Is math really that important to do physics?

[Avaro667]

Hello everyone .
Recently i have been thinking about the importance of math in physics . There are many times that i read books or papers that contain tons of hard math and yet i feel it doesn't really add that much to my understanding . On the other hand there are sources that provide a very good description of the physical situation , use a bit of algebra and a bunch of pictures and diagrams and seem much more intuitive and easier material to go through .

I remember for example taking a course in vector calculus just before taking any EM related course . But what i realised later on was that i could be introduced to the same ideas and understand the physical laws without knowing how to calculate say surface and volume integrals etc. Sure vector calculus provided the language to express problems in three dimensions but it was more messy than insightful .

Now i know that there are some areas like quantum mechanics where having a solid knowledge in linear algebra etc. is really helpful , but what's your experience ? Do you think an explanation that uses the least and simplest possible math is the best ?

 

[Frabjous]

No. For example, a physicist will learn E&M at least three times (freshman, junior, graduate school). Each time with increasing abstraction (more math). Simple math leaves out a lot of understanding.

 

[Dale]

With only the insight and without the math could you determine what a theory predicts for a particular experiment and compare the theory to the experiment? Without the math could you design a useable device and have it work according to your design?

Do you think an explanation that uses the least and simplest possible math is the best ?

Yes. But you need to be very clear on what criterion you are considering for “simplest possible”. If you are doing science or engineering the simplest possible will be more than if you are just looking for insight.

 

[Avaro667]

With only the insight and without the math could you determine what a theory predicts for a particular experiment and compare the theory to the experiment?

Probably not . The powerful property of math to predict interesting phenomena is indeed well known . If i am not mistaken , Maxwell predicted the existence of EM waves without doing an experiment . That was incredible . I don't know , maybe after all these math courses i just underestimate their impact to my understanding of physics...I guess i take some stuff for granted ?

Without the math could you design a useable device and have it work according to your design?

Ultimately having a good grasp of computation is important if you want to design anything . So yeah , in this case you absolutely need to know how things would actually be calculated ( or even measured ) , which require maths . But isn't understanding physics more about understanding the physical laws than doing calculations ?

Yes. But you need to be very clear on what criterion you are considering for “simplest possible”. If you are doing science or engineering the simplest possible will be more than if you are just looking for insight.

Simplest possible would mean not introducing more advanced math than what you really need to understand and solve a problem .

 

[Avaro667]

No. For example, a physicist will learn E&M at least three times (freshman, junior, graduate school). Each time with increasing abstraction (more math). Simple math leaves out a lot of understanding.

It's true , the less math you use , the more detail you lose . But is it true for all branches of physics ? For example what about classical mechanics ? Wouldn't just some basic differential equations knowledge and calculus suffice ?

On the other hand classical mechanics is supposed to be a special case of quantum mechanics . So i guess it kinda makes sense for a more general ( and therefore more complex ) theory you need more math .

 

[marcusl]

You could solve simple problems like dropping a ball but you couldn’t handle solid body motion (precession of a top) or even practical ballistics problems like targeting an artillery shell (Coriolis force).

 

[fresh_42]

This could be a good opportunity to recall Feynman's video on magnetism:


It shows the significance of the problem: What is a valid answer? What is an insight?

On the other hand there are sources that provide a very good description of the physical situation , use a bit of algebra and a bunch of pictures and diagrams and seem much more intuitive and easier material to go through .

I do watch Morgan Freeman, too, trying to explain the universe. It is entertaining. And it is wrong. This is an extreme example, yes, but it shows what you give up if you choose the easy way.

I wrote an article about mathematics and nature ...
https://www.physicsforums.com/insights/differential-equation-systems-and-nature/
... and explained the two-species predator-prey model of Lotka and Volterra. It is simple, one can learn a lot even without calculating an explicit solution, and it is two-dimensional so we can draw it. But when do we have only two species that interact? I also mentioned a dissertation about a model of regional timber management. The author set up 221 differential equations, 182 parameter functions, and 371 single parameters. And that's only his standard, i.e. basic version.

Thus, mathematics is necessary in depth as it is in volume if we want to describe nature accurately. Mathematics and physics were developed along with each other for centuries. I doubt that Gauß or Maxwell made a distinction. And here comes the real reason for mathematics in physics: you need it to talk to each other and communicate. You need this blackboard and write down formulas if you want to express an idea such that others can follow you. It is far more a language than it is an encyclopedia of formulas.

