I How do you answer "So what's the practical application....?"

[dkotschessaa]

I suppose you recognize, by title, the situation I am referring to. I don't know if physics people get it as often as math people.

The situation of course is that I tell somebody that I am studying math, and if I mention some specifics, like mention Topology or Algebra, (which I have to sort of explain is not "college algebra"), or whatever. Then comes the question "So what's this used for in..you know, real life?"

As I see it there are two extremes to answer this question:

a) A speech or possible tirade about how this question is not really relevant. Possible comparison of science to art, i.e. "Well, what's the practical application of music?" Trying, perhaps in vain to explain how mathematics doesn't always seek applications but that they often find their uses later, then tell a story about number theory and cryptography. Another variant is that for me, I've studied mathematics for the joy of it and because I think the thinking skills I learned can be applied to anything.

b) Just say some stuff I heard about what people might be using this for. "Topological data analysis!" "Cryptography" (again). "Something in physics!"

 

[.Scott]

Of course, Algebra is a critical tool in all engineering ... and it doesn't take much for a situation to involve "college algebra" versus the introductory stuff one gets in High School. So, for Algebra, it could be to assist almost any kind of engineering design team. Topology is another tool - not as often used as, say, trig. But you would expect a carpenter to have a hammer even in these days of nail guns - and know how to use it.

 

[jedishrfu]

G H Hardy wrote a book on it called A Mathematician's Apology where he discusses this very topic.

https://en.wikipedia.org/wiki/G._H._Hardy

https://en.wikipedia.org/wiki/A_Mathematician's_Apology

 

[jedishrfu]

A third option is to say we live in a world of magic and mystery where mathematics helps us make sense of the chaos and then while saying this do a cool magic trick based on some sort of mathematics principle.

Here's a book where you can find some cool math based card tricks:

https://www.amazon.com/dp/0691151644/?tag=pfamazon01-20

 

[dkotschessaa]

G H Hardy wrote a book on it called A Mathematician's Apology where he discusses this very topic.

https://en.wikipedia.org/wiki/G._H._Hardy

https://en.wikipedia.org/wiki/A_Mathematician's_Apology

I'm aware of it, but I don't think this is his target audience. :)

 

[dkotschessaa]

Of course, Algebra is a critical tool in all engineering ... and it doesn't take much for a situation to involve "college algebra" versus the introductory stuff one gets in High School. So, for Algebra, it could be to assist almost any kind of engineering design team. Topology is another tool - not as often used as, say, trig. But you would expect a carpenter to have a hammer even in these days of nail guns - and know how to use it.

Well, but when I say "algebra" i mean group theory, rings fields, Galois theory. I don't know how people use this outside of mathematics.

 

[lavinia]

Modern mathematics is widely applied in many fields, Physics included. If you are interested in those areas of Physics where its application is common then you probably would want to learn it. Modern Differential Geometry is intensely topological. Here is a quote from a physicist,

"
The beauty and profundity of the geometry of fibre bundles were to a large extent brought forth by the (early) work of Chern. I must admit, however, that the appreciation of this beauty came to physicists only in recent years.

— CN Yang, 1979 "

Chern was a pure mathematician, primarily a differential geometer. One can not understand much about fiber bundles without some knowledge of topology.

Here is some reading that may give you an idea of how mathematics is used in modern physics:

- 'A First Course in String Theory" by Zwiebach. Brian Greene recommended this book to me.

Or maybe you would like to read this review article.

https://www.maths.ox.ac.uk/groups/m...eas/calabi-yau-manifolds-and-particle-physics

- Here is a Wikipedia article on topological Quantum Field Theories.

https://en.wikipedia.org/wiki/Topological_quantum_field_theory

- If you are interested in the physics of stars you may wish to learn about Knot Theory which is applied to understanding the formation of magnetic filaments.

Munkres book is an elementary topology book. One way or the other you will need to know what is in it if you are interested in the mathematically intense areas of Physics. On the other hand you may wish to pick the math up as you go along rather than take time out. That is a matter of intellectual style.

 

[Dr. Courtney]

There is nothing wrong with inventing new tools before their specific applications are recognized.

Math has a long history of examples where the tool is invented and the applications follow. Mention some examples.

