Math Elitists: A Controversial Opinion

[micromass]

That probably was a bad example I just came up with that off the top of my head. Just look at the questions on this thread https://www.physicsforums.com/showthread.php?t=312799&page=9. Those are the typical questions a scientist or a engineer doesn't need to know at that level yet unless they want to go into math.

Let me look at the first file I come across.

The first question is a definition of integrability. So you say that people don't need to know the definition of an integrable function?? Really?

Questions (2) is calculating integrals without the fundamental theorem of calculus. Be sure that every engineer needs to be able to do this.

Questions (3)-(7) are really easy questions on calculating integrals. I don't see why an engineer won't need to knowthis

Only the last three questions are proof questions. Which I do think everybody should be able to do if they know the theory. Understanding calculus is very different from being able to make stupid calculations...

 

[micromass]

Well world news and other statistical organizations think so there is a reason why so many foreign students come here to study. In our colleges including MIT and Harvard the classes are separate because both address totally different needs. http://www.usnews.com/education/worlds-best-universities-rankings/top-400-universities-in-the-world If you look most of the top schools are American or British not a single Belgium school is on the list. We do though have a crappy high school system which is why a lot of our students are foreign and not domestic.

OK, that only shows that SOME colleges are extremely good. That doesn't mean that the entire college system is good.
Secondly, I would like to see the criteria on how they ranked these schools.

I really wouldn't know why the US colleges are so much better than European colleges. Please tell me what is so much better about US schools??

 

[xdrgnh]

Those are pretty standard questions actually but check the PDF questions Mathwong posted and you'll see what I am talking about.

 

[micromass]

Those are pretty standard questions actually but check the PDF questions Mathwong posted and you'll see what I am talking about.

These are the questions by Mathwonk... I find them pretty standard. Can you give me an exact question that you're complaining about?

 

[micromass]

OK, I looked at how the rankings of the universities was obtained. Look at this site:

http://www.usnews.com/education/wor.../21/worlds-best-universities-the-methodology-

Well, to be honest, the 6 criteria they used isn't really indicative on the teaching quality of the university. I mean, what does "citations per faculty member" has to do with how good the university is at teaching??

This survey lists the most WELL-KNOWN or FAMOUS universities. It doesn't necessarily list the BEST universities for students...

 

[Bourbaki1123]

Mathematics is formal logic,

As someone with a reasonable amount of knowledge in the major areas and history of mathematical logic, I can tell you that this is incorrect. It is neither historically true nor presently true.

Formal logic was developed tangentially, and although mathematics can be described by logical systems, it is not the be all end all of mathematics.

therefore to assert something you must prove it.

Generally correct, but in the real world there is a place in mathematics for conjecture, and even proofs based on the assumption that something is false (such as P = NP).

In addition to this, mathematics cannot be captured by a single formal logical system unless perhaps you allow for certain as-yet-to-be-fully-specified fuzziness (like with the human mind, which appears to be somewhat statistical in nature).

Also, many mathematical ideas and intuitions precede proof; proof simply helps to ensure that the intuition is consistent by tying it down to things that are simple, "obvious" and well-tested (often first order predicate calculus, but other logics can and have been used, and mathematical results have successfully been embedded in them).

If you look at the reverse mathematics programme (which aims to embed certain areas of pre-set theoretic mathematics in the simplest subsystems of second order arithmetic possible), it becomes exceedingly clear that logic provides a rich tool set for exploring mathematical ideas rather than providing a magical Platonic foundation for all of mathematics. There is no such foundation; at least, there is no such foundation that we know of.

 

[xdrgnh]

OK, that only shows that SOME colleges are extremely good. That doesn't mean that the entire college system is good.
Secondly, I would like to see the criteria on how they ranked these schools.

I really wouldn't know why the US colleges are so much better than European colleges. Please tell me what is so much better about US schools??

Regardless if you believe it or not, schools like Harvard and MIT are better in math and science then anything in Belgium and they separate the classes. But if I had to guess our college system is better then Europe's because we just put so much money into it.

 

[xdrgnh]

I can't see the questions because I am using a 11 year old computer on dial up right now but I imagined they would be pretty tough considering he is talking about lipschitz continuous function in calc II. Lipschitz continuous in calc II alone proves my point

 

[micromass]

I can't see the questions because I am using a 11 year old computer on dial up right now but I imagined they would be pretty tough considering he is talking about lipschitz continuous function in calc II. Lipschitz continuous in calc II alone proves my point

No it doesn't. Why wouldn't physicists and engineers have to use Lipschitz continuity??Lipschitz continuity is the easiest form of continuity out there, so the question probably won't be too hard.