While writing that article I recognized the beauty of
$$F\sim \ddot x$$
However, if you really want to use it, you first add arrows and introduce three coordinates $\mathrm{i\, , \,j\, , \,k}$, then insert the mass as the proportion factor, and you can do a lot of classical physics with it. In the end, you will learn that it cannot explain everything that we see in the universe. Even the mass becomes a problem! The formula explains a lot, but general relativity requires some differential geometry that is not a simple parabola.

 

[Dale]

Simplest possible would mean not introducing more advanced math than what you really need to understand and solve a problem .

Scientists usually already do that

 

[Frabjous]

It's true , the less math you use , the more detail you lose . But is it true for all branches of physics ? For example what about classical mechanics ? Wouldn't just some basic differential equations knowledge and calculus suffice ?

On the other hand classical mechanics is supposed to be a special case of quantum mechanics . So i guess it kinda makes sense for a more general ( and therefore more complex ) theory you need more math .

In mechanics, calculus and PDE’s will take you far, but you can go even farther (for example Arnold’s Mathematical Aspects of Classical Mechanics)

Thinking of classical mechanics as a special case of quantum mechanics will rarely get you anything.

 

[Avaro667]

In mechanics, calculus and PDE’s will take you far, but you can go even farther (for example Arnold’s Mathematical Aspects of Classical Mechanics)

Thinking of classical mechanics as a special case of quantum mechanics will rarely get you anything.

I have read that book . While i know the answer depends on the level you want to study the subject , don't you think for most people in stem this book would be an overkill ?

You could solve simple problems like dropping a ball but you couldn’t handle solid body motion (precession of a top) or even practical ballistics problems like targeting an artillery shell (Coriolis force).

Just to be clear about that , i'm not supporting by any means that someone could actually understand physics by just reading plain text and algebra . I'm just wondering whether someone could really get something out of knowing more math than the basic ones ( real analysis , vector calculus , diff.eq. , linear algebra , probability/Statistics , numerical Analysis and maybe some differential geometry and group theory etc. ) . But i guess what you guys say is that if you know only algebra and trigonometry for example , then you could study physics in a basic level and later on when you have a greater understanding of more advanced math , you could go more in depth .

This could be a good opportunity to recall Feynman's video on magnetism:


It shows the significance of the problem: What is a valid answer? What is an insight?

Hmm , i see your point . Different people would be satisfied with different answers . So a person who doesn't know lots of math would be happy with a simple , rough and descriptional answer . Someone else would be happy with an equation and so on , based on what is their context of understanding .

I do watch Morgan Freeman, too, trying to explain the universe. It is entertaining. And it is wrong. This is an extreme example, yes, but it shows what you give up if you choose the easy way.

It's true . There are many great videos lately trying to explain physics and math but personally most of the times i have hard time understanding them and i think it's normal because their purprose is entertain not teach . There are people who think they actually learn stuff , and they might learn a little , but i'm pretty sure it's just an impression they have than an a fact . Watching is far different than doing .

I wrote an article about mathematics and nature ...
https://www.physicsforums.com/insights/differential-equation-systems-and-nature/
... and explained the two-species predator-prey model of Lotka and Volterra. It is simple, one can learn a lot even without calculating an explicit solution, and it is two-dimensional so we can draw it. But when do we have only two species that interact? I also mentioned a dissertation about a model of regional timber management. The author set up 221 differential equations, 182 parameter functions, and 371 single parameters. And that's only his standard, i.e. basic version.

I will take a look on that , it seems pretty interesting , thanks . For simple cases you can get away with intuition but indeed after a point you need math to generalize successfully that intuition .

 

[Frabjous]

Freshman physics becomes easy after taking junior physics so knowing the more advanced math provides a deeper understanding of what the simpler math is saying.

 

[Avaro667]

Freshman physics becomes easy after taking junior physics so knowing the more advanced math provides a deeper understanding of what the simpler math is saying.

Still at least in engineering i see people being able to tackle really hard problems with very simple math and lots of diagrams . And they seem to have a fairly good understanding of the problems . That's mainly the reason i wonder how much they really add to our understanding . I'm trying to figure out what would be the line that separates math knowledge providing more insight and making things more beautiful from math math knowledge being completely necessary to have a good understanding of a problem .

 

[phinds]

And the deeper you get into the math and the physics that it explains, the more you will understand the answer to that question. I think at the moment your understanding and your belief are more shallow than you think.

 

[Avaro667]

And the deeper you get into the math and the physics that it explains, the more you will understand the answer to that question. I think at the moment your understanding and your belief are more shallow than you think.

But there are people who are really good at math and still trouble with physics . So...it seems while math is crucial , it's not the whole story . What lots of physicists tell me themselves from time to time is that many times people are so lost in math sometimes that they miss the really important aspects of physical theories .

