Mathematical purity vs real world understanding

[Hurkyl]

Things like this is what I meant, to good physicists it is obvious in which way the limits are to be taken since they know what the integral and the limit represents.

I think you are reading too much into it. It's more like flautists using the term "middle C" to refer to the pitch at 523 Hz, and momentarily confusing a piano player who is used to the term referring to 261 Hz.

Before I made my realization, I had simply thought the authors were honestly uncaring about the ordering of limits and integrals and derivatives, and thus wrote them in any order they pleased and interchanged them at whim.

intuition is the natural state for humans. A large part of mathematics courses is even built specifically to tear down as much ties you have with your intuition as possible!

It is the natural state of humans to think they know much more than they really do. One of the main points of any course is to develop and refine your intuition about the subject, and mathematics is included. Mathematics is just more dramatic because it is far more likely to deal with subject matter where the student cannot be expected to have much intuition prior to the class, or worse have genuinely wrong intuition.

AFAIK, among all subjects, mathematics has far, far more words for "something that behaves as we would intuitively expect" than any other, and spends more effort trying to find ways to refine muddled intuitive notions into something that clear, precise, and explicit.

Related to this, I fully believe that the legendary claim that nobody can understand quantum mechanics just stems from a bias that a person should already have an intuitive understanding of a subject before they have started studying it.

 

[deRham]

AFAIK, among all subjects, mathematics has far, far more words for "something that behaves as we would intuitively expect" than any other, and spends more effort trying to find ways to refine muddled intuitive notions into something that clear, precise, and explicit.

The power of this precision should also be mentioned. Why do we even bother doing that? Why not just be a physicist and learn the meaning behind things, and use mathematics to make things precise?

Two issues:

1) The meaning may not be clear. Yet something can behave in a way that parallels our intuitive understanding of something else, which has clearer meaning.

2) By making our intuition precise, we open up clear avenues for seeing the same things come up over and over again, and developing further theories that apply to other scenarios. And here, precision really is important, because it keeps us honest about what distinctions and similarities exist between the scenarios.

 

[Functor97]

I really still am unsure about what this physical intuition really is? I mean Einstein got started on his road to relativity, by conducting thought experiments, but i think theoretical physicists follow a more mathematical style, then say philosophic, not a bad thing, that is just my observation. If you look back a couple of hundred years, men were researching at the boundary of physics without knowing too much mathematics (Faraday for instance), yet physicists have become slowly more and more mathematical.
The purpose of physics is to understand the world around us, and if we are to ever reach a final understanding (which i think impossible) we would have effectively turned physics into pure mathematics, due to the nature of flawless systems. Yet i do not see this occurring ever, i see pure mathematics as the limit of physics, they will never meet. It makes me sad to wonder where our final understanding will rest. So it begs the question which side of the process is more important in understanding our universe? The answer is obviously both. So that is what i intend to study. Or at least try to.

 

[Functor97]

Am I right in stating that rigour is just the continuous flow of intuition? I mean we must start our mathematics from some axioms, and they are intuitive by definition, thus intuition matters in mathematics too! So in the end mathematicians and physicist deal with the same "thing" it is just the approach or nature of the flow of intuition that changes. A physicist will be happy with more advanced postulates, whereas a mathematician, in an attempt at continuous intuition will seak for the simplest axioms.

Just some ponderings. Feel free to tear then apart!

 

[Klockan3]

And then you deal with the issue that the intuition that your brain has from 4 billion of years of evolution will hardly be of any assistance when dealing with the quantum world, or even relativity.

Quantum and relativity are still quite intuitive, you just need to redefine your definition of a particle/time. There are still some quirks with quantum but there are quirks in classical mechanics which are strange as well.

But at the research level, mathematics involves mixing crystal clear rigor with great intuition, and that's really hard.

I know that, but at least I was discussing how things are in the coursework. You need a great deal of intuition to be a good mathematician, but it isn't taught in the courses. I am still relying almost solely on my intuition when I do maths so it is possible, it is just that most maths student don't do that.

Related to this, I fully believe that the legendary claim that nobody can understand quantum mechanics just stems from a bias that a person should already have an intuitive understanding of a subject before they have started studying it.

That claim is quite fuzzy since they don't define what it means to understand quantum mechanics. I could say that none understands classical mechanics either which is true at some level, fluid mechanics still got people stumped today.

