Skills Difference between Theoretical Physicists and Mathematicians

Question

As I am preparing to start my PhD in theoretical physics, I have come to the realization that the more math you know (as a physics person), the better. But, I also realized that my physics brain is sometimes struggling to fully appreciate the subtle points in abstract math, in part because I was taught as a physicist to not pay too much attention to mathematical details. I am currently trying to get rid of this bad habit.

However, I am curious to know, what do you think are some skills that theoretical physicists have (beyond facts, formulas etc.), that mathematicians would have difficulty learning? In other words, if a mathematician attempted to become a theoretical physicist, and since physics is described through a mathematical framework, do you believe that the mathematician would have a straightforward transition?

My Opinion: I think that theoretical physics is a combination of making good assumptions and mathematics. So I would assume that mathematicians would have a relatively hard time making sense of the assumptions in physics.

I am very interested to hear opinions from people that transitioned between the fields.

Answer 1

My opinion departs rather significantly from the main line seen in most of the answers here, so I'll write another answer.

I have a PhD in physics and transitioned between physics and math many times, finally settling down with physics. I always say, jokingly, that there are no two fields as far away from each other as physics and mathematics. While this is obviously an exaggeration, let me tell a true story that happened to me in 2012:

I decided to take a course on measure theory while transitioning to the applied math program. I liked (and still like) mathematics a lot more than physics, loved the teacher and the subject, and was finally able to understand what was going on behind the Lebesgue integrals used in quantum mechanics. The first exam contained a question that looked absolutely trivial to me, regarding properties of a set in view of Carathéodory's criterion (if I remember well). My solution was only three or four lines long. Once I finished and left the room I met with math students arguing about how hard that question was, to which I replied "well, you know that the set does this and this, so there's the theorem that allows you to do that and that". Everyone stopped talking and looked at me, so I felt like a genius. Then one of them said "of course, but did you prove that the set exists?".

I got a zero for this solution. The teacher knew I was coming from physics, so he left a comment: "no pieces can be left hanging". It took me some time to understand that mathematics is about flawlessly connecting A and B, often with little regard to either A or B - the connection is the most important thing, and needs to be made with no pieces left hanging. In physics one is also connecting A and B, but often little regard is given to the connection - the focus is mostly placed on A and B. The mathematician will therefore spend their life perfecting their connection skills (think of a gardener adorning and taking care of a path with beautiful flowers), while the physicist will dedicate their time to coming up with interesting events that seem connected (think of an explorer looking for the highest peak in the middle of a dense forest).

Epistemologically I believe these differences stem from the fact that physics is a science, while mathematics isn't. Mathematics is not concerned with nature (although the mathematician might be); it is concerned with following a logical path (any path!) and figuring out its consequences. These consequences might be aligned with something useful to the study of nature (like the case of many PDE's or ODE's), but can also be completely disconnected from reality (like some weird field in propositional logic where negation is not defined [1]). This means that the As and Bs from the previous paragraph can be literally anything in math, while in physics they will most of the time be connected to an experiment [2], with the whole process fitting within the paradigm of the scientific method (which is irrelevant in math). Physicists, therefore, have a luxury unavailable to mathematicians: They can cheat and circle around a rigorous proof by directly comparing their guesses to outcomes of experiments. In mathematics the equivalent of this process would be to declare that your statement was elevated to the status of theorem by the will of a higher power and that's it, period.

Practically, a big consequence of these extremely different approaches is that mathematicians understand a few things in great detail, while physicists tend to have a much wider view (albeit coarse grained). A mathematician will also become stuck once the desired level of rigor cannot be sustained, and will waste time with minor details that a physicist will often skip over due to some glorified, hand waving argument that cannot be put into rigorous form. The physicist, on the other hand, will get stuck trying to connect events that, after talking to a mathematician, turns out could never be connected due to some obscure theorem no one knows about in physics.

---

[1] Funny enough it's hard to find a field in mathematics that is not relevant to physics, no matter how useless it looks. Turns out explaining nature requires a very high level of abstraction.

