With all due respect, no, it isn't. His drivel against set theory shows that he didn't even read the basic axiomatic set theory texts. In one of his papers, he is ranting against the axiom of infinity saying that 'there exists an infinite set' is not a precise mathematical statement. However, the axiom of infinity does not say any such thing! It precisely states the existence of some object than can be thought of as infinite but does not assign any semantics to it. Ironically, if he looked deeper, he would realize that the most interesting set theoretic proofs (independence results) are really the results in basic arithmetic (although covered in a lot of abstractions) and thus no less 'constructive' than his rational trigonometry.
Almost every critique of the axiom of infinity is philosophical. I don't think you can just say "the axiom is sound, so what's your point". And you don't even get to claim that because of Godel's incompleteness theorem.
The axioms were not handed to us from above. They were a product of a thought process anchored to intuition about the real world. The outcomes of that process can be argued about. This includes the belief that the outcomes are wrong even if we can't point to any obvious paradox.
If with an axiomatic system there are undecidable propositions, that is not the same with the axiomatic system being contradictory, i.e. where you can prove that a proposition is both true and false.
An undecidable proposition is neither true nor false, it is not both true and false.
A system with undecidable propositions may be perfectly fine, while a contradictory system is useless.
Thus what the previous poster has said has nothing to do with what Gödel had proved.
Ensuring that the system of axioms that you use is non-contradictory has remained as useful today as by the time of Euclid and basing your reasoning on clearly stated non-contradictory axioms has also remained equally important, even if we are now aware that there may be undecidable things (which are normally irrelevant in practice anyway).
The results of Gödel may be interpreted as a demonstration that the use of ternary logic is unavoidable in mathematics, like it already was in real life, where it cannot always be determined whether a claim is true or false.
Indeed. Soundness and completeness are different things.
There are two well accepted definitions of soundness. One of them is the inability to prove true == false, that is, one cannot prove a contradiction from within that axiomatic system.
> It precisely states the existence of some object than can be thought of as infinite but does not assign any semantics to it
Can you elaborate on this? I think many understand that the "existence of some object" implies there is some semantic difference even if there isn't a practical one.
I really enjoyed Wildberger's take back in high school and college. It can be far more intuitive to avoid unnecessary invocation of calculation and abstraction when possible.
I think the main thrust of his argument was that if we're going to give in to notions of infinity, irrationals, etc. it should be when they're truly needed. Most students are being given the opposite (as early as possible and with bad examples) to suit the limited time given in school. He then asks if/where we really need them at all, and has yet to be answered convincingly enough (probably only because nobody cares).
I have a couple of EB6s to show to my students but ever since scientific calculators were allowed in written tests, I have never used one myself. Law of cosines is good enough for wind triangles :). Worked for a commercial test as well as the ATP. It is a beautiful device though ...
I am not sure why you are being downvoted but you are absolutely right: it is even in the name (WD stands for 'Water Displacement'). My reaction to this article was a huge: 'why?'. WD-40 is at best mediocre at everything it is used for. Wurth makes much more capable compounds for the came purposes. Their penetrating oil is unmatched. I guess as part of the popular culture, WD-40 has its value but I am not sure its chemical properties are all that unique.
>Also, possessive pronouns are exactly like in English, concording in gender with the owner, not the object.
This is only true in third person singular. For example, in first person singular: 'моя чашка' (my cup, 'cup' is feminine) vs. 'мой ключ' (my key, 'key' is masculine). Third person plural: 'ихнее дело' (their business, neuter) vs. 'ихняя забота' (their concern, feminine) although most educated Russian speakers would object to these pronouns as a bit too colloquial (although not as colloquial as 'евойная'). Same in second person singular: 'твой друг' (your friend, masculine) vs. 'твоя подруга' (your friend, feminine). In all of the examples above, the gender of the speaker/owner cannot even be determined (grammatically speaking).
This is a fine opinion, here is another one. Calling everyone who uses LaTeX cultists is a bit insulting. I do not particularly like LaTeX and use it only when a journal requires it in which case it is pretty much painless as I simply follow the provided template. TeX, on the other hand pretty much satisfies all my typesetting needs and I use it daily. I wish everyone would use TeX or LaTeX but I would not force it on everyone. Typst seems like a fine if a bit immature system but I truly hope it goes away, the sooner the better. Not because it is bad at what it does but because it solves a problem that does not exist and distracts from a standard that may not be perfect but has proved itself again and again. I am also weary of any piece of software that is controlled by a private entity, no matter how good the initial intentions are. Being open source is not really a guarantee of anything.
I did not call everyone that uses LaTeX a cultist, I called everyone that is a fan of it a cultist.
> because it solves a problem that does not exist
No, the problems with LaTeX are real, objective, and undeniable.
> I am also weary of any piece of software that is controlled by a private entity, no matter how good the initial intentions are. Being open source is not really a guarantee of anything.
That is valid, but it does not excuse the accumulation of bad technical quality in LaTeX. Living in the stone age is not an efficacious solution to the problem of Typst's governance.
With all due respect to your perspective on TeX, isn't it the whole point? There are weaknesses in TeX (like using $ for both beginning and ending the math mode) but they are quite minor. Quirkiness is not a weakness but being a standard (since 1986!) is a major strength. The fact that it does not change is a blessing, truly. Python may be a fine language with a great community of supporters and I use it because I pretty much must but programming in something that is essentially a moving target is no fun. And I would take TeX macros over Python ugly syntax any day, no offense. Not to say I am right just to point out that taste is not necessarily a reliable guide.
On the other hand here is a Long-EZ that gets 40 mpg flying at over 250 mph (so not quite 250 knots but still). The actual savings will be higher since one can fly in a straight line most of the time. It also avoids congestion and ... is simply more fun than driving. Also an auto-pilot in aircraft has been a real thing for quite some time while same in a car ... is still quite pathetic. I have eaten plenty of lunches comfortably while flying a plane while I would not be comfortable doing it while driving.
That’s impressive but it hardly sounds like a comfortable experience for passengers, and while I get the whole argument that overregulation has made light aircraft less safe, i wouldn’t get in that plane thanks.
I was wandering the same thing. Quite often the insurance sets a much higher standard than what is legal. I had a lot of fun flying a local police helicopter as a commercial helicopter pilot with the police pilot (who only had a private certificate at the time) simply because insurance required a commercial pilot to be present in the cockpit.
In my experience a more appropriate title for this book would be: 'Linear algebra done ok if this is your second time doing it'. I have seen way too many students who, after having taken a course that used this textbook, could not give an example of a linear operator (yes, I literally asked, show me an example of a linear operator in R^3) because they literally do not have the language for it (because 'matrices are bad').
While there may be genuine issues with the book, especially when used as a first text, this strikes me as evidence either that the book was only taught through a few chapters or that the students simply didn't understand anything from it (which may be the fault of the book, of course -- but it also may be the fault of the instructor, or the preparedness of the students for the course).
Chapters 5-8 are all on operators (i.e., the entire second half of the book!). One of the most common exercises in the book is "give an example of..." And chapters 7 and 8 are literally titled "Operators on Inner Product Spaces" and "Operators on Complex Vector Spaces." If you can complete the homework with a passing grade and then pass an exam covering that material, there's no way you don't know examples of operators. Possibly you forgot the definition, but a quick, one-sentence reminder of that should make it easy to list plenty of examples)