Am I right in stating that rigour is just the continuous flow of intuition?

No, rigor when you use as little intuition as possible. Rigor is to make sure that there are no objections whatsoever to what you say, since it is made to follow rules which just about everyone can agree are true. Intuition however is very different from person to person.

Or you could say that rigor is a continuous flow of intuition since you take so small steps which anyone would find intuitive and could thus agree of the truthfulness of the whole process. You could say that rigor is the limit when the amount of intuition required for each step goes to zero. Of course mathematics aren't usually that rigorous but it is a lot closer than things like physics.

I mean we must start our mathematics from some axioms, and they are intuitive by definition, thus intuition matters in mathematics too! So in the end mathematicians and physicist deal with the same "thing" it is just the approach or nature of the flow of intuition that changes. A physicist will be happy with more advanced postulates, whereas a mathematician, in an attempt at continuous intuition will seak for the simplest axioms.

Just some ponderings. Feel free to tear then apart!

Yes, mathematics is ultimately an intuitive science as well. To get away from intuition you need to go study pure logic at the philosophy department.

 

[Functor97]

Quantum and relativity are still quite intuitive, you just need to redefine your definition of a particle/time. There are still some quirks with quantum but there are quirks in classical mechanics which are strange as well.

I know that, but at least I was discussing how things are in the coursework. You need a great deal of intuition to be a good mathematician, but it isn't taught in the courses. I am still relying almost solely on my intuition when I do maths so it is possible, it is just that most maths student don't do that.

That claim is quite fuzzy since they don't define what it means to understand quantum mechanics. I could say that none understands classical mechanics either which is true at some level, fluid mechanics still got people stumped today.

No, rigor when you use as little intuition as possible. Rigor is to make sure that there are no objections whatsoever to what you say, since it is made to follow rules which just about everyone can agree are true. Intuition however is very different from person to person.

Or you could say that rigor is a continuous flow of intuition since you take so small steps which anyone would find intuitive and could thus agree of the truthfulness of the whole process. You could say that rigor is the limit when the amount of intuition required for each step goes to zero. Of course mathematics aren't usually that rigorous but it is a lot closer than things like physics.

Yes, mathematics is ultimately an intuitive science as well. To get away from intuition you need to go study pure logic at the philosophy department.

I think all of our knowledge, in every field is based upon intuition. All pure mathematics was discovered or developed in our mind, the same mind that had evolved to hunt and survive the cold winters. It would be great to have something else to put our trust in, but we have evolved to exploit certain patterns, there maybe "truths" to reality which we will never grasp due to it being "outside" our intuition and thus understanding. I mean surely the theories of physics we have developed have been a subset of our greater potential intuition, which means we may develop it, but to an extent. I do not think that rigour alone can lead our quest for knowledge or understanding. Rigour is a tool we use to better use our intuition. I am unsure wether intuition is limited at all, it may be boundless in potential?

 

[Functor97]

I think all of our knowledge, in every field is based upon intuition. All pure mathematics was discovered or developed in our mind, the same mind that had evolved to hunt and survive the cold winters. It would be great to have something else to put our trust in, but we have evolved to exploit certain patterns, there maybe "truths" to reality which we will never grasp due to it being "outside" our intuition and thus understanding. I mean surely the theories of physics we have developed have been a subset of our greater potential intuition, which means we may develop it, but to an extent. I do not think that rigour alone can lead our quest for knowledge or understanding. Rigour is a tool we use to better use our intuition. I am unsure wether intuition is limited at all, it may be boundless in potential?

I think my claim there is wrong now. We seem to use our intuition to push for interesting mathematics. The rigour should take care of itself, as mathematics is an axiomatic system.

 

[Hurkyl]

No, rigor when you use as little intuition as possible.

I think you mean something very different when you use the word "use intuition" than what I mean. (And what I think most people mean) Rigor and intuition are not exclusive.