Kinda like Viagra: no one knew how useful it would be until after it was invented.

https://en.wikipedia.org/wiki/Sildenafil#History

Sildenafil (compound UK-92,480) was synthesized by a group of pharmaceutical chemists working at Pfizer's Sandwich, Kent, research facility in England. It was initially studied for use in hypertension (high blood pressure) and angina pectoris (a symptom of ischaemic heart disease). The first clinical trials were conducted in Morriston Hospital in Swansea.[38] Phase I clinical trials under the direction of Ian Osterloh suggested the drug had little effect on angina, but it could induce marked penile erections.[3][39] Pfizer therefore decided to market it for erectile dysfunction, rather than for angina. The drug was patented in 1996, approved for use in erectile dysfunction by the FDA on March 27, 1998, becoming the first oral treatment approved to treat erectile dysfunction in the United States, and offered for sale in the United States later that year.[40] It soon became a great success: annual sales of Viagra peaked in 2008 at US$1.934 billion.[41]

 

[dkotschessaa]

Modern mathematics is widely applied in many fields, Physics included. If you are interested in those areas of Physics where its application is common then you probably would want to learn it. Modern Differential Geometry is intensely topological. Here is a quote from a physicist,

"
The beauty and profundity of the geometry of fibre bundles were to a large extent brought forth by the (early) work of Chern. I must admit, however, that the appreciation of this beauty came to physicists only in recent years.

— CN Yang, 1979 "

Chern was a pure mathematician, primarily a differential geometer. One can not understand much about fiber bundles without some knowledge of topology.

Here is some reading that may give you an idea of how mathematics is used in modern physics:

- 'A First Course in String Theory" by Zwiebach. Brian Greene recommended this book to me.

Or maybe you would like to read this review article.

https://www.maths.ox.ac.uk/groups/m...eas/calabi-yau-manifolds-and-particle-physics

- Here is a Wikipedia article on topological Quantum Field Theories.

https://en.wikipedia.org/wiki/Topological_quantum_field_theory

- If you are interested in the physics of stars you may wish to learn about Knot Theory which is applied to understanding the formation of magnetic filaments.

Munkres book is an elementary topology book. One way or the other you will need to know what is in it if you are interested in the mathematically intense areas of Physics. On the other hand you may wish to pick the math up as you go along rather than take time out. That is a matter of intellectual style.

I personally have no problem with any of this. I am asking how people deal with being asked this question by the uninitiated.

-Dave K

 

[lavinia]

I personally have no problem with any of this. I am asking how people deal with being asked this question by the uninitiated.

-Dave K

I posted this on the wrong thread. Someone asked a question about Munkres book. that said maybe it is OK to leave it here.

People often ask what is the usefulness of pure mathematics. To me this is a biased attitude that asserts that nothing is worth anything unless it has a practical application. That attitude rules out the importance of art, music, literature, much of philosophy, charity and compassion (since they lead to economic inefficiency) to name a few useless enterprises. Can you make a widget with a Rembrandt portrait?

When the proof of Fermat's Last Theorem was announced on the front page of the New York Times a mortgage securities strategist at an investment bank said to me, " How much money did he make spending his whole life on this?" I said "None. He didn't do it for money." He shook his head and said,"What a waste." and walking away - no doubt to go do something practical.

 

[dkotschessaa]

I posted this on the wrong thread. Someone asked a question about Munkres book. that said maybe it is OK to leave it here.

Yeah, it was still good stuff. :)

People often ask what is the usefulness of pure mathematics. To me this is a biased attitude that asserts that nothing is worth anything unless it has a practical application. That attitude rules out the importance of art, music, literature, much of philosophy, charity and compassion (since they lead to economic inefficiency) to name a few useless enterprises. Can you make a widget with a Rembrandt portrait?

Of course, I agree with you, and this is part of option (a). The question is, given that this attitude is so ingrained, and so prevalent, how should we respond?

Clearly this bias is taught from the beginning. We are taught that we need to do math, because things can be numbered, thus counted, thus added, subtracted, multiplied, and divided. We create "word problems," idealized imaginary scenarios about things that people are doing in the world, in order to give the impression that arithmetic is a practical skill.

