The math level of computer scientists and physicists

[Ricster55]

I am wondering, what is the math level of your average CS or physics major? Like how much math do these two take beyond stuff like the Calculus sequence, differential equations and linear algebra?

I was having a discussion with one of my old math professors from my community college (I'm in a four year university by the way) about my major in the STEM field. I asked her about how much math do computer scientists and physicists take compared to a mathematician. She told me that the only important math classes that those two majors need are differential equations and linear algebra, and discrete math for CS majors (which I understand). She tells me that all those math classes (Calculus, Diffy Q, Algebra) are all baby stuff and that real mathematicians take/use Analysis, Abstract Algebra, Topology, etc and you won't be seeing CS or physics majors using abstract algebra or any other high level math course (complex analysis may be of some importance to physics majors). Linear Algebra is probably closer to what pure mathematicians do since its not just about matrices and eigenvalues, but proving abstract theorems if they are true or exist, she told me and I have taken a linear algebra class before so I understand.

But i'd imagine that some CS and physics majors take more higher level math courses. I think some CS majors who take advanced math courses take more purer math courses as they deal with logic and some physics majors take more applied math courses like probability or partial diffy-q, is that right?

Also, how important would abstract algebra be to a CS major? I've always wondered if that would be more important than say discrete math, which most CS majors take, and discrete math is very logical and proof based, just like abstract algebra.

 

[jedishrfu]

She gave you the straight answer. Of course you can always find someone myself included who has taken some pure math courses. However while I enjoyed struggling through them, I’ve never had to use them to do my job.

If you’re programming visual things or doing computer simulation work or even game programming then you might use vector algebra and rotational matrices or even real or complex analysis but again the pure math topics are not used.

One exception might be number theory and elliptical functions ...in computer based cryptanalysis though.

 

[MidgetDwarf]

I am wondering, what is the math level of your average CS or physics major? Like how much math do these two take beyond stuff like the Calculus sequence, differential equations and linear algebra?

I was having a discussion with one of my old math professors from my community college (I'm in a four year university by the way) about my major in the STEM field. I asked her about how much math do computer scientists and physicists take compared to a mathematician. She told me that the only important math classes that those two majors need are differential equations and linear algebra, and discrete math for CS majors (which I understand). She tells me that all those math classes (Calculus, Diffy Q, Algebra) are all baby stuff and that real mathematicians take/use Analysis, Abstract Algebra, Topology, etc and you won't be seeing CS or physics majors using abstract algebra or any other high level math course (complex analysis may be of some importance to physics majors). Linear Algebra is probably closer to what pure mathematicians do since its not just about matrices and eigenvalues, but proving abstract theorems if they are true or exist, she told me and I have taken a linear algebra class before so I understand.

But i'd imagine that some CS and physics majors take more higher level math courses. I think some CS majors who take advanced math courses take more purer math courses as they deal with logic and some physics majors take more applied math courses like probability or partial diffy-q, is that right?

Also, how important would abstract algebra be to a CS major? I've always wondered if that would be more important than say discrete math, which most CS majors take, and discrete math is very logical and proof based, just like abstract algebra.

I know someone who is about to get their PHD in physics, and the person's area is Group Theory. This person can pass as a mathematician. I think it depends how theoretical you get into CS/Physics.

 

[Crass_Oscillator]

Physicists typically know more math/understand the math more deeply than CS majors since they are more likely apply it to reality, but know less discrete math and algorithms, since their training does not typically include it.

Goofy math like topology, real analysis, abstract algebra etc are basically just academic subjects that exist for fairly contrived, formal reasons. Occasionally academics looking to get their jollies off will "apply" them to problems in physics, but usually with inscrutable or irrelevant consequences.

 

[Ricster55]

Physicists typically know more math/understand the math more deeply than CS majors since they are more likely apply it to reality, but know less discrete math and algorithms, since their training does not typically include it.

Goofy math like topology, real analysis, abstract algebra etc are basically just academic subjects that exist for fairly contrived, formal reasons. Occasionally academics looking to get their jollies off will "apply" them to problems in physics, but usually with inscrutable or irrelevant consequences.

As I stated before, I'd imagine that computer scientists know more of the abstract kinds of math that deal with logic and algorithms while physicists know more of the applied math topics that apply to everyday phenomenon. Those pure math courses (real analysis, abstract algebra, topology) seem like they would be relevant to say, a theoretical computer scientist (I would also like to add number theory, and data structures). While for a theoretical physicists, the advanced math courses that would seem relevant to them are (complex analysis, partial differential equations, probability, Fourier analysis, etc), and these courses fall into the more applied math (save for complex analysis). But I really don't know. All I know is that physicists use math to model physical and natural phenomenon, and computer scientists use math to understand and solve algorithms, make use of logic, computer graphics and all. So I think both are equally adept at using math.