For example, when faced with

\lim_{x \rightarrow \infty} \frac{1 + x^2}{x^2}​

I intuitively understand that, in the numerator, x^2 is the only important term, and so

\lim_{x \rightarrow \infty} \frac{1 + x^2}{x^2} = \lim_{x \rightarrow \infty} \frac{x^2}{x^2}​

Now, if I decide to write

\lim_{x \rightarrow \infty} \frac{1 + x^2}{x^2}<br /> = \lim_{x \rightarrow \infty} \frac{x^2}{x^2} \cdot \lim_{x \rightarrow \infty} \left(1 + \frac{1}{x^2}\right)<br /> = \lim_{x \rightarrow \infty} \frac{x^2}{x^2}<br />​

(or one of the other variations on the idea) to be more rigorous, I haven't changed the fact that I'm still making the same intuitive argument. The difference is that I've written my intuition in a symbolic fashion rather than in words.

 

[Klockan3]

I think all of our knowledge, in every field is based upon intuition. All pure mathematics was discovered or developed in our mind, the same mind that had evolved to hunt and survive the cold winters. It would be great to have something else to put our trust in, but we have evolved to exploit certain patterns, there maybe "truths" to reality which we will never grasp due to it being "outside" our intuition and thus understanding. I mean surely the theories of physics we have developed have been a subset of our greater potential intuition, which means we may develop it, but to an extent. I do not think that rigour alone can lead our quest for knowledge or understanding. Rigour is a tool we use to better use our intuition. I am unsure wether intuition is limited at all, it may be boundless in potential?

I am of that opinion myself, but we aren't talking about how the fields were built but how they are taught.

I think you mean something very different when you use the word "use intuition" than what I mean. (And what I think most people mean) Rigor and intuition are not exclusive.

For example, when faced with

\lim_{x \rightarrow \infty} \frac{1 + x^2}{x^2}​

I intuitively understand that, in the numerator, x^2 is the only important term

I wouldn't call that intuition, intuition would for example be to see that the terms gets more equal the larger x gets so the limit should be 1 or some other more innovative approach. What you are talking about is utilizing rules to compactify rigor. The rules you learn isn't intuition, constructing new rules solely using old rules isn't intuition either, following rules is never intuition, intuition is when you make your own rules without having tested or been told if they work.

Physics teaches intuition since often the problems have many vague statements, there are no hard rules how to interpret them but you need to do so anyway. That is how the real world is, vague. There is intuition in maths as well, of course. It is just that the curriculum often tries to downplay it there while in physics it is usually praised.

Rigor do coexist with intuition but not in the way you describe. Intuition points the way while rigor tests the path. Without intuition you would need to brute force like a computer and without rigor you would never really know if you are correct or not. When you see that you see a limit and a quote, which directly leads to finding dominant terms. No intuition at all, that is a solution by the book.

 

[clope023]

intuition is when you make your own rules without having tested or been told if they work.

 

[Ryker]

The purpose of physics is to understand the world around us, and if we are to ever reach a final understanding (which i think impossible) we would have effectively turned physics into pure mathematics, due to the nature of flawless systems. Yet i do not see this occurring ever, i see pure mathematics as the limit of physics, they will never meet. It makes me sad to wonder where our final understanding will rest.

Didn't you scorn the whole of liberal arts just a couple of pages ago, saying you can't abide by what you've said just now?

The rules you learn isn't intuition, constructing new rules solely using old rules isn't intuition either, following rules is never intuition, intuition is when you make your own rules without having tested or been told if they work.

I don't know, to me intuition is unwittingly following internalized rules, whereas rigor is deliberately following external rules. It's hard to make a clear distinction or definition of what either is, but I think both approaches are about following rules, it's just in a different manner.

 

[Hurkyl]

I wouldn't call that intuition, intuition would for example be to see that the terms gets more equal the larger x gets so the limit should be 1 or some other more innovative approach.

I'm boggled, because you described exactly the same thing I did, just using different words. (or, at least, those words can be used to describe the same thing I described -- I can't actually know if the idea in your head is the same)

Anyways, intuition is not innovation. Google search gives a good definition:

intuition - noun - The ability to understand something immediately, without the need for conscious reasoning​

Maybe your response is because you haven't really developed a strong intuition for asymptotics -- that the idea of replacing an expression with something asymptotically equivalent is something you still have to consciously think about?

Or maybe it's just another variant on the old joke that if you really understand something, you are inclined to think it too trivial to be worth noting.

The latter is more likely -- I can't imagine someone getting very far in physics without having the notions like "first-order approximation" drilled deeply into their subconscious.

 

[Klockan3]

I'm boggled, because you described exactly the same thing I did, just using different words.

