Do physics books butcher the math?

[atyy]

Firstly, many theoretical physicists such as Martin Luscher are increasingly adopting this mathematical culture (which I think can be described as Platonist, as it is in the article posted by martinbn). I suspect this will retard the development of theoretical physics, although many problems at its precipices seem fundamentally intractable from an observational/experimental standpoint anyway. Of course if this attitude bears fruit (e.g if the string theory gubbins ever get around to actually making a testable prediction which turns out to be true) I will change my mind.

Well, then your point is lost. Luscher and the others I mentioned mostly study the path integral, discretizing it and numerically studying to see if it gives results that match experiment. I thought that's what you said physicists do. Now physicists are mathematicians?

Here's Jansen's lectures:
http://www-zeuthen.desy.de/~kjansen/lattice/qcd/talks/dubna1.pdf
http://www-zeuthen.desy.de/~kjansen/lattice/qcd/talks/dubna2.pdf

Hopefully it's obvious these are physics questions by the standard you've been proposing, approached by path integral you favour.

You seem to have become interested in a relatively small point that I made previously in the thread, which is that one can formulate quantum mechanics in a "more" intuitive manner without reference to objects such as Hilbert spaces. We should probably just agree that intuitive is a subjective point of view. I can write an essay on why I think the path integral formalism is more intuitive and it would probably still not convince you. Since nobody understands quantum mechanics, of course you will find bizarre qualities such as complex time; I am making a relative, not absolute argument.

I'll take this as a concession from you.

Incidentally, I do understand quantum mechanics - an increasing number have since 1952

 

[1MileCrash]

Right. Some of my teachers didn't even use terms like n-tuple. Instead they said e.g. that a tensor is something that transforms according to the tensor transformation law, without explaining what sorts of "something" the definition can be applied to, or what they mean by "transforms". Some of them said that a vector is 3 numbers that transform according to the tensor transformation law. I could have forgiven them if they had at least defined the vector as the function that associates triples with coordinate systems, but no, they made it sound like it's a specific triple...if it "transforms" correctly.

The thread title talks about "butchering" mathematics. This sort of thing is a great example of what that means to me. I have no problem with non-rigorous arguments. But stupid definitions like this really irritate me.

Is (1,2,3) a tensor according to this "definition"? How about {1,2,3}? How about a cow? (Yes, the farm animal). Does the definition tell us that a cow isn't a tensor?

I eventually found the multilinear algebra definition of "tensor" in Schutz, and the definition of tensor field in Spivak. When I read the extremely clear presentation of tensors in Schutz, I couldn't believe that all my teachers had been peddling that painfully inadequate definition in class.

I learned the same definition initially, and had the same experience as you (though you are vastly more well-read).

I'm certainly extremely interested in whether or not the two definitions are actually even equivalent. I've heard that they are, but when I consider that the "transformation" definition leads to pseudotensors (which do not meet the transformation definition), I start to think the definition is just flat out wrong. But it's just a passing curiosity for another thread.

 

[jergens]

Mathematics is a tool. If some "mathematics" never permeates into other disciplines, especially given that mathematics is a tool, it is questionable whether or not it is mathematics. One could argue that it is a tool in the same way a square wheel is a tool, but then you're in the unflattering position of having invented a useless tool. As an example, the epsilon delta formalism for proving that a limit exists is not something people actually use in other disciplines*

A much greater portion of modern mathematics (including the stuff developed by and for mathematicians) has applications in other disciplines than you give it credit for. Just to give a few examples:

1. Donaldson and Freedman made some revolutionary advances in 4-manifold theory during the 1980s and these have found a number of applications in understanding gauge theories. More recent advances like those in Seiberg-Witten theory fall into the same category.
2. Homotopy groups of spheres and of the classical groups seem to come up occasionally in condensed matter theory publications, which means these researchers are relying on computations from algebraic topology.
3. Stochastic calculus is another relatively recent subfield of mathematics with some pretty serious applications in statistical physics. Lots of early results in this direction came from physicists, but increasingly important things like the Ito rule and the connection with random walks have come from mathematicians.
4. Algebraic topology and functional analysis (in the form of fixed-point theorems) are important in economics and finance. Stochastic calculus is also super prominent in this field.
5. Stochastic calculus underlies much of machine learning algorithms and data analysis. Algebraic topology is finding applications in this area too with things like persistent homology.