[2] There are many physicists working with As and Bs that are not directly connected to any experiment. Some of them are very close to mathematicians, but most of them are not. They are still using the often lax lines of justification adopted in physics, but now dealing with more abstract matters.

Answer 2

This is probably a better fit for Mathematics Stack Exchange or Physics Stack Exchange, and might be too opinion-oriented anyway, but for what it's worth I'll give some comments about my experience of mainly studying pure mathematics (i.e. not any mathematics-for-physicists courses [*]) with substantial physics taken both as an undergraduate and during my first (of several) graduate enrollments. However, I've never actually transitioned between the fields.

[*] I have taken an undergraduate "separation of variables" PDE course and undergraduate complex variables, but no vector analysis (I often wanted to remedy this, but never really did) or numerical methods.

When reading physics texts I had much more difficulty in determining whether some random statement being made was supposed to be taken on face-value as an assumption/comment/observation or to be taken as something the reader is expected to logically deduce from previous results.

I also found that certain things that are of primary concern in mathematics (e.g. manipulation of infinite series) were almost always passed over without comment in physics texts. It would have helped me a lot if physics authors had sometimes inserted a few words here and there, such as "formal series manipulation results in", to let the reader know at what level of rigor the reader is expected to use in following a derivation of something. Moreover, I remember one explanation given by an author for truncating an infinite series to get an approximation was that the n-th term approaches zero -- had the author never heard of the harmonic series that is so prominent in every beginning calculus text? This puzzled me for a long time until I realized it was a Taylor series expansion, and the error in dropping (for example) all terms past the 2nd degree (quadratic order of smallness) is roughly speaking proportional to a cubic order of smallness. So the approximation was OK, but the reason for why it was an approximation was very misleading, at least to me initially.

I also had a lot of difficulty early on with the fact that words in math often have very specific meanings (e.g. series is a sum, sequence is an ordered list, vector is an element of a vector space that has been explicitly defined, etc.) and in physics these meanings were often different and certain distinctions were ignored without comment.

Answer 3

Here are my opinions based on my experience as a student and as a professor.

For as long as I can remember, I always wanted to understand how things worked at a fundamental level. Naturally this led me to study physics in university. I had an epiphany in my first abstract algebra course: the process of starting from a collection of axioms, building a rich theory out of them, and the ensuing sense of certainty gave me immense intellectual satisfaction. At the same time I was struggling in my physics courses, things went too fast, and my understanding never felt quite rock solid. I switched to pure math over the course of a year and I am glad I did.

Skill difference #1: Although it's not a skill per se to succeed in either math or physics you need to feel inspired and motivated by the field.

As a professor, I've had graduate physics students attend advanced math courses. Typically, such students struggled with writing proofs. It was not their fault though: unlike their math student counterparts, they had no previous proof writing experience. The ability to perform long and complex derivations, as is typical in physics, doesn't immediately transfer to proof writing.

Skill difference #2: Proof writing (as a skill) is important in mathematics and doesn't necessarily follow from being good in physics.

Another reason I quit physics was that I could not carry through the complicated derivations (I'm wired to make frequent calculation mistakes.) Also physics is taught the way it is for a reason: the underlying math of even the simplest situations gets difficult very quickly. Let me illustrate. I taught a mathematics course in Functional Analysis and, after one semester, I got to the point where I could discuss the theory of self-adjoint bounded linear operators. Now in quantum theory, the simplest model is the wave function for single particle moving in 1D, but the most basic operators, the position and momentum operators, are unbounded. Basically, I would have needed to give two whole Functional Analysis courses to adequately (for a pure mathematician) address the mathematics of the simplest model in quantum theory.

I can totally understand that a physicist may have different priorities. For a mathematician transitioning to physics, to do it right would requires a tremendous amount of effort. There would be just too much stuff to learn for me to make a transition.

Skill difference #3: The ability to correctly perform complicated calculations and go with the flow (I really can't think of a better way to say this) is crucial to doing well in physics.

