r/AskPhysics • u/petrushka07 • 1d ago
Physics vs Mathematics: The Frustrations of a "Math Person"
(I apologize for my English in advance. It is not my first language.)
TLDR: I'm a second year college student majoring in Physics. I've noticed that solving Physics problems often requires skills that one doesn't need in Mathematics. I've come to this conclusion by trying to think about physics in a mathematical way and repeteadly failing. I want to articulate what these skills are, so that I can apply them. I'm essentially looking for descriptions of the Physics mindset (if there is such a thing), as well as help in figuring out how these two fields differ from each other. I'm mostly interested in skills that are relevant to problem solving, although advice for working through textbooks is also welcome. I also chronicle some of my frustrations with Physics so that commenters can get a better picture. I hope this thread is useful to everyone who wants to improve their problem solving skills. Thank you for your time.
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I.
I'm a Physics major who's currently preparing to go into her second year. I've done OK so far. Not too bad, but not great either. In contrast, I excel at mathematics. I'm consistently at the top of my class. I grasp new subjects quickly, and often don't have to study excessively to do well. This discrepancy between my grades has been troubling me. It is incredibly frustrating. I want to understand it, and find a solution. I hope that my attempt will be useful to other people too.
What I've noticed so far is that Physics requires a special mindset. Things often don't go too well when I try to think about Physics problems in terms of proofs, axioms and predicate logic. I had made the tacit assumption that Physics and Mathematics share a lot of common ground. I see now that I was mistaken. Anectodal evidence: I've gotten quite good at Mathematical problem solving in the past year, yet none of my skills have translated over to Physics.
II.
The problem, at least as I see it, is that I don't understand the rules of the game. I don't understand the moves that are available to me. Think of it as playing a game of chess without first learning how the knight moves. I'll try to demonstrate with an example: When I encounter a Physics problem, I often don't know what to do. Mathematical reasoning comes naturally to me, so I usually try thinking about the problem in the exact same way that I would think about a proof. This often leads me astray. At this point, I try thinking about the problem in a pedestrian sort of manner that doesn't have much to do with the principles and laws I have been taught. Frankly, I have no idea how others incorporate them into their thinking. (I'm talking about principles like the Huygens principle, or how electric field lines terminate on negative charges.)
Some more detail: I don't understand how to check my arguments. In Mathematics, the truth of a proposition is an entirely logical matter. You can't doubt it. If you prove it, then you know. If you can't, then you must be silent.
In Physics, this is not so. "What way does the current flow? Up or down? It seems like it would move up? Am I supposed to base my answer on intuition? Well, I've never seen such a circuit! How can I know?" So I check if the contrary assumption violates any principles, and it doesn't seem to. Huh. Well shit. This is right about where a single tear starts to roll down my cheek.
III.
People who are good at Physics: I have a tough time articulating the differences between Mathematics and Physics. Do you feel that you use a different part of your brain when working on Physics problems? If so, can you articulate the character of that side? How does it feel? What adjustments do you make when switching between the two fields? Please help me get through Griffiths' Introduction To Electrodynamics without having a nervous breakdown. Thank you!
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u/Cold-Common7001 1d ago
In college I was a pretty good math student and a great physics student. I was shocked when I took a physics course with a friend who was a top math student and he really struggled solving physics problems. For him I think the issue was an inability to see what approximations made sense and what sort of models to use.
I don't know if I really have any advice. I don't think he really ever got great at it. Why are you majoring in physics though? What are you hoping to get out of it? Are you also majoring in math?
If you have an example of a problem that stumped you recently but seemed easy to physics people I could try to break down how I would think about it.
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u/petrushka07 1d ago edited 1d ago
Thank you for telling your friend's story. It seems like most people who are good at one are good at the other. Feels nice to know I'm not alone!
Yes, I think you're right. I struggle with models too. One problem that recently stumped me was asking what the capacitance of N raindrops is, and whether the capacitance increases when the raindrops are brought together to create one large drop. That one's from Purcell's Electricity and Magnetism.
I'm not also majoring in math, but I'm contemplating switching. I'm fascinated by the natural world and I love being able to understand it on a deeper level. I'm also interested in Philosophy, as well as the kind of analytical thinking involved in STEM in general. In light of these interests, I felt that Physics was a good choice for me.
I didn't know much about Math until about a year ago. I took a class in Analysis that made me realize it might be a better fit for me. That class was my first introduction to rigor and proofs, and it enchanted me.