Now while the epsilon-delta definition of limits will likely appear nowhere in the textbooks on 4-manifold theory and algebraic topology and stochastic calculus, this definition underlies all of those fields. A rigorous footing for analysis was absolutely crucial for developing these results.

It is also worth mentioning how bizarre your conception of mathematics as purely a tool really is---honestly it is much like the people who slag modern math with the tagline that it has gotten away from its roots in computation (the irony there being that huge swaths of it are actually devoted to computation in some form or another). But getting back to the point this notion of yours has no real basis in the historical record since even the Greeks considered mathematical problems of intrinsic interest. This has remained unchanged through the rise of physics in the 17th century onwards as evidenced by the continued interest in solutions to polynomial equations and Fermat's Last Theorem (among other things) during that time. Now you are certainly entitled to conceive of mathematics however you wish, but you have to recognize your ideas on this issue are not widely held by anyone.

 

[WannabeNewton]

Obviously a lot of pure math is useless but so is quite a bit of physics. Right now I am doing a project involving the pole-dipole approximation of spin evolution in GR with external fields. Suffice to say it will probably not be of any use in applications that don't care about extremely precise measurements. More illustrative is the case of Marek Abramowicz and the series of papers he coauthored on a covariant formulation of inertial forces in curved spacetimes. As fun as it is to read the papers, it is safe to say that they will have no applications whatsoever. With that said, math and physics are certainly different in that most of physics can have applications in principle, regardless of whether or not it is realizable or of any interest, which is certainly not true of math by any stretch of the imagination. But that's ok. It's clear from the ridiculously abstract problems tackled by pure mathematicians of various modern fields that even the slightest of application is the farthest thing on the agenda.

In the end, I find physics infinitely more interesting than pure math, except for differential topology of course, while others are of the opposite persuasion. I wish we could just leave it at that.

 

[AnTiFreeze3]

I would tend to agree were it not for the fact that there is a disdainful and pernicious attitude presented by mathematicians towards physicists/physics students which ought to be quashed ...

And the following denotes your... friendly disposition towards mathematicians?

I don't know, I'm not one to say that we should stop funding all math departments, collect all of the the wrinkly math professors and throw them unceremoniously from the top of their ivory towers to a mob of torches and pitch forks below. However I think any academic discipline should be subjected to criticism about its relevance.

Beneath your pseudo-philosophy of mathematics done by mathematicians not being mathematics unless the mathematics is used by physicists as a tool (i.e. masonry done by masons forging bricks isn't masonry unless that brick is picked up by a construction worker who uses it for a building), I sense just a little bit of overwhelmingly distasteful solipsism and arrogance.

No one cares if you don't like mathematicians. But when you're making claims that mathematicians don't know math, or that math isn't math unless a physicist blesses it with his reverent hands and uses it for a failed theory, it becomes hard to take you seriously.

And I'm sure you're probably smarter than I am, and as of now the rigorous discussions of mathematics and physics and their entwined existence has flown beyond my comprehension. Regardless, it doesn't take a genius to see your brooding chauvinism and your distaste for anything or anyone outside your own domain. The topic was, and has been, whether physicists, in their textbooks, "butcher" and overly simplify mathematics. They do. No one ever claimed this was an inherently bad thing. As WBN points out, this allows physics texts to hone in more so on the important physical theory and intuition than on usually unnecessary logical formulations. But this fact does not nullify the importance of the existence of rigorous mathematics, nor the usefulness of understanding rigorous mathematics when approaching your oft-repeated "esoteric" physical theories.

As Razumikhin says in C&P:

Talk nonsense, but talk your own nonsense, and I'll kiss you for it. To go wrong in one's own way is better than to go right in someone else's. In the first case you are a man, in the second you're no better than a bird.