It is because I lack this third skill that I have so much admiration for physicists. That said, there is no reason why a particularly talented person can't have all these skills and it is also wrong to think that there is a fundamental incompatibility between mathematics and physics.

For my part, over the years, as I pick up more math as a researcher, I frequently find myself revisiting concepts in physics with new insights. Each time I do this, I find physics even cooler.

Answer 4

You've already gotten a lot of answers, but many seem to be from the point of view of mathematicians, or people who do both mathematics and physics. This answer is from the point of view of someone who did a PhD and some postdocs in theoretical physics, and very strongly believes "physics is not math" and shouldn't be math.

Here are some skills a good theoretical physicist needs:

• Look at a real physical system and identify which features are key to understanding the system and which are second order details.
• Develop toy models that can be solved analytically, or approximately, and give insight into how those key features explain some phenomena.
• Understand the ways in which the toy model gives insight, and which ways the toy model oversimplifies the real world. And, understanding how complications in the real world that cannot be treated analytically affect the behavior of the real system.
• A sense of numbers and scales -- being able to quickly do dimensional analysis and mental arithmetic to determine if some effect is likely to be important or not.
• The ability to calculate -- correctly perform a long sequence of steps to start from first principles to the answer in a usable form. Calculation is different from proof -- for example, the goal is usually to understand in detail a special case that is relevant to a specific physical problem, rather than make a general statement about all cases in some class.
• Intuition and ability to guess new laws. New scientific laws cannot be derived logically from first principles, they need to be guessed, usually based on keeping some physical principles from previous theory and modifying others in a specific, motivated way.
• Knowledge of phenomenology (specific experiments and bounds) and a sense of which bounds/results are the most likely to be relevant to constrain some new model.
• Ability to keep in mind multiple possible contradictory explanations for a currently unexplained phenomena, and maintain a "scorecard" as new discoveries happen and evidence builds for or against different theoretical explanations.

Now, I suspect that mathematicians also do many of these same kinds of things when they are working at an intuitive level before they settle on a particular theorem they want to prove.

I think much of the difference comes down to what physicists are trying to communicate to each other, vs what mathematicians are trying to communicate.

I think mathematicians want to convey absolute certainty about a logically well defined statement, with some kind of optimal tradeoff between having general assumptions and a powerful conclusion that is determined by the specific field being studied. The "absolute certainty" means a proof is necessary, and finding this "optimal tradeoff" means that a lot of work is put into defining precisely the assumptions that are being made.

Physicists are more interested in describing Nature than in making absolute mathematical statements. A general assumption about Nature is that various functions (like density of a fluid) are smooth (for example as a function of space and time). Therefore, when doing calculus, often proof the exact assumption on a function needed to perform some step is justified is not so important, because functions are usually "sufficiently smooth" than any reasonable assumption is going to end up being satisfied if you check it.

I understand why that probably sounds maddening to a mathematician. But there is a physical reason for it. Suppose that it turns out that this assumption is not correct, that is, that the density of a fluid behaves in a way that we can no longer assume it is arbitrarily smooth. From a mathematical point of view, we can assume that we are working in some well-defined mathematical framework, like the Navier-Stokes PDE, and carefully study solutions which do not obey typical smoothness requirements. From a physics point of view, the lack of smoothness may be pointing to a breakdown in the mathematical model we are using to describe the situation. It may point to the fact that in this case, we cannot treat the fluid as a fluid, and may need to consider the particle nature of the underlying atoms, governed by some other equations. Of course, you'll find exceptions -- physicists who are interested in the mathematical details. But, historically, understanding approximate solutions and physical principles has proven much more fruitful than understanding exact solutions or physically unmotivated but logically possible pathological cases.