For the record, I have seen engineering articles on image processing that uses Lipschitz continuity and even Holder continuity. I don't see why it shouldn't be useful to know...

 

[thegreenlaser]

we have the best college system in the world.

Regardless if you believe it or not, schools like Harvard and MIT are better in math and science then anything in Belgium

Wait, you started a thread because you're upset that people are being elitist about the way math should be taught?

 

[Kevin_Axion]

OK, that only shows that SOME colleges are extremely good. That doesn't mean that the entire college system is good.
Secondly, I would like to see the criteria on how they ranked these schools.

I really wouldn't know why the US colleges are so much better than European colleges. Please tell me what is so much better about US schools??

I actually read an article recently that showed that Canadian and British universities are better than American universities in general when the population and other factors are taken into account.

 

[xdrgnh]

Easiest form of continuity is that you can draw the function with a single stroke of a pen. Lipschitz continuity is something you would never see in a standard calculus class in college, probably in a honors class but not a regular one. These more abstract idea scientist and engineers need but not in a calc 1 and 2 class it's very extreme. I had to look it up to know what it meant because in all of the math and college lectures I watched I never saw it. Of coarse the easiest form isn't good enough for any college level class, there are other conditions that should be mentioned also.

 

[Hurkyl]

That probably was a bad example I just came up with that off the top of my head. Just look at the questions on this thread https://www.physicsforums.com/showthread.php?t=312799&page=9. Those are the typical questions a scientist or a engineer doesn't need to know at that level yet unless they want to go into math.

Really? Looking at the first pdf I saw, I see:

1. A question of knowing words and basic ideas
2. Some computations
3. A cookbook calculation
4. A calculation
5. A calculation that is straightforward, but tests a students ability to break a problem into parts and tackle them one at a time
6. A calculation
7. Some calculations
8. A combination of pattern recognition and an ability to apply convergence tests
9. A calculation
10. A test of a student's ability to work with a function defined via integrals
11. A test of a student's ability to work with a function defined via series
12. A calculation (but done in two different ways)
13. A test of problem solving skills (decide on your own how to check a fact, then use your method to actually check something)

Which of these are things are you proposing that an engineer or a scientist doesn't need to know?

 

[xdrgnh]

I actually read an article recently that showed that Canadian and British universities are better than American universities in general when the population and other factors are taken into account.

If you factor in America's terrible secondary education then yes you could argue that Canadians schools and British schools are better.

 

[micromass]

Easiest form of continuity is that you can draw the function with a single stroke of a pen. Lipschitz continuity is something you would never see in a standard calculus class in college, probably in a honors class but not a regular one. These more abstract idea scientist and engineers need but not in a calc 1 and 2 class it's very extreme. I had to look it up to know what it meant because in all of the math and college lectures I watched I never saw it. Of coarse the easiest form isn't good enough for any college level class, there are other conditions that should be mentioned also.

Well, mathwonk said that it IS a honours course test, so yeah.
But still, I don't see why people won't need Lipschitz continuity.

And really "you can draw the function with a single stroke of a pen". Is this really what you want the students to have as the definition of continuity? This is silly.

 

[xdrgnh]

Yes he did and he also said that his objection to AP calculus is because it doesn't substitute a honors class. The math elitists want the honors to become the standard for all students to take. I don't want students to have that definition but in truth a student could get away with that definition till they take more abstract classes. Something like Lipschitz continuity which is from my understanding that the absolute value of the slope of the secant line of a function can't exceed a certain number is very technical and isn't necessary for a class meant for scientists and engineers. This is why in America we separate the classes.

 

[Hurkyl]

Lipschitz continuity ... These more abstract idea

It is not abstract at all. It is merely something that an elementary calculus class can usually deal with in an ad-hoc manner (e.g. cases where we can simply invoke the mean value theorem).

Easiest form of continuity is that you can draw the function with a single stroke of a pen.

Now this is something that is actually quite abstract (assuming you meant it seriously and weren't just being glib).

