The uniform infinite Newtonian universe (the standard solution to the Seeliger paradox)

When Newton contemplated what the universe is like, he considered three possibilities:

• Aristotle's idea: the universe is finite 
• Stoicism's idea: the universe is a finite island of stuff floating in an infinite vacuum.
• Epicurus's idea: the universe is infinite and uniformly filled with stuffs.

He settled on the infinite uniform universe.

Now he faced a question: what is the gravitational force felt by a test mass in this infinite uniform universe? He thought that, by spherical symmetry, it has to be zero, because if the force isn't zero, it would break the spherical symmetry.

Now, a quick thought would show that there is a serious problem. It's known that a uniform, infinite slab attracts a mass with the same force, no matter how far.

So cut the universe into a infinite series of equally thick slabs, on the left and right of the test mass. We see that to calculate the total force on the test mass, we must do this infinite sum:
$$F + F - F + F - F ...$$

which is indeterminate.

In short, these three conditions are incompatible:

• The universe is infinite and uniform
• The universe is Newtonian (Newton's three laws of motion, plus Newtonian gravity)
• The Newtonian laws of physics should fix a unique solution.

This was what Seeliger argued in 1895. Since then, many solutions have been proffered, by giving up some of the conditions. One interesting solution is by Milne and McCrea in 1934, see Newtonian Cosmology (1955). It gives up the uniqueness, but shows that there are some very interesting solutions: an expanding and contracting universe with finite lifespan and a "center point" (despite being infinite and uniform), something that Newton could have discovered!

A very unexpected 333 that I found in math

While studying about dynamical systems theory for my ergodic theory course, I came across the extremely unexpected number 333. From Resonances and small divisors (Etienne Ghys, 2007), chapter 10 of Kolmogorov’s Heritage in Mathematics:

In 1962, Moser succeeded in accomplishing the feat of proving the theorem in the space of infinitely differentiable functions [Moser, J. On invariant curves of area-preserving mappings of an annulus (1962)]. 

In fact, Moser used functions which are 333 times differentiable and the topology of uniform convergence on these 333 derivatives... The mere fact that it is necessary to use as many derivatives shows the difficulty of the proof. Nowadays, it is known that the theorem is true with 4 derivatives and false with 3 [Sur les courbes invariantes par les difféomorphismes de l'anneau Herman, Michael R. Astérisque, no. 144 (1986)].

333 is half the number of the devil. Moser probably cut half of a deal with the devil.

Accelerate without humanity: Summary of Nick Land's philosophy

I took note of the philosopher Nick Land from reading about posthumanism on Wikipedia.

A more pessimistic alternative to transhumanism in which humans will not be enhanced, but rather eventually replaced by artificial intelligences. Some philosophers, including Nick Land, promote the view that humans should embrace and accept their eventual demise. This is related to the view of "cosmism", which supports the building of strong artificial intelligence even if it may entail the end of humanity, as in their view it "would be a cosmic tragedy if humanity freezes evolution at the puny human level".

I was intrigued by such boldness, so I read more. And turns out Nick Land's writing is sometimes easy to read but most of the times extremely hard to read, and probably garbage. I wrote this post so that you don't have to waste time wading through the garbage, looking for fragments of good poetry.

Nothing human makes it out of the near-future.

Causality is not fundamental in the world

Today we tear down the illusion of causality, with quantum mechanics.

This started when I was reading The Order of Time (2018) by the rather poetic physicist Carlo Rovelli, and read that he said that causality is not certain, and two events can have a superposition of causality: a superposition of two possibilities: A causes B, and B causes A.

Quantum corelations with no causal order

The paper that started this seems to be a highly cited (over 200 currently) Quantum correlations with no causal order (2012), Ognyan Oreshkov, Fabio Costa, Časlav Brukner.

The abstract starts with a question: is causality fundamental?

Fermi or DIE! More studies on Fermi paradox.

In this post we review some new (after 2010 at least) solutions to the Fermi paradox, or "the Great Silence".

Aestivation hypothesis

Proposed in That is not dead which can eternal lie: the aestivation hypothesis for resolving Fermi’s paradox (2017), by Anders Sandberg, Stuart Armstrong, Milan Ćirković. 

[Aestivate: to sleep through the summer, or a period too hot for living. The opposite of "hibernate".]

Idea: Aliens are sleeping until the far future, when the universe is cold enough, to maximize computation. The reason is that cold = more efficient use of energy for computation, according to the Landaur limit: $E\ge kT\ln(2) \text{Joule}$ need to be dissipated for an irreversible change of one bit of information.

... this can produce a $10^{30}$ multiplier of achievable computation. We hence suggest the “aestivation hypothesis”: the reason we are not observing manifestations of alien civilizations is that they are currently aestivating...

The phrase "That is not dead which can eternal lie" comes from a poem by H. P. Lovecraft, a philosopher ahead of his time who really understood how human thoughts matter little in a cosmic viewpoint.

A civilization would aestivate if:

Let's read: $C^*$-algebra, quantum mechanics, Pascual Jordan

A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details. ---- Hermann Weyl

After 3 courses in analysis I have finally reached Banach algebras, I feel exhausted by the climb and yet there's more theory ahead until the true pinnacle: the theory of $C^*$-algebra, for quantum mechanics!

Let's read, what is being promised?

$C^*$-algebras were first considered in quantum mechanics to model algebras of physical observables... began with Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of $C^*$-algebras which are now known as von Neumann algebras.

So experiments in quantum mechanics → weird behavior of observables in quantum mechanics → Heisenberg matrix mechanics → Jordan purified it into Jordan algebra → von Neumann generalized to rings of operators → von Neumann algebra generalized to $C^*$-algebras.

Let's Read: Nielsen and Ninomiya (2007)'s backward causation, LHC, and the end of mathematics

I considered writing a blog because it's better than writing things on the wood paper, which is hard to search, even more prone to be forgotten than electronic writing, and almost impossible to be read by anyone but me.

I mean, there's a slight chance that my papers will get buried by volcanic ash and become dug up by future archeologists (like the scrolls in the Pompei library), while there's pretty much no hope of any server surviving such a treatment, but that's too unlikely to be considered a valid argument for using wood paper.

Today we write about LHC, SSC, and Nielsen and Ninomiya's crazy proposal in 2008 about why LHC shut itself down. Then it digresses to a depressing thought about the ends of scientific and mathematical fields.