You build the machine, or you use the machine, because otherwise you are trying to be the machine
Schooling in mathematics spends a lot of time on:
(1) carrying out arithmetical or algebraic procedures that a tool like wolfram alpha can perform automatically.
-> There is no job where you will ever be required to manually carry out procedures that a computer can carry out.
(2) memorizing proofs
-> Proofs are named after the person who discovered it. That is how rarely it occurs. You won't discover new theorems or proofs by memorizing existing ones. Furthermore, you can only memorize existing proofs because you won't discover them by yourself. If it were that easy, the person who originally came up with it, would not have made it into the history books.
Neither activity is meaningful in any shape or fashion. That is, however, what mathematics education is all about.
(1) carrying out arithmetical or algebraic procedures that a tool like wolfram alpha can perform automatically.
-> There is no job where you will ever be required to manually carry out procedures that a computer can carry out.
(2) memorizing proofs
-> Proofs are named after the person who discovered it. That is how rarely it occurs. You won't discover new theorems or proofs by memorizing existing ones. Furthermore, you can only memorize existing proofs because you won't discover them by yourself. If it were that easy, the person who originally came up with it, would not have made it into the history books.
Neither activity is meaningful in any shape or fashion. That is, however, what mathematics education is all about.
Comments (50)
Trains the mind in the significance, functions and manipulation of numbers, of quantitative relationships and proportions.
Quoting Tarskian
It's not the memorizing that matters; it's the understanding of how they were derived and why they are valid.
While everyone needs arithmetic to navigate life successfully, few people need mathematics. But they won't know who they are until after they're introduced to the concepts. Therefore, every secondary school students should be given a basic education in maths and science.
Learning to formalize a real-world problem into a mathematical model is indeed more meaningful. Next, you can give it to a computation engine to solve. However, that is not mathematics. That is an activity downstream from mathematics. As soon as the problem has any real-world semantics, it is not mathematics but something else:
Quoting Mathematical formalism
There is nothing wrong with training people in using a spreadsheet or other computation engines. However, that is not an exercise in mathematics in any shape or fashion. It could be one in accounting, or engineering, or any other downstream discipline, but not one in mathematics.
In fact, you do not learn to use mathematics by studying mathematics. You must learn that elsewhere.
Quoting Vera Mont
And this is tested by the education system by asking the student to repeat the proof from memory. That does not require understanding. That merely requires rote memorization.
Quoting Vera Mont
They won't know who they are until they are confronted with a real-life problem that they need to solve and that requires them. Hence, only exposure to real-life problems will make people learn what they need. Therefore, the legitimate starting point is not the concepts or the tools. The legitimate starting point is a problem that you need to solve. no matter what. Next, you try to figure out what concepts and tools could help you doing that. It is of no use to learn concepts or tools unless you are going to actually use them. This can only be guaranteed when you start from the problem to solve and not from arbitrary tools that could possible be useful in solving some future nondescript problem.
Quoting Vera Mont
I don't think so. In practice, it merely leads to memorizing concepts that the student will never use. In my opinion, it is a waste of time. If you some day you really need it, you can learn it then and there. Employers do not value it either. If they did, youth unemployment would not be a thing.
This waste of time is not without consequences. Around the globe, hundreds of millions of young people graduate from secondary schools every year. There is no job that they will be hired for that requires anything they have learned in high school. Starbucks will hire them, but they will also happily hire someone who did not graduate from high school at all. For most of them, it would even be better to get something like a truck driving license instead of a high school diploma, which only takes a month or two, instead of six years. Still, they will much more easily find a job and make a much better living that their high-school peers.
Learning something that you may possibly some day need, has turned out to be a losing strategy. Instead, get straight into a professional environment, and try to solve the problem at hand. Only then, learn what you need to solve it.
They used to start working at around fourteen.
Back then, there was no problem of youth unemployment. Employers were perfectly fine to hire apprentices. Nowadays, it is illegal. Around the globe, child labor laws prevent teenagers from working. It proves again that there is no problem in the world that the government won't make worse.
If someone is interested in academic subjects, he will undoubtedly find his way to youtube and start viewing introductory material. If they are not interested, then why would they have to learn it?
There are not that many jobs in the modern world that are suitable for children between six and fourteen. In the past, they could help out with subsistence farming, but that option is not available for most families. But then again, even today there are still lots of jobs could be done by a fourteen-year old.
Everybody will automatically learn how to use the simplest version of the computer, which is the mobile phone. In fact, children learn by themselves how to use it, long before they graduate from primary school. That is also how most jobs will end up using computers (by using apps on a mobile phone).
That being said, I don't think math should be taught for more than a half hour a day. Advanced maths also should not be required in high school. They should be electives. There really is no point anymore in forcing the average student to learn algebra and geometry.
What about having HS students take an "intro to maths" class, where the textbook covers "consumer math" (fraction, decimals, percents, etc.) and briefly touches on more advanced concepts. After that, they're done with math requirements. If the student wants to, they can take advanced math as an elective. I would rather require philosophy classes in this day and age than math. Too many people can't think.