Another version of this idea is that one of Hilbert's problems from the early 20th century was to "formalize physics." Most theoretical physicists I know would consider this a waste of time. The reason isn't that it's bad to have a logical foundation for your subject. The reason is that the physics Hilbert was talking about was classical mechanics. All of that physics was replaced, since the he proposed that problem, by quantum mechanics. We never know if our theories are complete, we typically believe our theories are effective theories that are superseded at some deeper level of understanding, and may even contain logical inconsistencies that we punt to a future unknown theory to resolve (like singularities in GR). So, from a physicist's perspective, it's much more interesting to calculate consequences of theories in regimes where everything is well behaved, then to consider the kinds of "pathological" cases you would need to make the theories fully mathematically rigorous, given that these cases might not be of interest physically and even if they do represent something that occurs in Nature, might need to be treated in a different framework.

To try to summarize this in one overly-simplified sentence: I believe a lot of the difference between math and physics boils down to the fact that mathematics is deductive -- it starts from a known, logical starting point and studies the consequences in a logically rigorous way -- whereas physics (like any natural science) is inductive -- the basic principles are always provisional and most of the work is either trying to calculate consequences of the basic principles in specific situations or trying to find holes in the currently accepted framework to get clues on what might lie beneath it.

Answer 5

I sit (as an applied mathematician) on a few PhD committees in Physics and I would say the line is quite blurred, many times. The main difference, I would say, lies in the "requirement for proof". As a mathematician, I want things to be nice and tidy. My physicist colleagues focus more on results: who cares about ensuring all cases are covered, if the one that's of main focus to you is.

So I don't think it is very hard for math people to do work in theoretical physics, if they can overlook some untidy ends. And it's not hard for theoretical physicists to do work in math if they dot their i's. As a matter of fact, you often find a high level of interactions between researchers in both fields, locally, such as sitting on one another's students committees, attending seminars, etc.

Answer 6

As a math person, I do get pretty frustrated trying to read physics material by all the unstated assumptions and vague definitions floating around the work (much as the OP says). Here's an article I found to be very insightful on the subject:

Redish, E.F., Kuo, E. Language of Physics, Language of Math: Disciplinary Culture and Dynamic Epistemology. Sci & Educ 24, 561–590 (2015). https://doi.org/10.1007/s11191-015-9749-7

In this paper, we suggest that a fundamental issue has received insufficient exploration: the fact that in science, we don’t just use math, we make meaning with it in a different way than mathematicians do. In this reflective essay, we explore math as a language and consider the language of math in physics through the lens of cognitive linguistics. We begin by offering a number of examples that show how the use of math in physics differs from the use of math as typically found in math classes.

The standout example is a fairly simple question to write a function which most mathematicians answer one way, and most physicists a different way. This is from an online primer by Dray and Manogour (2002), now called Corinne’s Shibboleth (and more commentary by Gethyn Jones here):

Suppose the temperature on a slab of metal is given by T(x, y) = k(x ² + y ²) where k is a constant. What is T(r, θ)?

There's a somewhat related question on SE Math Overflow regarding Examples where physical heuristics led to incorrect answers. One of the answers sparked this incisive comment by Noah Snyder:

This is a good example of what seems to be quite common with results based on physical intuition: they're true in many cases but ultimately rely on an additional technical assumption. This is the tradeoff inherent in having a lower standard of rigor, you get better results faster but often then have to go back and rethink the exceptional cases.

For me, having physics material that doesn't bother to mention what those exceptional cases are, or what the prerequisites or limitations are, is overwhelmingly hard to digest. Or things that are literally mathematically false statements. I can't even tell if the authors knew but didn't bother to express them properly, or were actually unaware of them altogether. Either way it's hard to know to what extent we can trust a work like that.

Answer 7

A question near and dear to my heart. I arrived at college self-taught in several subjects which, except to those with self-discipline of a type unknown to me, meant my skills were strong in those particular subareas that caught my imagination and much less sturdy in others. I qualified for advanced placement in Physics, but had to grind through Calculus in spite of being bored with significant chunks. Still, I pushed on through both in Physics and Math. I found my interests in Math gradually increasing beyond those in Physics but the tension between the disciplines was a serious issue for me. As mentioned above, I was expected to make essential unstated assumptions in Physics and to rigorously justify every step in Math. It wasn't until after I graduated that I really grasped that I should have explicitly developed two different methods to approach each field.