 

[ZombieFeynman]

A cursory glance at the modern theoretical physics literature would probably further help you to disabuse yourself of this absurd notion. Beware of citing articles in string theory or topological matter, for instance, since neither of these fields constitute breakthroughs yet

you ought to do some more glancing. Bernevig Hughes and Zhang decidedly predicted the quantum spin hall effect in HgTe quantum wells in their landmark 2006 science paper. The molenkamp groups subsequent observations convincingly made HgTe the first 2D Topological insulator.

Topological quantum matter definitively is a breakthrough, one guided by rather esoteric field theory along with some astoundingly creative physical insights.

 

[Char. Limit]

You know, now that I think about it... where are the applications for things like high-energy particle physics? I mean, you can't make power plants out of it, you can't design new materials from it. It's almost like people are doing it for the sake of... knowledge... you know, just like mathematics...

 

[1MileCrash]

You know, now that I think about it... where are the applications for things like high-energy particle physics? I mean, you can't make power plants out of it, you can't design new materials from it. It's almost like people are doing it for the sake of... knowledge... you know, just like mathematics...

Stop being so reasonable.

 

[AnTiFreeze3]

You know, now that I think about it... where are the applications for things like high-energy particle physics? I mean, you can't make power plants out of it, you can't design new materials from it. It's almost like people are doing it for the sake of... knowledge... you know, just like mathematics...

To be fair, though, most new fields of research are deemed too abstract to be applied to anything. It's not until a bit later that some intelligent engineers and entrepreneurs figure out how to give the research a meaningful role in society.

Take, for instance, our journeys to space. Sure, we do some relatively important experiments up at the ISS, but landing on the moon was largely done just to do it (not that it's a bad thing at all). We didn't learn anything really important except that the moon's rocks were just like the Earth's rocks.

But the technologies developed to reach the moon have had a huge impact on modern civilization. As with HEP, when we're forced to create novel machinery and techniques that greatly strain our abilities, there are often more implementations for these technologies than what their original design might indicate. With Fermi and CERN, lots of medical applications have arisen due their research, i.e. neutron therapy at FermiLab.

When, for a current events class sophomore year in HS, you choose NASA as your subject of study, you're forced to find reasons why it deserves funding, and when you have an internship with a minor role in HEP at Fermi/CERN, you tend to find ways to combat the "Well, it's cool, but why is it useful?" question

 

[Char. Limit]

To be fair, though, most new fields of research are deemed too abstract to be applied to anything. It's not until a bit later that some intelligent engineers and entrepreneurs figure out how to give the research a meaningful role in society.

Take, for instance, our journeys to space. Sure, we do some relatively important experiments up at the ISS, but landing on the moon was largely done just to do it (not that it's a bad thing at all). We didn't learn anything really important except that the moon's rocks were just like the Earth's rocks.

But the technologies developed to reach the moon have had a huge impact on modern civilization. As with HEP, when we're forced to create novel machinery and techniques that greatly strain our abilities, there are often more implementations for these technologies than what their original design might indicate. With Fermi and CERN, lots of medical applications have arisen due their research, i.e. neutron therapy at FermiLab.

When, for a current events class sophomore year in HS, you choose NASA as your subject of study, you're forced to find reasons why it deserves funding, and when you have an internship with a minor role in HEP at Fermi/CERN, you tend to find ways to combat the "Well, it's cool, but why is it useful?" question

That's a very good point, Anti, but it's actually similar to the point I was making. Just as you said that most new fields of research are deemed to be too abstract to really be applied to anything, the same could be said about new fields of mathematics when they are first introduced. While subjects like calculus and differential equations saw use in physics and other disciplines as soon as they were created, I'd be willing to bet large sums of money that many of the more modern mathematics were of the style to see no real practical use at first, but then physics "caught up", if you will. And the same could be said about many of the highest features of theoretical mathematics today.

The point I was trying to make was that it's folly to discard a discipline because "there aren't any applications for it right now", especially if you're going to embrace things like high-energy particle physics.

 

[micromass]

That's a very good point, Anti, but it's actually similar to the point I was making. Just as you said that most new fields of research are deemed to be too abstract to really be applied to anything, the same could be said about new fields of mathematics when they are first introduced. While subjects like calculus and differential equations saw use in physics and other disciplines as soon as they were created, I'd be willing to bet large sums of money that many of the more modern mathematics were of the style to see no real practical use at first, but then physics "caught up", if you will. And the same could be said about many of the highest features of theoretical mathematics today.