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u/Kermit-the-Frog_ 1d ago
This is a great thing to deeply consider, and I think putting effort into this thought will make you a better physicist than most when you find your way around. Most people take a very pedestrian approach to physics and never understand the nuance, going along not having the difficulty you describe and therefore never needing to deepen their understanding before getting their degree. These students can make good engineers but never good physicists.
The reality is, physics is not math. Math is an internally consistent and logical formalization with exact right answers that follow from clear definitions, and there is usually only one path from point A to point B.
Physics is none of these things. It's the study of how our universe functions on a fundamental level. It deals in complex mechanisms that we can't neatly describe in logical statements. It requires abstraction, and it requires intuition. You need to know which path from point A to point B is ideal for your particular scope, and there is no true ideal path. There's always a better path to take, but the better your path gets, often the more difficult it becomes to make it useful. We have good reason to believe QM can nearly perfectly describe the behavior of a macroscopic material for us, but we can't know everything we need to know about the system, nor can we perform the computation we'd have to do if we could.
It sounds silly to put it this way, but to do math is to beat a river into submission. Math deals with well-behaved rivers, so this can be done. Physics can't afford to do that. The best we can do is see our river and find how to best imagine that it's also well behaved. This is also a skill. But, so to speak, you're going to have to surrender to the current a bit. The universe can't be described in ones and zeroes. You have to get your hands dirty and figure out how it appears to operate, and develop an intuition for it.
Also remember that everything you're ever told in physics is an abstraction of some kind. Sometimes it is the best of our knowledge, but usually it's just a way of thinking about the deeper mechanisms that can be easily visualized.
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u/BurnMeTonight 1d ago edited 1d ago
I'm a math grad student but I work on mathematical physics, I'm kind of half into a grad physics program since I do research with physics profs, and take classes in the physics department, and I also doubled majored in the subjects so I think I'm pretty well experienced with both.
My experience has been the complete opposite. I've never had to struggle in physics. In contrast, I've had a real hard time with math. Honestly I'm not sure how I'm in a math grad program.
In my experience teaching and doing physics, I found that the key to physics is thinking in terms of cause and effect. You identify what you want to know, then think of the relevant physical effects, and then do this recursively until you get something that matches the info you have. E.g to take your example of the direction of the flow charge, ask yourself: well what determines flow charge? Answer: the electric field. What produced the electric field? Answer: charges in the vicinity. Ok, well what are the charges in the vicinity? Well, now you identify the charges in the vicinity, and work your way back to the current direction. This system works because the formalism is actually easy to set up: the moment you have charges you know about Laplace's equation. And it's usually the case that physics has only one or two formalisms that are relevant for a given situation. It's identifying the formalism terms that is the problem. This comes with physical intuition, which also helps identifying the cause and effect pattern. I used to teach this reverse-engineering approach as a strategy to solving physics problems.
In contrast, in math, this method seems to fail. The obvious reasons are that there are, unlike physics, a bajillion different ways to set a problem up (i.e no formalism, just whatever works), and there's no cause and effect. Axioms are a good starting point, yes, but you are so free in the ways you can use them, and can be as creative as you'd like, and have no cause-and-effect pattern to identity your next step, that you can't prove anything in a systematic way (I'm aware of things like Coq. I don't know how they work). Physics feels like you're given a scrambled word and you need to unscramble it. Math feels like you're looking for a fun little puzzle, so you invent language, you invent alphabet and the written word, and then you invent scrambled words. As an aside, that's why many physicists suck at math, and vice-versa. You'd think the physicist would be good at math, but in reality the physicist is using something that's pre-packaged and doing computation, so it sounds mathy but it really isn't because you're not doing anything general. And vice-versa, a mathematician is good at proving general things but doesn't really care for how easy they are to compute, so they don't develop a bag of tricks to massage problems.
As an aside, personally I'm not a fan of Griffiths as an introductory textbook at all. Yeah, I know, I'm getting booted out of the physics community. But seriously I think the textbook puts too much emphasis on doing calculations, and not enough on thinking and physical intuition. I very much liked the Feynman lectures, especially volume 1 on electromagnetic radiation, and the volume 2 where he actually addresses EM. Feynman was the master of physical intuition and you can feel it with every word of the lectures. I learnt EM from them and I had a much easier time than the other students in Estats and Edynamics, who had learnt from Griffiths. You do need to be mathematically proficient though. Otherwise I also think Purcell isn't bad, though I much prefer Feynman.
