Computer Science and Math
deanzchen.com
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For example, the "Concrete mathematics" book by Knuth that is praised so much is mainly a book about generating functions in the form of power series of a complex variable. Despite the problem being discrete, you only get to get answers by setting up those power series with the discrete coefficients in front of the complex powers, and then obtain answers by differentiation and whatnot.
You do not get any "mathematical thinking" without understanding calculus, this is non-sense. The theory of real variables is not only one of the most ubiquitously practical one, but it is also the model for many other theories. Modern probability theory is largely advanced calculus in a particular setting, for example.
That is not to say that calculus is all there is. You just need to know a whole lot of math to apply it fruitfully in modern complicated settings, and calculus is some of the math that you need to know if you want to work on machine learning or robotics. If you want to work on webapps on the other hand, you do not need almost any math at all, but then you also do not need a university to learn in in the first place.
I protest that formal power series and formal derivative should not be counted under calculus or continuous math. You can define and prove properties of formal derivatives without any reference to continuum whatsoever.
Math has unity, real numbers are great, but I think people are overselling continuous-discrete connections. Dividing line between continuous math and discrete math is one of the most clear boundary in math, and there are huge amounts of useful discrete math which does not require continuous math.
If you work on machine learning, learn continuous math. On the other hand, if you work on programming language, learn logic and discrete math instead of continuous math. If you work on web applications, well, good luck. :)
Writing code to implement error-correcting codes doesn't use continuous math, but some error-correcting codes are best understood as working in polynomial rings over finite fields, and many of the ideas and tools are directly linked to similar structures in infinite continuous spaces.
Limiting yourself to learning only about discrete math that does not require any continuous math is like limiting yourself to using only one style of programming, or one language, or one operating system, or one hardware platform. If you don't mind being so limited, fine. If you want to understand things in greater depth and thus give yourself more opportunities and have more tools in your arsenal, learn both.
They support each other in ways you can't imagine without actually learning them.
While I don't have PhD in math, I did learn calculus and probability. I am still not convinced that they should come before logic.
It's the blub[0] paradox all over again. I would suggest that you haven't (yet) learned enough to realize that the connections are much deeper, stronger, and important than you realize.
It may well be that you don't need to know that the derivative of 3x^2 is 6x, but the influence on working with discrete math of a deeper understanding of continuous math is real, but subtle and hard to explain. Like Spock said when McCoy asked what it was like to be dead:
McCoy: Come on Spock this is me, McCoy!! You really
have been where no one has been before, can't
you tell me what it felt like?
Spock: It would be impossible to discuss the subject
without a common frame of reference.
McCoy: You're joking - you mean I have to die before
we can discuss your insight on death?
Similarly other subjects in math that do not, on the surface, seem to be relevant.
http://www.inference.phy.cam.ac.uk/itila/
See how far you'll get without calculus.
If throwing a tome is your style, I suggest going through http://www.cis.upenn.edu/~bcpierce/tapl/ or http://www.decision-procedures.org/ -- you can go through entire books without calculus.
All this is tangential in the end, there are certainly some specialized areas of CS that require less calculus than others, in general though you encounter calculus a lot in almost all areas of science and engineering, and the university is supposed to give you a broad foundation for your future career, so it should teach calculus to prospective scientists and engineers.
> For too long, we have taught computer science as an academic discipline (as though all of our students will go on to get PhDs and then become CS faculty members) even though for most of us, our students are overwhelmingly seeking careers in which they apply computer science.
This attitude strikes me as fundamentally wrong - universities are not and should never be trade schools. The point of a university should be to put you in contact with the cutting edge, and allow you to broaden your horizons, not train you for some specific software development job!
There should be computer science, and then there should be software engineering. One is in the sciences school, one is in an engineering school.
I bet the engineering enrollment would be significantly higher.
Or, one can get a university degree in engineering, which is basically training in the practical application of physics to solving real problems. A physicist can be a good engineer, but they would probably need an on-the-job apprenticeship first.
I think there is a parallel with the study of computer science vs. software development.
I work as a researcher in the computer science department at the University of Alabama, and I can tell you this is an ongoing conversation/debate not only here, but at many schools across the southeast. Unlike the valley, we have a tremendous shortage of developers, and that pressure ends up being felt at the university level.
To combat this, classes have been opened that expose the students to software engineering principles while not losing the theory that differentiates computer science from programming. This is largely helping, but is still, in many respects, an experiment.
Also at play is the notion that there are people in computer science that fall more into the design disciplines (i.e. HCI, UI, UX people). Right now, we have nothing for these people except double majoring with art/phycology and the occasional HCI class, but this may change soon. There are certainly frontiers in this area yet to be explored!
Either way, it's a big problem worth discussing, so I am happy I see it here.
*Edited for clarity
It's a bit worse than that. Imagine selecting that first language based on past popularity (Latin), universality (Esperanto), internal consistency (Lojban), precision and concision (Ithkuil), etc.