The point I was trying to make was that it's folly to discard a discipline because "there aren't any applications for it right now", especially if you're going to embrace things like high-energy particle physics.

Yes, there are plenty examples of things without applications in mathematics that ended up being useful. For example, functional analysis was of limited use. But then it suddenly saw applications in the fundamentals of QM and QFT, and even image processing. Number theory was long said to be completely useless (and the mathematician Hardy was extremely proud that he did such a useless thing!), but now it has found applications in cryptography. Mathematical logic was also of limited use outside of mathematics, but is now important in computer science.

One mathematicians whom I personally know did research on units in group rings. A completely useless topic. Until she got an email from an engineer designing telephone wires or something and was asked to explain some things about her research and perhaps even collaborate.

There are really myriads of example of math which eventually ends up being applied in something completely unexpected.

That said, many research in mathematics is done because it are fundamental questions which are good to know, not just because it is applicable. For example, Liouville's theorem answers a very natural question on which functions have elementary antiderivatives on which do not: http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra )
I would think this is good to know regardless of applications. It did eventually find applications in computer software which use it to evaluate many antiderivatives.

Many mathematics is also of historical interest. Things like squaring the circle are not immediately useful, but are historical questions which are good to know. Still, it ended up developing group theory (by Galois), which is now used in physics and in much more.

Differential geometry was completely useless when it was first invented. They used it to settle historical questions about the parallel axiom. Guess what? It is now used in relativity and even aerospace engineers use it.

 

[AnTiFreeze3]

That's a very good point, Anti, but it's actually similar to the point I was making. Just as you said that most new fields of research are deemed to be too abstract to really be applied to anything, the same could be said about new fields of mathematics when they are first introduced. While subjects like calculus and differential equations saw use in physics and other disciplines as soon as they were created, I'd be willing to bet large sums of money that many of the more modern mathematics were of the style to see no real practical use at first, but then physics "caught up", if you will. And the same could be said about many of the highest features of theoretical mathematics today.

The point I was trying to make was that it's folly to discard a discipline because "there aren't any applications for it right now", especially if you're going to embrace things like high-energy particle physics.

Gotcha, looks like I didn't really read your post too closely (my natural tendency is to disagree first). Glad to see that we agree on this.

 

[Arsenic&Lace]

I have not forgotten this thread, I am just currently pondering some of the mentioned applications of algebraic topology.

A few miscellaneous thoughts:
Particle physics does have useful applications in medicine, and accelerator physics has many industrial applications. Major discoveries in fundamental quantum field theory so far have no applications that I am aware of, but field theory itself is very helpful in condensed matter physics; ironically, then, theoretical physicists as mathematicians are helpful to other theoretical physicists.

Have I actually said that pure mathematics has no applications ever? If so that was a mistake and unintentional; I was rather attempting to argue that the assumptions that a). mathematical objects, ontologically speaking, can be distinguished meaningfully from their applied origins as a formal language and b). extremely high levels of rigor as an artifact of assumption a) significantly reduce the usefulness of the pure discipline. If I haven't mentioned it already, it is exceedingly clear that the mathematical culture of the 18th/19th century was incredibly productive and useful to others.

The latter point has been greeted with much derision and leads to the common analogy "math is to physics as physics is to engineering" as a counterargument, but this reaction is caused by assumption a). Physics is not merely a tool for engineering; it is a science. Mathematics is a formal language invented by humans to solve problems. The process of pure mathematics seems to be the process of studying extremely formalized arguments in this language in fastitidous detail. I would not say that they shouldn't do this, but I do not understand what the point is of inventing a set of premises and then studying highly formalized arguments which do not apparently pertain to anything. When I stated that one should look at the literature to see that this underlying machinery is useless to physicists, this is precisely what I meant; some good ideas have come from the math department, such as continuous groups, but the accompanying freight train of theorems and their meticulous proofs built in highly formal terms out of sets and stilted mathematical prose does not seem to come along for the ride.