Well, the difference is that the way you described it would probably not be impossible for someone who hadn't solved limit problems before while the way I described mine would, since you described the algorithm taught. Of course all manners of solutions are "similar" in the sense that you do really take the same steps since the problem is so simple, but there is a great difference between following an algorithm or finding the path yourself.

Anyways, intuition is not innovation. Google search gives a good definition:

intuition - noun - The ability to understand something immediately, without the need for conscious reasoning​

But with that definition all acts of memorization would be called intuition. Would you call it intuition when someone solves a second order polynomial equation by using the standard formula? That description do not satisfy me, at least not when talking about scientific subjects.
Here is a more encompassing one:

Intuition is the ability to acquire knowledge without inference or the use of reason.

http://en.wikipedia.org/wiki/Intuition_(knowledge )

Maybe your response is because you haven't really developed a strong intuition for asymptotics -- that the idea of replacing an expression with something asymptotically equivalent is something you still have to consciously think about?

I saw the whole solution the instant I saw the problem, I have taught the whole calculus sequence and linear algebra so I have a quite firm grasp of elementary maths.

 

[clope023]

Well, the difference is that the way you described it would probably not be impossible for someone who hadn't solved limit problems before while the way I described mine would, since you described the algorithm taught. Of course all manners of solutions are "similar" in the sense that you do really take the same steps since the problem is so simple, but there is a great difference between following an algorithm or finding the path yourself.

You and that excluded middle you love to dig yourself into, I've called you out on it so many times and it's almost like you ignore it. I'm finding a pattern in all of your subjective reasoning and that whenever someone attempts to write down a solution via algebra you're saying they don't understand what they're doing and have only memorized a solution algorithm. However when one describes what they're doing via pictures and words than that's 'real' understanding. Did it not occur to you that he'd made the same pictoral leap that you did and simply wrote it down in symbols to describe the presentation better since that's what the symbols are for?


When you say you understand something, do you really know it or have you simply memorized the definitions and the connections between the pictures and what they mean symbolically? My guess is it's closer to the later than most with your attitude like to admit.

 

[Hurkyl]

I saw the whole solution the instant I saw the problem, I have taught the whole calculus sequence and linear algebra so I have a quite firm grasp of elementary maths.

That doesn't mean you have a strong intuition for any particular aspect. I was quite proficient at dealing with limits (enough both to solve problems and to tutor the subject), and could intuitively recognize what techniques might be useful on any given problem.

However, it took be some time I really had an intuitive notion of the "important part" of an expression, more time before I developed techniques to systematically convert my intuition into rigor, and more time before my general problem solving intuition adapted to quickly spot when doing this may be useful.

But with that definition all acts of memorization would be called intuition. Would you call it intuition when someone solves a second order polynomial equation by using the standard formula?

Maybe -- I'd have to mull it over.

But the reason for you're dissatisfied, I think, is that there is a lot more to it. A person with proficiency in solving "find the solutions to this quadratic equation" problems might still:

• Fail to recognize that this skill can be used in other problems when the issue of solving a quadratic equation comes up
• Fail to connect the solutions to the quadratic equation back to the original problem
• Fail to recognize that reducing a problem to a quadratic equation is a fruitful manipulation
• Fail to recognize quadratic equations presented in non-canonical forms
• Start using the quadratic formula to solve problems that involve quadratics but without needing to solve them
• ...

IMO, to honestly say "I have an intuitive grasp of solving quadratic equations", one really not have any of the above shortcomings.

 

[Klockan3]

You and that excluded middle you love to dig yourself into, I've called you out on it so many times and it's almost like you ignore it. I'm finding a pattern in all of your subjective reasoning and that whenever someone attempts to write down a solution via algebra you're saying they don't understand what they're doing and have only memorized a solution algorithm. However when one describes what they're doing via pictures and words than that's 'real' understanding. Did it not occur to you that he'd made the same pictoral leap that you did and simply wrote it down in symbols to describe the presentation better since that's what the symbols are for?

I don't know, I just assume of course. When it sounds like they have thought for themselves it sounds better to me than when they repeat something which could be found in a random textbook.

When you say you understand something, do you really know it or have you simply memorized the definitions and the connections between the pictures and what they mean symbolically? My guess is it's closer to the later than most with your attitude like to admit.