Whenever I read articles about social network-type sites that are doing UI changes it seems like there's pretty extensive use of data analysis based on A/B testing for different changes that are being made. Wouldn't a practitioner still need solid coding skills and a strong base in statistics?
I'm continuously frustrated by the number of people on the "software engineering" side of the fence who confuse the two, and who demand computer science skills in applicants to develop web apps.
By the same token, it appears here that we have someone on the other side of the fence (computer science in the academy) making the same error. He refers repeatedly to "programming" and its application as a job skill. He then rightly states that few programmers need calculus for their job. But from there he extrapolates that computer scientists don't need calculus for their job, which is not only a huge logical jump, but also makes this age-old error of confusing computer science and "programming" as a job skill.
With more experience, I think you can become a better software engineer/developer. With computer science, it seems to be more of an "either you get it or you don't" situation, rather like higher level math. This is why you will get some really great developers that were just willing to put in the sweat equity, but maybe have a hard time with the only tangentially relevant math requirements.
The requisite calculus could very well be covered in an Intro to CS class itself "hey by the way there are these numerical functions and they have rates of change. Moving on..."
Much more important mathematics for CS is Logic. Linear algebra. Combinatorics. It's true at some point "it all ties together" and there are some combinatorial problems that can be solved with complex analysis.
But to me, calculus as it is generally taught (heres a polynomial calculate its derivative using the symbol-manipulation rules you memorized), is the complete opposite of what CS and actual mathematics is. I think logic or maybe combinatorics are the easiest-to-grasp domains that capture the actual type of reasoning problems CS people encounter.
The lecture was being delivered by John Hughes, one of the researcher & designer of Haskell. In 1 hour class he blew my mind and I was sold to Computer Science in general and Programming Languages in specific. Next day I registered myself in Introduction of Functional Programming class. The Haskell language broke me from inside out and I failed the course. But I got one of the biggest pleasures of my life. It was about thinking in terms of mathematics, lambda calculus and building a life long love for mathematics. I then studied courses in Logic, compilers, and frontiers of programming language technologies. I started job during the education and couldn't complete my masters degree but that 1 year of studies in CS changed my mind forever. And I'm very thankful for the 1 hour I spend with John Hughes.
If people want to start programming trade schools, that's fine, but that's not "Computer Science", or "Software Engineering" for that matter. Engineering means being able to solve novel problems in an applied manner. You can't do that without mathematics.
Calculus is ultimately symbolic manipulation, as another commenter said. Mathematica / Matlab can do calculus. So should you.
However, mathematics is really all about reasoning and dealing with abstractions. A proof in linear algebra is not much different than proofs in real analysis or topology, in my opinion (the content might be very different, but the methods of reasoning are the same). So I think it really comes down to the answer to two questions: why study CS and why study math?
If we are a little more precise, and split "CS" into "computer science - the study of the theory of computation" and "programming" while also splitting "math" into "mathematics - the art of mathematical reasoning" and "a tool to solve problems" we can cross those two sets (to be a little math-nerdy :) and we get the following tuples:
(computer science, math for reasoning), (computer science, math as a tool), (programming, math for reasoning), (programming, math as a tool)
The problem is that people who have little or no exposure (and some people who have lots of exposure) may not think about/be aware of the difference between each of the tuples. Furthermore, by the time you understand the differences and what your personal preference is, you may have invested too much time into one particular school/program of study/however you're learning to switch.
You don't need a college degree for that, you just want to be a software bricklayer-monkey, learn it by yourself.
anyways, CS is full of guys like that. ...I should have gone full math.
Consider this: many great "theoretical" discoveries in history, including those in CS, were not theoretical at their inception. They were the result of people trying to solve practical problems -- not the product of a bunch of debt-ridden students trying to maximize GPA in an ivory tower. This cannot be overstated.
At the end of the day, the distinction between academic and practical is largely self-imposed, at least partially ego-driven, and probably highly inefficient for society as a whole. If you want "practical" people to engage "theoretical" problems, then make it practical for them.
10. Determination of the solvability of a Diophantine equation. Given a Diophantine equation with any number of unknown quantities and with rational integral coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers. The Entscheidungsproblem is solved when we know a procedure that allows for any given logical expression to decide by finitely many operations its validity or satisfiability ... The Entscheidungsproblem must be considered the main problem of mathematical logic.
Computers may seem obvious in hindsight but it was formally defined to solve a theoretical mathematics problem by the mathematician Alan Turing.
I believe I share this view on maths and cs. There appears to be a gap between computer science and software-making (writing, coding, engineering, "bricklaying", "applied cs"). The first thing is academic and scientific, the second is a craft. I know educated computer scientists who couldn't program to save their mothers and I know programmers in the industry who never went to CS class and taught themselves, and they keep reinventing the wheel over and over again (poorly). I believe these are different disciplines, although they are related. The academic part surely benefits from understanding maths, but for the "making" part, where people write rails apps, rest apis or websphere portlets, these skills aren't necessary (although I'd say one always benefits from deeper knowledge).