Treating it as a science results in misleading statements such as "Physicists butcher the mathematics." No, they do not. The mathematics that they use is a formal language in which to express their ideas; if they are satisfied that these ideas have been clearly communicated then that should be the end of the story. In fact, there is a distinction between the ideas and the language used to express them. I can use an informal language to describe Newton's law "The rate of change of the rate of change of the position of an object is inversely proportional to its mass and directly proportional to the net force" or a much more efficient formal expression (a=F/m). Complaining that the precision of the language used is not high enough is irritating as it is, but even worse, mathematicians will state that the expressions do not communicate what physicists think they do. An integral to a physicist is not actually the exact same object as an integral to a mathematician, although they overlap and occasionally coincide.

The reason I dislike the notion of mathematicians as experts in mathematics is because it implies that these linguistic structures belong to the mathematicians and that the technical rules and formal arguments mathematicians concoct regarding them in lieu of application should bear some importance to people trying to use them. It is not apparent to me that the technical rules and formal arguments have any bearing whatsoever on people trying to use these constructions, and it is this aspect of the enterprise I see no evidence for having been consulted by individuals outside math departments. I don't need to open up an advanced calculus book when I'm playing with limits... ever. Mathematicians do engage in the process of abstraction though, which occasionally is useful.

An analogy would be like somebody saying "I'm feeling good today.", which makes perfect sense to most people, and having someone else respond by telling them that this statement is meaningless; according to their technical rules composed in a bubble somewhere at a university, yes, this statement is meaningless. In the real world, it isn't.

 

[Fredrik]

...misleading statements such as "Physicists butcher the mathematics." No, they do not. The mathematics that they use is a formal language in which to express their ideas; if they are satisfied that these ideas have been clearly communicated then that should be the end of the story.

No one is saying that every non-rigorous argument is "butchering" the math, or that every physics book is doing it. The objection is against those who dumb things down to the point where the ideas aren't even close to being clearly communicated.

 

[1MileCrash]

You make it sound as if mathematical rigor is some sort of arbitrary convention.

Sure, you may not need an advanced calculus book to use limits. That's the difference between using math and doing math. The people who would read an advanced calculus book are not doing it so they can use anything. The "tool" that you call mathematics wasn't handed to Moses on a stone tablet. It was developed by people, the same type of people that would read an advanced calculus book. New mathematics can never be developed if we are only concerned with "using" it.

 

[rubi]

I don't need to open up an advanced calculus book when I'm playing with limits... ever.

And why don't you need to do that? Because you have learned the rules for limits like for example $\lim a_n \cdot b_n = a \cdot b$ whenever $a_n\rightarrow a$ and $b_n\rightarrow b$. And why can you be sure that this rule always works? Because some mathematician (probably Weierstrass or Cauchy) has proved it using the $\epsilon$ definition of a limit. A rigorous proof is absolutely necessary to establish the truth of that limit rule. You have been taught this rule, because it has been proved to work, not the other way around.

 

[dkotschessaa]

Another thing that is rarely mentioned is that some math has "applications" to other math, which in turn may have "real world" applications. A lot of math is building bridges, finding connections, seeing how a problem in one place can be solved in another.

-Dave K

 

[WWGD]

As a mathematician in training, I wish I knew enough physics to be able to butcher it. Because that would imply I knew something beyond basic undergrad level.

WWGD:
What Would Gauss Do?

 

[WannabeNewton]

Honestly this is all a matter of taste. I don't see much else to it. There are some "physics for mathematicians" books that completely butcher physics, like "General Relativity for Mathematicians"-Sachs and Wu, but that's ok because the goal of the book isn't to do physics justice. Indeed often times they don't. It is to present the underlying formalism of the physical theory in a rigorous way accessible to mathematicians. The way the aforementioned book presents GR is almost comical because of its high level of rigor-that just isn't the way physics is done. But regardless I still managed to learn a lot from it because it presents ideas in a very clear, rigorous, and logical fashion that quite literally all true GR books with the exception of Wald fail to do. A more extreme example is Spivak's book on physics which is just absolutely laughably terrible. I don't even know why it was written.