To those asking the question, it's a simple question. They are not looking for a lecture. Is it a totally unfair question? Not really.

We don't teach math the same way we teach art. We teach it as a means to an end, and so naturally people want to know what that end is. Of course, not all people appreciate art and music either, and will often question the legitimacy of studying either.

When the proof of Fermat's Last Theorem was announced on the front page of the New York Times a mortgage securities strategist at an investment bank said to me, " How much money did he make spending his whole life on this?" I said "None. He didn't do it for money." He shook his head and said,"What a waste." and walking away - no doubt to go do something practical.

The news cycle can be a big problem. I once saw a very amusing talk by a mathematician who worked on the Pizza Theorem, which is a very interesting problem in geometry with a long history. There was an article published about it in a mathematics magazine, and it eventually made it to a more mainstream journal New Scientist, with the title The perfect way to slice a pizza!

The comments section has since been closed, but as you can imagine, it was littered with comments to the effect of WHO FUNDED THIS RESEARCH?

-Dave K

 

[FactChecker]

Abstract math like topology and abstract algebra give you general rules that help you understand a large variety of specific mathematics and physics subjects more quickly and easily.

That being said, if the person is asking how you use it, then the answer is that you actively seek out subjects to understand and abstract math helps. If he is asking how he will use it, then the answer is that if he sits at home watching TV and drinking beer, it is unlikely that applications will come knocking on his door.

 

[dkotschessaa]

Abstract math like topology and abstract algebra give you general rules that help you understand a large variety of specific mathematics and physics subjects more quickly and easily.

Thank you for saying that concisely. I've known this is true but my explanation was much more long winded.

That being said, if the person is asking how you use it, then the answer is that you actively seek out subjects to understand and they help. If he is asking how he will use it then the answer is that if he sits at home watching TV and drinking beer, then it is unlikely that applications will come knocking on his door.

It is times like this that I wish life really were more like a musical. Then a band would start playing, I would sing a song called "math is everywhere" and then everybody would understand by the end.

-Dave K

 

[Dr. Courtney]

I'm not big on trying to justify funding to non-experts. I'd just say the research had the full approval and support of those who decided to fund it. And of course, some of the best work is done for love rather than for money.

https://www.physicsforums.com/insights/science-love-money/

 

[dkotschessaa]

I'm not big on trying to justify funding to non-experts. I'd just say the research had the full approval and support of those who decided to fund it. And of course, some of the best work is done for love rather than for money.

https://www.physicsforums.com/insights/science-love-money/

Indeed. The pizza theorem guys spent something like 10 years working on the problem, but the length of time is owed to the fact that they did so mostly in their spare time.

-Dave K

 

[lavinia]

Indeed. The pizza theorem guys spent something like 10 years working on the problem, but the length of time is owed to the fact that they did so mostly in their spare time.

-Dave K

Did the pizza get cold?

 

[dkotschessaa]

Did the pizza get cold?

After 10 years I would not like to imagine what it got.

 

[fresh_42]

This problem already occurs in school math. When tutoring I sometimes just answered: because you need it for the next test, your school qualification or similar. I mean as long as things like "10 construction workers build a house in 20 days, how long ...?" can be found in school books, can we really expect to be taken seriously? The real question is: Why doesn't this question about profits arise in fields like history? As if mankind ever had learned something from past events.

I plead to return to the original meaning of mathematics. Let's strip it off the natural sciences and regard it as a relative of philosophy again.
That doesn't solve the problem (what is it good for?), but nobody will expect an answer anymore. I mean, we've done it before: AC, the barber problem and we buried Hilbert's program.

It's a bit like CERN. Many people (if they even know about it) consider it as a giant loss of money but at the same time, they are proud of the fact that mankind has achieved something like this.

 

[dkotschessaa]

I don't think history is immune from the question actually, but people seem to relate to it better.

 

[FactChecker]

People know that every historical fact had at least one significant application -- when it was a current event. Fewer people will know any application of abstract algebra.

 

[ZapperZ]

This is taken from the section titled "To The Student" from Mary Boas's excellent text "Mathematical Methods in the Physical Sciences":

There is a story about a young mathematics instructor who asked an older professor "What do you say when students ask you about the practical applications of some mathematical topic"? The experienced professor said "I tell them!"

