Related
Closed. This question is opinion-based. It is not currently accepting answers.
Want to improve this question? Update the question so it can be answered with facts and citations by editing this post.
Closed 9 years ago.
Improve this question
I would like to learn a functional language in order to broaden my horizon. I have knowledge of Python and C/C++ and I want a language to be easy to learn from someone who comes from the imperative domain of languages. I don't care if the language is powerful enough. I just want a language in order to learn the basic of functional programming and then I will try for a more difficult (and powerful language).
Thanks
I recommend pure-lang for these pedagogical ends. It's also plenty powerful. If you want something more popular / with more community support, then I'd recommend Scheme or OCaml, depending on whether you'd rather deal with unfamiliar syntax (go with Scheme) or deal with unfamiliar typing (go with OCaml) first. SML and F# are only slightly different from OCaml. Others have or will mention Clojure, Scala, and Haskell.
Clojure is a variant of Scheme, with its own idiosyncracies (e.g. no tail-call optimization), so using it would be a way of starting with Scheme. I'd expect you'd have an easier time with a less idiosyncratic Scheme implementation though. Racket is what's often used for teaching. Scala looks to be fundamentally similar to OCaml, but this is based on only casual familiarity.
Unlike Haskell, the other languages mentioned all have two advantages: (1) evaluation-order is eager by default, though you can get lazy evaluation by specifically requesting it. In Haskell's the reverse. (2) Mutation is available, though much of the libraries and code you'll see doesn't use it. I actually think it's pedagogically better to learn functional programming while at the same time having an eye on how it interacts with side-effects, and working your way to monadic-style composition somewhat down the road. So I think this is an advantage. Some will tell you that it's better to be thrown into Haskell's more-quarantined handling of mutaton first, though.
Robert Harper at CMU has some nice blog posts on teaching functional programming. As I understand, he also prefers languages like OCaml for teaching.
Among the three classes of languages I recommended (Pure, Scheme and friends, OCaml and friends), the first two have dynamic typing. The first and third have explicit reference cells (as though in Python, you restricted yourself to never reassiging a variable but you could still change what's stored at a list index). Scheme has implicit reference cells: variables themselves look mutable, as in C and Python, and the reference cell handling is done under the covers. In languages like that, you often have some form of explicit reference cell available too (as in the example I just gave in Python, or using mutable pairs/lists in Racket...in other Schemes, including the Scheme standard, those are the default pairs/lists).
One virtue Haskell does have is some textbooks are appearing for it. (I mean this sincerely, not snarkily.) What books/resources to use is another controversial issue with many wars/closed questions. SICP as others have recommended has many fans and also some critics. There seem to me to be many good choices. I won't venture further into those debates.
At first, read Structure and Implementation of Computer Programs. I recommend Lisp (for, example, it's dialect Scheme) as first functional programming language.
Another option is Clojure, which I'm given to understand is more "purely" functional than Scheme/Racket (don't ask me about the details here) and possibly similar enough to let you use it in conjunction with SICP (Structure and Interpretation of Computer Programs, a highly recommended book also suggested by another answer).
I would like to learn a functional language in order to broaden my horizon. I have knowledge of Python and C/C++ and I want a language to be easy to learn from someone who comes from the imperative domain of languages. I don't care if the language is powerful enough. I just want a language in order to learn the basic of functional programming and then I will try for a more difficult (and powerful language).
Great question!
I had done BASIC, Pascal, assembler, C and C++ before I started doing functional programming in the late 1990s. Then I started using two functional languages at about the same time, Mathematica and OCaml, and was using them exclusively within a few years. In particular, OCaml let me write imperative code which looked like the code I had been writing before. I found that valuable as a learner because it let me compare the different approaches which made the advantages of ML obvious.
However, as others have mentioned, the core benefit of Mathematica and OCaml is pattern matching and that is not technically related to functional programming. I have subsequently looked at many other functional languages but I have no desire to go back to a language that lacks pattern matching.
This question is probably off-topic because it is going to result in endless language wars, but here's a general bit of advice:
There are a class of functional programming languages which are sometimes called "mostly functional", in that they permit some imperative features where you want them. Examples include Standard ML, OCaml, F#, and Scala. You might consider one of these if you want to be able to get a grip on the functional idiomatic style while still being able to achieve things in reasonably familiar ways.
I've used Standard ML extensively in the past, but if you're looking for something that has a bit less of a learning curve, I'd personally recommend Scala, which is my second-favourite programming language. The reasons for this include the prevalence of libraries, a healthy-sized community, and the availability of nice books and tutorials to help you getting started (particularly if you have ever had any dealings with Java).
One element that was not discussed is the availability of special pattern-matching syntax for algebraic datatypes, as in Haskell, all flavors of ML, and probably several of the other languages mentioned. Pattern-matching syntax tends to help the programmer see their functions as mathematical functions. Haskell's syntax is sufficiently complex, and its implementations have sufficiently poor parse error messages, that syntax is a decent reason not to choose Haskell. Scheme is probably easier to learn than most other options (and Scheme probably has the king of all macro systems), but the lack of pattern matching syntax would steer me away from it for an intro to functional programming.
Closed. This question needs to be more focused. It is not currently accepting answers.
Want to improve this question? Update the question so it focuses on one problem only by editing this post.
Closed 6 years ago.
Improve this question
I haven't taken any math classes above basic college calculus. However, in the course of my programming work, I've picked up a lot of math and comp sci from blogs and reading, and I genuinely believe I have a decent mathematical mind. I enjoy and have success doing Project Euler, for example.
I want to dive in and really start learning some cool math, particularly discrete mathematics, set theory, graph theory, number theory, combinatorics, category theory, lambda calculus, etc.
My impression so far is that I'm well equipped to take these on at a conceptual level, but I'm having a really hard time with the mathematical language and symbols. I just don't "speak the language" and though I'm trying to learn it, I'm the going is extremely slow. It can take me hours to work through even one formula or terminology heavy paragraph. And yeah, I can look up terms and definitions, but it's a terribly onerous process that very much obscures the theoretical simplicity of what I'm trying to learn.
I'm really afraid I'm going to have to back up to where I left off, get a mid-level math textbook, and invest some serious time in exercises to train myself in that way of thought. This sounds amazingly boring, though, so I wondered if anyone else has any ideas or experience with this.
If you don't want to attend a class, you still need to get what the class would have given you: time in the material and lots of practice.
So, grab that text book and start doing the practice problems. There really isn't any other way (unless you've figured out how osmosis can actually happen...).
There is no knowledge that can only be gained in a classroom.