Because you have to imagine that your pen is drawing a mathematical function. Oh, and that we're supposed to imagine the pen marking "continuously" on the paper. Oh, and by "stroke" you didn't really mean one stroke, but that many strokes can be taken as long as the pen keeps pressed against the paper. Oh, and that you're imagining a person can sketch out arbitrarily fine detail. And...

If you seriously tried to teach that as a definition, you're going to be faced with a lot of people who really don't get the difference between "continuous" and "smooth".

 

[atyy]

How about Ramanujan's formulas - were his divinations mathematics - or only his proofs?

I guess more generally, can the formulation of a conjecture be considered mathematics?

 

[xdrgnh]

It is not abstract at all. It is merely something that an elementary calculus class can usually deal with in an ad-hoc manner (e.g. cases where we can simply invoke the mean value theorem).



Now this is something that is actually quite abstract (assuming you meant it seriously and weren't just being glib).

Because you have to imagine that your pen is drawing a mathematical function. Oh, and that we're supposed to imagine the pen marking "continuously" on the paper. Oh, and by "stroke" you didn't really mean one stroke, but that many strokes can be taken as long as the pen keeps pressed against the paper. Oh, and that you're imagining a person can sketch out arbitrarily fine detail. And...

If you seriously tried to teach that as a definition, you're going to be faced with a lot of people who really don't get the difference between "continuous" and "smooth".

An ideal pen that doesn't leave any gaps, I'm not trying to be a teacher here but I'm sure you guys get the idea. If you are talking about the proper definition of a smooth function then no standard calculus class would talk about that either because it's not necessary. The purpose of a calc 1 and 2 for scientist and engineers is to show them the tools they will be using for there 1st and 2nd year classes, bringing in topic from analysis only gets in the way and makes the class harder.

 

[Fredrik]

I guess more generally, can the formulation of a conjecture be considered mathematics?

Of course. It's an essential part of the process that ends with the theorem being proved. I think that without non-rigorous arguments, mathematicians wouldn't be able to guess what statements they should try to prove. I'd be pretty surprised if the first person who proved the chain rule didn't do something like this first:

It follows immediately from the definition of the derivative that when h is small, $$f(x+h)\approx f(x)+hf'(x).$$ Let's just use this formula twice, once on g and then once on f. $$f(g(x+h))\approx f\big(g(x)+hg'(x)\big)\approx f(g(x))+hg'(x)f'(g(x))$$ This implies that $$\begin{align}(f\circ g)'(x) &\approx \frac{f(g(x+h))-f(g(x))}{h}\approx \frac{f(g(x))+hg'(x)f'(g(x))-f(g(x))}{h}\\ &\approx f'(g(x))g'(x).\end{align}$$ What's missing here is of course a proof that the error in this approximation really goes to zero when h goes to zero.

But still, I don't see why people won't need Lipschitz continuity.

I think that's actually a good example of something that only a very small number of people ever need. I only recall seeing it in an existence and uniqueness theorem for differential equations. Experimental physicists and engineers would be satisfied knowing that someone has proved the theorem. (Actually, most of them wouldn't even care about that). And a lot of people would be satisfied with a proof of a stronger version of the theorem, where the function is assumed to be continuous.

 

[micromass]

An ideal pen that doesn't leave any gaps, I'm not trying to be a teacher here but I'm sure you guys get the idea. If you are talking about the proper definition of a smooth function then no standard calculus class would talk about that either because it's not necessary. The purpose of a calc 1 and 2 for scientist and engineers is to show them the tools they will be using for there 1st and 2nd year classes, bringing in topic from analysis only gets in the way and makes the class harder.

Indeed, and the tools are easiest to learn with a good deal of theory. How can one ever understand calculus without understanding the theory?? We don't want people to become like brainless calculating monkeys do we?

 

[xdrgnh]

I guess this is the difference between the Belgium system and the American system, in the American one the classes are separate while in Belgium they are together. Again I'm not saying take out the theory I am just saying don't make the main emphasis theory for a non honors calculus class.

 

[micromass]

I think that's actually a good example of something that only a very small number of people ever need. I only recall seeing it in an existence and uniqueness theorem for differential equations. Experimental physicists and engineers would be satisfied knowing that someone has proved the theorem. (Actually, most of them wouldn't even care about that). And a lot of people would be satisfied with a proof of a stronger version of the theorem, where the function is assumed to be continuous.

