Ben Galehouse

@kevinr@masto.free-dissociation.com @lcamtuf@infosec.exchange And if you ask it to write a detailed spec based on its implementation, and then separately to write an implementation of that spec? https://www.allaboutcircuits.com/news/how-compaqs-clone-computers-skirted-ibms-patents-and-gave-rise-to-eisa/

@lcamtuf@infosec.exchange

Indeed. More generally you need to be very careful when using self-reference in mathematics.

IIRC, you can only define a set in terms of what is defined prior to the set's definition. In specific cases you can create self-reference in a more roundabout way though, Gödel's fame stems from this.

@lcamtuf@infosec.exchange
Suppose that not all positive integers are definable in under 11 words. Then there is a smallest integer \(n\in \mathbb{N}\) which is not definable in under 11 words.

But then the statement "The smallest natural number not definable in under eleven words." is only 10 words and unambiguously defines \(n\). This contradicts the choice of \(n\).

Therefore all natural number are definable in under 11 words.

Flatland and Minimum Action [Addendum] A few more observations about minimum action. The process of minimizing the action actually leads to the conservation of energy, at least in cases where \(V\) is only a function of position. That is, you need both endpoints, and \(L\) to perform the minimization process, and as a result of that process \(T+V\) is constant over the resulting curve. So the global minimization process leads to a quantity which is both local and constant. I find it interesting to compare this to the role of energy in quantum wave equations, e.g. the frequency of a photon is a local property, constant through a long narrow region of spacetime. This emergent conserved quantity \(T+V\) also explains why cutting up the path of a particle doesn’t really work. If you fix \(x(0), x(1), x(2)\) minimizing over the two segments \([x(0), x(1)]\) and \([x(1), x(2)]\) does not in general lead to a valid path for \([x(0), x(2)]\). In order to conserve energy at \(x(1)\), we need to choose \(x(1)\) carefully - essentially by solving the longer segment first.

Flatland and Minimum Action [2 of 2] Now let’s consider a simple but non-trivial example of how the minimum action principle works. Assume a uniform potential \(V=-kx\) and a kinetic energy \(T=\frac{1}{2} m \dot x^2\). Finally, assume that \(x(0)=x(1)=0\) Then we consider all sufficiently differential paths \(x(t)\) with those endpoints, and find one which minimizes the path integral \[\int_0^1 L =\int_0^1 T-V= \int_0^1 \frac{1}{2} m \dot x^2 + kx\] The Euler-Lagrange equation tells us that our solution will be such that \(m \ddot x = k\) and then integration plus our boundary conditions give \(x = \frac{k}{2m} (t^2 - 1)\). Some might argue that it was a choice to fix the endpoints. But a minimization process in which either endpoint moves would not give the same answer. Furthermore, the derivation of the Euler-Lagrange equation only proves that the equation holds on an interval \((a,b)\) under the assumption that \(x(a)\) and \(x(b)\) are fixed. And finally, this gives a bit of intuition about the negative sign in \(L=T-V\). During the minimization process, a potential trajectory is pulled towards a higher potential field. But then when we consider the effect on a particle over time, we see it as accelerating towards a lower potential field. Geometrically, this is because the endpoints are, ahm, nailed down. And so the action minimization principle feels a little like an artifact from outside of flatland.

Flatland and Minimum Action
[1 of 2]

Suppose we have a long board, and a flatland-inspired denizen which moves along it, perceiving at each point in time a cross section of the board. Now, let’s put nails into the board somewhat randomly and then tie wires between some of the nails.

Our flatlandian therefore occasionally sees nails appear for a moment, and sees bits of wire seem to move from one nail to the next. In fact they might come to call these wire bits “particles”, and that these particles are created by nails, and then move at a constant speed until they happen to hit another nail.

Now suppose that we place a strong magnet on one side of the board, so that a wire forms a bow shape, bending towards the magnet. From our point of view, the wire is attracted to the magnet, but what about our flatlandian? They see a particle heading towards the magnet, then bending away, and then hitting a nail. So the magnet is attractive from our point of view, but repulsive from the flatlandian’s point of view.

In this setup, the different viewpoints give the flatlandian the same predictive power, and the flatlandian might be very hesitant to move away from their initial interpretation. After all, if a flatlandian cannot see a nail before it appears, how could the particle?

@_dm@infosec.exchange With or without term limits, it seems dangerous to let anybody who actually wants to job to have it.