While I was wrestling with the problems of trying to use a single framework for both Physics and Math, I was also expanding on my computer hardware and software skills, also self-taught. This field was much more direct than the other two and so less painful. I still love Physics and try to follow enough Math to respect a deeply felt love for pure Math as a spectator, but I eventually wound up combining much of my STEM skills and passion working in the space industry, particularly on lunar landers.

I suppose it is too much to expect educators to recognize oddball students struggling with this kind of cognitive dissonance; I think not too many people have so much interest in both subjects that they get into this kind of trouble. Still, I wonder if I couldn't have felt less of a drowning sensation if someone had spent an hour helping me to understand explicitly to think this way over here and think this other way over there.

Answer 8

Since no one seems to have defended the physics approach yet let me just add that giving up some rigor does have its advantages. Theoretical physics is usually very much ahead of rigorous math. The most famous example is of course that the path integral in quantum field theory which is rarely well defined. Yet when we perform logically seeming manipulations on it we manage to make precise and accurate predictions that have never turned out to be wrong. (Except in cases where they were wrong ;), but in those cases we have again - in our physicsy way - learned why the specific manipulations were incorrect.)

There is more of an attitude of doing something because it looks right and then checking the result (against experiment or consistency with principles we hold dear). This probably comes from the fact that historically physicist were able to check any computation/result against experiment. Of course modern theoretical physics does not always have that luxury anymore (mainly because it has outraced experiment quite a bit with little expectation that experiments will for example reach the energy scales that are of most interest to (high energy) theoretical physicists.

If you are interested in finding the right answer (rather then proving that it is the right answer) the physics approach is usually the faster one. (Of course you might sometimes, I would say very rarely, be eventually surprised that you found the wrong answer.)

---

I see you wanted replies from people who switched between fields and I actually did not. (Although I am now in a Math department, I'm very much still a physicist). Perhaps there is something telling that all the answers from people who moved between fields say they now identify as a mathematician.

Answer 9

I have not studied math or physics at anything like an advanced level. But I have read a lot of biographies of mathematicians and physicists. Of course, physicists vary. But quite a few of the very best have a visual and spatial intuition about real objects that many mathematicians don't have. Mathematicians are better at imagining things that don't exist, while physicists are better at imagining things that do exist!

Answer 10

There are many great comparisons given, and I would like to add another along side them. In physics many assumptions are often made to simplify things. That is necessary, and perfectly reasonable. HOWEVER, one thing that has been driving me crazy is those assumptions are often not listed and tracked. That makes it much easier to forget something is an assumption. Also, math is valid if it holds together logically. In physics reality can break your beautiful model.

Answer 11

This answer isn't complete and is only meant to be complementary to the other answers here.

One thing I have noticed is that physics curricula, on average, do a much better job emphasizing geometric visualisation than mathematics curricula. Mechanical engineering curricula, are similarly strong as physics programs for this.

IMO it's really a shame how poor mathematics curricula tend to be when it comes to geometry.

Mathematics programs tend to emphasize the literal, so they tend to attract people who are verbal thinkers, and scare-away geometric/spatial thinkers. This is much in-line with the criticisms Temple Grandin has of curricula in western nations.

Answer 12

I report a conversation I had with an exceptionally talented physics professor.

He said, at some point you realise that it is not possible to know everything and the skill set to build proof in mathematics is very different from the kind of intuitive understanding of phenomena you need to do physics. There is only one person who knows everything and is (insert name of the head of department).

He would also say that one has to have the courage to drop irrelevant terms like flies (eg in an expansion) and make bold approximations without fear.

We're talking about a guy who does work with very abstract topological stuff in soft matter physics, for context.