Check out the MIT Courseware for Mathematics
Also their YouTube site
Project Euler is also a great way to think about math as it relates to programming
Take a class at your local community college. If you're like me you'd need the structure. There's something to be said for the pressure of being graded. I mean there's so much to learn that going solo is really impractical if you want to have more than just a passing nod-your-head-mm-hmm sort of understanding.
Sounds like you're in the same position I am. What I'm finding out about math education is that most of it is taught incorrectly. Whether a cause or result of this, I also find most math texts are written incorrectly. Exceptions are rare, but notable. For instance, anything written by Donald Knuth is a step in the right direction.
Here are a couple of articles that state the problem quite clearly:
A Gentle Introduction To Learning
Calculus
Developing Your Intuition For
Math
And here's an article on a simple study technique that aims at retaining knowledge:
Teaching linear algebra
Consider auditing classes in discrete mathematics and proofs at a local university. The discrete math class will teach you some really useful stuff (graph theory, combinatorics, etc.), and the proofs class will teach you more about the mathematical style of thinking and writing.
I'd agree with #John Kugelman, classes are the way to go to get it done properly but I'd add that if you don't want to take classes, the internet has many resources to help you, including recorded lectures which I find can be more approachable than books and papers.
I'd recommend checking out MIT Open Courseware. There's a Maths for Computer Science module there, and I'm enjoying working through Gilbert Strang's Linear Algebra course of video lectures.
Youtube and videolectures.com are also good resources for video lectures.
Finally, there's a free Maths for CS book at bookboon.
To this list I would now add The Haskel Road to Logic, Maths, and Programming, and Conceptual Mathematics: A First Introduction to Categories.
--- Nov 16 '09 answer for posterity--
Two books. Diestel's Graph Theory, and Knuth's Concrete Mathematics. Once you get the hang of those try CAGES.
Find a good mentor who is an expert in the field who is willing to spend time with you on a regular basis.
There is a sort of trick to learning dense material, like math and mathematical CS. Learning unfamiliar abstract stuff is hard, and the most effective way to do it is to familiarize yourself with it in stages. First, you need to skim it: don't worry if you don't understand everything in the first pass. Then take a break; after you have rested, go through it again in more depth. Lather, rinse, repeat; meditate, and eventually you may become enlightened.
I'm not sure exactly where I'd start, to become familiar with the language of mathematics; I just ended up reading through lots of papers until I got better at it. You might look for introductory textbooks on formal mathematical logic, since a lot of math (especially in language theory) is based off of that; if you learn to hack the formal stuff a bit, the everyday notation might look a bit easier.
You should probably look through books on topics you're personally interested in; the inherent interest should help get you over the hump. Also, make sure you find texts that are actually introductory; I have become wary of slim, undecorated hardbacks labeled Elementary Foobar Theory, which tend to be elementary only to postdocs with a PhD in Foobar.
A word of warning: do not start out with category theory -- it is the most boring math I have ever encountered! Due to its relevance to language design and type theory, I would like to know more about it, but so far I have not been able to deal...
For a nice, scattershot intro to bits of many kinds of CS-ish math, I recommend Godel, Escher, Bach by Hofstadter (if you haven't read it already, of course). It's not a formal math book, though, so it won't help you with the familiarity problem, but it is quite inspirational.
Mathematical notation is is akin to several computer languages:
concise
exacting
based on many idioms
a fair amount of local variations and conventions
As with a computer language, you don't need to "wash the whole elephant at once": take it one part a at time.
A tentative plan for you could be
identify areas of mathematics that are interesting or important to you. (seems you already have a bit of a sense for that, CS has helped you develop quite a culture for it.)
take (or merely audit) a few formal classes in this area. I agree with several answers in this post, an in-person course, at local college is preferable, but, maybe at first, or to be sure to get the most of a particular class, first self-teaching yourself in this area with MIT OCW, similar online resources and associated books is ok/fine.
if an area of math introduces too high of a pre-requisite in terms of fluency with notation or with some underlying concept or (most often mechanical computation and transformation techniques). No problem! Just backtrack a bit, learn these foundations (and just these foundations!) and move forward again.
Find a "guru", someone that has a broad mathematical culture and exposure, not necessarily a mathematician, physics folks are good too, indeed they can often articulate math in a more practical fashion. Use this guru to guide you, as he/she can show you how the big pieces fit together.
Note: There is little gain to be had of learning mathematical notation for its own sake. Rather it should be learned in context, just like say a C# idiom is better memorized when used and when associated with a specific task, rather than learned in vacuo. A related SO posting however provides several resources to decipher and learn mathematical notation
Project Euler takes problems out of context and drops them in for people to solve them. Project Euler cannot teach you anything effectively. I think you should forget about it, if it is popular it does not mean anything. You cannot study Mathematics through Project Euler as it contains only bits and pieces(and some pretty high level pieces) that you're supposed to know in order to solve the problems. Learning mathematics means to consider a subject and a read a book about it and solving exercices or reading solutions, that's how you learn math. If it so happens that through your reading you find something that is close to some project euler thing, your luck , but otherwise Project euler is a complete waste of time. I think the time is much better invested choosing a particular branch of mathematics and studying that. Let me explain why: I solved 3 pretty advanced Projec Euler problems and they were all making appeal to knowledge from Number theory which I happened to have because i studies some part of it. I do not think Iearned anything from Project Euler, it just happened that I already knew some number theory and solved the problems.
For example, if you find out you like number theory, take H. Davenport -> Hardy & Wright -> Kenneth & Rosen's , study those.
If you like Graph Theory take Reinhard Diestel's book which is freely available and study that(or check books.google.com and find whichever is more appropriate to your taste) but don't spread your attention in 999999 directions just because Project Euler has problems ranging from dynamic programming to advanced geometry or to advanced number theory, that is clearly the wrong way to go and it will not bring you closer to your goal.
This sounds amazingly boring
Well ... Mathematics is not boring when you find some problem that you are attached to, which you like and you'd like to find the solution to, and when you have the sufficient time to reflect on it while not behind a computer screen. Mathematics is done with pen and paper mostly(yes you can use computers .. but that's not really the point).
So, if you find a real-world problem, or some programming problem that would benefit from
you knowing some advanced maths, and you know what maths you have to study , it can be motivating to learn in that way.
If you feel you are not motivated it is hard to study properly.
There is also the question of what you actually mean when you say learn. Does the learning process stop after you solved the problems at the end of the chapter of a book ? Well you decide. You can consider you have finished learning that subject, or you can consider you have not finished and read more about it. There are entire books on just one equation and variations of it.