OK, fine, lipgarbagez continuity isn't the most useful concept around (although it's just saying that derivatives, if they exist, are bounded...). But I still think it's useful enough to include in a calculus class. It gets you thinking about the slope and about how fast the function changes.

I think it's quite ok for a calculus class to use such a advanced definitions. That way, students learn to deal with abstract things.

 

[micromass]

I guess this is the difference between the Belgium system and the American system, in the American one the classes are separate while in Belgium they are together. Again I'm not saying take out the theory I am just saying don't make the main emphasis theory for a non honors calculus class.

You keep saying that, but you always fail to say why you think that.
Obviously, applications should get attention, but there are ways to teach both applications and theory. If it can happen here in my country, it can happen everywhere. There's no real reason to separate theory and applications, other then to dumb down the course...

 

[xdrgnh]

I'm not a educator so I don't know why to do that. All I know is that the top schools in the world Harvard, MIT Cambridge have a theoretical path and a applied path in there math and physics studies. The reason is so that not all engineering and science students need or want that kind of rigor in there math class. It works very well in America if you look at every statistic for colleges.

 

[micromass]

I'm not a educator so I don't know why to do that. All I know is that the top schools in the world Harvard, MIT Cambridge have a theoretical path and a applied path in there math and physics studies. The reason is so that not all engineering and science students need or want that kind of rigor in there math class. It works very well in America if you look at every statistic for colleges.

Well, yeah, of course you should have a theoretical path and an applied path. But that doesn't mean that we don't have to confront the applied people with proofs. And that also doesn't mean that we don't have to confront the pure people with applications. Both are very much needed.

 

[thegreenlaser]

Yes he did and he also said that his objection to AP calculus is because it doesn't substitute a honors class. The math elitists want the honors to become the standard for all students to take.

I don't understand how the second sentence at all follows from the first. Since when was AP calculus designed to be a standard course that all students take? AP courses are supposed to be for honors students, that's why they're called Advanced Placement.

 

[xdrgnh]

Well, yeah, of course you should have a theoretical path and an applied path. But that doesn't mean that we don't have to confront the applied people with proofs. And that also doesn't mean that we don't have to confront the pure people with applications. Both are very much needed.

I'm not saying don't confront them with proofs I'm saying the main emphasis shouldn't be on proofs. Do they have applied math or physics classes in Belgium, in those classes you see very little proofs and it's mostly applied work. Proofs are needed to understand the material but shouldn't be the main emphasis especially when the people in that class just want to know the math so they can build a bridge or airplane.

 

[micromass]

I'm not saying don't confront them with proofs I'm saying the main emphasis shouldn't be on proofs. Do they have applied math or physics classes in Belgium, in those classes you see very little proofs and it's mostly applied work. Proofs are needed to understand the material but shouldn't be the main emphasis especially when the people in that class just want to know the math so they can build a bridge or airplane.

Again: teaching proofs is not mutual exclusive with teaching applications.

 

[xdrgnh]

I never said that, I just said it shouldn't be the main emphasis of a non honors math class.

 

[micromass]

I never said that, I just said it shouldn't be the main emphasis of a non honors math class.

You can have multiple main emphasises...

 

[xdrgnh]

Doesn't work like that in real life, you say it works in Belgium but in America where it is different our system works better. It's not a bad way of doing things but seperating the classes has just proved to be better.

 

[micromass]

Doesn't work like that in real life, you say it works in Belgium but in America where it is different our system works better. It's not a bad way of doing things but seperating the classes has just proved to be better.

Please provide factual evidence for this claim. (and no I won't accept that site that you linked earlier, for reasons I already explained).
Please provide evidence that Belgian engineers are worse than american engineers.

 

[xdrgnh]

Worse is a immature word really to describe two engineers that went to very reputable universities. You can't really compare them because they are both trained differently. The only proof I can show you is more statistics and if you don't accept that then I don't know much else. I can say though that America has more Nobel prize winners then Belgium.

 

[micromass]

Worse is a immature word really to describe two engineers that went to very reputable universities. You can't really compare them because they are both trained differently.
The only proof I can show you is more statistics and if you don't accept that then I don't know much else.

I accept statistics. But I might not accept the criteria which decide the quality of universities. Number of publictions is not a criteria on which we can decide that one university is better, for example.
So, find me a study with decent criteria.

I can say though that America has more Nobel prize winners then Belgium.

Yeah... Do I really need to answer this? Compare the population number...