 

[Crass_Oscillator]

The number of computer scientists with familiarity in topology, real analysis, and abstract algebra is a minuscule minority. In fact, even though they are a minority, there are probably more physicists who know about such things.

As far as mathematical aptitude, computer scientists in general are trained with significantly less 19th century math than physicists, and focus more on algorithms. I was once in an interdisciplinary computational biology program and the computer science students often hadn't even seen calculus since they took it as first years. Instead they were more interested in data structures and algorithms, which I guess you can call a different branch of math.

Both groups are skilled in different mathematical disciplines.

 

[RedDelicious]

Just a note on your terminology, the title of computer scientist or physicist or mathematician almost always implies a PhD in the subject. When you say physics major or math major, that means someone with an undergraduate level of education. It's a bit confusing when you keep switching between the two because they don't usually refer to the same group. It will help just generate more consistent responses by making it clear.

am wondering, what is the math level of your average CS or physics major?

As far as I know, for physics majors (undergraduates) the calc sequence, diff eq, and matrix/linear algebra are usually what's required for the degree and so I imagine that's where the average is. You get some exposure to more specialized topics in the mathematical methods course (using Boas's book for example) and obviously as it comes up in your work, but that's the extent of the full on math courses.

In my experience, however, a number of physics majors, including myself, also tacked on math minors, which only meant taking a few more courses. Of those, PDEs, ODE II, and probability were probably the most popular.

 

[newjerseyrunner]

Statistics is an important course from computer scientists. You need to be able to analyze and understand very large amounts of data, so you have to be good at modeling them.

 

[FactChecker]

CS and physics of the future are way too broad to give such specific answers. With quantum computers, theorem provers, code breakers, data mining, medical imaging, robotics, artificial intelligence, etc., there is a tremendous amount happening in the next several decades. To put a limit on the amount of math required would be wrong.

 

[QuantumQuest]

I also want to add that although in CS, Calculus, (some) Linear Algebra and Discrete Math is what is basically required, this does not tell the whole story about the complexity - especially in discrete math, that any CS major will have to deal with - particularly if he/she follows Theoretical CS. A quite demonstrative and important example is Analytic Combinatorics - (the link is to the reference book of 2009 by the leaders in the field, Philippe Flajolet and Robert Sedgewick, legally available for research purposes (non-commercial single-use)), which enables precise quantitative predictions of the properties of large combinatorial structures. Quoted from the preface of the book

With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps.

It is quite obvious that it is far more involved than the usual Calculus and discrete math that we use in Algorithms.

As already mentioned by jedishrfu, there is also nothing to stop you from taking pure math courses (I personally took Topology and Functional Analysis and a more thorough treatment of Linear Algebra). Besides the obvious beauty of taking more abstract math, in the long run, it will pay off in various even non-obvious ways.

 

[Ricster55]

Just a note on your terminology, the title of computer scientist or physicist or mathematician almost always implies a PhD in the subject. When you say physics major or math major, that means someone with an undergraduate level of education. It's a bit confusing when you keep switching between the two because they don't usually refer to the same group. It will help just generate more consistent responses by making it clear.
As far as I know, for physics majors (undergraduates) the calc sequence, diff eq, and matrix/linear algebra are usually what's required for the degree and so I imagine that's where the average is. You get some exposure to more specialized topics in the mathematical methods course (using Boas's book for example) and obviously as it comes up in your work, but that's the extent of the full on math courses.

In my experience, however, a number of physics majors, including myself, also tacked on math minors, which only meant taking a few more courses. Of those, PDEs, ODE II, and probability were probably the most popular.

What about for computer scientists? As far as I have seen in a few colleges with different requirements, it seems that discrete math is the only required math course. Some colleges I've looked at also require students to take Linear Algebra (especially those specializing in graphics) and two semesters of Calculus.

I was just thinking, would an upper level math course like abstract algebra be any use to a CS major, that is if CS majors decided to take additional math courses? My old professor mentioned complex analysis being of important to Physicists, that is of course of some physics majors take additional math courses? I'd imagine that if CS and physics majors decided to take upper level math courses, CS majors would tend to take purer math courses while physicists would tend to take more applied math courses, though applied math courses like probability and statistics are also important to CS majors.