End of story.

Zz.

 

[dkotschessaa]

People know that every historical fact had at least one significant application -- when it was a current event. Fewer people will know any application of abstract algebra.

I suppose the complaints I have heard tend to come from something-teen year olds, but they complain about everything.

 

[lavinia]

There are areas of mathematics that historically had no application that became useful later. There are areas now with no obvious application although Mathematics and Physics have merged in recent decades to the point where it would be difficult to isolate an area of modern mathematics with no application whatsoever. I sat in on a course in PDE's with Richard Hamilton and he said 'Every differential equation has or someday will have a use. So go solve it.'

 

[Stephen Tashi]

I personally have no problem with any of this. I am asking how people deal with being asked this question by the uninitiated.

An honest answer to many questions is "I don't know".

Of course an answer of "I don't know" might lead to follow-up questions like "Then why do we have to study it?".

The requirement that someone must study something is the outcome of a complex social pheonomena - you could give that answer to the follow-up.

 

[ZapperZ]

An honest answer to many questions is "I don't know".

Of course an answer of "I don't know" might lead to follow-up questions like "Then why do we have to study it?".

The requirement that someone must study something is the outcome of a complex social pheonomena - you could give that answer to the follow-up.

But here's the situation. I highly doubt that such a question will be asked at higher-level courses and advanced mathematics topics. Students who are taking such classes either have an innate interest in the subject, or already have a clue on why they need that type of mathematics. So the question will prop up most likely at those in lower level classes, even in high school and beginning undergraduate.

Now, if an instructor cannot come up with simple, direct applications for that level of mathematics, then there is a problem with that instructor! It is why I quoted Mary Boas's take on this. You tell them! And you need to know what these applications are, because they are numerous!

As a physics instructor, I am faced with the same type of questions, because my students are predominantly not physics majors. I tell them the practical applications or the importance of understanding the physics that they are being forced to study. If these are pre-med majors, I tailor my instructions and examples to include things that they might encounter in the medical fields. If they are engineers or engineering-tech majors, I slant my content towards that direction of application.

As instructors, we can't simply put blinders on and teach the material without looking at the students and what they need or want.

Zz.

 

[dkotschessaa]

An honest answer to many questions is "I don't know".

The truth, yes. Usually this answer is passed off as meaning I don't care, or that I am blowing the person off.

-Dave K

 

[dkotschessaa]

Now, if an instructor cannot come up with simple, direct applications for that level of mathematics, then there is a problem with that instructor! It is why I quoted Mary Boas's take on this. You tell them! And you need to know what these applications are, because they are numerous!

Well, I hang out with physics peeps such as yourself online, but the truth is I really do not know how most of the math I study is applied, and I only sort of care, because I'm studying math because I like math.

BTW the audience I am referring to in my situation is not students or a math literate audience.

-Dave K

 

[.Scott]

Well, but when I say "algebra" i mean group theory, rings fields, Galois theory. I don't know how people use this outside of mathematics.

Galois theory might be useful in terms of keeping you from working on a problem that is known to be fruitless. You should be able to authoritatively say that only a practical solution should be sought for certain engineering problems.
That being said, there is an awful lot of Math that is only Math. People do Math for Math's sake - and in some cases a current of future practical problem is addressed in the process.

I would be surprised if Appel and Haken were thinking that their efforts in proving the four-color theorem would have practical value - and it certainly wasn't the motivation behind their considerable effort.

An oft-quoted statement in this Forum: "If all of mathematics disappeared, physics would be set back by exactly one week.” - Richard Feynman.

That said, there are definitely parts of Math that have immediate application in Physics. For example, PDE's have already been mentioned in this thread.

 

[dkotschessaa]

So maybe I just need specifics, and then as @ZapperZ said, I can just tell them. Here is what we have so far:

Chern was a pure mathematician, primarily a differential geometer. One can not understand much about fiber bundles without some knowledge of topology.

I don't know what a fiber bundle is, but if it's something physicists want to know about, then I know topology --> physics. Now are you mostly talking about point set topology?

- If you are interested in the physics of stars you may wish to learn about Knot Theory which is applied to understanding the formation of magnetic filaments.