The amount of programming-related math that you can learn without formal training is limited, but it's more than enough. But maybe you can self-teach yourself.
It all boils down to your resources and motivation.
To know mathematics you have to do mathematics not programming(project euler).
For beginning to learn category theory I recommend David Spivak's Category Theory for the Sciences (AKA Category Theory for Scientists) because its relatively comprehensible due to many examples that enable understanding by analogy and which quickly builds a foundation for understanding more abstract concepts.
It requires the ability to reason logically and an intuitive notion of what is a set. It proceeds from sets and functions through basic category theory to adjoint functors, categories of functors, sheaves, monads and an introduction to operads. Two main threads throughout are modeling databases in terms of categories and describing categories with annotated diagrams called ologs. The bibliography provides references to more advanced and specialized topics including recent papers by Dr. Spivak.
An expected outcome from reading this book is the capability of understanding category theory texts and papers written for mathematicians such as Mac Lane's Category Theory for the Working Mathematician.
In PDF format it is available from http://math.mit.edu/~dspivak/teaching/sp13/ (the dynamic version is recommended since its the most recent). The open access HTML version is available from https://mitpress.mit.edu/books/category-theory-sciences (which is recommended since it includes additional content including answers to some exercises).
Closed. This question is off-topic. It is not currently accepting answers.
Want to improve this question? Update the question so it's on-topic for Stack Overflow.
Closed 11 years ago.
Improve this question
This is my very first question so I am a bit nervous about it because I am not sure whether I get the meaning across well enough. Anyhow, here we go....
Whenever new milestones in programming have been reached it seems they always have had one goal in common: to make it easier for programmers, well, to program.
Machine language, opcodes/mnemonics, procedures/functions, structs, classes (OOP) etc. always helped, in their time, to plan, structure and code programs in a more natural, understandable and better maintainable way.
Of course functional programming is by no means a novelty but it seems that it has experienced a sort of renaissance in recent years. I also believe that FP will get an enormous boost when Microsoft will add F# to their mainstream programming languages.
Returning to my original question, I believe that ultimately programming will be done in a natural language (English) with very few restrictions or rules. The compiler will be part of an AI/NLP system that extracts information from the code or should I say text and transforms it into an intermediate language which the compiler can compile.
So, does FP take programming closer to natural-language programming or is it rather an obstacle and mainstream OOP will lead us faster to natural-language programming?
This question should not be used to discuss the useability or feasability of natural-language programming because only the future will tell.
Sorry, I don't agree at all. Code is ultimately a blueprint for making things (objects), so it has to be very precise and rule-governed in order to function reliably. Natural language won't take over programming any sooner than sketching ideas on napkins will take over mechanical engineering.
I personally have come to the conclusion natural language programming is somewhat crack.
English is not exactly suited to be used fully as a programming language, too many abstract words that have no-correlation in programming, such as emotive terms and other abstract notions that have no place in programming, so to say programming could ever be "natural language" would follow, that "natural language" could be programming, but it isn't.
Now while I get what you're saying here, the problem is the english language has too many scrap terms and repeated names for the same things, so we'd be using something that isn't even specific to the domain of programming, for the task of programming.
I think its more suited that people understand that programming is in fact a highly specialized language, and use their brains and learn to code in a language, which is simple, declarative, and has a consistent definition, unlike English, where definition is highly subjective.
Once you learn the ins and outs of a language, and learn its schematics and behaviors, you can combine them to do new things.
Take Perl, everyone lambasts it for being line noise, but when you know many programming languages, once you get past the initial hurdles of "OMG LINE NOISE", there is a degree of intuitiveness about it where you can make stuff up you never read about and then see it magically works just as you expected.
And IMHO, domain specific languages trump spoken ones for targeted problem solving.
"So, does FP take programming closer to natural-language programming or is it rather an obstacle and mainstream OOP will lead us faster to natural-language programming?"
Neither. Both operate on the same principle that you have to be specific about what you want the computer to do. There must be no room for uncertainty, and neither paradigm has anything to do with natural languages. They tackle an entirely different problem: That of managing and structuring complex code and large codebases.
The big obstacle in natural languages is the parsing. It is impossible to unambiguously parse natural language. Even humans can't do it without a lot of context information (facial expressions, tone of voice), and even then, we still get it wrong quite often.
OOP and FP are only about what happens after parsing. Which meaning is assigned to each semantic element, once it's been identified and parsed.
Perhaps we'll one day be able to program in natural language. I doubt it'll happen within the next couple of decades, but it may happen one day. But today's programming paradigms will neither speed up this process or delay it. They simply have nothing to do with it, and won't help solving the parsing problem.
I don't think that functional programming is any closer to natural language programming than OO programming. Functional programming has a very verb-oriented syntax. When you program in Lisp or Scheme, you spend a lot of time thinking about functions and what actions you want to take on your data. In OO programming, you spend most of your time thinking about objects, hence it seems very noun-oriented. However, in Smalltalk, C++, and Java, you also have methods, which allow you to apply verbs to all of your nouns (so to speak).
I don't think that OO programming will necessarily lead us to natural language programming, but from my point of view it's a little bit closer than functional programming. Functional programming, to me, seems a little bit closer to math than to natural language. That's not such a bad thing, since maybe math is the language we should be thinking in when we program anyway.
Just FYI, Inform 7 is probably the closest anyone has gotten to natural-language programming. It is a language for a very specific domain: writing interactive fiction, the kind of software that began with "adventure games".
The current spurt of interest in Functional Programming result primarily of C# 3.0's cool new features is basically to enable parallelism and denotes a shift towards multi-core computing. IMHO, I don't think we can consider this a next step towards 'natural language programming'
If you are looking for the next evolution in programming languages, I would look to DSLs. DSL allows for highly customized languages that enable sophisticated biz users to configure a system without having to worry about coding details such as datatypes, threads and UI widgets.
Functional languages will have their place in "highly parallel processing" space.
Do you think subjective questions will get this here order for "Windows Internals the 5th Element" added to the database and shipped to my address? If so, natural language programming will be very close to functional programming, since I asked my question in a somewhat functional manner. If not, then natural language programming won't get my order shipped, will it? Functional programming can work because it still has nothing to do with natural languages.
No. Functional programming will take us closer to proving compilers. That is compilers that prove more assertions about your code. The more compilers can prove for us, the closer software development comes to be engineering rather than art.
A NLP programming language is probably more of a "do what I mean not what I say" style language. That is probably the opposite of the direction functional languages go.
"All programming languages are converging towards LISP."
As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.
Closed 11 years ago.
Locked. This question and its answers are locked because the question is off-topic but has historical significance. It is not currently accepting new answers or interactions.