 

[FactChecker]

I can't imagine a person actively doing physics on a job who does not know complex analysis fairly well. Same with differential equations. And statistics is fundamental for any experimental work.

 

[Ricster55]

I don't know why I think purer math is more important for CS majors while applied math is more important for Physics majors. Only because for CS, you need to be able to do logic and think logically and be able to do proofs (but then again, I'm not a CS major, nor have I taken discrete math, but I have taken a mathematical proof workshop class), while for physics majors, they have to apply the math they learned in order to describe and model some natural phenomenon, so proving stuff is something most physicists won't even bother doing.

 

[radium]

For the more advanced math, you will generally learn it as you need it during physics courses. For example, if you take quantum you will learn some things about group theory (relevant for discussing things like spin or symmetries), if you take general relativity you will learn about different geometry, etc.

Abstract algebra is relevant to many areas in physics. It is essential to understand groups when learning about symmetries in quantum mechanics (which are very profound), spin, crystal structures, molecules in chemistry, and many other things. It also has applications in computer science like cryptography (I’m sure there are many others). Number theory is of course very related to group theory.

Topology is relevant when studying things like defects and vortices, I think this may apply to some areas of fluid mechanics (you probably wouldn’t see that in undergrad.

 

[Dr. Courtney]

I can't imagine a person actively doing physics on a job who does not know complex analysis fairly well. Same with differential equations. And statistics is fundamental for any experimental work.

I don't know most of the material taught in most undergrad courses in complex analysis, only the stuff picked up along the way in diff eq and quantum mechanics courses. And most of the experimentalists I've known in my career are in the same boat.

 

[Crass_Oscillator]

There are people assuming that the typical physicist is a theorist, which is probably false. Experimentalists are the largest category, and their knowledge base is not the same.

 

[StatGuy2000]

The number of computer scientists with familiarity in topology, real analysis, and abstract algebra is a minuscule minority. In fact, even though they are a minority, there are probably more physicists who know about such things.

As far as mathematical aptitude, computer scientists in general are trained with significantly less 19th century math than physicists, and focus more on algorithms. I was once in an interdisciplinary computational biology program and the computer science students often hadn't even seen calculus since they took it as first years. Instead they were more interested in data structures and algorithms, which I guess you can call a different branch of math.

Both groups are skilled in different mathematical disciplines.

@Crass_Oscillator , when you talk about computer scientists, are you referring only to those who complete a BS in the field, or are you taking into account those who are pursuing a PhD program? Because whether or not computer scientists are familiar with fields like topology, real analysis, and abstract algebra will vary greatly between these two answers.

There are fields of research within computer science (e.g. theoretical computer science, machine learning, numerical analysis, etc.) that are very heavy users of mathematics and most people I know who are specializing in these areas have very advanced mathematical expertise, including strong familiarity in topology, real analysis, algebra, etc. (in fact, the majority of people who are pursuing grad studies in these fields often have a double major in computer science and math). There are other people who are more like the students you encountered, where the level of math background is less stringent.

So we need to be careful on not lumping all computer science students in the same bucket.

 

[SchroedingersLion]

I don't know why I think purer math is more important for CS majors while applied math is more important for Physics majors. Only because for CS, you need to be able to do logic and think logically and be able to do proofs (but then again, I'm not a CS major, nor have I taken discrete math, but I have taken a mathematical proof workshop class), while for physics majors, they have to apply the math they learned in order to describe and model some natural phenomenon, so proving stuff is something most physicists won't even bother doing.

This is wrong. First of all, physicists need to be knowledgeable in complex analysis, differential equations, Fourier analysis, even stuff like topology or group theory. This is certainly not less pure than the stuff a CS focuses on, which is mainly discrete mathematics.
Secondly, if you are a theorist, you need to be proficient in proving statements because otherwise you would not be able to derive new mathematical models of nature. At least you need to be able to see the validity of the statements in a model which can't be done if you can't even see at which point mathematically simplifying assumptions had been made.

In my country, Germany, a major in physics contains more maths than CS on average.

 

[FactChecker]

This is wrong. First of all, physicists need to be knowledgeable in complex analysis, differential equations, Fourier analysis, even stuff like topology or group theory.

I agree. So much of this mathematics was developed by physicists, for physicists. It's hard to imagine a physicist doing much without a working knowledge of it.

That being said, @Dr. Courtney points out that it can be learned "as needed" and that a math class may include a great deal that is not as useful. I think it depends on how much a person can learn on his own, without a class. And if there are courses for physicists, that would be ideal.