 

[Frabjous]

Understanding is more than being able to calculate from a formula.

 

[Avaro667]

Understanding is more than being able to calculate from a formula.

Yet lots of high school and university courses embrace this approach . Some calculations are useful because they reveal important physical facts and help familiarize yourself with the theory , but having the equations for many students is the means to the end for solving a problem . They focus more on how to use the equations and handle the math than trying to think about the results they get and what they mean , and also what the equations mean and how they were derived .

 

[PeroK]

But there are people who are really good at math and still trouble with physics . So...it seems while math is crucial , it's not the whole story . What lots of physicists tell me themselves from time to time is that many times people are so lost in math sometimes that they miss the really important aspects of physical theories .

Logic is also important. And there's a common logical fallacy there:

Proposition: advanced mathematics is necessary to understand advanced physics.

This proposition is not refuted by some people being lost in mathematics. That, to some extent, refutes:

Proposition: advanced mathematics is sufficient to understand advanced physics.

The fundamental problem with your argument, IMO, is that the mathematics that is understood by physicists is a tiny fraction of the mathematics studied by mathematicians in each area. Take Group Theory, for example. A maths student would delve into much greater detail on finite groups and progress into Ring Theory and Abstract Algebra more generally. The physics student must get to grips with the basics of finite groups for particle physics and Lie Groups (and even Lie Algebras) more generally. But, if they pick up a graduate textbook on Lie Algebras, for example, they are likely to be lost in the sheer abstraction of it.

In other words, physicists are only learning the minimum mathematics required to study physics.

 

[Avaro667]

Logic is also important. And there's a common logical fallacy there:

Proposition: advanced mathematics is necessary to understand advanced physics.

This proposition is not refuted by some people being lost in mathematics. That, to some extent, refutes:

Proposition: advanced mathematics is sufficient to understand advanced physics.

You are correct , of course . What they supposed to mean i guess is that lengthy and complex problems( which are quite common in exams) don't necessarily require that much of physical insight and therefore are not that good indicators of how solid your physics knowledge is .

The fundamental problem with your argument, IMO, is that the mathematics that is understood by physicists is a tiny fraction of the mathematics studied by mathematicians in each area. Take Group Theory, for example. A maths student would delve into much greater detail on finite groups and progress into Ring Theory and Abstract Algebra more generally. The physics student must get to grips with the basics of finite groups for particle physics and Lie Groups (and even Lie Algebras) more generally. But, if they pick up a graduate textbook on Lie Algebras, for example, they are likely to be lost in the sheer abstraction of it.

In other words, physicists are only learning the minimum mathematics required to study physics.

I know that my phrasing of the question right now won't be the best here , since knowing lots of math can never be bad , but...is there a good rule of thumb about when someone delves too much into the math , that it turns out counter-productive for studying physics ?

I'm not denying that this question is inherently problematic since as already mentioned math and physics are heavily related to the extent there can't be really any boundary between the two , and therefore being a good mathematician lots of times proves helpful for being a good physicist , and vice verca...but yeah , sometimes i feel like math is the priority , and that doesn't feel right to me .

 

[PeroK]

I know that my phrasing of the question right now won't be the best here , since knowing lots of math can never be bad , but...is there a good rule of thumb about when someone delves too much into the math , that it turns out counter-productive for studying physics ?

It varies from student to student. For me, the two go hand in hand: modern physics has a mathematical basis. I find I've never really grasped something until I've got the mathematical basis. That said, I often spend a good deal of time thinking about the concepts.

I'm not denying that this question is inherently problematic since as already mentioned math and physics are heavily related to the extent there can't be really any boundary between the two , and therefore being a good mathematician sometimes proves helpful for being a good physicist , and vice verca...but yeah , sometimes i feel like math is the priority , and that doesn't feel right to me .

The mathematics is rarely the priority. There are some brilliant lectures online on mathematical physics by Carl Bender. I watched these during lockdown. He emphasises that you don't necessarily need to prove everything rigorously for the mathematics to be extremely powerful. These are graduate level lectures and some of his students are very sharp! Here's the first:

 

[Avaro667]

It varies from student to student. For me, the two go hand in hand: modern physics has a mathematical basis. I find I've never really grasped something until I've got the mathematical basis. That said, I often spend a good deal of time thinking about the concepts.

Sounds fair . If you can't handle the language with ease , then it won't be easy dealing with the actual physics .