What strikes me, is that they still test students while imposing the following conditions:
- you cannot use a calculator
- you cannot use your books
- you cannot use google search
Students learn exactly the opposite of what makes you productive in a professional environment.
If you can find a solution by using a piece of software (such as a calculator) or by using a search engine, but you don't, you are considered to be highly unproductive. You won't be able to compete with people who are good at doing exactly the opposite.
In that sense, schools mostly teach students how to be unproductive in a professional environment. The students do not learn concepts, especially not the concept of doing whatever it reasonably takes to figure out the solution, because in fact, this is not allowed.
But then again, if the students are simply allowed to use a tool such as wolfram alpha, then it would become patently clear that they are not learning concepts at all. They are in fact not learning anything that a machine cannot do better than them. On the contrary, all they are learning is to try to be the machine, but miserably fail. That is the main reason why a high school diploma is completely worthless in the labor market. University degrees are rapidly going in the same direction, for exactly the same reasons.
If you are competing for a job with another person, and the other person isn't an HS graduate and you are, isn't that going to be a significant advantage for you?
I'm okay with that. I'm actually a huge fan of a rounded education, rather than one aimed at a 'career path' (which in my experience is a futile enterprise, often as not, because things change and keep changing.) I would be grateful if you could also squeeze in a bit of history and geography, but for pity's aske,* don't stint on sciences!
Quoting Tarskian
Oh, really? And here I thought I was being facetious. Pretty soon, with increasing automation, there won't be (m)any jobs for adults, either. The 'modern world' is a fragile and volatile thing. Why assume it will continue as it is?
Quoting Tarskian
Right up until the power grid and internet break down. After that, when there are no professional environments, it's the ones who don't rely on devices who will have to solve the real life problems.
Ed *or keyboarding skills!
Only if that is the only requirement (which it never is because a HS diploma does not reflect any ability to solve any particular problem). Ok, let's take an example and let's look at the following job:
First remark: you don't learn any of the following in high school:
- development and implementation of APIs, microservice contracts, data structures and interfaces, along with other relevant tools and technologies
- native, react native, hybrid, PWA, and/or mobile-responsive apps
- development and implementation of APIs, microservice contracts, data structures and interfaces, along with other relevant tools and technologies
You do not even learn this at university. Universities don't teach it. They would not even be able to teach it. So, a candidate can have a masters degree from MIT or Stanford or even a Ph.D, but he will still not qualify for the job. Seriously, this company won't hire that candidate. Hence, you won't be competing with any graduate from any level, because they are simply not qualified for the job.
So, how do you learn it?
Well, I always had to learn this kind of things by googling for something like "tutorial react native", and then start from there. I've had to do that a lot of times in my career.
The people who would compete with someone like me for this job, are people who have used these technologies in their previous job. They actually stand a much better chance at getting the job, but they are probably not applying for the job, because they get called by recruitment companies who offer them jobs, just like they used to call me when I was still working. So, they do not have to look for jobs. So, they are unlikely to apply.
Does this employer really care about your university degree? Probably not. Almost surely not.
I have a degree, but nobody has ever asked me about it. I am semi-retired now. So, nowadays I don't need to deal with this kind of things anymore, but I can guarantee to you that you are better off with a 3-month bootcamp in "react native" than with a Ph.D in computer science:
So, no, in my experience, having a high school degree won't make any difference. You are not competing on that basis. You really don't need one. You are competing in the area of having practical hands-on experience in a subject that no school has ever taught or will ever teach.
By the way, "react native" may be popular today, but I can guarantee that a few years from now, it will be something else.
While math knowledge isnt important for people who wont use it, and people who will use it probably still dont need the knowledge (although it is helpful to have a baseline knowledge in case common tools fail to perform a specific task), it is really for the people making the tools. While most people will not use it, the people that will make a huge impact, and progress as a species depends of some group knowing these things.
You might ask why teach everyone, then? It is true that this is inefficient, and if it was feasible to offer students only the education they need, Im sure we would.
This idea has failed because of two problems.
One: the resources required. You cant have a specialized path for every student, so you end up grouping students in large groups based on what you think they will end up doing, which we already do, to an extent.
Two: Its hard to know what a person will do, and you cant have everyone decide at an early age, when education starts. Therefore, its necessary to teach a baseline in many topics and then later allow options for specialized learning.
Current education does this fine.
Im sure there could be more efficient ways to do this, but there always are.
This is a side topic to the main post, so Ill summarize the relevant point here: the knowledge of basic math must be taught to make students, to assure that the people who would make advancements in fields that require math have basic knowledge to build upon in order to eventually make contributions.
After a few decades of making money by writing software, I can guarantee to you that the claim that "AI will replace all jobs" is complete bullshit. I really don't know why the mainstream media are pushing that stupid narrative. Maybe because some people will believe it anyway?
