Typical = Random

Round 1

Reviewer 1 Report

This is a very good and interesting paper that should be published in the present form. It gives a very deep, rigorous, and quite readable account of relations between probability and randomness (in particular, whether randomness can be defined without invoking the concept of probability), connections between randomness and irreversibility, and related problems. A comprehensive bibliography makes the article also a very good source as shoul be expected from a review paper.  

Author Response

Thank you, no reply needed

Reviewer 2 Report

The author review some theorems, presents their revision in the context of P-randomness and indicate some applications to statistical and quantum mechanics. The review is very well written, and the presented concepts are strongly supported by the cited literature. Taking this into consideration, my suggestion is that this review should be accepted for publication.

The following suggestions are made exclusively in order to try to improve the text in some points:

1- The second sentence in the abstract could be written as: To this end, various theorems from mathematics and physics are reviewed. Their original version states that some property $\Phi(x)$ holds for $P$-almost all $x \in X$, where $P$ is a probability measure in some space $X$. These theorems more refined (and typically more recent) formulations shows that $\Phi(x)$ holds for all $P$-random $x \in X$.

2- In the first line of the introduction, the text "and subsequently Boltzmann" should be enclosed by commas. Another possibility is to remove the word "subsequently".

3- The diagram in page 3 does not include a line for $a_1$, while eq (2.1) states its existence. Should this line be included in the diagram or this is not necessary?

4- Since eq (2.16) ends in a comma, the following sentence should begin with a lowercase "t".

5- In the first line of section 4, "review" should be "reviewed".

6- In the last quotation of the text, dealing with the Bohmian explanation of randomness in the context of quantum mechanics, it is written: "[...] s is random because s is random". Shouldn't the last s be s'?

7- In the following, scenario's should be scenarios.

8- Ref [17] lacks a dot at the end of its title. [35] lacks a dot at the end of the reference. [40] has a space preceding the colon. [61] has a comma at the end of its title, where it should be a dot or nothing. [63] number and page are separated by a colon, while it tshould be a comma. [68] has a comma at the end of its title, where it should be a dot.

The suggestions above are not mandatory. My recommendation is that the article should be accepted for publication.

The text is written in excellent scientific english. Some suggestions were made only in order to improve readability.

Author Response

Thank you, I have taken all of your suggestions into account in the revision, 

Reviewer 3 Report

The "Typical = random" article is quite an interesting review of various aspects of probability, trying to compare the interplay between probability and randomness in several factors. The article, in general, is a bit difficult to read; given the large number of footnotes it introduces, I would recommend the author remove some of these footnotes and include them naturally in the main text, making the reading more continuous and enjoyable. On the other hand, the article does not have conclusions. However, there are enough results throughout the paper that it would be very nice to see them summarized at the end.

Some additional points on notation: 

1. The author uses the same letter \rho for particle density and the density operator.
2. Page 20, below Eq. (6.1), says: in experiment no. n -> in experiment No. n

Author Response

Thanks you, I have loved at all footnotes and taken some into the main text as you suggest. I have also added a final section as a Conclusion. 

Reviewer 4 Report

This manuscript was introduced to review the interplay between probability, (algorithmic) randomness, and entropy via examples from probability itself, analysis, dynamical systems and (Boltzmannstyle) statistical mechanics, and quantum mechanics. Some basic relations were explained. Furthermore, the author reviews the algorithmic randomness beyond binary sequences. Moreover, he introduced the key “intuition pumps”: these are results in which ‘for P-almost every x’ in some “classical” result can be replaced by ‘for all P-random x’ in an “effective” counterpart thereof (this replacement may even be seen as the essence of algorithmic randomness). Hence, this idea was applied to statistical mechanics. Finally, some brief comments on quantum mechanics were given.

The manuscript was written in excellent way. The idea was clearly introduced. The author is well known in his field. However, there is no conclusion,  if he end his review with a conclusion I think it would be a good idea.  

For example in the second page, the line before the last: " be " was missed 

for the sentence :" replacement may even seen"   

Author Response

Thank you, suggestions taken into account including adding a final concluding section.