book will be relevant: An Elementary Treatise on Determinants: With Their Application to Simultaneous Linear Equations and Algebraic Geometry:

 Chapter VII in particular:

See also my comment on this video:

9:49 You could do it in 3D space like Euclid did. He studied the ruled surfaces generated by mappings of a circle under f(z) between the two parallel planes, ... see https://livelogic.blogspot.com/2019/09/a-concrete-way-to-see-what-computable.html

At 14 mins 22 secs "Artists in the 15th and 16th century were interested in how things really looked, when you drew them accurately, ..." See the remark on Plato's allegory of the cave and Platonic reality in Wildberger on Rational Parameterization. The Eye, ... see 18 minutes 50 seconds:

The point at infinity in the eliptical projection of the parabola is actually a fixpoint: see this, at 14 minutes 53 seconds. Here's a friendly introduction to fixpoints:

What we are aiming to be able to do is to find all the fixpoints Sharkovsky's. theorem tells us exist, ...

... but which don't show up in dynamical systems we compute in practice, such as the Logistic Map, which shows up as a Period-doubling bifurcation, in which all the other solutions (fixpoints of period 3 or greater) are unstable. So we should find that the Feigenbaum constant is just one term of a much more interesting sequence. So, if we do this right, we will be well on the way to being able to control chaotic systems, ... and then we may even understand the axiom of choice, ... well, one can but hope, ...

Note that in intuitionistic systems like Martin-Löf Type Theory, choice is deterministic, so the choice function is just a program, and so it has a fixpoint which can be deterministically calculated by a fixpoint combinator, Y, say ... see A Concrete Way to See What Computable Functions Actually Are for some ideas about how to handle this in theorem proving systems. See also this brief note on constructivism, intuitionism and foundations. Maybe that explains the problems this guy has understanding MLTT:

That is from "Institute of Advanced Study", but they don't say which one. Therefore I assume it's Dublin. That conference in Sienna must've been interesting, if this is anything to go by:

And for why this matters, see This Is What We Should be Thinking About, and Near Earth Asteroid Apophis and Really Weird Stuff Going On, ... and Putin Inspects New Vostochny Cosmodrome. Dude, this is very uncommon sci-fi!