 

[Crass_Oscillator]

I was at a grad school with a fancy machine learning department and my experience was that the student's backgrounds varied widely, and I would not say that the majority of them had backgrounds in goofy bourgeois math ala topology or real analysis, although real analysis was probably the most ubiquitous form of mathematics sourced from the math department that wasn't applied. In fact, I was often the only one with such experience, and I was in applications (computational biology) not theory. I did not overlap much with the theoretical CS people so I can't judge them.

In your defense however, there is a very strong trend for machine learning students to develop more mathematical rigor and delve into such topics, more than I've seen in physics departments. The difference is between rigor and novel mathematical expressivity. Theory students in physics still, thankfully, see little use for mathematical rigor, but are more obsessed with mathematical novelty ("How can I apply academic math to a problem whether the problem calls for it or not, since I like convoluted academic math?") whereas my impression of CS students is that it was more about rigor ("How can I foist mathematical rigor on the algorithm, the irrelevance of mathematical rigor to successful techniques such as Nelder-Mead optimization or convolutional neural networks notwithstanding?").

Suffice it to say, whichever department you ask about, having experienced both, they spend too much time on the wrong math, but in different ways, so I became an engineer.*

*At least that's what I tell myself regarding why I became an engineer.

 

[FactChecker]

I think that there are many cases when a physicist or a computer scientist works, as needed, with a mathematician. Even Einstein, who was excellent in mathematics, got help occasionally from mathematicians. But I think that it is important to have a "working knowledge" of a lot of mathematics, even if one does not have complete rigorous, formal knowledge. That would certainly open a lot of doors to one that are otherwise closed.

 

[Crass_Oscillator]

I've worked with mathematicians; some of them are very useful, especially if they focus on numerical methods.

I just think this discussion should concentrate on the stuff that connects with successful applications, rather than focusing on the rarefied intellectual atmosphere of pure academia, where reality can be a distant figment. Otherwise, it's probably not relevant to the OP, since the majority of physicists are applied physicists (either in academia or industry) and the majority of computer scientists are applied computer scientists (either in academia or industry).

 

[gmax137]

And if there are courses for physicists, that would be ideal.

I remember learning complex analysis in the math department and then learning why anyone cared about it in my physics classes. Actually the mathematicians like if for its own beauty, something I was unlikely to see as a student.

 

[StatGuy2000]

I was at a grad school with a fancy machine learning department and my experience was that the student's backgrounds varied widely, and I would not say that the majority of them had backgrounds in goofy bourgeois math ala topology or real analysis, although real analysis was probably the most ubiquitous form of mathematics sourced from the math department that wasn't applied. In fact, I was often the only one with such experience, and I was in applications (computational biology) not theory. I did not overlap much with the theoretical CS people so I can't judge them.

In your defense however, there is a very strong trend for machine learning students to develop more mathematical rigor and delve into such topics, more than I've seen in physics departments. The difference is between rigor and novel mathematical expressivity. Theory students in physics still, thankfully, see little use for mathematical rigor, but are more obsessed with mathematical novelty ("How can I apply academic math to a problem whether the problem calls for it or not, since I like convoluted academic math?") whereas my impression of CS students is that it was more about rigor ("How can I foist mathematical rigor on the algorithm, the irrelevance of mathematical rigor to successful techniques such as Nelder-Mead optimization or convolutional neural networks notwithstanding?").

First of all, I object to your characterisation of "goofy bourgeois math" (unless you were being sarcastic), since topology and real analysis form the foundation of much of the work done in a variety of fields.

As far as your point about developing mathematical rigour among machine learning students -- you say that this is irrelevant, but I think you are gravely mistaken. The goal of mathematical rigour is to develop a theoretical foundation so that we understand why the algorithms work as well as they do, in what problems they work in, and suggest what limitations they impose (I'm thinking of the theory behind, say, boosting, regularization, some of the theoretical work on deep learning, etc.)

Suffice it to say, whichever department you ask about, having experienced both, they spend too much time on the wrong math, but in different ways, so I became an engineer.*

*At least that's what I tell myself regarding why I became an engineer.

Sounds like the words of a philistine engineer!

 

[StoneTemplePython]

as long as people have a sense of humor, I thought "goofy bourgeois math" could be the phrase of the day.

 

[radium]

I think people need to be very careful when they offhandedly dismiss something as being irrelevant or useless. To be a successful scientist requires one to develop a unique perspective that gives them an advantage when looking at certain problems. Most people are very specialized these days, but there are certain people (I can think of several physicists) who are able to make significant contributions in several different areas. They are able to do this because they have a wide base of knowledge and are able to use it to think of new creative ways to approach problems that never occurred to anyone else. For example, people have recently found connections between gravity and several different areas, which many people would first assume make no sense as they appear entirely unrelated. But actually there are very beautiful and intuitive arguments why this should be the case in many instances.