In that same light, physics books don't cater to mathematicians. They try to present things in a way that is accessible to physicists and for such a purpose rigorous mathematics is absolutely useless. But that is not to say they shouldn't present mathematics clearly. Fredrik gave one example of this but there are countless in the physics literature. There's a difference between physics books not presenting math rigorously, which is fine, and presenting it in an unclear fashion, which is not fine because this has the terrible effect of making the underlying physics less coherent and more confusing to understand. Nowhere is this more pronounced than in typical QM books and in typical QFT books, the latter with regards to representation theory in particular. In this respect I almost always find myself turning to math books for clearer expositions and with good consistency find the physics much more understandable after getting a more lucid understanding of the math from proper pure math texts.

To say pure math is not useful to a physics student is just patently false unless you're of the persuasion who just takes mathematical results for granted and doesn't get frustrated by the cryptic explanations of math found in many physics books. The first chapter of Maggiore's QFT book was so frustrating for me because its development of the representation theory of the Poincare group was so unclear and laden with such poor physics terminology to the point of making me cry.

 

[atyy]

But regardless I still managed to learn a lot from it because it presents ideas in a very clear, rigorous, and logical fashion that quite literally all true GR books with the exception of Wald fail to do.

How about Hawking and Ellis, or Straumann? Or even Weinberg?

OK, maybe not Weinberg, but he's clear and logical, and has his eye on the quantum theory.

Incidentally, I would be interested to know when differential geometry books became mathematical. Did Riemann or Levi-Civita use the modern definitions, or did they use the physics "a tensor is a thing that transforms as ..." ?

A more extreme example is Spivak's book on physics which is just absolutely laughably terrible. I don't even know why it was written.

To show that it can be done

 

[atyy]

Arsenic&Lace is not qualified to defend his point, because he is a sophisticated mathematician in disguise, claiming that imaginary time and Grassmann numbers are intuitive. He is probably trying to bring a Trojan horse into the domain of physics.

Nonetheless, he is correct.

Typical condensed-matter physicist’s opinion:
“C’mon, there always is a speed of sound!”

Typical mathematical physicist’s opinion:
“Hmmm, interesting, but very difficult”

http://profs.if.uff.br/paraty07/paraty09/palestras/Eisert_2.pdf (p49, but their best slide is p57)

 

[WannabeNewton]

How about Hawking and Ellis, or Straumann? Or even Weinberg?

Hawking and Ellis is so dense I honestly can't tell if it is logical and clear or just pretentious. Straumann is unequivocally logical, clear, and rigorous: it's a brilliant physics book. I don't know why I forgot to mention it. And Weinberg is...well Weinberg. His books are about as clear as the air in China.

Did Riemann or Levi-Civita use the modern definitions, or did they use the physics "a tensor is a thing that transforms as ..." ?

I have no idea to be honest.

 

[atyy]

Hawking and Ellis is so dense I honestly can't tell if it is logical and clear or just pretentious. Straumann is unequivocally logical, clear, and rigorous: it's a brilliant physics book. I don't know why I forgot to mention it. And Weinberg is...well Weinberg. His books are about as clear as the air in China.

I've always wanted to visit.

 

[Matterwave]

Hawking and Ellis is so dense I honestly can't tell if it is logical and clear or just pretentious. Straumann is unequivocally logical, clear, and rigorous: it's a brilliant physics book. I don't know why I forgot to mention it. And Weinberg is...well Weinberg. His books are about as clear as the air in China.

Hey...hey hey... the air in Beijing and Shanghai and other major cities in China is polluted...but not through all of China! Especially not true in the forest and dessert areas.

 

[atyy]

Hey...hey hey... the air in Beijing and Shanghai and other major cities in China is polluted...but not through all of China! Especially not true in the forest and dessert areas.

I'd especially love to visit the dessert areas! http://en.wikipedia.org/wiki/Dessert

 

[Matterwave]

 

[WannabeNewton]

Hey...hey hey... the air in Beijing and Shanghai and other major cities in China is polluted...but not through all of China! Especially not true in the forest and dessert areas.