Actually, I usually do understand why the media push a particular bullshit narrative. There are so many, and their motivation is usually very transparent. In the case of the AI nonsense, however, I do not even see what they are trying to achieve by making people believe it !?
No, it doesn't. For example, if you want to figure out how to write a mobile app, no school will ever help you. I don't say that anybody should learn this, though. But then again, take whatever real-life example of how something works, and you will quickly understand that the school is totally ignorant of how it works. So, how can they teach it?
Quoting Igitur
Not true. You definitely can.
For example, the best way to get into software engineering, is to do a 3-month bootcamp. There is no need for 6 years like in high school. I would say that the only way to get people started in their career is a specialized path for every student. It is possible and it is being done already.
It takes two months to train a truck driver and to get the license, if that is what the student wants to do.
Quoting Igitur
Well, he still has to start somewhere. Not having any starting point at all, is not the solution either.
Quoting Igitur
Not true. Baseline generalities do not prepare for anything at all. We already know that. That is why youth unemployment is a reality nowadays.
Quoting Igitur
No, current education is pretty much a complete failure. I am surprised that any graduate finds any job at all. If it goes on like this, they will all end up slinging coffee at Starbucks. That is the true career for which they are being prepared.
Quoting Igitur
No, because Starbucks et alii do not require it. The cash till can perfectly handle all arithmetic. The cash till is a computer. Starbucks will not allow its staff to do any math. Again, either you build the machine, or else you use the machine, because in all other cases, you are trying to be the machine.
Yes.
We might still need a few doctors and architects....
Quoting Tarskian
Good. So that's where all the 34 million 14-year-olds dropouts should be heading. (Except those two dozen football players, five rappers and one stand-up comic.)
I said it was an attempt, not that it was completely successful. Also, teaching someone how to make an app wouldnt be the same as teaching them how apps work. I wasnt saying that it teaches general information about everything, thats obviously not true, but instead that it teaches a set of basics that people think are necessary.Quoting Tarskian
Im not talking about programs that teach more specialized subjects, but instead specialized paths built into public education systems. Totally agree with this though, just not that some organization could create separate paths for every student in a largeish country.Quoting Tarskian
True. I was mainly talking about public education systems, and how they usually dont have that many options to fully commit to a certain path because younger students arent trusted to make good decisions for themselves. The starting point would be higher education.Quoting Tarskian
Maybe. I was mainly saying the baseline education
was necessary for students who wish to go into jobs that have to do with them. Kind of like a way of introducing a lot of jobs that need to be done, but otherwise might not. (Like math related ones.)
Are you saying we need to shift the baseline to something more applicable for the majority of students likely future careers, or just get rid of baseline education altogether?Quoting Tarskian
This is very pessimistic. As long as there are experts who were able to make progress because of their education (talking mostly about experimental fields of science here), progressing the species (or fulfilling roles like doctors) the education system has not failed completely. While its stir that this doesnt happen to the majority of people, how else do you propose we teach the people who end up being the ones who play important roles in the success of humans as a species?
They may come up anyway, but this education system probably either helps them learn about such subjects or helps more students to explore possibilities that they might not have without education.Quoting Tarskian
Not for the people using the computer, but for the people designing new, better computers, eventually making life better for the average worker by automating more things/making current systems work more efficiently.
I can only talk from my own experience. It is just one example of how things work in practice. The labor market consists of an almost infinite myriad of different jobs, of which I only know 0.1%.
Personally, I did not learn anything in school between the ages of 14 and 18 that was later on helpful in any way. In fact, I did not learn anything at university either that turned out to be useful. I guess that 14-year-olds instinctively sense this.
Therefore, I can understand why a 14-year-old feels like dropping out. In fact, dropping out is actually not the problem. The real problem is that they need to do something else instead. Something that does make sense.
That is where the system fails, and by design so.
With all the child labor laws, it is illegal to hire a 14-year-old. So, he cannot become an apprentice with someone who can teach him something useful. Society has simply outlawed the solution. The government does not solve problems. Instead, the government pretty much always makes the problem unsolvable.
Before around 1850 (before the school system was introduced), there were no "drop outs". There were no demotivated youngsters who felt useless and lost. At the age of 12, you could join the crew of a sail ship and travel the world. Look for example at the fifth officer of the Titanic, Harold Lowe:
This system worked much better. Today, Harold Lowe would be a directionless dropout because nowadays nobody can take him on as an apprentice.
The current system is objectively worse than what they had back then.
In practice, there are no "architects". In practice, there are only users of architect software. So, either you build the machine, or else you use the machine, because in all other cases, you are trying to be the machine. The last bit, is what architects learn at university. Employers want users with years of experience with the software that they happen to use, while the people who graduate from universities are trying themselves to be the software. That is why they are unemployed when they graduate.
Concerning doctors, the entire medical industry is actually a gigantic scam. Quite a few people became more aware of that during the past scamdemic. Doctors do not learn at university what to prescribe to the patients. They learn that from the pharmaceutical mafia. It is one of the most regulated professions and therefore the profession that tops the list in terms of mafia behavior. If things go on like this, they will manage to bankrupt every modern western country -- if something else does not bankrupt them first. (There is a long list of situations that are busy bankrupting the West, such as the banking system, and so on).