I am making the same experience in another field: Psychology. Most of my costudents wish to become Psychotherapists. Yet the bachelors degree is mostly limited to scientific examinations of the field. There is a gap between "psycho-science" and psychotherapy as well. I believe one can become a very good, practical, experienced and effective therapist without reading hundreds of papers from all different fields such as social, cognitive, work. Instead they could specialize in the things that require treatment such as depression etc. Again, knowing the science behind the practical stuff might help a therapist to gather deeper insights, but aren't always necessary.
The seats to study Psychology in Germany are scarse and limited to students with the very best grades. Most students appear to get very bored by the scientific aspects but have to go through the whole system in order to get a degree, which is required if you wish to work as a therapist. Therefore they keep blocking the spaces for people who are genuinely interested in the scientific aspects.
There - at what point in time may a practical discipline emancipate itself from its scientific origin, thus freeing up resources and gaining through specialization? Should programmers really have to study cs, or whould a new "software crafting" subject be sensible?
Mathematics professors can't do accounting, because they don't know the relevant regulations etc.
Similarly, A CS professor may not know the popular javascript library of the day etc, but I find it to be a myth and stereotype that educated computer scientists can't program. When pressed, no one is able to give real world examples. Every actual CS professor is a counterexample though. To give a famous one, Donald Knuth is prolific for instance:
And the problem is that second semester college Freshmen aren't really in a position to know what will really matter in the long term. Professors of course aren't either, it's just that personal study and research and the 900 years of institutional experience inherent in the university meme are likely to produce slightly better intuitions.
Can math be taught better? Hell yes, and we will see a significant change as Common Core bubbles upward. We'll also see changes as university education continues to become less elite and the institutions more 'customer' driven to less rigorous requirements. I mean the gist of the article is: standards are too high.
There's some truth to that. Widening the funnel inherently means lowering the standards - business schools used to mostly train accountants, now they produce a lot of marketing degrees. What is really needed is colleges of computing.
Of course those 900 years come back to bite. Someone loses the turf war: Engineering, Mathematics, or Business. Maybe the best place to teach the computing crafts would be the studios of the College of Art, particularly if mathematics is pushed to the periphery.
Why would a computer scientist need calculus? Because what calculus provides is training in replacing equations from one domain with equations of another - it provides a background in high level symbol manipulation. Doing it well requires competence in algebra, and algebra is the cornerstone of analysis of algorithms, and analysis of algorithms is what distinguishes computer science as a subset of all programming related activities.
The question is where does the line get drawn between programmer and software user when we talk about the person at the keyboard - or which side dominates when we are talking about building something with Rails: knowledge of relational algebra or ~rake db:migrate~?
If you however were going into a 3½ IT programmer degree, it is a whole other story.
And calculus, calculus is just calculations on functions. Not very profound, but not very difficult either. Interestingly, I find the procedures of calculus to be very similar to iterative programming procedures. A limit is kind of like a while loop, a sequence is an iterator, etc... For example
lim_{n->∞} n <==> while(true){ n++; }
in reality, the point of introductory level math and science courses in college is to show you a rather narrow skillset: you are given what amounts to a toolbox with a couple dozen tools and asked to use this small set of tools to solve problems that require application of, at most, 2 or 3 of these tools in succession. a fair analogy is that it is a multistep "put the round peg in the round hole" problem.
if assembling a small toolbox of skills and trying to apply the correct tools in short succession is too much for you, just quit university and code some stupid app. is it really that hard to understand why basic math and science is relevant?
DISCLAIMER: i was a teaching assistant for undergrads for 3 years during undergrad and grad school.
The article is not arguing "math is too much". It is arguing "if math is for problem solving, why not choose math with some relevant math content instead"?
But I have done just fine without calculus tools in my computer science degree (not purely computer science or math, though), which I'll be starting on the fifth year of this fall. This is isn't to say that you don't need calculus in computer science; it is to say that you don't seem to need calculus for basic and introductory computer science courses, and things related to that; discrete math has been much more relecant.
In spite of that, my department has calculus as its math course (you can also take stats in parallell), while discrete math is in the second semester. They also advice students that haven't taken the most advanced math in high school against taking the most theoretical/mathematical computer science degree. This was due to the calculus course building on that high school math experience. But then the discrete math course - which is much more directly relevant - does not build much or at all on calculus, and you can do just fine without the most advanced high school math in that course. Maybe the only thing you'll suffer from is having a bit less mathematical maturity at the offset.
But since they start with calculus, they discourage people like me who, while not having had the most advanced calculus (or pre-calculus) in high school from taking a more math-heavy degree, even though I've found to like a lot of the math involved in some subfields of computer science!