For many students, it is not so simple to grasp the concept of an abstract vector space. They could be taking linear algebra as college freshmen, without having seen any formal algebraic structures before. Many are unfamiliar with the formal notion of a set (and certainly have not seen the actual axioms of a set before). Most linear algebra students are not actually math majors; they are typically studying engineering, computer science, or some other physical science. Examples of abstract vector spaces are most often function spaces of some form (for example, polynomials of at most a given degree). These examples are not so motivating for non-math students.
The main reason why people care about linear algebra is that it lets you solve linear systems of equations (and perform related operations, such as projections). A linear system of equations has an immediate correspondence with a matrix of coefficients, a right-hand side vector, and a solution vector. For this reason, it is very natural to first talk about matrices and vectors (they can be used to represent concretely a linear system of equations), and then introduce the concept of vector space in cases where the abstract view can be clarifying or help with understanding.
From my perspective, the "right" way to teach linear algebra depends on the mathematical maturity of the students. If they are honors math majors, they can easily handle the definition of an abstract vector space right away. If they have less mathematical maturity, the abstract viewpoint isn't helpful for them (at least not without first familiarizing themselves with the more concrete concepts). Think about it this way: we don't teach school children about natural numbers and arithmetic by first listing the Peano axioms.
I think at least in the UK the lack of "mathematical maturity" among early undergraduates is partly the result of this very coyness about mathematical concepts. Enormous time at A-Level is spent rote learning algorithms, and very little on grasping the basic concepts of mathematics, so it's hardly surprising students turn up unprepared for such simple notions as "vector space".
I don't have first hand experience of the French system, but from what I understand the approach there is more along the lines I'm thinking of, and the relative over-representation of French graduates among my more mathematical colleagues suggests it may be rather effective in practice.
Listening to audiobooks is included in the survey (but they also mention that most people who listen to audiobooks also read print books; you are likely an outlier in that regard).
Since other commenters seem to think that the passage is just the first paragraph of chapter 1 (the fact of which suggests its own meta-commentary on the content of the article), it's worth mentioning that the passage is the first seven paragraphs of chapter 1, in which there are definitely some challenging sections, particularly in the later paragraphs.
To understand the passage and what follows, it is actually important to know what "Lincoln's Inn Hall" is. And to be clear, it's not an "inn" in the standard modern usage of the term, but rather a rather a professional association for lawyers.
You can figure out from later context that it's some kind of legal institution with an associated court. The students were also allowed to look things up: "Facilitators also provided subjects with access to online resources and dictionaries and told them that they could also use their own cell phones as a resource" [0]
Yes, agreed completely. You can get a lot (but not everything) from context. The text will be clearer if you look up unfamiliar terms (which they were allowed to do). But if you gloss over Lincoln's Inn Hall as "obviously some kind of inn", you won't have a full understanding of what follows.
This was strange and disappointing thing to read. Why such a conspiratorial tone ("we are being lied to", "we have enemies", etc.)?
Since this is written by Marc Andreessen, I did a search for "crypto" and "web 3", but no results came up. Isn't that strange coming from someone who was so prominent in promoting and advocating for a crypto-based future? I guess that one didn't pan out…
"Our enemy is the ivory tower" is a bit rich coming from a billionaire who lives in one of the most exclusive California communities (and vociferously opposes any development in that community, despite loudly proselytizing to others that "it's time to build").
Looking at the "Patron Saints", after skipping the meme-posting pseudonymous Twitter accounts, I am pretty sure university professor is the most common profession. The ivory tower is the enemy indeed. And it does seem distasteful to use the names of dead scientists and intellectuals like John Von Neumann and Richard Feynman to burnish the image of this polemic.
The Patron Saints section is where the truth is laid bare because it contains Filippo Tommaso Marinetti. He was quite in the business of writing Manifestos. Check which one he co-authored in 1919.
Politically, the Manifesto calls for:
* Universal suffrage with a lowered voting age to 18 years, and voting and electoral office eligibility for all ages 25 and up;
* Proportional representation on a regional basis;
* Voting for women;
* Representation at government level of newly created national councils by economic sector;
* The abolition of the Italian Senate (at the time, the Senate, as the upper house of parliament, was by process elected by the wealthier citizens, but were in reality direct appointments by the king. It has been described as a sort of extended council of the crown);
* The formation of a national council of experts for labor, for industry, for transportation, for the public health, for communications, etc. Selections to be made of professionals or of tradesmen with legislative powers, and elected directly to a general commission with ministerial powers.
In labor and social policy, the Manifesto calls for:
* The quick enactment of a law of the state that sanctions an eight-hour workday for all workers;
* A minimum wage;
* The participation of workers' representatives in the functions of industry commissions;
* To show the same confidence in the labor unions (that prove to be technically and morally worthy) as is given to industry executives or public servants;
* Reorganization of the railways and the public transport sector;
* Revision of the draft law on invalidity insurance;
* Reduction of the retirement age from 65 to 55.
In military affairs, the Manifesto advocates:
Creation of a short-service national militia with specifically defensive responsibilities;
Armaments factories are to be nationalized;
A peaceful but competitive foreign policy.
In finance, the Manifesto advocates:
* A strong extraordinary tax on capital of a progressive nature, which takes the form of true partial expropriation of all wealth;
* The seizure of all the possessions of the religious congregations and the abolition of all the bishoprics, which constitute an enormous liability on the Nation and on the privileges of the poor;
* Revision of all contracts for military provisions;
* The revision of all military contracts and the seizure of 85 percent of the profits therein.
After Wikipedia has been caught to be influenced by the Polish, Hungarian and Croatian far right -- and you bet there are more -- I would be very slow to trust a summary of that manifest from it. It's one of those topics where a little omission here and there can make quite a difference and you don't even need to insert a lie into Wikipedia. Not that's hard, just put down an obscure book as a source, no one checks whether your quote is in it.
Your comment primed me for some wikipedia shenanigans, but when I looked up and translated the 1919 original, it’s more or less exactly the same as the Wikipedia summary.
Marc has been meeting up with neoreactionaries, this is a normal viewpoint in SV VC circles. See how his article recommends the co-author of the Fascist Manifesto.
The main reason why people care about linear algebra is that it lets you solve linear systems of equations (and perform related operations, such as projections). A linear system of equations has an immediate correspondence with a matrix of coefficients, a right-hand side vector, and a solution vector. For this reason, it is very natural to first talk about matrices and vectors (they can be used to represent concretely a linear system of equations), and then introduce the concept of vector space in cases where the abstract view can be clarifying or help with understanding.
From my perspective, the "right" way to teach linear algebra depends on the mathematical maturity of the students. If they are honors math majors, they can easily handle the definition of an abstract vector space right away. If they have less mathematical maturity, the abstract viewpoint isn't helpful for them (at least not without first familiarizing themselves with the more concrete concepts). Think about it this way: we don't teach school children about natural numbers and arithmetic by first listing the Peano axioms.