I happened to debate with a friend during college days whether advanced mathematics is necessary for any veteran programmer. He used to argue fiercely against that. He said that programmers need only basic mathematical knowledge from high school or fresh year college math, no more no less, and that almost all of programming tasks can be achieved without even need for advanced math. He argued, however, that algorithms are fundamental & must-have asset for programmers.
My stance was that all computer science advances depended almost solely on mathematics advances, and therefore a thorough knowledge in mathematics would help programmers greatly when they're working with real-world challenging problems.
I still cannot settle on which side of the arguments is correct. Could you tell us your stance, from your own experience?
To answer your question as it was posed I would have to say, "No, mathematics is not necessary for programming". However, as other people have suggested in this thread, I believe there is a correlation between understanding mathematics and being able to "think algorithmically". That is, to be able to think abstractly about quantity, processes, relationships and proof.
I started programming when I was about 9 years old and it would be a stretch to say I had learnt much mathematics by that stage. However, with a bit of effort I was able to understand variables, for loops, goto statements (forgive me, I was Vic 20 BASIC and I hadn't read any Dijkstra yet) and basic co-ordinate geometry to put graphics on the screen.
I eventually went on to complete an honours degree in Pure Mathematics with a minor in Computer Science. Although I focused mainly on analysis, I also studied quite a bit of discrete maths, number theory, logic and computability theory. Apart from being able to apply a few ideas from statistics, probability theory, vector analysis and linear algebra to programming, there was little maths I studied that was directly applicable to my programming during my undergraduate degree and the commercial and research programming I did afterwards.
However, I strongly believe the formal methods of thinking that mathematics demands — careful reasoning, searching for counter-examples, building axiomatic foundations, spotting connections between concepts — has been a tremendous help when I have tackled large and complex programming projects.
Consider the way athletes train for their sport. For example, footballers no doubt spend much of their training time on basic football skills. However, to improve their general fitness they might also spend time at the gym on bicycle or rowing machines, doing weights, etc.
Studying mathematics can be likened to weight-training or cross-training to improve your mental strength and stamina for programming. It is absolutely essential that you practice your basic programming skills but studying mathematics is an incredible mental work-out that improves your core analytic ability.
While advanced mathematics may not be required for programming (unless you are programming advanced mathematics capability) the thought process of programming and mathematics are very similar. You begin with a base of known things (axioms, previously proven theories) and try to get to someplace new. You cannot skip steps. If you do skip steps, then you are required to fill in the blanks. It's a critical thought process that makes the two incredibly similar.
Also, mathematicians and programmers both think critically in the abstract. Real world things are represented by objects and variables. The ability to translate from concrete to abstract also links the two fields.
There's a very good chance that if you're good at one, you will probably be good at the other.
computer science != programming
OK, seriously, I know good and bad programmers who were English and Psychology majors and some that were Computer Science majors. Some very famous guys that I admire as developers didn't have a CS background. Larry Wall(Perl), for example, was a linguist.
On the other hand, it helps to know something about the domain you are working on because then you can at least see if your data makes sense and help your customer/users drill down to what they really want.
And yes, there's the issue of computational complexity and efficient data structures and program correctness. That's stuff you learn in Computer Science and that's useful to know in almost any domain, but it's neither necessary nor sufficient.
I guess I am going to be the first person to say you do need math. As others have said math is not all that important for certain aspects of development, but the fundamentals of critical thinking and structured analysis are very important.
More so, math is important in understanding a lot of the fundamentals that go into things like schedulers, optimizations, sorting, protocol management, and a number of other aspects of computers. Though the math involved from a calculation level is not complex (its mostly High school algebra) the theories and applications can be quite complex as a solid understanding of math through calculus will be of great benefit.
Can you get by without it, absolutely, and you shouldnt let a less then thorough knowledge of math hold you back, but if you had the chance, or the inclination I would study as much math as you could, calculus, numeric theory, linear algebra, combinatorics, practical applications, all of it has both practical and theoretical applications in a wide range of computer science.
I have known people who were highly successful on both sides of the fence (those without a strong focus on math, and those who went to school for physics or math), but in both groups they enjoyed numerical problems and learning about algorithms and math theory.
I have a maths degree, but I can't remember requiring that maths a single time in my career. It was useful in terms of training my mind for logical thinking, but I've not written any code using fluid dynamics, quantum theory or Markov Chains. (The last is the most likely to come up, I suspect.)
Most line-of-business developers won't need advanced maths most of the time. Sometimes knowing trigonometry can help, and certainly being able to understand enough maths to implement algorithms described mathematically can be important - but beyond that? Nah.
Don't forget that most programmers aren't advancing computer science - they're building applications. I don't need to know advanced engineering to drive a modern car, even though that car has almost certainly been improved through advanced engineering.
I would argue that having advanced logic (discrete) math can really help. That along with set theory. When dealing with common computer programs, these disciplines can help a lot. However, a lot of the other math I took in university was calculus, which as far as I can see, had very limited usage. Since 90% (or something like that) of programming is doing business apps with very simple math, I would say that for the most part, you can get by with very little math knowledge. However, a good understanding of boolean algebra, logic, discrete math, and set theory can really put you up to that next level.
I'll go against the grain here and say "Yes"
I switch from Civil Engineering to programming (Concrete Sucks!). My math background consists of the usual first year stuff, second and third year Calculus(Diff EQ, volume integrations, Series, Fourier and Laplace transforms) and a Numerical Analysis course.
I find that my math is incredibly lacking for computer programming. There are entire areas of Discrete math and logic that I am missing, and I only survive due to an extensive library of textbooks, Wikipedia and Wolfram. Most advanced algorithms are based on advanced math, and I am unable to develop advanced algorithms without doing extensive research (Essentially the equivalent to a half-course worth of work.) I am certainly unable to come up with NEW algorithms, as I just don't have the mathematical foundations as the shoulders of giants upon which to stand.
It depends on what you are doing. If you do a lot of 3D programming, knowledge of 3D geometry is certainly necessary, don't you agree? ;-) If you want to create a new image format like JPG or a new audio format like MP3, you are also pretty lost if you can't understand a cosine or fourier transformation, as these are the basics most lossy compression are based on. Many other problems can be resolved better if you know your math rather well.
There are also many other programming tasks you will find do not need much math.
If you find the subject fascinating enough to post this, just go ahead and start learning. The rest will come naturally.
Yeah, there is no need for advanced mathematics - if you are programming commercial - off the shelf software.