The mathematics is rarely the priority. There are some brilliant lectures online on mathematical physics by Carl Bender. I watched these during lockdown. He emphasises that you don't necessarily need to prove everything rigorously for the mathematics to be extremely powerful. These are graduate level lectures and some of his students are very sharp! Here's the first:

Seems quite interesting , i will take a look on that .
So if you just study and learn the necessary math tools , is it enough , even though you can't prove many of the theorems rigorously ?

 

[berkeman]

Thread closed temporarily for Moderation...

 

[berkeman]

Sounds fair . If you can't handle the language with ease , then it won't be easy dealing with the actual physics .

Thread reopened provisionally.

@Avaro667 -- perhaps I misinterpreted your post. Who is the "you" in your post above?

 

[Avaro667]

Thread reopened provisionally.

@Avaro667 -- perhaps I misinterpreted your post. Who is the "you" in your post above?

I wasn't refering to someone specifically .

 

[topsquark]

So if you just study and learn the necessary math tools , is it enough , even though you can't prove many of the theorems rigorously ?

I've been reading this with a question in mind: What does it matter how much Mathematics that you are putting in? Obviously, you want to focus on the Physics and not the Mathematics, so you will occasionally be able to "circumvent" some of the more abstract Mathematical issues (such as when can you switch integrals and summations, and whether you can assume that a horse is a spherical object.) But why are you so concerned with the amount that is used? Higher level Mathematics helps give us a picture of what the Physics is talking about... it's not just about numbers. I've found that the more Mathematics I learn, the more Physics that I understand because Mathematics gives me tools I can use to exploit the Physics.

Rigor can be a good thing! For example, I knew that there was a problem with simply calling a Quantum state a function over the complex numbers, but I never knew why until recently. The knowledge comes from a level of Mathematics that I didn't really have much of a grasp on, and I'm glad I did that study: it helped me understand the Physics better. Can you get away with calling a Quantum state a function? Sure, most of the time. But the Mathematics gives me an understanding of why we can make those definitions, and where those definitions might fail. I have a better understanding of what a Quantum state is because I learned the extra Mathematics.

You don't have to learn all of Mathematics to be a good Physicist, but the Mathematics helps you understand it better. Why would you consider this sort of thing to not be helpful?

-Dan

 

[Avaro667]

I've been reading this with a question in mind: What does it matter how much Mathematics that you are putting in? Obviously, you want to focus on the Physics and not the Mathematics, so you will occasionally be able to "circumvent" some of the more abstract Mathematical issues (such as when can you switch integrals and summations, and whether you can assume that a horse is a spherical object.) But why are you so concerned with the amount that is used? Higher level Mathematics helps give us a picture of what the Physics is talking about... it's not just about numbers. I've found that the more Mathematics I learn, the more Physics that I understand because Mathematics gives me tools I can use to exploit the Physics.

Rigor can be a good thing! For example, I knew that there was a problem with simply calling a Quantum state a function over the complex numbers, but I never knew why until recently. The knowledge comes from a level of Mathematics that I didn't really have much of a grasp on, and I'm glad I did that study: it helped me understand the Physics better. Can you get away with calling a Quantum state a function? Sure, most of the time. But the Mathematics gives me an understanding of why we can make those definitions, and where those definitions might fail. I have a better understanding of what a Quantum state is because I learned the extra Mathematics.

You don't have to learn all of Mathematics to be a good Physicist, but the Mathematics helps you understand it better. Why would you consider this sort of thing to not be helpful?

-Dan

To be 100% honest with you i feel this question have been bothering me because i have always tried to focus so hard on getting as solid physics and math background as possible , mostly because it was fun and intriguing , but what i observe is that these skills don't really pay off that much as i thought they would( at least not in my country ) . Other people who have tried much less than me are still sucessful in their areas and that kinda depress me . So for months now i'm wondering "what was really the point then ? Yeah i gained some deeper understanding on some topics but it doesn't really make any difference to the world" . I should also mention that my background is in engineering(EE - electromagnetics) , which falls closer to applied physics , and in a few months i will be graduating ( Msc in engineering ) . I just find it so dissapointing that my career options will be so limited .

 

[gleem]

I should also mention that my background is in engineering(EE - electromagnetics) , which falls closer to applied physics , and in a few months i will be graduating ( Msc in engineering ) . I just find it so dissapointing that my career options will be so limited .

Well, I didn't see that coming. It seems that you are mentally sidetracked. Sure physics is important but how does your insecurity with your present understanding of physics affect your career options? Why are you in EE?

 

[Avaro667]

Well, I didn't see that coming. It seems that you are mentally sidetracked. Sure physics is important but how does your insecurity with your present understanding of physics affect your career options? Why are you in EE?