 

[FactChecker]

I think people need to be very careful when they offhandedly dismiss something as being irrelevant or useless.

I agree -- especially when a fast-changing field like computer science is involved. Who can anticipate what will happen in the next 40 years?

That being said, there is a difference between saying that something like theoretical math will open doors for a person to do advanced research, and saying that it will be necessary to work anywhere in the field. There will probably be a lot of work and jobs in computer science that will not require advanced math. But several of the advanced research jobs will be off-limits.

 

[Crass_Oscillator]

I am indeed a proudly philistine engineer.

I'm being snarky but I'm also making a point; I think it is hard to decide what is the correct barometer for how much math one needs. If you use theoretical physicists at the Institute for Advanced Study, you will probably decide that modern physics is very much dependent upon exotic, complicated, inscrutable mathematical ideas. If you looked at what DFT people do, you would think that workhorse computer science and 19th century mathematics was representative. If you look at some machine learning theorists like Jordan, you might think that the subject is indistinguishable from mathematical statistics. If you read Hinton's AMA responses, you will think that you had better think intuitively like an old school physicist and that complicated mathematics is perhaps not that important.

If you talk to a QA engineer at facebook, you would think that you need a stash of standard algorithms and very up to date knowledge on the relevant, specific software tools.

 

[bhobba]

Even Einstein, who was excellent in mathematics, got help occasionally from mathematicians.

Actually he wasn't - he was downright sloppy - competent - but excellent - no. In later years he hired assistants to do that tiresome stuff.

Compare him to an actually great mathematician like Von-Neumann and he was way ahead. What set Einstein apart from Von-Neumann, and Von-Neumann was one of the greatest that ever lived at this - in fact he was so great a guy like Poyla was genuinely scared of Von-Neumann (he mentioned in a class some theorem that nobody had yet proved - at the end of the class Von-Neumann gave him the solution) is his ability to see to the heart of a problem. Von-Neumann, like Feynman was simply a magician at it, but Einstein was supreme. He saw to the heart of things better than anyone - but yet his math was rather ordinary. It's this ability that is crucial - not mathematical expertise.

That's not to say we should confuse students by leaving important concepts unexplained - in fact I think it crucial we don't - its just it's not the key thing in making progress.

Thanks
Bill

 

[bhobba]

I think it is hard to decide what is the correct barometer for how much math one needs.

What one needs actually isn't much, and unless you are a mathematician you don't have to be good at it either. Like I said Einstein wasn't.

But what we can't do is confuse thinking students - we must make sure some rather obvious inconsistencies are not left without comment - at least - yes this is non-sense but can be explained although it will take us to far from our main aim to do it.

Thanks
Bill

 

[FactChecker]

Actually he wasn't - he was downright sloppy - competent - but excellent - no. In later years he hired assistants to do that tiresome stuff.

Compare him to an actually great mathematician like Von-Neumann and he was way ahead.

That's harsh. If we compare everyone with geniuses, then no one was excellent except Gauss.

 

[bhobba]

That's harsh. If we compare everyone with geniuses, then no one was excellent except Gauss.

Maybe - but I think people get he gist.

How true is it - make your own mind up - see the following:
https://www.amazon.com/dp/0393337685/?tag=pfamazon01-20

Thanks
Bill

 

[StoneTemplePython]

Maybe - but I think people get he gist.

How true is it - make your own mind up - see the following:
https://www.amazon.com/dp/0393337685/?tag=pfamazon01-20

Thanks
Bill

I think having von Neumann as your bar is too high -- he was at once brilliant in pure and applied mathematics and fast. Outside of Olympiad settings its not at all clear that being so fast matters that much. (Gowers has written about this, I can dredge something up a link.)

Einstein's Mistakes is a really enjoyable book (though people should be aware that the author, a physicist, takes cheap shots at engineers from time to time).

 

[bhobba]

he was at once brilliant in pure and applied mathematics

Its not his speed that made him great - it was his ability to penetrate a problem. For example he solved many of the problems on the atomic bomb project such as using a conventional bomb around the atomic material to reach critical mass and hold it there long enough for explosive fission to occur. Natuarally that's just one of many things eg he practically invented game theory.

But yes he was so fast it was said he was the only person fully awake.

Thanks
Bill