However when dealing with hardcore stuff such as:
Calculating trajectories to control
a robot
Creating AI-like applications to
support uncertainty and automatic
reasoning
Playing with 3-D motion and graphics
Some advanced mathematics knowledge might come in handy. And it's not like they are "out-of-this world" problems.
I had to create a software to try to "predict" the necessary amount of paper for an office (and it was hell just to find out the best way to approximate values).
You have to be careful, though, because it is easy to get lost when using advanced things - there is a friend of mine who resorted to using Turing to store the state of a dynamic menu just to display it correctly - humm... perhaps he wnet too far in his imagination.
What type of programming?
In my commercial experience, I have needed no advanced mathematics, but this is heavily dependent on the field you are in.
Computer graphics require a large amount of advanced mathematics. A lot of academic computer programming requires advanced mathematics.
So saying there tends to be a correlation between people who are good at mathematics and people who are good at programming.
I hope this wishy-washy answer helps.
Mathematics are needed for developers in some fields but are almost useless in others.
If you are a game developer and have to work with physics a lot - understanding of math is crucial. If you are working with advanced visual controls - you could not do much without geometry. If you're planning to do some financial calculations - it would REALLY help to have solid knowledge of statistics.
On the other hand over last 5 years I had only 2 or 3 projects where ANY amount of math was required at all. Of these there was only 1 occurrence when a Google search did not help.
At the end of the day even financial calculations are very often something your clients do for you and give you formulas to implement.
So if you're in 'applied software' business you are likely to never use your math degree. If you're in academic software maths are crucial.
I agree with Chris. I would say "Yes", also. But this depends on your market as stated above. If you are simply creating some basic "off-the-shelf" applications or writing tools to help your everyday work...then math isn't nearly as important.
Engineering custom software solutions requires lots of problem solving and critical thinking. Skills that are most definitely enhanced when a mathematics background is present. I minored in Math with my Computer Engineering degree and I give credit to all of my math-oriented background as to why I'm where I am today.
That's my 2 cents, I can tell from reading above that many would not agree. I encourage all to consider that I'm not saying you can't have those skills without a math background, I'm simply stating that the skills are side-effects of having such a background and can impact software positively.
In my experience math is required in programming, you can't get away from it. The whole of programming is based on math.
The issue is not black and white, but more colorful. The question is not whether or not you need math, but how much. The higher levels of math will give you more tools and open up your mind to different paths of though.
For example, you can program if you only known addition and subtraction. When multiplication is required, you will have to perform a lot of additions. Multiplication simplifies repetitive additions. Algebra allows one to simplify math before implementing it into programs. Linear Algebra provides tools for transforming images. Boolean Algebra provides mechanics for reducing all those if statements.
And don't forget the sibling to mathematics, Logic and Philosophy. Logic will help you make efficient use of case or switch statements. Philosophy will help you understand the thinking of the guy who wrote that code you are modifying.
Yes, you don't need much math to write programs. Some programs may require more math than others. More math knowledge will give you an advantage over those who have lesser understanding. In these times, people need every advantage they can get to obtain those jobs.
I've been programming for 8 years professionally, and since I was 12 as a hobby.
Math is not necessary, logic is. Math is horribly helpful though, to say it's not necessary is like saying that to kill a man, a gun isn't necessary, you can use a knife. Well, it is true, but that gun makes it a lot easier.
There are a couple bare minimums, which you should already meet. You need to know basic algebraic expressions and notation, and the common computer equivalents. For example, you need to know what an exponential is (3 to the 3rd is 27), and the common computer expression is 3^3. The common notations for algebra does change between languages, but many of them use a somewhat unified methodology. Others (looking at you LISP) don't. You also need to know order of operations.
You need to understand algorithmic thought. First this, then this, produces this which is used in this calculation. Chances are you understand this or you don't, and it's a fairly hard hurdle to jump if you don't understand it; I've found that this is something you 'get', and not really something you can learn. Conversely, some people don't 'get' art. They should not become painters. Also, there have been students in CS curriculum who cannot figure out why this does not work:
x = z + w;
z = 3;
y = 5;
It's not that they don't understand addition, it's that they aren't grasping the requirement of unambiguous express. If they understand it, the computer should too, right? If you can't see what's wrong with the above three lines, then don't become a programmer.
Lastly, you need to know whatever math is under your domain of programming. Accounting software could stop at basic algebra. If you are programming physics, you'll need to know physics (loosely) and math in 3-dimensional geometry (Euclidean). If you're programming architecture software, you'll need to know trigonometry.
This goes farther then math though; whatever domain you are programming for, you need to soundly understand the basics. If you are programming language analysis software, you'll need to know probability, statistics, grammar theory (multiple languages), etc.
Often times, certain domains need, or can benefit from, knowledge you'd think is unrelated. For example, if you were programming audio software, you actually need to know trigonometry to deal with waveforms.
Magnitude changes things also. If you are sorting a financial data set of 1000 items, it's no big thing. If it was 10 million records, however, you would benefit greatly from knowing vector math actually, and having a deep understanding of sorting at the binary level (how does a system sort alphabetically? How does it know 'a' is less than 'b'?)
You are going to find that as a programmer, your general knowledge base is going to explode, because each project will necessitate more learning outside of the direct sphere of programming. If you are squeamish or lazy about self-learning, and do not like the idea of spending 10+ hours a week doing essentially 'homework', do not become a programmer.
If you like thought exercises, if you like learning, if you can think about abstract things like math without a calculator or design without a sketchpad, if you have broad tastes in life and hobbies, if you are self-critical and can throw away 'favorited' ideas, if you like perfecting things, then become a programmer. Do not base this decision on math, but rather, the ability to think logically and learn. Those are what is important; math is just the by-product.
Of course it depends on what kind of programmer you want to be, or better what kind of programmer your employers want you to be. I think calculus and algebra are essentials, statistic and linear programming is indeed a good tool to have in your briefcase, maybe analysis (derivative, integrals, functions...) could be done without. But if you want to know how things work skin-deep (electronics, for example, or some non trivial algorhytms) "advanced" math is something you'd better not go without anywhere.
Most of the programming I have done involved physics simulations for research including things like electromagnetism, quantum mechanics and structural mechanics. Since the problem domains have advanced mathematics associated with them I would be hard pressed to solve them without using advanced mathematics.
So the answer to your question is - it depends on what you are trying to do.
Advanced maths knowledge is vital if you're going to be writing a new programming language. Or you need write your own algorithms.
However, for most day-to-day programming - from websites to insurance processing applications - only basic maths are necessary.
Someone with a solid mathematical (which is not merely arithmetic) or logic background will cope well with algorithms, variable use, conditional reasoning and data structures.
Not everyone can design a UI.