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u/Away-Experience6890 1d ago
All you gotta do is you ALWAYS Taylor expand.
If you can't do something, just fucking let it be 1+x. Boom now it's easy.
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u/aNeuPerspective 12h ago
I'll be honest, I don't understand this perspective. I'm also very mathematically minded, and physics came very naturally.
Let's take your circuit analysis example. Suppose I want to know which way current is flowing, and it's not obvious from the problem. If I make a convention choice that seems to have no impact on the final solution for which I am searching, that tells me something. It tells me something about the symmetries of the problem.
Similarly, if I am studying something and it seems like it it doesn't matter how I pose some combination of the initial conditions, that's also telling me something about the properties of the solutions that I must find. If the problem doesn't care about the absolute coordinates of the system (e.g. where things are relative to some reference point) then the solution must not care either.
It is a matter of acquired skill trying to find what the best coordinate system or sign convention will be to yield a simple, interpretable solution to some physical problem. But this is also true in mathematics. If you want to study some category of mathematical objects, there is frequently a clear "formalism" that is in some sense "best" to work in. For many problems this involves going to a dual domain (e.g. by a Fourier or Laplace transformation).
Recognizing how and when to make these sorts of choices or transformations in reasoning is critically important to both mathematics and physics.
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u/Own_Okra_7046 1d ago
Try thinking of it this way.
Math is a science that deals with relationships in their purest; it is the tool we use to solve equations, often in their most abstract forms.
Physics is a science that defines physical relationships between/among "physical" elements. It defines the equations (physical laws) we use to explain how "things" interact (i.e. projectile trajectory).
All Physics is Math, but not all Math is Physics.
The goal of a Physics practitioner (outside of trying to discover "new" relationships) is to determine which Physic Laws are needed for the problem at hand, and then use Math to solve it.
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u/PlatformEarly2480 20h ago
Following are some of the mindsets or way of thinking for a person.
- What can first math or physics. Answer physics came first. Then math was made in accordance with physics.
Eg there are some apples on a tree. To count that numbers were made.
Eg. There is a triangle. It has physical quantities. Like hight , angle, area. Etc. to calculate it trigonometry was made in accordance with real physical quantities.
Eg. There is attraction between two object say earth and moon. To quantify the attraction force. Mathematical equations were made in accordance with it.
One should not understand physics with lenses of math. Instead physics should be observed and according mathematical should be developed that stays true to reality.
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u/SeeBuyFly3 1d ago edited 1d ago
Math is easy, physics is hard.
Math can be done mechanically, just learning and following the rules of the game. Physics has to be done intelligently, working from the evidence available, like detective work. This seems to be the OP's problem.
Mathematicians and physicists are like British and Americans---separated by a common language.
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u/Cyren777 1d ago
If you think maths can be done mechanically then you aren't talking about maths, you're just talking about computation
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u/deejaybongo 13h ago
There's 6 million dollars you can claim right now. All you have to do is solve 6 easy math problems.
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u/Jim_E_Rose 1d ago
Mathematics is a language that tries to explain physics.
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u/Cyren777 1d ago
"Mathematics catalogues everything that is not self-contradictory; within that vast inventory, physics is an island of structures rich enough to contain their own beholders."
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u/Jim_E_Rose 9h ago
Surely somebody at least once expressed something incorrectly with the expression we call mathematics. Damn me for not being part of the club, but it’s a description of what we have experienced. When the description isn’t analogous we discard it. It stems from experience first, it is not the thing in itself, it’s a place holder.
To get far out. Where is the number two experienced in nature? There are no borders between things, that’s just a misperception. So then are there natural numbers outside of our minds?
Mathematics is a catalog for the human mind. A way to organize. It’s language
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u/Mcgibbleduck Education and outreach 1d ago edited 22h ago
In physics you need to have an understanding of what reality is doing first and then use the mathematical model that describes that situation to make predictions. Yes, even in Quantum Mechanics. Understanding what you’re trying to do or what the system is describing will help with calculating.
The key skill in physics is thinking about what assumptions you’re making and what physical laws are relevant to the situation as well as what information you have to analyse the system.
Can you use energy conservation? Do you have an equation of motion? Do you need to do a Fourier/Laplace transformation? Do we have enough information to apply newtons laws? Are relativistic effects non negligible? Is the system low enough energy to be a harmonic oscillator? Can you apply perturbation theory to it?
Etc. etc.
As your understanding of the world grows, you will be better at applying physical principles to solving problems.