Mathematical purity vs real world understanding

[nlsherrill]

Thanks for the advice deRham. I have some time to make up my mind, but i may be leaning towards physics at this stage. Is it just me though, or do the most intelligent/talented people tend to choose pure mathematics over physics, maybe that is just my incorrect observation, but can anyone point to why?

I guess there is a bit of ego involved too. Many mathematicians often look down on physicists in the way physicists look down on chemists. I do not personally do this, but i am worried that if i take a physics phd, i will be deemed subpar compared to the purists...

For example xkcd:

Yes, its just you.

Just kidding. What about Einstein, Tesla, Hawking, Susskind? You may perceive this because many of the child prodigies that people are aware of tend to go toward physics and mathematics because of the abstractness and because they simply accel in these subjects compared to others(because they are more difficult than say, sociology). People usually tend to go for what they are more inclined to do, which for most humans isn't math or physics because our brain did not evolve to think like you have to think with these subjects.

Also, like greenlaser said, people in all departments think they are the best. This is very general, but I have seen this and can verify that that mentality seems to really exist. For example, my school has a nationally ranked engineering program. So the engineering students here have a huggge chip on their shoulders and tend to flaunt that they are in engineering. However, as a physics major, the physics kids regularly talk down on the engineering students as if what they are doing is "simple" compared to physics coursework, and frequently make jokes about it too, which gets annoying. Then you have the math people. I am also a math major, and in math major courses you hear the math students saying something like "oh well the engineers think they are so smart but what they are doing is easy compared to real analysis." There really is no end to it. If you have to bash other areas of study to make yourself feel good about what you are doing, then you have a bigger problem to solve then what's in front of you on paper.

In regards to your concern whether to do physics or mathematics to understand the "true nature of reality", it seems pretty clear to me that physics is the first choice. You can learn all the branches of mathematics as well as anyone and still wouldn't have a clue about how the physical world really works if you hadn't have learned the physics. Understanding how to do the math is crucial, but what you really need to understand is what the mathematics tells you. If you are still indecisive, then just double major in physics and math and you will probably have a good idea of where you want to go graduation time.

 

[Functor97]

"To understand the universe" might mean that you want to be able to converse intelligently with people who talk about cosmology or particle physics. That wouldn't imply that you could rebuild a automobile engine or design a amplifier. It would be nice to dream that studying some particular field (like math) would be the key that unlocks everything, but unfortunately that is not the case. Studying math makes it easier to learn the parts of other subjects that involve math, but it doesn't give you an instant grasp of other fields. Plus, the mathematics necessary to understand the great questions of our time (such as how the brain works, how economies work etc) may not be invented yet.

Your best best is to trim down your ambition to a reasonable size, pick a specific interest and study it. Statistically speaking, I don't think many math graduate students understand Quantum physics and I don't think many math professors do. You might get lucky and find a math department that has people who are interested in the applications of math to theoretical physics. Otherwise, I don't think you will learn theoretical physics by being a graduate student in math.

Thanks for the advice stephen!
Many people i speak to seem to be of the opinion that the skills of the physicist are a subset of the skills of a mathematician. Rather, a mathematician would find it easier to understand physics, than the converse. That said, I have read on other sites, such as math overflow that some mathematicians do not take undergrad physics courses and find it difficult to learn say Quantum field theory. Some mathematicians put it down to the lack of rigor and apparent motivation, rather then the difficulty of perception and working.

Anyway, I am currently a math/physics undergrad, but i am only really starting. Would it be advisable to take a masters in pure mathematics, in an area such as Geometric analysis or Topology, before taking a Phd in theoretical physics? I am aware that physicists do not NEED to understand all the rigor and proofs behind the mathematics, but i would be interested to do so! I should be able to cover most of it in my undergrad degree, but i will probably miss a few of the higher end graduate courses, such as morse theory for instance, which is very applicable to physics, or so i hear.

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.