Not everyone can make efficient code.
Not everyone can comment and document clearly.
Not everyone can do a good algorithm
Mathematics will help you to a point, but only to a point.
I dont think advanced mathemetics knowledge is a requirement for a good programmer, but based on personal experience I think that programmers who have a better grasp at advanced maths also make better programmers. This may simply be due to a more logical mind, or a more logical outlook due to their experiences of solving mathematical problems.
The fundamental concept of maths is the following, devising, understanding, implementation, and use of algorithms. If you cannot do maths then it is because you cannot do these things, and if you cannot do these things then you cannot be an effective programmer.
Common programming tasks might not need any specific mathematical knowledge (e.g. you probably won't need vector algebra and calculus unless you're doing tasks like 3D graphics or physics simulations, for example), but the underlying skillsets are identical, and lack of ability in one domain will be matched by a corresponding lack of ability in the other domain.
Math is a toolbox for creating programs. I recommend Cormen's Introduction to Algorithms. It touches on the more "mathy" stuff.
- Greatest lowest limit (managing resources)
- Random variables (game programming)
- Topological sort (adjusting spreadsheets)
- Matrix operations (3d graphics)
- Number theory (encryption)
- Fast fourier transforms (networks)
I don't think that higher math is a requirement for being a good programmer - as always it depends on what you are coding.
Of course if you are in 3D graphics programming, you'll need matrices and stuff. As author of business software, you'll probably need statistics math.
But being a professional programmer for almost 10 years (and another 10 years amateur) "higher math" is not something that I needed regularily. In about 99.8% of all cases it's just plus, minus, division and multiplication in some intelligent combinations - in most cases it's about algorithms, not math.
Learning higher math, for most programmers, is important simply because it bends your brain to think logically, in a step-by-step manner to get from one thing to another.
Very few programming jobs, though, require anything above high school math. I've used linear algebra once. I've never used calculus. I use algebra every day.
Mathematical knowledge is often useful to a programmer, as are graphic design skill, puzzle-solving ability, work ethic and a host of other skills and traits. Very few programmers are good at everything that a programmer can possibly be good at. I wouldn't agree with any statement of the form "you're not a real programmer unless you can {insert favorite programming ability here}".
But I would be wary of a programmer who couldn't do Math. More so than of one who couldn't draw.
I think it really depends on what you're trying to do, but IMHO, the CS and OS theory are more important than math here, and you really need only the math that they involve.
For example, there's a lot of CS background of scheduling theory and optimization that stands behind many schedulers in modern OSs. That is an example of something that would require some math, though not something super complicated.
But honestly, for most stuff, you don't need math. What you need is to learn the ability to think in base 2 and 16, such as the ability to mentally OR/AND. For example, if you have a byte and within that byte there are two 3-bit fields and 2 wasted bits, knowing which bits are in which fields are active when the byte value is something like 11 will make things slightly faster than having to use pen and paper.
I began programming about the same time I entered my pre-algebra class.. So I wouldn't say math is all that important, though it can help in certain types of programming, especially functional.
I haven't taken Discrete Math yet, but I see a lot of theory stuff with programming written in a math-notation that is taught in this class.
Also, make sure you know how to calculate anything in any base, especially base 2, 8, and 16.
Also, one class that really brought home some concepts for me was this pre-programming class. We got taught unions, intersections, and all that happy stuff and It almost exactly parallels bitwise math. And we covered boolean logic very heavily. What I considered the most useful was when we learned how to reduce complex boolean statements. This was very handy:
(x|y) & (x|z) & (x|foo)
can be simplified to
x | (y & z & foo)
Which I previously did not quite grasp.
Well you generated a number of responses, and no I did not read them all. I am in the middle on this, no you certainly do not need math in order to be a programmer. Assembler vs device drivers in linux are no more or less complicated than the other and neither require math.
In no way shape or form do you need to take or pass a math class for any of this.
I will agree that the problem solving mindset for programming is quite similar to that of math solutions, and as a result math probably comes easily. or the contrary if math is hard then programming may be hard. A class or a degree or any pieces of paper or trophies are not required, going off and learning stuff, sure.
Now if you cannot convert from hex to binary to decimal quickly either in your head, on paper, or using a calculator you are going to struggle. If you want to get into networking and other things that involve timing, which kernel drivers often do but dont have to. You are going to struggle. I know of a very long list of people with math degrees and/or computer science, and/or engineering degrees that struggle with the rate calculations, bits per second, bytes per second, how much memory you need to do something, etc. To some extent it may be considered some sort of knack that some have and others have to work toward.
My bottom line is I believe in will power, if you want to learn this stuff you can and will, it is as simple as that. You dont need to take a class or spend a lot of money, linux and qemu for example can keep you busy for quite some time, different asm langauges, etc. crashable environments for kernel development, embedded, etc. You are not limited to that, but I dont believe that you have to run off and take any classes if you dont want to. If you want to then sure take some ee classes, some cs classes and some math classes..
You need math. Programming is nothing more than math. Any findings of theoretical physics does not become a practical(applicable) implication, unless they are explained in terms of mathematical solutions. None of those can be solved computationally if they cannot be interpreted on computers, and more specifically on programming languages. Different languages are thus designed to solve specific problems. But for the general purpose and wide spread programming languages like java, c, c++ much of our programming tasks involve repetitive(continuous) solution to same problems like extracting values from database, text files, putting them on windows(desktop, web), manipulating same values, sometimes accessing some data from similar devices(but given different brand names, different port and a headache) etc which does not involve more than unitary method, and algebra(counter, some logic), geometry(graphics) etc. So it depends what you are trying to solve.
IMO, you probably need an aptitude for mathematics, without necessarily having much knowledge in the field. So the things you require to be good at maths are similar to the things you require to be good at programming.
But in general, I can't remember the last time I used any sort of advanced maths in day-to-day programming, so no.
As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.
Closed 10 years ago.
I've heard many times that all programming is really a subset of math. Some suggest that OO, at its roots, is mathematically based, but I don't get the connection, aside from some obvious examples:
using induction to prove a recursive algorithm,
formal correctness proofs,
functional languages,
lambda calculus,
asymptotic complexity,
DFAs, NFAs, Turing Machines, and theoretical computation in general,
and the fact that everything on the box is binary.
I know math is very important to programming, but I struggle with this "subset" view. In what ways is programming a subset of math?
I'm looking for an explanation that might have relevance to enterprise/OO development, if there is a strong enough connection, that is.
It's math in the sense that it requires abstract thought about algorithms etc.
It's engineering when it involves planning schedules, deliverables, testing.