I'd say that I understand something when I have translated it properly to my minds natural language, ie I have made my own "picture" of it which explains everything. Doesn't have to be an actual mental picture but something which you feel naturally leads to those conclusions. I can assure you that it is not just the act of memorizing a connection between a picture and a formula, it is so much more than that. For pictures to be useful you need to be able to work with them, constructing a picture which works exactly like the mathematical concept isn't a trivial thing. But when I got my pictures I can work lightning fast with them, I can identify them anywhere and there is low risk of making errors. A good sign for that is when I for example figure out the content of the next lesson during this one.

The fact that I have done most of my exams without having done a single practice problem before it should also mean something, when I got my pictures I can do whatever they throw at me. But yeah, I do got many holes in my understanding, at several points I have been lazy and just memorized things which becomes a disaster afterwards. Another point is that I for example don't remember the strict definitions for for things like pointwise/uniform convergence, cauchy sequences, uniform continuity or equicontinuity but I can write them down by translating my pictures.

I can discuss this all day, just ask away and I will answer to the best of my ability. Sometimes during discussions I do of course assume things about the person I am talking to but if you don't do that it is hard to talk at all. Also I prefer to spark a discussion rather than getting ignored posts, going slightly over line does just that. I am visiting forums for the discussions and I won't challenge myself if I don't try to argue for something which isn't obvious or common knowledge at that forum. I believe that people aren't learning in an optimal way, I theorize on how to improve on it and parts of that is discussing with people. Getting criticism for your ideas is the best learning method ever and you get way more criticism on the net than in real life. I have gotten the idea that in general my learning have been more efficient than the learning for most else, so I figured that it could partly be because I do it so differently. I don't like to assume that others can't do what I can, they would have to convince me of it before I believe them. When I see a reasonable explanation to why it wouldn't work I will shut up, or when I am sufficiently sure that it would work I will also stop since then the discussion is over and I don't like the role as a "prophet".

That doesn't mean you have a strong intuition for any particular aspect. I was quite proficient at dealing with limits (enough both to solve problems and to tutor the subject), and could intuitively recognize what techniques might be useful on any given problem.

However, it took be some time I really had an intuitive notion of the "important part" of an expression, more time before I developed techniques to systematically convert my intuition into rigor, and more time before my general problem solving intuition adapted to quickly spot when doing this may be useful.

When I tutored I winged my classes, I have no problems solving them real time. Could mean an endless amount of memorizing, of course, but given that I had solved less relevant problems on the subject than many of my students I doubt it.

 

[Functor97]

Klockan, the problem is you are assuming the stance of the opposition in the argument, ergo you are arguing with yourself. It is frustrating to the people having a discussion, when you do not even read their points. I am not saying you always do, but you just admitted yourself you assume things about the opposition, well in this case they were non trivial.

Also, unless you have a fields medal hidden away, i would not be ready to assume that either you or what your learning style has led to, are very different from everyone else.

 

[Functor97]

Didn't you scorn the whole of liberal arts just a couple of pages ago, saying you can't abide by what you've said just now?

In the liberal arts, you can sit down at the end of the day and say, well there is no such thing as truth, so we are both right! That is what i cannot abide by.

Just because we will never reach absolute truth, does not mean we do not have the workings of it in our physics.

 

[Ryker]

In the liberal arts, you can sit down at the end of the day and say, well there is no such thing as truth, so we are both right!

This is of course ridiculous and far from the truth. Also, for your own sake, google the term "liberal arts".

The only reason I'm being so aggresive here is because you seem to have taken a condescending and elitist view towards sciences and fields you obviously don't know.

 

[Functor97]

This is of course ridiculous and far from the truth. Also, for your own sake, google the term "liberal arts".

The only reason I'm being so aggresive here is because you seem to have taken a condescending and elitist view towards sciences and fields you obviously don't know.

I was providing a hyperbole to convey my point. You brought this back up, and i explained why the failure to reach an eternal truth does not forgo approximate ones. I did not claim that the Liberal arts were worthless or inferior in general to science, i claimed that they do not study subjects of interest to me and that their attempt at discrediting science as just another postmodern theory is baseless. I did refer to them as junk, but it was the process i was speaking of, not the content. I do not see any reason for you to be offended. This is a Physics board, maybe a literature one would be more to your liking.