 

[kramer733]

I don't know why people make these threads. Honestly for me it can be summed up as the following:

Just do what you're interested in. Anything else, you can read about it or actually pick up a textbook. It doesn't have to be this complicated at all.

 

[Functor97]

I don't know why people make these threads. Honestly for me it can be summed up as the following:

Just do what you're interested in. Anything else, you can read about it or actually pick up a textbook. It doesn't have to be this complicated at all.

Cool so, if you don't want to read it, no one is forcing you to.

 

[Stephen Tashi]

(These thoughts are based on what I know about USA programs. I don't know about the UK or European situation.)

My advice on a masters degree (if you pursue one) is to get one that will help you get a job if you are unable or disinclined to go further in graduate school. Do something in engineering, computer programming or applied math. As to whether to get a masters degree at all, ask someone at the graduate schools you plan to attend what they think. But don't ask until you are almost ready to begin such a program - if you ask 2 years in advance, you'll come across as a kid who is just a big dreamer.

As to how geniuses can be so versatile - I haven't the faintest idea! Unless you are one, it isn't relevant to your career planning.

 

[Functor97]

(These thoughts are based on what I know about USA programs. I don't know about the UK or European situation.)

My advice on a masters degree (if you pursue one) is to get one that will help you get a job if you are unable or disinclined to go further in graduate school. Do something in engineering, computer programming or applied math. As to whether to get a masters degree at all, ask someone at the graduate schools you plan to attend what they think. But don't ask until you are almost ready to begin such a program - if you ask 2 years in advance, you'll come across as a kid who is just a big dreamer.

As to how geniuses can be so versatile - I haven't the faintest idea! Unless you are one, it isn't relevant to your career planning.

Haha, no i doubt i will end up like witten.

Do many institutions let you take relevant pure mathematics courses during the first years of a theoretical physics degree? I mean say if you want to work in Quantum gravity or String theory.

 

[yenchin]

Haha, no i doubt i will end up like witten.

Do many institutions let you take relevant pure mathematics courses during the first years of a theoretical physics degree? I mean say if you want to work in Quantum gravity or String theory.

I think in most places it is up to you on what courses you want to take. But seriously, I have learned (and have also been advised by profs) that there is no need to take so many courses in grad school! This may seem counter intuitive, but in graduate school your priority is in research, and taking too many courses means you are not getting enough time to think and do research. The exceptions are when the teaching staff is a very prominent expert in the field, or if they teach more about their understanding instead of merely from the texts.

As many have mentioned before in this thread, just pick up any book you like and learn it on your own as and when necessary along the road. If there is anything I consider the most important from my undergrad study, it is that after a 4-year-education, it has taught me enough basics to go on to self-study whatever I need to study in the future, math or physics.

 

[Klockan3]

Thanks for the advice stephen!
Many people i speak to seem to be of the opinion that the skills of the physicist are a subset of the skills of a mathematician. Rather, a mathematician would find it easier to understand physics, than the converse.

Were they perhaps people who studied maths? People who studies mostly maths gets too reliant on that rigor to be able to study physics. In physics graduate courses you often go through more maths than in a mathematics graduate course, except that you don't do it thoroughly in the physics one. Mathematicians aren't used to this, they can't keep up using only the skills they developed at the maths department.

However if you reverse this the physics student would face just as much problems. Take some proof based courses such as real analysis and abstract algebra during your physics degree and you won't run into that problem though. After you have figured out how to construct proofs the study of mathematics degenerate to memorizing axioms and derivations of theorems. That was an overstatement but still, maths isn't the divine science some wants you to believe it to be. I have studied both pure maths and theoretical physics at a graduate level and I can say that in general the maths classes are much easier to follow and requires less work, at least if you want to understand what you are doing which I assume applies to you.

 

[nlsherrill]

Thanks for the advice stephen!
Many people i speak to seem to be of the opinion that the skills of the physicist are a subset of the skills of a mathematician. Rather, a mathematician would find it easier to understand physics, than the converse. That said, I have read on other sites, such as math overflow that some mathematicians do not take undergrad physics courses and find it difficult to learn say Quantum field theory. Some mathematicians put it down to the lack of rigor and apparent motivation, rather then the difficulty of perception and working.