Answer 13

I have not looked at all the all the answers, but I will give my story. The high school and undergraduate skills are about the same. It is not until you take the advanced undergraduate or graduate courses in real and complex 'analysis' that you know whether you want to be a physicist or a mathematician. First year graduate courses in electromagnetism and 'mathematical physics' will also help you decide. I think I liked math more than physics until I took courses in 'analysis'. The point is that the distinguished math teacher felt insulted when I asked, "What is the point? How can this be used?" A good example is the three names Poincare, Einstein, Minkowski. All three independently derived special relativity within a couple of years of each other. Poincare was the first. He wrote down most of the important equations, which is why the name 'Poincare group' is used. He didn't care much about using it. As you have guessed, he was a mathematician. Einstein wrote as a physicist, using gedanken experiments as examples. A problem with this method is that all the pseudo paradoxes of special relativity are based on gedanken experiments. Minkowski was a mathematics professor who possibly would have been the greatest physicist of the 20th century had he not come down with appendicitis at an early age. He invented and applied Minkowski space to develop the special relativity we use today. However, apart from him, the more modern example of von Neumann, and possibly Dirac, there is a wide gulf between physicists and mathematicians. From what you have written, you sound like a physicist. It would be a good idea for you to audit, rather than take, the mathematics analysis courses, because you will find the detours into unimportant theorems will annoy you. As a physicist, start with a purpose, develop it and work it out, and then present your conclusions.

Answer 14

In my experience as a mathematician who has started to study physics at various points, there is a level of "physical intuition" that I lack compared to my colleagues who studied physics the whole time. Like, I can see all the equations, and understand them in a vacuum, but it's unclear to me which situations warrant which equations.

Another similar issue I have is trying to pin down the exact "axioms" or assumptions that are being made in each context. I tend to go down rabbit holes chasing details that were never there to begin with, because (as you say) physicists tend to ignore them.

My whole ability to understand math comes from my ability to trace everything down to its foundation, or at least a foundation that I'm comfortable with; in physics, this often eludes me, causing me to feel as though my understanding of the subject is fractured and shaky.

Answer 15

I think there's a big difference between mathematical detail and pedantry. Very often the validity of a theorem relies on many details, and one has to be very careful with the assumptions. Sometimes as a physicists we assume things that may be obvious, but it's better to spell them out. As concerns the question, pure mathematicians with no training in physics may have hard time in developing physical intuition, notably if they don't know the fundamental principles of physics (which are legion). Mastering these principles and developing the physical intuition requires a lot of thinking and training. Theoretical phycisists may invent whatever maths are needed to solve a problem, even if those maths are not superrigorous or lack formalization. This is the way physics has been done for years or centuries. In my opinion, (hyper)formalization of a physical idea sometimes hinders the progress of it. I think that once the general framework (or theory) proves to be on the right path, one can start formalizing things. Otherwise it can be a total waste of time from the physics point of view. Moreover, if one has a strong physical intuition one can guess the answer of a problem even without solving it in detail. And even if we make many mistakes in a particular calculation we can get back to them and correct them if the outcome contradicts the physical intuition. I agree that the more maths a theoretical phycisist knows the better, but knowing all the maths of the world is NOT a guarantee to become a good physicist.

Answer 16

It is not so much a difference in skills, but the philosophical approach that determines their use.

A physicist would primarily be interested in questions about what reality is and how we perceive it.

A mathematician would primarily be interested in how we reason about things: in abstraction and logic, in linguistic and non-linguistic mental constructs.

This difference in philosophy changes the kinds of questions one asks and the kinds of answers that are considered "satisfactory".

It also leads to differences in the aspects of the skills (be they computational, relational, spatial or some other) that one hones even when the basic skills are from the same set.

In summary, the difference is not so much in the skill sets, but in the value/onus placed by the "owner" of these skills on them. As a consequence, their use can be different.

---

Update: Looking through the answers, I realised that Andrew has given a somewhat similar, but more detailed answer. However, he has written it as a physicist, and I have written it as a mathematician.

---

An Example: Occam's Razor is commonly used in physics research and in mathematical thinking.

In physics, it would typically be used to remove things which have low impact on the phenomenon under consideration. To find out "which terms to neglect".

In mathematics, it would be used to remove hypothesis that are not used in a certain proof or in typical uses of a certain definition.

In the former case, the reality being described determines what can be "shaved off". In the latter case, the argument or calculation determines what can be excised.