 

[Klockan3]

Klockan, the problem is you are assuming the stance of the opposition in the argument, ergo you are arguing with yourself. It is frustrating to the people having a discussion, when you do not even read their points. I am not saying you always do, but you just admitted yourself you assume things about the opposition, well in this case they were non trivial.

Maybe you misunderstood, they assume just as much about me as I assume about them. You can't have a discussion without assumptions, as I said. Maybe the assumption I made in this topic was a bad one, but after a few posts I almost always agree with my opponent. As I said, I do not discuss to try to sway people to my cause but to further my understanding, I have no problem with losing arguments. I do however try to argue till my opponent shows his point clearly, the best way to do that is to attack the holes which I don't see how he would fill. Then if he fills them I have learned something new. In this case I wanted to make sure that we used the same definitions, talking like he did it wasn't obvious.

Also, unless you have a fields medal hidden away, i would not be ready to assume that either you or what your learning style has led to, are very different from everyone else.

I managed to take courses for a dual masters + the necessary undergrad courses in 4 years with barely any studying outside of classes. I can bet quite a lot that you and most others who study these subjects won't manage to do the same, but that would solve all your problems if you could, right?
Why do I need to have a fields medal by the way, can't I discuss things relevant for education without one? Then, considering that it is just 4 years since I took my first college course requiring a fields medal would seem a bit excessive for anything at all considering that none have ever gotten one that quick. I probably won't ever get one, but considering that at most one per year gets one it isn't that strange.

Lastly, unless you have managed to do anything noteworthy at all I would advice you to take a softer approach with your replies. What we know of you so far is that you haven't even started with your higher education and it shows.

I was providing a hyperbole to convey my point. You brought this back up, and i explained why the failure to reach an eternal truth does not forgo approximate ones. I did not claim that the Liberal arts were worthless or inferior in general to science, i claimed that they do not study subjects of interest to me and that their attempt at discrediting science as just another postmodern theory is baseless. I did refer to them as junk, but it was the process i was speaking of, not the content. I do not see any reason for you to be offended. This is a Physics board, maybe a literature one would be more to your liking.

You are aware of the fact that mathematics and theoretical physics are parts of liberal arts? And that there are philosophy classes which are more rigorous than any mathematics class will ever be? Liberal arts is more than book reviews and political discussions.

 

[Functor97]

Maybe you misunderstood, they assume just as much about me as I assume about them. You can't have a discussion without assumptions, as I said. Maybe the assumption I made in this topic was a bad one, but after a few posts I almost always agree with my opponent. As I said, I do not discuss to try to sway people to my cause but to further my understanding, I have no problem with losing arguments. I do however try to argue till my opponent shows his point clearly, the best way to do that is to attack the holes which I don't see how he would fill. Then if he fills them I have learned something new. In this case I wanted to make sure that we used the same definitions, talking like he did it wasn't obvious.

I managed to take courses for a dual masters + the necessary undergrad courses in 4 years with barely any studying outside of classes. I can bet quite a lot that you and most others who study these subjects won't manage to do the same, but that would solve all your problems if you could, right?
Why do I need to have a fields medal by the way, can't I discuss things relevant for education without one? Then, considering that it is just 4 years since I took my first college course requiring a fields medal would seem a bit excessive for anything at all considering that none have ever gotten one that quick. I probably won't ever get one, but considering that at most one per year gets one it isn't that strange.

Lastly, unless you have managed to do anything noteworthy at all I would advice you to take a softer approach with your replies. What we know of you so far is that you haven't even started with your higher education and it shows.

You are aware of the fact that mathematics and theoretical physics are parts of liberal arts? And that there are philosophy classes which are more rigorous than any mathematics class will ever be? Liberal arts is more than book reviews and political discussions.

Klock, i do appreciate the fact that you responded to my thread, but your attitude has impacted my view upon some of your points. I understand your English is not the best, so maybe you are coming off as more arrogant than you really are.
I have looked through most of your posts and your responses seemed to be reserved for fairly trivial areas. By that i mean general topics, and fairly basic undergrad questions, there are very few graduate mathematics or physics topics you engage in. If you really are the prodigy you claim to be, i am sure many on this site would appreciate the guidance from someone so talented. Instead i see you making a goose out of yourself in many threads, mostly because you have resorted to ad hominem attacks or baseless ones.
Anyone can claim to be anything on the Internet, and i would question how experienced you are with the broader mathematical or physics community, if you are so quick to place yourself above others. If your masters courses were at a prominent research university such as Harvard, Princeton, caltech or say Cambridge in Britain, i would be more inclined to take your claims seriously. I do not deny it is possible, i simply doubt that you are all you claim to be.
Yes i am only an undergraduate, i try not to take myself too seriously, we are all human, and for the sake of good conversation it helps to act a tad modest.