You really haven't taken much physics or math, have you? They are more different than you think. Remember, a physicist is a scientist. They must use scientific equipment, such as particle accelerators, to test their theories. This isn't required of a mathematician at all. Their work is verified in a different way...except I suppose for a mathematical physicist.


Well of course they might find QFT difficult if they hadn't taken undergraduate physics. Its extremely difficult as is for most physics students even with the undergrad. training.

Anyway, I am currently a math/physics undergrad, but i am only really starting. Would it be advisable to take a masters in pure mathematics, in an area such as Geometric analysis or Topology, before taking a Phd in theoretical physics? I am aware that physicists do not NEED to understand all the rigor and proofs behind the mathematics, but i would be interested to do so! I should be able to cover most of it in my undergrad degree, but i will probably miss a few of the higher end graduate courses, such as morse theory for instance, which is very applicable to physics, or so i hear.

Most people here would say no, its not advisable, its advisable to get your masters in physics before a PhD in physics. I think you are clinging to the idea that being better at math will automatically make you better at physics. This isn't really true.

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.

Maybe because they are geniuses? I am familiar with Witten, and I would say he clearly seems to be highly inclined for either field. Hes just really, really smart. If you aspire to be on his level, then you need to get off the forums and study every bit of physics and math you can, because he's at the very top of the mountain as far as theorists go.

Were they perhaps people who studied maths? People who studies mostly maths gets too reliant on that rigor to be able to study physics. In physics graduate courses you often go through more maths than in a mathematics graduate course, except that you don't do it thoroughly in the physics one. Mathematicians aren't used to this, they can't keep up using only the skills they developed at the maths department.

However if you reverse this the physics student would face just as much problems. Take some proof based courses such as real analysis and abstract algebra during your physics degree and you won't run into that problem though. After you have figured out how to construct proofs the study of mathematics degenerate to memorizing axioms and derivations of theorems. That was an overstatement but still, maths isn't the divine science some wants you to believe it to be. I have studied both pure maths and theoretical physics at a graduate level and I can say that in general the maths classes are much easier to follow and requires less work, at least if you want to understand what you are doing which I assume applies to you.

Oddly enough, I have heard this as well. I know a mathematics graduate student who is specializing in algebra, and I asked him "wow those courses must be extremely difficult, right?" While he said they were indeed dense subjects, he said that as long as you tried, even if you did poorly on some tests, you would get a B and that is was very very rare to see someone get a C. This is probably just a special case though. On the other hand, I know many physics grad students. All of them say that classes such as the legendary "Jacksons E&M" can be a complete nightmare to get through. This could possibly be just a departmental difference, though.

 

[Hurkyl]

n physics graduate courses you often go through more maths than in a mathematics graduate course, except that you don't do it thoroughly in the physics one.

Well, yes and no. The topic of the courses are really very different and only superficially related. It's like the difference between introductory calculus courses, and a real analysis course -- although superficially the same subject, the topics of the courses are actually very different.

The calculus class is more geared to how to do calculations with derivatives, integrals, series, and approximation methods, and how to use these tools to solve problems.

The real analysis class, on the other hand, is more focused on how to build the tools of calculus rather than how to use them. Real analysis has its own collection of tools and techniques that are useful to that purpose, and can be used to construct new sets of tools in novel situations.


However, in texts where it is appropriate, it is common to see things like:

Lemma: blah blah blah
Proof: See [some other text]​

and then that lemma is subsequently used in the next argument. Sometimes you even see appendices that boil down to things like "homotopy theory in five pages!" Such a thing would:

• Give names to the objects of homotopy theory that will be used
• State how to manipulate those objects. (often in theory form)
• State a few proofs / exercises to give the flavor of the techniques used to manipulate those objects to a fruitful purpose
• Give references for further reading

People who studies mostly maths gets too reliant on that rigor to be able to study physics. I Mathematicians aren't used to this, they can't keep up using only the skills they developed at the maths department.