This is very good to know. I did a bit of knot theory as a seminar research topic. Anything else you know people are using knot theory for?

There are areas of mathematics that historically had no application that became useful later. There are areas now with no obvious application although Mathematics and Physics have merged in recent decades to the point where it would be difficult to isolate an area of modern mathematics with no application whatsoever. I sat in on a course in PDE's with Richard Hamilton and he said 'Every differential equation has or someday will have a use. So go solve it.'

PDE's definately. No problem there.

Galois theory might be useful in terms of keeping you from working on a problem that is known to be fruitless. You should be able to authoritatively say that only a practical solution should be sought for certain engineering problems.

Has anyone run into this situation directly who is working on an actual physical problem?
Some others: Graph Theory - obviously very useful in networking, sometimes called "networking theory."
Complex Analysis: I know you folks are using it, but I don't know for what exactly. I'm not a big analysis guy.

Combinatorics? My favorite area of math. I thought it made it a lot easier to study statistics, but I don't know what else people are using it for. Fun as heck though.

Mathematical logic? All I can say is maybe computer science.

Linear Algebra: I think I once heard someone say that if you can't translate a question into linear algebra terms it's not even worth asking. A bit of an exaggeration perhaps, but we know it is extremely useful.

-Dave K

 

[dkotschessaa]

Oh, forgot to add (though I might have mentioned earlier) that algebraic topology is finding a lot of uses in the buzzwordy world of Data Science right now. I was taking a course on Topological Data Analysis before I had to bail on my semester. Cool stuff.

 

[lavinia]

http://www.pnas.org/content/111/43/15350.full

- point set topology is assumed without comment. It is too basic to be sufficient to understand much of topology. The topology of fiber bundles is a subject in itself. The first book on it was probably Steenrod's book.

 

[Stephen Tashi]

But here's the situation. I highly doubt that such a question will be asked at higher-level courses and advanced mathematics topics.

Such a question often came up in my mind in higher-level and advanced courses - I just didn't dare ask it!

 

[dkotschessaa]

http://www.pnas.org/content/111/43/15350.full

- point set topology is assumed without comment. It is too basic to be sufficient to understand much of topology. The topology of fiber bundles is a subject in itself. The first book on it was probably Steenrod's book.

I didn't recognize anything there from my knowledge of topology or (my admittedly elementary knowledge of) knots. But I believe you. :)

 

[dkotschessaa]

Such a question often came up in my mind in higher-level and advanced courses - I just didn't dare ask it!

There is analogue to the "what good is this for" question within math itself, of course. To come back to algebra, I wish I had known from the start that our goal was to arrive at the classification of finite simple groups. That was the "Why are we doing all of this?" and it made sense when we finally got to that chapter. I mean, I thought algebra was beautiful, but the questions would have made more sense if I knew that was the end goal.

-Dave K

 

[TeethWhitener]

Has anyone run into this situation directly who is working on an actual physical problem?

I'm a chemical physicist and my Erdos number is 5 because of a paper on Galois theory. It involves a collaboration between the mathematician Harold Shapiro and his grandson who is a chemist, looking at exact solutions of high-order polynomial equations that appear in some obscure area of chemical kinetics.

 

[dkotschessaa]

I'm a chemical physicist and my Erdos number is 5 because of a paper on Galois theory. It involves a collaboration between the mathematician Harold Shapiro and his grandson who is a chemist, looking at exact solutions of high-order polynomial equations that appear in some obscure area of chemical kinetics.

A fantastic anecdote is what you are.

 

[Ygggdrasil]

Well, but when I say "algebra" i mean group theory, rings fields, Galois theory. I don't know how people use this outside of mathematics.

Group theory, as I am sure mathematician are aware, is the language of symmetry. This turns out to be extremely important in chemistry and biology because we use the diffraction patterns from crystals to study the structure of molecules at the atomic level. Concepts from group theory are important for interpreting the diffraction data so that we can turn a series of spots on an piece of film into a three-dimensional model of an important biological macromolecule.

tl;dr: if it weren't for group theory, we wouldn't know what molecules looked like.