S0. A nation of innumerate illiterates who can't find North on a compass just need to be trained in which buttons to push. Maybe a chimp can instruct them.
Then they can get jobs.
What more can there possibly to human life?
Someone with a degree in swimmology has read a lot of books about swimming, but he does not necessarily swim himself. That does not matter, because he can teach swimmology!
Quoting Igitur
It used to be high school. Now, they delay it to higher education. But then again, if higher education were such a fantastic starting point, then why do so many of its graduates end up slinging coffee at Starbucks? The proof is always in the pudding, isn't it?
Quoting Igitur
In fact, there are no math-related jobs. There is only math-containing software. There simply is no job where you have to manually compute math results. These students do not learn how to build such software. They also do not learn how to use such software. Instead, they learn how to fail at being themselves the software.
Quoting Igitur
That's just another scam. The government spends money on "scientific research". Next, when there is scientific progress somewhere, the government is quick to claim credit for it, and then wants some more money for "scientific research".
There is an interesting German woman, Sabine Hossenfelder, who spent decades in scientific research and who explains in excruciating detail how the scam works. One of her videos:
What's Going Wrong in Particle Physics? (This is why I lost faith in science.)
Quoting Igitur
You cannot make progress inside the system, because that will almost always be shut down. Every innovation is in one way or another a threat to existing interests. That is why all progress is made outside the system. Example: cryptocurrencies. They dangerously threaten the monopoly of the banking system and associated government power and control. That is why they could never have emerged out of the academia or even the corporate world. The powers that be would have immediately nipped it in the bud. Same for the medical industry. Don't cure a disease that currently makes the pharmaceutical mafia billions already by not curing it.
Wrong again.
Software needs to be maintained. If nobody knows how it works anymore, the software is simply dead. These people need to know how to do it manually. If you don't know how to do it manually -- excruciatingly and step by step -- you can't touch one line in the software either.
You see, the specialized knowledge is massively important to some people. However, shoving it down the throat of everyone else, is not the solution. They first need to learn how to use the software.
But then again, after using the software for years, there are always people who want to know more, and who want to know how it works and why it works. That is how I spent years developing small toy compilers and interpreters, because I wanted to know what my compiler or script engine was doing under the hood. That allowed me to read the source code of the tools I was using. That brought me to learning the computer-science math with regards to compiler construction.
You see, it is absolutely wrong to start the journey there. Just learn to use the tool first. Only if it truly piques your interest, it makes sense to look under the hood. Computer science graduates may learn some of it, but they learn it at the wrong time in their lives. They learn it at a point at which it simply does not matter to them. That is why they suck at it.
We used to be hunter-gatherers. So, don't grow food. Hunt it instead.
There is just one problem with this view. There was a reason why we started growing food instead of hunting it. This planet was able to support less than a million hunter-gatherers. We are 8 billion now.
Mutatis mutandis, our current head count of 8 billion does not allow us to get rid of the machines. If we do that, we need to get rid of billions of people too. Who volunteers to leave first? Not me.
Look forwards, not backwards.
Quoting Tarskian
I will be leaving soon enough. So will you. That is going to happen, and the population will reduce to a sustainable level. Because an unsustainable level is "unsustainable". So your fatuous argument is to exclude the proposed middle of an agrarian society because 'no reason' in favour of a complete collapse to an imagined hunt that neglects the accompanying gathering. Or else the machine paradise...
Maybe true, but it could also be that most people arent interested in the jobs that education helps with attaining, and so for the majority it is not that helpful. What would you propose?Quoting Tarskian
The students arent learning how to do math themselves, but how math works. (Which you need to know to know how you increasingly complex science and math work.) While you may need to know how to make software to make a software that does math, you also need to know how to do math, and knowing how to do math is more helpful towards making a software than the other way around.
There are also ways people can learn to make software within public education programs too, they just arent ass common (by all means, we should have more though).Quoting Tarskian
Theres scams everywhere. Just because the government wastes money on this doesnt mean we dont need scientists to actually make progress.Quoting Tarskian
Just because making progress in the system is hard doesnt mean it doesnt give students opportunities. I wasnt talking at all about innovation in the part you responded to.
Quoting Igitur
"Child labor". Like it used to be.
Harold Lowe was one of the officers on the Titanic:
And how can you pick the correct toll, if you don't know the arithmetical and algebraic procedures themselves? By at least learning to do them yourself, you understand them.
Quoting Tarskian
I disagree. The problem is that there's simply too much math to study at a slow pace. So teachers in school and in the university don't have the time to go really through how some "proof" finally got to be what is now. The pace is so quick it favours memorization and simply those who can use various algorithms quickly.
Inevitably!