Mmm dessert.

 

[Matterwave]

Desert*...dang...

 

[micromass]

The reason I dislike the notion of mathematicians as experts in mathematics is because it implies that these linguistic structures belong to the mathematicians and that the technical rules and formal arguments mathematicians concoct regarding them in lieu of application should bear some importance to people trying to use them. It is not apparent to me that the technical rules and formal arguments have any bearing whatsoever on people trying to use these constructions, and it is this aspect of the enterprise I see no evidence for having been consulted by individuals outside math departments. I don't need to open up an advanced calculus book when I'm playing with limits... ever. Mathematicians do engage in the process of abstraction though, which occasionally is useful.

OK, then who are the experts in mathematics? Are you an expert in mathematics?

An analogy would be like somebody saying "I'm feeling good today.", which makes perfect sense to most people, and having someone else respond by telling them that this statement is meaningless; according to their technical rules composed in a bubble somewhere at a university, yes, this statement is meaningless. In the real world, it isn't.

Funny, I always thought that physicists were as bad as mathematicians in this respect. Look at some threads started in the physics section of PF, many start out by saying that their question is ill-posed.

Or see this video of a poor reporter asking what you feel when you put two magnets together.

https://www.youtube.com/watch?v=36GT2zI8lVA

 

[Arsenic&Lace]

OK, then who are the experts in mathematics? Are you an expert in mathematics?



Funny, I always thought that physicists were as bad as mathematicians in this respect. Look at some threads started in the physics section of PF, many start out by saying that their question is ill-posed.

Or see this video of a poor reporter asking what you feel when you put two magnets together.

https://www.youtube.com/watch?v=36GT2zI8lVA

Experts in differential and integral calculus can be found in math departments, engineering departments, physics departments and elsewhere. Once you've taken the first 3 semesters of calculus (and gotten A's, well maybe a B... at least if you've solved lots of elementary calc problems correctly!), you're an expert in elementary calculus. A math professor with a specialty in analysis who's taken two semesters of advanced calculus, four semesters of undergraduate and graduate real analysis, and publishes papers in I don't know PDE's or something, doesn't really know anything more than you do about calculus, because the knowledge s/he possesses cannot be meaningfully be described as "more calculus" since calculus is a tool for solving engineering/physics/other types of problems and the vast majority of what s/he uses cannot be used for this purpose and never will be.

Earlier in the thread somebody stated, essentially, that my usage of elementary calculus is contingent upon all of those very precise and rigorous proofs of theorems using epsilon's and deltas, Cauchy sequences and what not. It isn't. The rules of calculus, from the product rule to L'Hospital's rule, were figured out and used as much as a thousand years (MVT/Rolle's theorem) before these proofs were written. It is completely unclear what purpose this rigor actually serves.

Of course this thread is actually about how authors can explain things poorly or well. I've read very lucid explanations of mathematics by mathematicians and physicists, and very poor explanations of mathematics by mathematicians and physicists. I am attempting to define what butchering actually means. The best explanations of mathematics, pedagogically speaking, are among the least rigorous. If every engineering calculus student began in a bog of Dedekind cuts, set theory, and basic topology, it would do nothing for them as far as actually performing calculus except to confuse them; likewise, they thankfully spend hardly any time at all with epsilons and deltas.

I will say this though: I actually have a huge preference for learning mathematics over physics, generally speaking, and for solving mathematics problems. My favorite sections of physics textbooks are those which discuss theory; when matters turn to actually computing things, this can be somewhat interesting but is nevertheless often tedious , depending upon the problems you have to solve. I've been doing physics research in computation and theory for several years now, and there is a great deal of joy to be had before implementation (in my case through programming usually). I don't completely hate implementation and calculation and often enjoy it as well, but I have a vast preference for theory. In other words, if it had not occurred to me that pure mathematics was useless, I would have chosen to become a mathematician. I personally find a bog of Dedekind cuts, set theory, and basic topology to be a nice place to hang out.

Of course people have come up with cases where algebraic topology is not "useless"; I'm still exploring these, and if I'm convinced that they are right I might just consider trying to become a topologist.