My last point above, I think, is the key difference. (admittedly, I'm now speaking on little information)

A physics student might be trained to see new manipulations and learn how to repeat them, and to ignore things that don't quite add up. A mathematician, however, brings a different skill-set and is trained to be creative, and is likely to run into the things that don't work more quickly and become frustrated trying to wade through the white lies to see the truth. (e.g. treating position eigenstates as if they were actually quantum states)

For a personal anecdote, I can't express how irritated I was when I finally divined that most integrals and limits that physicists write are meant in the distributional sense, which boils down to being meant to be evaluated in the reverse of the order there written.

 

[Klockan3]

As I said I have studied maths on a graduate level, i got just a thesis left for a masters in the subject(When I come around to do that one is a different topic though). I know what those maths courses looks like.

For a personal anecdote, I can't express how irritated I was when I finally divined that most integrals and limits that physicists write are meant in the distributional sense, which boils down to being meant to be evaluated in the reverse of the order there written.

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. You are doing the typical error of looking at these things as they were purely mathematics, there is a lot of information that you miss by doing that. I believe that it is easier to teach rigor to a physicist than it is to teach intuition to a mathematician, because it is hard to go back to intuition once you are bogged down in rigor while 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!

 

[nlsherrill]

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.

 

[deRham]

I believe that it is easier to teach rigor to a physicist than it is to teach intuition to a mathematician, because it is hard to go back to intuition once you are bogged down in rigor while 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!

I would be more specific and replace the term intuition with physical intuition.

Mathematics hardly is bogged in rigor. If you read, say, Terence Tao's words on this matter, the whole point is that they teach you the rigor in mathematics courses during your undergraduate degree, so that you will be ready to state and do things precisely. Nevertheless, the ultimate goal is as long as you can presumably write things down rigorously , eventually most mathematicians will believe you without your reducing everything to symbolic manipulations.

Take the example of topology - they are perfectly content using pictures as proofs sometimes. Because that's really what is going on, and in principle they all are convinced their colleagues could write down the real rigorous mathematics behind what is going on. The same holds true of a small portion of analysis, although rigor is heavy there.

You are doing the typical error of looking at these things as they were purely mathematics

That isn't an error really - it is fine to view it as mathematics. It's whatever floats your boat. Intuition can mean many things. One can develop intuition for how purely symbolic things behave. In fact, that is what is known as Algebra.

The accusations you make do not really apply to analysts much, as they tend to think in terms of metrics and relatively physical quantities anyway.

However, you are right that it's unfair to criticize the physicist for how he/she writes and communicates, just because the mathematician doesn't get what picture the physicist has in mind.




I think the whole mathematics elitism against physics is in fact a load of nonsense, so don't get me wrong. Physics is probably a harder thing to grasp at the beginning, simply because math courses are so caught up in rigor, that once you have that down, you might not have to do much more. But at the research level, mathematics involves mixing crystal clear rigor with great intuition, and that's really hard.

 

[Functor97]

Thanks for the great responses guys!

 

[ecneicS]

It seems you have a misconception about what the title of your profession means. Your title doesn't define you, it doesn't categorize you and put you in a group of people who think this way, and have these views, and have these limitations. The only true limitation on your knowledge is yourself. You can be a mathematician who cares little about the nature of reality and much about the rigor of proof. You can also be a mathematician who cares much about the nature of reality and much about the rigor of proof. My point is to not let your profession characterize the way you think.

 

[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

$$\lim_{x \rightarrow \infty} \frac{1 + x^2}{x^2} = \lim_{x \rightarrow \infty} \frac{x^2}{x^2} \cdot \lim_{x \rightarrow \infty} \left(1 + \frac{1}{x^2}\right) = \lim_{x \rightarrow \infty} \frac{x^2}{x^2}$$​

(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.