Quoting Tarskian
Oh, goodie! The six people who still understand some aspect of 'manual' programming can teach it to their children, set up dynasties and rule the world. For the +/-30 years (at the rate it's progressing, probably low end - you're looking at 1-generation dynasties.) it will take AI to generate its own programs.
For all other kinds of work, paid or volunteer, zero education required. O, brave new world!
Sounds as if you are arguing for an intellectually impoverished populace, and I wonder why?
That would be the same question as how do you choose an encryption algorithm if you can't encrypt/decrypt manually?
I don't know anybody who actually can.
I can pretty much guarantee that almost nobody who uses libsodium can manually carry out any operation in xchacha20:
https://doc.libsodium.org/secret-key_cryptography/encrypted-messages
In fact, it is never seen as a requirement.
There are, of course, people like Daniel Bernstein who specialize in the knowledge of the level below libsodium but they are outnumbered 1 to 10000 by the people who just use libsodium.
Quoting ssu
You have to make choices.
But then again, you can only make those choices when faced with real-life problems to solve. Hence, it is the problem at hand that chooses what you should learn.
All other math is irrelevant in your particular context. It's too much anyway. Seriously, why even waste your time on that? In order to achieve what exactly?
No school on earth teaches you how to use libsodium. No university teaches you how xchacha20 works. They would not even be able to. Universities don't even teach you anything that is even remotely relevant in that respect. In fact, I spent most of my career -- I am semi-retired now -- picking up knowledge and using it, that no school or university ever even remotely mentions.
So yes, I used a lot of underlying math, hidden in programs and software libraries, some of which I somewhat investigated under the hood. It is totally unrelated to what universities teach. Universities are clearly not even aware of the existence of this kind of math.
Hence, if you are interested in relevant mathematics, you are wasting your time studying it at university, because in my decades-long experience, pretty much everything they do at university, is irrelevant to modern technology. These people cannot choose what to study and what not, simply because they don't use it themselves. They somehow believe that what they do, is meaningful, but it simply isn't.
They already are intellectually impoverished. They just don't know it.
In fact, university graduates are being placed in the worst situation possible. The university makes them believe that they know, but in fact, they know absolutely nothing of value.
Why would you even learn if you think that you know it all already? You can even prove that you know it all. Isn't that what your academic credential is for?
So, now you need to face potential employers. They perfectly well know that you don't know. They also know that you are convinced that you know, even though you don't. So, you still feel entitled. You think that you deserve the world, essentially for being useless and ignorant.
Instead of spending years regurgitating irrelevant trivia, do a 3-month boot camp. Employers will be more interested in hiring you.
Again wrong!
All the relevant software is free and open source. There is nothing hidden. It's all there for everyone to look at.
The question is rather: Why would you even look at it?
Well, you would need a problem to solve. That is the only legitimate reason to pick up any knowledge. Otherwise, you are just wasting your time.
So, how do you find a problem to solve?
Simple. Find someone who is even willing to pay you real money for solving the problem. That problem must be somehow real. Why else are they even willing to part with real dollars?
You will quickly discover that other people have similar problems and that they are also willing to pay real money for you to do that. It is always an entire market.
So, instead of looking at the curriculum of a university degree, instead look at what problems are mentioned in job adverts. Pick one. Figure out how much time it would take to give a meaningful response to the job adverts. It rarely takes more than a few months.
If they require credentials that take years to acquire, skip that job. They are clearly full of bullshit, and their scammy job advert is just the tip of the iceberg. You would be entering an overregulated market in which ability to solve problems takes a back seat on regulatory credentials and other nonsense. It is not a real job. Instead, you will be spending your days filling out meaningless paperwork.
You see, you don't need nine years to learn how to be a family doctor and hand out medical prescriptions to elderly patients. You'll just become a pawn in the hands of the government-orchestrated pharmaceutical mafia. If it takes years to merely join at the lowest level, it is always a scam. Don't spend your life ripping off other people and getting ripped off yourself.
Ask any world-class concert pianist if students should do their scales.
Ask a pro athlete if they do wind sprints and hit the weight room.
As one proceeds in their mathematical education, they do proofs. As they progress they learn to tackle harder problems, until they get to the point where they can prove things and discover things nobody's ever known before.
There's nothing unusual about this. Your complaints are vague and general, and little of what you say bears on actual math education.
You actually haven't studied much pure math, according to the background you described on the other forum where you posted this.
The concert pianist actually intends to solve a problem. So does the athlete.
What problem does the math graduate intend to solve except for teaching math?
Concerning my academic background in a branch of applied math, if it were still relevant after decades, it would mean that I wouldn't have done anything meaningful in the meanwhile.
If a degree matters after your first job, it simply means that your first job did not matter.
My stints in pure math came much later. Sometimes because I was looking under the hood of the software I was using. Sometimes just out of interest.
For example, I did my first foray in abstract algebra by looking under the hood of elliptic-curve cryptography. In fact, you understand abstract algebra much better if you have first been exposed to subjects like ECDSA and Shnorr signatures. The other way around is not true.