 

[houlahound]

I would not try to justify math on practical grounds. I would try and engage them in doing some math with you, passion is infectious.

My most influential teacher was a master at aligning math problems to the individual student. He always had students lining up for another problem, they stopped asking why and just asked for another problem. Its a positive feedback loop.

It is hard to match a student to a problem they will get into, most just assign/throw out dozens of problems and hope one sticks.

All people like games, math is just another game - that's your answer.

 

[symbolipoint]

An honest answer to many questions is "I don't know".

Of course an answer of "I don't know" might lead to follow-up questions like "Then why do we have to study it?".

The requirement that someone must study something is the outcome of a complex social pheonomena - you could give that answer to the follow-up.

The less advanced the level, the easier finding the applications will be. The higher the level of advancement, the more difficult it is identifying the applications and reporting this to the student who asks.

 

[symbolipoint]

I would not try to justify math on practical grounds. I would try and engage them in doing some math with you, passion is infectious.

My most influential teacher was a master at aligning math problems to the individual student. He always had students lining up for another problem, they stopped asking why and just asked for another problem. Its a positive feedback loop.

It is hard to match a student to a problem they will get into, most just assign/throw out dozens of problems and hope one sticks.

All people like games, math is just another game - that's your answer.

Last part NOT true. A few people dislike games; although some of these few people do really like studying and finding understanding. Not everyone will view learning Mathematics as a game. Some people take it as the struggle to understand.

 

[mathman]

Well, but when I say "algebra" i mean group theory, rings fields, Galois theory. I don't know how people use this outside of mathematics.

Group theory is widely used in physics. Symmetry plays an important role.

 

[lavinia]

Gauss considered mathematics to be a science like other sciences although he called it "the queen of sciences." Here is a purported quote from him from the Wikipedia article.

"Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank."

It would be interesting to know what he meant by this. It differs from the modern point of view that mathematics is not a science but is merely a "language" used for Physics.

Gauss was an astronomer and did early research on electricity and magnetism.

 

[dkotschessaa]

I would not try to justify math on practical grounds. I would try and engage them in doing some math with you, passion is infectious.

My most influential teacher was a master at aligning math problems to the individual student. He always had students lining up for another problem, they stopped asking why and just asked for another problem. Its a positive feedback loop.

It is hard to match a student to a problem they will get into, most just assign/throw out dozens of problems and hope one sticks.

All people like games, math is just another game - that's your answer.

Clearly the answer is going to vary depending on the audience. I've met the achievement of getting people excited about math, but usually they were intelligent people who were already passionate about *something,* computers, music, the arts, whatever.

-Dave K

 

[dkotschessaa]

Gauss considered mathematics to be a science like other sciences although he called it "the queen of sciences." Here is a quote from the Wikipedia article.

"Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank."

It would be interesting to know what he meant by this. It differs from the modern point of view that mathematics is not a science but is merely a "language" used for Physics.

Gauss was an astronomer and did early research on electricity and magnetism.

Yes, I've encountered the quote and have used it on occasion.

The view that math is a "language" used for physics is really only something I've heard from people doing physics. :D Mathematicians might agree that it's a language, but when you are immersed in pure mathematics it has much more the feeling of being a universe unto itself.

-Dave K

 

[houlahound]

Clearly the answer is going to vary depending on the audience. I've met the achievement of getting people excited about math, but usually they were intelligent people who were already passionate about *something,* computers, music, the arts, whatever.

-Dave K

The big thrill is helping someone who thinks they can't do it.

 

[atyy]

G H Hardy wrote a book on it called A Mathematician's Apology where he discusses this very topic.

https://en.wikipedia.org/wiki/G._H._Hardy

https://en.wikipedia.org/wiki/A_Mathematician's_Apology

I've never read Hardy's book, but he has the reputation of defending pure mathematics. Unfortunately, that's been completely destroyed. I think Hardy himself started the rot, with his efforts in genetics (Hardy-Weinberg), and with modern applications of number theory to cryptography (which were not his fault).

 

[Dr. Courtney]

If the laws of nature were not written in mathematics, I wouldn't bother.

Math is a necessary evil to do science and engineering. I won't pretend that I haven't learned to like it, but I won't pretend I would have ever bothered to learn it if not for my love of physics.