 

[Klockan3]

Klock, i do appreciate the fact that you responded to my thread, but your attitude has impacted my view upon some of your points. I understand your English is not the best, so maybe you are coming off as more arrogant than you really are.

I am tired, I really shouldn't have reacted like that.

I have looked through most of your posts and your responses seemed to be reserved for fairly trivial areas. By that i mean general topics, and fairly basic undergrad questions, there are very few graduate mathematics or physics topics you engage in. If you really are the prodigy you claim to be, i am sure many on this site would appreciate the guidance from someone so talented. Instead i see you making a goose out of yourself in many threads, mostly because you have resorted to ad hominem attacks or baseless ones.

Look through my arguments with jostpuur in this thread, it was 1 year ago:
https://www.physicsforums.com/showthread.php?t=360250&page=3
And these ones were 2 years ago:
https://www.physicsforums.com/showthread.php?t=318507
https://www.physicsforums.com/showthread.php?t=340886&page=2
Not much but something. But your post I was responding to was an ad hominem as well. An ad hominem is not a bad thing in all occasions, especially when discussing things strongly related to peoples own experiences.

Anyone can claim to be anything on the Internet, and i would question how experienced you are with the broader mathematical or physics community, if you are so quick to place yourself above others.

I don't place myself above others, there is probably plenty of people out there who had it much easier than myself although I haven't met them. But I am certain that my time have been much easier than for a majority out there and that is what is important, I might not have been to Harvard but I have studied with the best students of my country. I'd like to think that others could do the same thing or at least get closer to what I do if they just changed some of their habits.

And I don't claim to have mastered many things, I have a quite rudimentary understanding of several of the courses I have taken. I have a really hard time motivating myself, can barely keep a book open for minutes most of the time. I learn most of the things in the classrooms, so to get as much lecture time as possible I took many extra courses but that gives several holes since I miss things, I often manage to repair many of them afterwards but I don't know many things as well as I would have liked. As I said before a reason for me to discuss things is to fill in holes like that, although I don't really go searching for discussions pertaining directly to those subjects, I prefer discussing things like education and learning processes.

 

[MathematicalPhysicist]

And I don't claim to have mastered many things, I have a quite rudimentary understanding of several of the courses I have taken. I have a really hard time motivating myself, can barely keep a book open for minutes most of the time. I learn most of the things in the classrooms, so to get as much lecture time as possible I took many extra courses but that gives several holes since I miss things, I often manage to repair many of them afterwards but I don't know many things as well as I would have liked. As I said before a reason for me to discuss things is to fill in holes like that, although I don't really go searching for discussions pertaining directly to those subjects, I prefer discussing things like education and learning processes.

This has a huge drawback, cause coursework are usually short in scope, so if you want to broaden your horizons, this will hold you back.

 

[lostcauses10x]

Interesting thread.
Yet it does come back to personal choice of were the original poster goes with their career.
One of the most basic questions that should be being asked is simply: Which will get you further in the world the way, and what you want from it?

Again the question of mathematics and physics is brought up.
Again I go with mathematics is unto itself a study of itself.
How mathematics fits into the study of the physical universe we live in, is still a good question. Still so far math has managed to keep ahead of the physics so it can be used in the physics world.

In math so many times discoveries are made that have been said to be only of interest to the mathematician. Of course later situations arose in the real world, that such can be used for have arisen. It would not be unreasonable to expect the same still occurring today.

Again concerning the original question this thread asks, does come to a personal choice.
As also been stated, to go with one subject, and still perusing the other is an option.
Not an easy one per say, but if a person is motivated, well it has been done.
Again personal choice and personal limitations.
The question comes down to: Were do you want to be, and what do you want??

 

[deRham]

But with that definition all acts of memorization would be called intuition. Would you call it intuition when someone solves a second order polynomial equation by using the standard formula?