You have a credentialist view on knowledge. That is typical for teaching associates at university. They think that credentialism matters. Well, they have to, because their hourly rate clearly does not matter. The academia are full of postdocs and other idiots who think they know but who in reality have nothing to show for. Furthermore, the relevant math is elsewhere. They really do not understand, not even to save themselves from drowning, which areas in pure math power technology. That is why they are stuck in areas that are irrelevant.
Universities that focus on technology and engineering would then be the places you would refer?
Quoting Tarskian
You could generalize to a lot of what is taught and studied in universities here. Not only math.
Perhaps the real problem is that vocational education is so deprecated and doesn't drawn in the kind of people it should and doesn't go to the level it should.
My own personal interest is also pure math rather than applied math. However, it has to be somehow relevant. The subjects I end up investigating are ultimately still inspired by practical use. For example, zero-knowledge arguments. If you dig under the hood, you end up investigating the properties of Weil and Tate pairings:
The most interesting materials in the field are actually written by people like Vitalik Buterin, the founder of the Ethereum cryptocurrency. In order to implement zero-knowledge proofs in the Ethereum blockchain, he also ended up figuring out pairings:
Vitalik's articles on the subject are much better than what you could ever find at any university.
A bit like Bill Gates (Microsoft) or Steve Jobs (Apple), Vitalik had to stop wasting his time and drop out of his university undergraduate in order to do something more important:
Hence, Vitalik is not just some credentialist postdoc idiot. The Ethereum market cap is now well over $400 billion. It is exactly because he really uses elliptic curve pairings, which is pure math, that he is much better at explaining the subject than anybody in the academia.
Vitalik has absolutely no university degree whatsoever but he wipes the floor in terms of knowledge with anybody who does. Credentialists are simply born idiots. Have always been. Will always be.
The worst thing you can do for the personality and mentality of any individual, is to give him a piece of paper that says that he now knows everything better than everybody else. I spit, pee, and shit on these people.
The academia claim that they lack practical experience but that they would somehow still be good at teaching theoretical subjects such as pure math. In reality, they aren't good at that either. The truth is that they are actually good for nothing.
By enlarging the academia you simply make all the negative aspects of people working in huge groups more apparent. Fist and foremost, you cannot have a constructive interaction with thousands or tens of thousands of people. Herding, negative affects of group think, bureaucracy, everything that happens with large groups makes it's far less productive. However innovative and exciting Universities try to be, just because of the sheer numbers people they are falling down into something that they were in the Middle Ages. Perhaps it's not so bad, but anyway. Haven't been there for decades now.
Mathematicians solve problems. They solve problems in math. Like Wiles solving Fermat's last theorem, a problem pure mathematicians had been struggling to solve for 357 years.
For some reason your posts seem to dismiss or denigrate this type of work. I am perhaps not understanding your thesis, but I do feel that some of the things you say about math aren't actually true.
Quoting Tarskian
Wiles had to slog through all the proofs before him. He was already an established professional mathematician when he started his seven year quest to solve FLT. Imagine how many boring and tedious proofs he's gone through in his life.
This statement, that math grad students are there to become math teachers, is so wrong that it's the reason I say you haven't much understanding of math. That is not credentialism. That's just reading your posts, here and on the other forum. You have an ax to grind with math, and you are not articulating it very well, particularly as many of your premises are false.
Quoting Tarskian
I have no doubt that you have many meaningful accomplishments. I can only go by what you write. So far you seem to be unhappy about something, but the things you're unhappy about are strawmen. You say grad students are training to be teachers. Nothing could be further from the truth. Grad students are training to do research math.
The math department needs to teach calculus as a service course for the engineers and physicists and economists and the like. They teach the undergrad math major curriculum as the introduction to pure math, as professional mathematicians see it. It's the start of training to do research in higher math, and grad school is that on steroids.
The math teaching curriculum is actually separate. The Ph.D. track is about research. If you're a good teacher and a lousy researcher, you're out. If you're a terrible teacher and a good researcher, you have a job for life.
Surely you must have a sense of this.
Quoting Tarskian
Not relevant to my point. I'm judging your thesis by your posts. I don't care about your resume. How could I? On the Interned we only have our words and our ideas, not our credentials.
Quoting Tarskian
Perhaps you have learned a lot but still don't know everything there is to know, and perhaps you have made some wrong assumptions.
I don't understand the level of your annoyance or pique or whatever with math. Of course I agree with you that the grade school, middle school, and high school math curriculum has been an unmitigated disaster for decades. But you seem unhappy with grad-level professional training in math research. That part I don't understand.
Quoting Tarskian
That's a very interesting background. Reminds me of how functional programmers are into category theory. There are many paths to abstract math.
I will certainly agree with you that when an earnest undergrad first takes groups, rings, and fields, it's like being thrown into the deep end of the pool. It takes a while to understand why they're doing all that. I agree that perhaps they should motivate it better.
Quoting Tarskian
No not at all. I am judging the posts you've written on this forum and the other one. I don't understand your antipathy to the math establishment, or the purpose of graduate school, or whatever.
