I am curious as to what publisher's book you used as your copy of Euclid's Elements. I'm curious because the first edition of Euclid's Elements that I read was the Barnes And Noble edition. In that edition, Book 1 runs from page 1 to page 217, and my experience was the same as yours. I didn't see the logic that I expected to see. That edition is full of commentary. I'm sure that commentary is extremely useful for some readers. To me, it was extraneous information.

After that, I read the Green Lion Press version of the Elements. Book 1 runs from page 1 to page 36. It's mostly a bare bones translation of Euclid's definitions, common notions and propositions - I say mostly because the book contains bracketed references to previous propositions, common notions, etc. as justifications for the arguments used in the proofs of the propositions. What I saw in that book was an elegant logical construction of a fairly complex argument, the proof of the Pythagorean Theorem and its converse.

The logic of Euclid's argument came through much more clearly in the uncluttered book.

That was a part of Orwell's newspeak in the novel 1984. From wikipedia

Newspeak has no antonyms, therefore the prefix "Un–" is used to indicate negation; the Standard-English word warm becomes uncold, and the moral concept communicated with the word bad is expressed as ungood. When appended to a verb, the prefix "un–" communicates a negative imperative mood, thus, the Newspeak word unproceed means "do not proceed" in Standard English.

"Plus–" is an intensifier that replaces more and the suffix –er; thus, plusgood replaced the English words great and better.
"Doubleplus–" is an intensifier that replaces plus– to communicate greater intensity; to that purpose, the Newspeak word doubleplusgood replaced the English words excellent and best.
"Ante–" is the prefix that replaces before; antefiling replaces the English phrase "before filing."
"Post–" is the prefix that replaces after.

Science, on the other hand, cannot even investigate if science would be complete. There are no equivalent scientific incompleteness theorems in science about the scientific method. That alone makes science very, very incomplete.

I'm not sure if that's completely accurate. In 2015, there was an article, Quantum physics problem proved unsolvable: Godel and Turing enter quantum physics, which claims that physicists have found a problem that can't be solved. An excerpt from the article:

A mathematical problem underlying fundamental questions in particle and quantum physics is provably unsolvable, according to scientists at UCL, Universidad Complutense de Madrid - ICMAT and Technical University of Munich.

It is the first major problem in physics for which such a fundamental limitation could be proven. The findings are important because they show that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.

That's not as broad as Godel's Theorem in mathematics; but it does imply that physics is capable of recognizing the existence of problems that it can't solve.

My understanding is when Riemann is talking about the 2nd derivative, he is talking about the 2nd derivative of the metric function, the function for ds:

ds is the square root of an always positive integral homogeneous function of the second order of the quantities dx

He uses the example of this function in Space:

$ds = \sqrt{ \sum (dx)^2 }$

Would matrix form work? Let x = $(x_1 x_2, ... x_n)$ be a row vector, $x^t$ the column vector, and P be the $p_{ij}$ matrix. Then, something like:

$ds^2$ = x P $x^t$

I have been reading along in this thread and I appreciate the effort that you and others are making in explaining Riemann's paper. So far, this has been a tremendous help to me. I have a question on your last post. Unfortunately, I can't just copy down the formula so I'll try to enter it in MathJax. My question is about this formula:

$\sum_{i=1}^n l_mx_m + \sum_{i = 1} x_i \sum_{i \neq j} p_{ij}x_j$

From your last post:

... If we have 3 variables x y z, there are 6 possible quadratic terms, x^2, y^2, z^2, xy, xz, yz, and so on. ...

That doesn't have any terms of the form:

$\sum_{m=1}^n l_mx_m$

But rather has something like:

$\sum_{m=1}^n l_mx_m^2$

I think that first term could be dropped and included in the second term if the second term is modified to be:

$\sum_{i = 1}^n x_i\sum_{j=1}^i p_{ij}x_j$

This actually gives you the right number of terms, namely 0.5n(n+1) - the $p_{ij}$ would essentially compose an n x n matrix with the only non-zero terms being in the lower triangular portion. The justification for the lower triangular matrix is that in metrical relations we would always want $x_{ij} = x_{ji}$. If we include the full n x n matrix (i.e. include non-zero terms from above the diagonal), I think we will be double counting the distance of the $x_{ij}$ entries.

I agree with Wittgenstein that not everything we call language is this system. But, I also agree with Augustine that for many of us, this is the way we learn our first language. And, while Augustine does appear to give words a one-to-one association with objects, he also notes that we express our wishes through language.

This is a translation of all of Book 1, Chapter 8 of Augustine's Confessions translated by Albert C Outler - it's just a bit more than what Wittgenstein cited:

13. Did I not, then, as I grew out of infancy, come next to boyhood, or rather
did it not come to me and succeed my infancy? My infancy did not go away (for
where would it go?). It was simply no longer present; and I was no longer an infant who could not speak, but now a chattering boy. I remember this, and I have since observed how I learned to speak. My elders did not teach me words by rote, as they taught me my letters afterward. But I myself, when I was unable to communicate all I wished to say to whomever I wished by means of whimperings and grunts and various gestures of my limbs (which I used to reinforce my demands), I myself repeated the sounds already stored in my memory by the mind which thou, O my God, hadst given me. When they called some thing by name and pointed it out while they spoke, I saw it and realized that the thing they wished to indicate was called by the name they then uttered. And what they meant was made plain by the gestures of their bodies, by a kind of natural language, common to all nations, which expresses itself through changes of countenance, glances of the eye, gestures and intonations which indicate a disposition and attitude--either to seek or to possess, to reject or to avoid. So it was that by frequently hearing words, in different phrases, I gradually identified the objects which the words stood for and, having formed my mouth to repeat these signs, I was thereby able to express my will. Thus I exchanged with those about me the verbal signs by which we express our wishes and advanced deeper into the stormy fellowship of human life, depending all the while upon the authority of my parents and the behest of my elders.

The original German is something like: Die Welt ist die Gesamtheit der Tatsachen, nicht der Dinge.

I think that gets around the ambiguity in English.

For clarity, he does go on to further define fact. From The Tractatus:

2.0 What is the case, the fact, is the existence of atomic facts.

2.01 An atomic fact is a combination of objects (entities, things).

You state: (V→T), (¬V→H), (¬T→¬H), Therefore: V

Based on the previous paragraph, I think you mean: (V→T), (¬V→H), (¬T→¬H), Therefore: T

Can you work the statements to get to: (V→T) & (¬V→T)

Ford claims that both Kavanaugh and Judge were stumbling drunk at the time. Judge has written a book, "Wasted", in which he claims to have been a drunk during his high school years. Kavanaugh was Judge's high school friend. Was Kavanaugh a black-out drunk in high school? If he was, he can't credibly deny the charges - he can't know that he didn't do it. If I were a senator on the Judiciary Committee, I would definitely ask Kavanaugh about his high school drinking.

If people are interested Rosen's essay is available on line here - Google books .

I'm interested in following this thread.

I thought "states of affairs" and "atomic facts" are different translations of "Sachverhalten." Or, are you saying that specifically for this discussion, you will use those terms in that way?