It's art when you have no idea how it's going to eventually turn out.
Programming is one of the most difficult branches of applied mathematics; the poorer mathematicians had better remain pure mathematicians.
--E. W. Dijkstra
Overall, remember that mathematics is a formal codification of logic, which is also what we do in software.
The list of topics in your question is loaded with mathematical problems. We are able to do programming on a fairly high level of abstraction, so the raw mathematics may not be staring you in the face. For example, you mentioned DFAs.. you can use a regular expression in your programs without knowing any math, but you'll find more of a need for mathematics when you want to design a good regular expression engine.
I think you've hit on an interesting point. Programming is an art and a science. There are a lot of "tools of the trade", and you don't necessarily sit down and do a lot of high-level mathematics in order to simply write a program. In fact, when you're programming, you many not really being doing much mathematics or computer science.
It's when we start to solve difficult problems in computer science that mathematics shows up. The deeper you go, the more it will flesh itself out.. often in lower levels of abstraction.
There are also some realms of programming that you don't necessarily have to work in, but they involve more math. For example, while you can certainly learn a language and write some apps without any formal mathematics, you won't get very far in algorithm analysis without some applied math.
OK, I was a math and CS major in college. I would say that if the set A is Math and the set B is CS, then A intersects B. It's not a subset.
It's no doubt that many of the fathers and mothers of computer science were Mathematicians like Turing and Dykstra. Most of the founders of the internet were either Phd's in Math, Physics, or Engineering. Most of the core concepts of computer science come from math, but the act of programming isn't really math. Math helps us in our daily lives, but the two aren't the same.
But there is no doubt that the original reasoning behind the computer was to well, compute things. We have come a long way from there in such a short time.
Doesn't mention programming, but idea is still relevant.
Einstein was known in 1917 as a famous mathematician. It wasn't until Hiroshima that the general public finally came around to the realization that physics is not just applied mathematics.
When people don't understand something, they try to understand it as a type of something that they do understand. They think by analogy. Programming has been described as a field of math, engineering, science, art, craft, construction... None of these are completely false; it borrows from all of these. The real issue is that the field of programming is only about 50 years old. People have not integrated it into their mental taxonomies.
There's a lot of confusion here.
First of all, "programming" does not (currently) equal "computer science." When Dijkstra called himself a "programmer" (more or less inventing the title), he was not pumping out CRUD applications, but actually doing applied computer science. Let's not let that confuse us-- today, there is a vast difference between what most programmers in a business setting do and computer science.
Now, the argument can be made that computer science is a branch of mathematics; but, as Knuth points out (in his paper "Computer Science and its Relation to Mathematics", collected in his Selected Papers on Computer Science) it can also be argued that mathematics is a branch of computer science.
In fact, I'd strongly recommend this paper to anyone thinking about the relationship between mathematics and computer science, as Knuth lays out the territory nicely.
But, to return to your original question: to a practitioner, "enterprise/OO development" is pretty far removed from mathematics-- but that's largely because most of the serious mathematics involved at the lower levels of operation have been abstracted away (by compilers, operating systems, instruction sets, etc.). Similarly, advanced knowledge of the physics of the internal combustion engine are not required for driving a car. Naturally, if you want to design a more efficient car....
if your definition of math includes all forms of formal logic, and programming is defined only by the logic and calculations extant in the code, then programming is a subset of math QED ;-)
but this is like saying that painting is merely putting colored pigments on a surface - it completely igores the art, the insight, the intuition, the entire creative process
one could argue that music is a subset of math by the same reasoning
so i'd have to say no, programming is not a subset of math. Programming uses a subset of math, but requires non-math skills/talent as well [much like music composition]
Disclaimer: I work as an IT consultant and develop mainly portals and Architecture stuff. I have a Psychology degree. I never studied Maths in University. And i get my job done. And usually well. Why? Because I don't think you need to know Maths (as in 'heavy' Maths stuff) to write code. You need analytical thinking, problem-solving skills, and a high level of abstraction. But Maths does not give you that. It's just another discipline that requires similar skills. My studies in Psychology also apply to my daily work when dealing with usability issues and data storage. Linguistics and Semiotics also play a part.
But wait, just don't flame me yet. I'm not saying Maths are not needed at all for computers - obviously, you need real Math skills when designing encryption algorithms and hardware and etc -- but if, as lots of programmers, you just work an a mid/low level language (like C) or higher level stuff (like C# or java), consuming mostly pre-built frameworks and APIs, you don't really need to understand the mathematical principles behind Fourier transforms or Huffman trees or Moebius strips... let someone else handle that, and let me build value on top of it. I am not stupid. I know the difference between linear and exponential algorithms and data structures and etc. I just don't have the interest to rewrite quicksort or a spiffy new video compression technique.
Well, aside from all that...!
Math is used for many aspects of programming such as
Creating efficient and smart algorithms
Understanding Big O notation
Security (such as RSA)
Many more...
I think that programming needs math to survive. But I wouldn't call it a subset. It's just like blowing glass uses properties of physics, but those artists don't call themselves physicists.
The foundation of everything we do is math.
Luckily, we don't need to be good at math itself to do it. Just like you don't need to understand physics to drive a car or even fly a plane.
The difference between programming and pure mathematics is the concept of state.
Have a look at http://en.wikipedia.org/wiki/Dynamic_logic_(modal_logic). It's a way of mathematically analyzing things changing through time. Also, Hoare triples is a way of formalizing the input-output behavior of programs. By having some axioms dealing with sequential composition of programs and how assignment works, you can perfectly well deal with state changing over time in a mathematically rigorous way.
If the math you know is insufficient, "invent" some new math to deal with what you want to analyze. Newton and Leibniz did it for analysis (aka calculus, I think). No reason to not do it for computation and programming.
I don't believe I've heard that programming is a subset of math. Even the link you provide is simply a proposed approach to programming (not claiming it's a subset of mathematics) and the wiki page has plenty of disagreements in it as well.
Programming requires (at least some) applied mathematics. Mathematics can be used to help describe and analyze programs and program fragments. Programming has a very close relationship with math and uses it and concepts from it heavily. But subset? no.
I'd love to see someone actually claim that it is one with some clear reasoning. I don't think I ever have
Just because you can use mathematics
to reason about something does not
imply that it is, ipso facto, a
mathematical object. Mathematics is
used to reason about internal
combustion engines, radioactive decay
and juggling patterns. Using
mathematics is not doing mathematics.
I would say...
It's partly math, especially at the theoretical level. Imagine designing efficient searching/sorting/clustering/allocating/fooifying algorithms, that's all math... running the gamut from number theory to statistics.