Well you are right . It's been a long journey of wrong choices and unlucky events . Due to a medical incident i had when i was younger , lack of motivation( i didn't know what i wanted to become ) and not being able to afford tutoring like everybody else did , i didn't perform well in the national university entrance exams . So i got in the EE department almost randomly ( the entrance system here is weird here - i would dare to say broken really ).

So , after a few physics courses in the first semesters in the EE department i found out that my passion was definitely physics . I tried to pick as many physics related courses i could , but reality is i always wanted to go deeper than what these courses intended . I tried to do an internship to see if the engineering job environment would be a better experience than school...but i can't say it was exciting . I was also thinking of studying physics , but i don't know , i'm 24 , so it feels kinda too late for that .

So i'm almost stuck with a degree i don't know how to utilize here ( because there are no electromagnetics related jobs ) , and i possibly wouldn't like how the job market would expect me to utilize it . I guess that's why i did this question in the first place . I feel like all my effort learning all this hard math and physics concepts so far is pointless . On the other hand i feel too old to go after a physics degree ( i could become a teacher for example , it would make me happy and there are relevant jobs ) .

 

[berkeman]

So , after a few physics courses in the first semesters in the EE department i found out that my passion was definitely physics . I tried to pick as many physics related courses i could , but reality is i always wanted to go deeper than what these courses intended . I tried to do an internship to see if the engineering job environment would be a better experience than school...but i can't say it was exciting . I was also thinking of studying physics , but i don't know , i'm 24 , so it feels kinda too late for that .

Well, I didn't see that coming

Thread moved from the technical Physics forums to the Academic Advising forum...

 

[Frabjous]

I find it hard to believe that in a place where there are no electromagnetics jobs that there would be physics jobs.
Stop caring about how successful other people are.
You do not like where you are at in life. What are YOU going to do about it? You need to look forward not backward. Stop being passive about your future. What are your actual options?
You‘ve been in school for five or six years. There has to be someone you can go to for career advice. Go talk to them.

 

[Avaro667]

I find it hard to believe that in a place where there are no electromagnetics jobs that there would be physics jobs.
Stop caring about how successful other people are.

Well that's what i thought too . But physicists here work mainly as software engineers , physics teachers , medical physicists and also as telecommunication engineers .

I'm ok with software engineering but as long as it is related to physics ( which is not possible here ) . While telecom companies hire physicists and EEs , i really doubt they utlilize their background . Most of them are in a management position probably . Energy related EE jobs on the other hand are in high demand , but it wasn't my concentration ( telecom engineering is ) .

You do not like where you are at in life. What are YOU going to do about it? You need to look forward not backward. Stop being passive about your future. What are your actual options?

I'm trying to but i don't know , i feel that whatever i choose to do next will be wrong or i might regret it . Maybe i should just back off a bit until everything is more clear ?

You‘ve been in school for five or six years. There has to be someone you can go to for career advice. Go talk to them.

I'll give it a shot i guess , nothing to lose at this point .

 

[symbolipoint]

Still at least in engineering i see people being able to tackle really hard problems with very simple math and lots of diagrams . And they seem to have a fairly good understanding of the problems .

Beware of letting that mislead you. Talent may vary, but everybody in the party needs to study. Some people would like to fool themselves by skipping the drawing and labeling of diagrams and setting-up expressions and equations. Such skipping will NOT be recommended for most people in the party.

 

[symbolipoint]

I wasn't refering to someone specifically .

That can be said as "third person, impersonal".

 

[symbolipoint]

I am at about post #25, #26, although been skipping many long parts on the way.

In case this was not expressed plainly in the topic, Mathematics is necessary as a language for Physics.

 

[symbolipoint]

So i'm almost stuck with a degree i don't know how to utilize here ( because there are no electromagnetics related jobs ) , and i possibly wouldn't like how the job market would expect me to utilize it . I guess that's why i did this question in the first place . I feel like all my effort learning all this hard math and physics concepts so far is pointless . On the other hand i feel too old to go after a physics degree ( i could become a teacher for example , it would make me happy and there are relevant jobs ) .

That is the most important identity for the started topic, and should help in knowing what help you need.

 

[symbolipoint]

I find it hard to believe that in a place where there are no electromagnetics jobs that there would be physics jobs.
Stop caring about how successful other people are.
You do not like where you are at in life. What are YOU going to do about it? You need to look forward not backward. Stop being passive about your future. What are your actual options?
You‘ve been in school for five or six years. There has to be someone you can go to for career advice. Go talk to them.

Main point is, he both needs and wants help. But this is not so much to tell him what job he must take.

Avaro667, ask yourself exactly what jobs and what exact duties and actions do you want to do as your employment. Is your education aligning with some of those?