ho.history Overview – Theorems that hampered progress

It may be that certain theorems, if they prove true, are intuitively delayed
Progress in specific areas. Lloyd Trefethen provides two examples:

  • Faber's theorem on polynomial interpolation
  • Squire's theorem on hydrodynamic instability

Trefethen, Lloyd N. "Inverse Yogiisms". Notes from the American Mathematical Society 63, no. 11 (2016).
Likewise: The best typeface on math 2017 6 (2017): 28.
Google Books Link.

In my own experience, I have seen the various negative results sets

Marvin Minsky and Seymour A. Papert.
Perceptrons: An Introduction to Computer Geometry, 1969.
WITH Press.

hamper progress in neural network research for more than a decade.1

Q, What are other examples of theorems, their (correct) evidence (possibly temporary)
Suppression of research in mathematical subfields?

Olazaran, Mikel. "A Sociological Study on the Official History of the Perceptron Controversy." Social science studies 26, no. 3 (1996): 611-659.
Abstract: "[…]I particularly focus on the evidence and arguments of Minsky and Papert, interpreted as meaning that further advances in neural networks are not possible and that this approach to AI has had to be abandoned.[…]"

ho.history – Overview of the entropy of the causal path

in the [1]The authors represent the Causal path entropy as follows:

For every open thermodynamic system like a biological organism we can do this
Treating phase space paths that the system takes over a period of time
$[0,tau]$ as microstates and divide them into macrostats
$ {X_i } _ {i in I} $ using the equivalence relationship:

begin {equation} x (t) sim x & # 39; (t) iff x (0) = x & # 39; (0) end {equation}

As a result, we can identify every macrostat $ X_i $ with a gift
system state $ x (0) $,

We can then define the causal path entropy $ S_c $ from a makrostat $ X_i $
assigned to the current system state $ x (0) $ as path integral:

begin {equation} S_c (X_i, tau) = – k_B int_ {x (t)} P (x (t) | x (0)) ln P (x (t) | x (0)) Dx (t) end {equation}

from where $ k_B $ is the Boltzmann constant.

Well, the authors of [1] I claim that this is a new kind of entropy, and this is a point that I would like to clarify because it looks like this is simply a conditional Boltzmann entropy.

Actually on the second page of [2] C. Villani introduces Boltzmann entropy using time-dependent density $ f $ on particles in phase space $ (x, v) in omega times mathbb {R} _v ^ 3 $:

begin {equation}
S (f) = – int _ { Omega times mathbb {R} _v ^ 3} f (x, v) ln f (x, v) dxdv
end {equation}

and we can analyze the dependence of the development of $ S $ under special initial conditions $ p (0) = (x_0, v_0) $ by the definition of:

begin {equation}
S (f | p (0)) = – int _ { Omega times mathbb {R} _v ^ 3} f (p (t) | p (0)) ln f (p (t) | p (0)) dp
end {equation}

from where $ p in omega times mathbb {R} _v ^ 3 $

I am relatively new to statistical mechanics, but I would be very surprised if it did not occur to him to analyze $ S (f | p_0) $, Is the causal path entropy actually conceptually new?

Note: Although I mention the Boltzmann entropy here, I have to say that the authors of [1] do not count on Boltzmann for their ideas. Meanwhile, in a relatively recent TED talk, Wissner-Gross claims that & # 39; E = mc ^ 2 & # 39; to have discovered for the intelligence.


  1. Gross, A. Wissner. (2013) Causal entropic forces. Physical overview letters.
  2. Villani (2007) H-Theorem and beyond: Boltzmann's entropy in today

ho.history – Why did Voevodsky give up his work on "singletons"?

In an interview (Link to the Google translation) Voevodsky talks about how he worked on the problem of "restoring the history of populations according to their modern genetic composition" in the late 2000s. Some of his unpublished publications on the subject are now available online. For example, an article titled "Singletons" is available on the IAS website. Why did Voevodsky give up the subject of this rather fleshy paper so suddenly?