 

[ShayanJ]

It differs from the modern point of view that mathematics is not a science but is merely a "language" used for Physics.

That point of view always makes me wonder, specially when mathematicians themselves state it.(You're a mathematician, right?)
Mathematics is more than a language for anything. A language is a tool to communicate something. But mathematics does far far more than just communication. We'd have no idea how to do physics without mathematics. Of course mathematics is something much much more than merely a language.

It was sometime ago when someone asked me about applications of physics. I could start with solid state and AMO physics and all the obvious applications with lasers, semiconductors,etc. But that approach always makes me feel like I'm betraying what I love. Of course there is nothing wrong with solid state physics, AMO physics,etc. They're also beautiful physics and its good that they have those applications. But when you go in that direction, the audience may get the impression that even a student of particle physics thinks what he's studying is useless!
Instead of that, I proposed three levels of applications:

Level 1) Parts of physics that are obviously there because of the applications. like the parts I mentioned above. But of course you can see that they can't be there without the other parts of that specific field of physics which brings me to level 2.

Level 2) Physical theories that explain a wide range of phenomena and if it wasn't because of them, we couldn't harness the potentials of that range of phenomena for applications we have today. Obvious example is QM.

Level 3) Physics in the sense of trying to understand nature in its deepest levels, is a thousand years old endeavor. But in the modern sense, its only a few centuries. This long history of the efforts of millions of people have given us a wide range of tools. Now one may ask why are we limiting these tools to their original applications? Why aren't we trying to find out more areas where we can use these tools? And this is what happened in the field of complex systems. Nowadays we have physicists working on traffic, medicine, biology,etc. And these applications are not because physics is the underlying theory of biology. People who are familiar with complex systems know what I mean.

Level 1 applications are more obvious but more specific and limited. Level 2 applications are as broad as the range of phenomena the theory is supposed to work for. And level 3 applications are as broad as human's ability to come up with applications for a tool.

But there is also another point of view to answer this question. Its like asking a carpenter why should I care about your electric saw? He would say that you have no reason to care, its for me to use so that I can make for you that book shelf. So he can come back at you by asking what's the point of that book shelf? Of course you want to put your books there and if you happen to be a physicist, those will be physics and mathematics books. But why are you studying those? part of it is for applications, like that electric saw, and other parts are for more theoretical parts that are farther from applications. Now if that carpenter thinks your job is useless, his job is useless too because he is doing it for you so that you can do your job. You can follow this kind of chain reasoning for many chains of jobs and you'll end up thinking all jobs are useless. The correct way to think about this, is that mankind wants to flourish and go forward. A really critical part of this flourishing is understanding what's going on in this world. Actually most jobs out there are there to keep people alive and amused. By saying that intellectual endeavors like theoretical physics and pure mathematics are not as important as those jobs, people are actually saying that the flourishing of mankind is just by living longer and enjoying more. This is just missing the point. Of course for some people life is doing a job so that you can have money to enjoy life. That's OK, no problem with that. But if all of mankind was to think like that, we wouldn't be here. So its undeniable that a really critical part of the flourishing of mankind is by intellectual endeavors. If someone asks me this question and I have enough time and I think that the person actually listens and thinks about what I say, this'll be my answer.

And a little point at the end: People don't ask for applications of art because it wasn't supposed to have applications in the first place. But science started as people's efforts to build something they needed. So some people still think that is what it is. And so something in physics that doesn't help you build something is useless because of that definition of physics. But if you can show them that physics and mathematics are partly efforts in the direction of mankind flourishing, they may understand.

 

[atyy]

The practical application of mathematics is to enable one to open threads about the practical application of mathematics :P

 

[atyy]

Abstract math like topology and abstract algebra give you general rules that help you understand a large variety of specific mathematics and physics subjects more quickly and easily.

Did you mean something like this? I suppose a schema is like a category.

Schemas and memory consolidation
Tse D, Langston RF, Kakeyama M, Bethus I, Spooner PA, Wood ER, Witter MP, Morris RG.
Science. 2007 Apr 6;316(5821):76-82.
https://www-ncbi-nlm-nih-gov.libproxy1.nus.edu.sg/pubmed/17412951