It's partly engineering. Complex systems can rarely achieve ideal levels of performance and reliability, and software is no exception. A lot of software development is about achieving robustness in the face of unreliable hardware and (ahem) humans.
And it's partly art. Creative and idiosyncratic software design often comes up with great new ideas... like assembly language, multitasking operating systems, graphical user interfaces, dynamic languages, and the web.
Just my 2¢...
Math + art + logic
You can actually argue that math, in the form of logical proofs, is analogous to programming --
Check out the Curry-Howard correspondence. It's probably more the way a mathematician would look at things, but I think this is hitting the proverbial nail on the head.
Programming may have originally started as a quasi-subset of math, but the increasing complexity of the field over time has led to programming being the art and science of creating good abstractions for information processing and computation.
Programming does involve math, engineering, and an aesthetic sense for good design and implementation. Algorithms are an extension of mathematics, and the systems engineering side overlaps with other engineering disciplines to some degree. However, neither mathematics nor other engineering fields have the same level of need for complex, flexible, and yet understandable abstractions that can be used and adapted at so many different levels to solve new and evolving problems.
It is the need for useful, flexible, and dynamic abstractions which led first to the creation of function libraries, then class/component libraries, and in more recent years design patterns and service-oriented architectures. Although the latter have more of a design focus, they are a reaction to the increasing need to build high-level abstractional bridges between programming problems and solutions.
For all of these reasons, programming is neither a subset nor a superset of math. It is simply yet another field which uses math that has deeper roots in it than others do.
The topics you listed are topics in Theoretical Computer Science, and THAT is a branch of Pure Mathematics. Programming is an applied science which uses theoretical computer science. Programming itself isn't a branch of mathematics but the Lambda Calculus/theory of computation/formal logic/set theory etc that programming languages are based on is.
Also I completely disagree with Dijkstra. It's either self-congratulatory or Dijkstra is being misquoted/quoted out of context. Pure mathematics is a very very very difficult field. It is so enormously abstract that no branch of applied mathematics is comparable in difficulty. It is one field that requires enormous leaps of imagination. I did my first degree in computer science where I focused a lot on theoretical CS and applied areas like programming, OS, compilers. I also did a degree in Electrical Engineering - arguably the most difficult branch of engineering - and worked on difficult areas of applied mathematics like Maxwell's equations, control theory and partial differential equations in general.
I've also done research in applied and pure mathematics, and to this day I find applied far easier. As for the pure mathematicians, they're a whole different breed.
Now there's a tendency for someone to study an year or two of calculus unhinged from application and conclude that pure mathematics is easy. They have no idea what they're talking about. Studying calculus or even topology unhinged from application does not give you any inkling of what a pure mathematician does. The task of actually proving those theorems are so profoundly difficult that I will defer to a computer scientist to point out the distinction:
"If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in 'creative leaps,' no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss..." —Scott Aaronson, (Theoretical Computer Scientist, MIT)
I think mathematics provides a set of tools for programmers which they use at abstract level
to solve real world problems.
I would say that programming is less about math than it used to be as we move up to 4th Generation Languages. Assembly is very much about math, C#, not so much. Thoughts?
If you just want the design specs handed out to you by your boss, then it's not much math but such a work isn't fun at all... However, coming up with how to do things does require mathematical ideas, at least things like abstraction, graphs, sometimes number theory stuffs and depending on the problems, calculus. Personally, more I've been involved with programming, more I see the mathematical side to it. However, most of the times IMO, you can just pick up the book from library and look up the basics of the thing you need to do but that requires some mathematical grasp upfront.
You really can't design "good" algorithms without understanding the maths behind it. Searching in google takes you only so far.
Programming is a too wide subject. Good software based not only on math (logic) but also on psychology, linguistics etc. Algorithms are part of math, but there are many other programming-related things besides algorithms.
As a mathematician, it is clear to me that Math is not equal to Programming but that the process which is used to solve problems in either discipline is extremely similar.
Solving a higher level mathematics questions requires analytical thinking, a toolbox of possible ways of solving problems, experience with the field, and some formalized ways of constructing your answer so that other mathematicians agree. If you find a particularly clever, abstract, or elegant way of solving a problem, you get Kudos from your fellow mathematicians. For particularly difficult math problems, you may solve the problem in stages, and codify your stage arguments using things called conjectures and proofs.
I think programming involves the same set of skills. In programming, the same set of principles applies to the solving and presenting of solutions to problems. When you have a partial solution to a programming dilemna, you include it as part of your personal library and use it as part of another bigger problem later. These skills seem very similar to the skills used in mathematics.
The major difference between Math and Programming is the latter has a lot more in common between different disciplines of programming than Math does. Two fields of mathematics can be very, very different in presentation and what is used to communicate the field. By contrast, programming structures, to me at least, look very similar in many different languages.
The difference between programming and pure mathematics is the concept of state. A program is a state machine that uses logic (maths) to transition between states. The actual logic used to transition between states is usually very simple, which is why being a math genius doesn't necessarily help you all that much as a programmer.
Part of the reason I'm a programmer is because I don't like math. I have no problem with math itself, and I'm fine with it conceptually, I just don't like doing calculations by hand. When I found I could tell a computer what the math problem is and let it do the calculating for me, a life-long passion and career was born.
To answer the question, according to my alma mater, math == programming since they allowed me to take Intro to C++ to fulfill my math requirement.
Edit: I should mention my degree is in telecommunications which, at the time, had only the standard liberal arts math requirement of one semester.
Math is the purest form of truth. Everything inherits from math.
Amen.
It's interesting to compare programming with music too. In UK, anyway, there are computing based undergrad university courses that will accept applicants on the bases of music qualifications as supposed to computing due to the logic, patterns, etc. involved.
Maths is powerful, programming is powerful, if maths is a subset of programming then it is equally true to state that programming is a subset of maths.
Maths is described using language, often written down. Therefore is maths a subset of writing too?
Historicly maths came before computer programming, but then lists and processes probably preceded maths, both of which could be equally thought of as mathematical or do with programming.
Cirtainly programming can be represented using maths, so there is some bases for it being true that programming is a sub-set of maths. However a computer program could also implement maths, representing information symbolically, as maths typically does when done on paper, including the infinite and only somewhat defined, from the fundamental axioms, as well as allowing higher level structures to be defined that use each other and other sorts of relationships beyond composition, supporting the drawing of diagrams and allowing the system to be expanded. Maths is equally a subset of programming.
While maths can represent structures such as words, maths is by design about numbers. Strings for example are more programmatic than mathematic.
It's half math, half man speak, duh.