Quoting Tarskian
I don't know if that is or isn't. What does it have to do with me?
Quoting Tarskian
They're just trying to survive, like grad students everywhere.
Quoting Tarskian
Yes, grad students are cheap labor.
Quoting Tarskian
This is certainly the case. But better a lame math postdoc than a plagiarizing dean. I mean, if we are talking rot in the academy, surely the math postdocs are among the least affected. You have to get a Ph.D. in math by doing a piece of original research. Then you have to find a job in a very overcrowded and competitive job market, and now you have to start your career on the publish or perish treadmill.
These are not soft sociology degrees in grievance studies, not to get political. These are early career researchers in pure math. Why do they upset you?
I will concede that, just like ninety-nine percent of every profession, most of them in the end are mediocre. Still, they do their research, they write books, they go to faculty tea.
Why are you angry at these low-paid hard working young research mathematicians?
Quoting Tarskian
Interesting point of view. The academic mathematicians are putzes and the real work's being done in AI and quantum cryptography at Google.
Well maybe. The tech companies certainly employ mathematicians, as do the insurance companies. But for pure math, the academy is the only place they're going to pay you actual money to do the kind of work most pure math researchers do.
Quoting Tarskian
So what? They're not doing applications. If they are they are, and if they aren't, they aren't. A lot of pure math doesn't find application for decades or even centuries. Number theory was supremely useless for two thousand years till public key cryptography came about in the 1980s.
Why does this bother you?
Quoting Tarskian
Irrelevant to you, you mean. Clearly not irrelevant to their university math departments.
So what's your beef? Why is this a particular concern of yours? Is this an objection to pure math, or universities, or grad school, or what exactly?
Quoting Tarskian
Oh he's a brilliant guy, no question. But we can't all be Vitalik. Why does it bother you that some mathematician is sitting in a cramped office working on his research into pure math?
Quoting Tarskian
Yes that's true. Newton had his miracle year when the plague shut down Cambridge university in London, and he spent a year and a half on his family farm. Einstein's miracle year was when he was working as a patent clerk after being unable to get an academic position. Perelman was only able to resolve the Poincaré conjecture by refusing tenure-track appointments so he could focus on his work and not be bothered having to publish anything till he was ready.
And of course Wiles himself had a full time day job as a tenured math professor at Princeton, while he worked secretly in his attic to solve FLT.
So yes, for some people. dropping out is the better path, or just not being at the university. For most of the rest of us who aren't geniuses, there's always grad school.
I take your point. But so what? What is the nub of your irritation about this?
Sour Grapes.
Stephen Wolfram writes on this subject:
So, in order to know everything there is to know about mathematics, you need to read 3 million papers. Did I read them? Did I ever said that I read them? Did I even read 0.1% of them?
Knowledge is a gigantic database of (claim,justification) two-tuples that is for 99.999% stale and irrelevant. The only meaningful way of finding out what is relevant, is to work your way back from solutions that solve problems all the way into the math that directly or indirectly facilitates the solution.
So, is knowledge a good thing? Possibly, but it is first and foremost, utterly useless.
The idea of feeding students with some arbitrary excerpt from such knowledge database, assuming that it will ever be useful to them, is misguided and nonsensical.
That is the reason why the education system fails. Its knowledge-acquisition strategy simply does not make sense.
The only way to pick the right things to learn, is by going in exactly the opposite direction. You start by trying to solve a practical problem, for which there exists someone willing to pay for the solution, and only then you learn knowledge as required for producing the solution.
What exactly would I envy? Dealings with the HR department of a university? I have never had to go through any HR department. I find the practice insulting. It says everything about your station in life. I am semi-retired now. If I was ever going to work again, I'd rather swear fealty as a serf to the lord of the manor than to deal with an HR department.
I think so. What's funny is that I'm the one who should have sour grapes. I'm a math grad school dropout. The fault was not in my stars, but in myself.
Quoting Tarskian
Can you separate out your thoughtful critique of how post-secondary education should be done (and I agree that the current system often fails the creative and the gifted) on the one hand, with your emotional resentment of ... HR departments? Having spent my life working for corporations, I share your detestation of HR. But you seem to take it personally. You hardly ever interact with them, you fill out some forms when you get hired and then have an exit interview when you leave. Hopefully you stay out of their clutches the rest of the time. Maybe you just find administration insulting. Bureaucracy, regimentation, rules and regulations, the trappings of organizational order.
Or the very trappings of modern life. Is that you're main driver?
You are unhappy with students being taught the state of the art in their field. AND you are unhappy, viscerally so, with the administrative structure that supports organizations. Both for the same reason? You have a lot of feelings invested in all this.
Can you see that resentment of grad school pedagogy and resentment of HR department are two entirely different resentments? Frankly you come across as resentful.
Quoting Tarskian
Not really. The president of the United States draws a paycheck. He has taxes taken out, someone cuts him a check or does a direct deposit. Maybe it's the payroll department, maybe it's HR. How does the president have a low station in life because he works in an organization that follows the law with regard to employment and taxes and compensation?