Well, in a sense yes. It depends what the nature of the solving involved. As another poster stated, when presented with an unfamiliar situation, being able to reduce it to a familiar situation by remembering the formula or something would be an act of using intuition.

As you yourself said - there are assumptions to be made always. One kind of assumption is what we take for granted when doing mathematics. Just because the intuition relies somewhat more on symbol recognition doesn't mean it's not intuition - simply a different kind.

I think your problem with the limits example was that you wanted to say intuition should involve some degree of greater understanding. But I think that is clearly unnecessary - in fact, when studying advanced material, it is customary to not fully understand what is going on in a full sense, but have a good idea relative to some reference point that is not yet well understood?

At least, this is all my reading of what we're struggling with in these later parts of the thread.

 

[The_Brain]

Finally, how is it that physicists such as Edward Witten and Clifford Taubes become so talented as mathematicians? I heave heard it said that Witten for instance relies on intuition and does not often provide proofs, and that is often devisive amongst the pure mathematics community, but Taubes for instance got a phd in physics and now seems quite the pure mathematician. Someone like witten is what i aspire to be, or at least the area of research, but i have trouble telling if he is more mathematician than physicist, imo most of his results have been derived from pure mathematics, in a fashion similar to Dirac, rather then in physical ponderings in the fashion Einstein.

I was an ambitious undergraduate who took several graduate courses in physics. I didn't have quite the same dilemma as you because I knew I didn't want to go into pure mathematics. Still, I once emailed Ed Witten and asked if I should continue taking graduate level physics courses or math courses, as I was interested in string theory and mathematical physics. His response was along the lines of suggesting to take as many physics classes as possible and that I would or could learn the mathematics I didn't know along the way. This suggestion has been confirmed by a couple of other prominent high energy and string theorists since.

 

[Functor97]

I was an ambitious undergraduate who took several graduate courses in physics. I didn't have quite the same dilemma as you because I knew I didn't want to go into pure mathematics. Still, I once emailed Ed Witten and asked if I should continue taking graduate level physics courses or math courses, as I was interested in string theory and mathematical physics. His response was along the lines of suggesting to take as many physics classes as possible and that I would or could learn the mathematics I didn't know along the way. This suggestion has been confirmed by a couple of other prominent high energy and string theorists since.

That is quite interesting, if this really is true advice. I mean imagine how much you impacted modern physics by sending that email, how much more could witten have done? That is a joke of course, its good advice!

I would like to know how strong witten is at the rigor side of pure mathematics, does he have the skill to think intuitivley and rigorously? If he has not taken any graduate pure mathematics courses, then that makes his accomplishments all the more impressive.

From my point of view, i am going to wait a bit longer before getting set on one path or another. I would love to do both, but we are all human.

 

[deRham]

I what Witten meant is that if you want to understand physics and are intelligent enough, you can learn the requisite mathematical language on your own.

Mathematics courses are not to learn the formalism only - presumably, they expose you to how you can move forward with mathematics itself, but that's not of primary interest to the physicist.

Whereas for physics, I doubt you need more than the language. Which, to someone as brilliant as Witten, is child's play to pick up - so he was probably saying to learn the real content, if interested in the physics.

Whereas to do mathematics, you need much more than the language - after all, if someone can pick up your work very easily and guess half the results themselves, you're not going to get a job as a mathematician really.

 

[Hughesy]

To functor97; I am getting an impression that you are very extrinsically motivated. For example, I feel that you are too inclined to base your personal decisions on what people think and say about particular things. You are very driven to become someone great, like Edward. For me, since I do physics, the thing with such extrinsic motivations is that you lose sight of what is important in the sense that you subordinate your intimacy with what you love to superficial things like receiving recognition for what you do or fear of repugnance. I just have an opinion that these extrinsic motivations are harmful to endeavours in fields such as physics, and this sort of mentality is common as muck in areas such as sport where there is so much empahsis on being the best.

And here is the most important and simplest thing I want to point out: You do things because you love to. You do physics to not become like Einstein, but you do physics because it appeals to you. You do not need to consider its applications, its practicality, its usefulness, its purity or even its general importance. Just do it because its damn fun. You do not need to impress everyone and be the greatest physicist to have a motivation for a deep appreciation of nature. For me, the joy of physics is always enough. You are irresponsible of the prejudice and expectations of other people just as much as you are responsible for your own prejudices and expectations.