I bet when the new president takes over he has to fill out a ton of mindless HR forms. Of course he has people to do it fo him. But you can't run an organization without an HR function.
According to Google, "... the HR (Human Resources) department is a group who is responsible for managing the employee life cycle (i.e., recruiting, hiring, onboarding, training, and firing employees) and administering employee benefits." In every organization someone's got to keep track of the legal employment paperwork, payroll, benefits administration.
Why on earth this upsetting to you?
Quoting Tarskian
I don't think I see the relevance of any of this. There's plenty to learn, more than anyone could master in a lifetime. Much of it is obsolete, but that's why you go to school. To have experts guide you along a productive path.
Quoting Tarskian
You need to acquire the basic skills first. That generally takes a lot of schooling. You seem to think we should educate everyone as if they were geniuses. But that wouldn't work very well, you'd leave almost everyone behind. You get a class of eight year olds. "Ok see if you can all develop quantum physics." Or even 18 year olds, if you only want to apply your "sink or swim" approach in college.
Is that what you're saying?
Quoting Tarskian
Newton mastered the work of the ancients. And people tell myths that Einstein failed math and whatnot, but he was excellent at math in school and was thoroughly steeped in the physics of Newton and Maxwell.
Quoting Tarskian
If it's arbitrary, yes. But if the knowledge is curated by leading-edge practitioners of the craft, that's exactly how you come to understand the modern state of science. That's what universities are for.
I hardly think you'd take a math undergrad and toss them a copy of Grothendieck's Éléments de géométrie algébrique, in the original French of course, and tell them to figure it all out or go screw themselves. I don't think you are making a rational proposal.
Grothendieck himself was one of your savants. From Wiki, quoting someone else talking about him: He was so completely unknown to this group and to their professors, came from such a deprived and chaotic background, and was, compared to them, so ignorant at the start of his research career, that his fulgurating ascent to sudden stardom is all the more incredible; quite unique in the history of mathematics.
But we can not all be Grothendieck. He was the greatest mathematician of the second half of the twentieth century.
Quoting Tarskian
I certainly agree that the system fails geniuses. But 99.999% of people are not geniuses. The system of training scientists and mathematicians is flawed, so you should work out your own theory and publish it. But you seem like you are not coming from, "How can I improve the system," so much as, "I have some kind of irrational and substantially unfounded resentment of the system," and that just leaps out of your own words. You don't like the professors, and you don't like the HR department. What about the cafeteria workers, are you ok with them?
Quoting Tarskian
All the 18 year olds are apprenticed out to people who will pay them even though they're completely ignorant? You can't be serious. What are you talking about?
Maybe you think all the geniuses should be issued family farms, like Newton; or be given jobs as patent clerks, like Einstein. Or live with their mother in Russia, like Perelman. But these are the one percent of the one percent of the one percent. Literally one in a million. What about everyone else?
Quoting Tarskian
So maybe you're against large organizations.
You're like Marlon Brando in the 1953 film, The Wild One. Girl asks Brando, "What are you rebelling against?" And he answers: "What have you got?"
Quoting Vera Mont
LOL.
It is not the state of the art in their field.
Quoting fishfry
Not because he wants to. There are so many people willing to pay a million dollars just for an 15-minute appointment with him, but he is not allowed to accept the money. He could easily put it up for auction. So, the paycheck is just part of a dog-and-pony show. Money does not matter to the people who print the money.
Quoting fishfry
That is how it used to work. They would become apprentices at the age of 14 and learn a job. This makes much more sense than keeping them in holding pens like cattle. There was no youth unemployment in past times.
Quoting fishfry
Not necessarily.
I have worked as a contractor and done lots of consulting work at large organizations.
I would never have wanted to be staff, though. When we talk about "bottom line", the only one that mattered to me was my own "bottom line". I was not interested in selflessly "sacrifice" myself for someone else's bottom line. I cannot identify with the profit of the company. I can only identify with my own profit. I understand that C-level execs somewhat care, since they receive payments for when the stock goes up, but the other salaried office drones? Seriously, why would anybody else care?
Come the revolution it will all be made right.
I worked in corporations and share some of your opinions about them. But why hate on the math professors? Math is so cool, why can't you let them do their thing?
I don't hate on individual math professors. They are just pawns in the game.
One (or rather two) of the things I don't like, is the combo of academic credentialism combined with the student debt scam. Like all usury, it is a tool to enslave people. The banks conjure fiat money out of thin air and them want it back along with interest from teenagers who were lied to and most of whom will never have the ability to pay back. The ruling mafia even guarantees to the bankstering mafia that they will pay in lieu of the student, if he ultimately doesn't. First of all, though, they will exhaust all options afforded by the use of violent threats of lawfare.
I bet you're not a fan of fractional reserve banking either.
I don't. I also have no problems with bank staff. They are just pawns. They are just trying to make a living.