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> Great mathematicians can understand the theory and just apply it to whatever problem they come across. As someone who's both a mathematician and a programmer

> Great mathematicians can understand the theory and just apply it to whatever problem they come across. As someone who's both a mathematician and a programmer, I think that analogy glosses over an important difference, which is also relevant to why many people refuse to memorize certain things. Good mathematical theories are internally consistent in a very strong sense. When you're learning them, you feel like you're learning something eternal that could not possibly turn out any other way. If you forget half of calculus, you can reconstruct it from the other half, and the reconstruction will be unique. The details of programming languages, like C and especially C++, are not like that. For many mathematicians who would otherwise make excellent programmers, the details of C will feel like something not worth memorizing, because they don't make sense - they are not an inevitable, provably unique solution to any problem. To a mathematically inclined mind, programming concepts form a hierarchy depending on how "inevitable" and worth memorizing they are. Concepts like computability, lambda calculus or big-O complexity are near the top, syntaxes of specific languages are lower down, and API details are at the bottom. That also reflects the rates of change: math doesn't change, languages change every decade, and APIs change every year. If I want to hire a programmer for the long term, as opposed to getting quick help on my current project, I will be mostly asking them about concepts from the top of the hierarchy, not what static means in C. Of course some people will disagree with me and demand detailed knowledge about static or templates or whatnot, while dismissing mathematical questions as "puzzles". Maybe that's also a valid style of programming, I'm not sure...

tel 13y ago
A thousand times this. Mathematicians are spoilt by maths riches of eternal problems. CS ought to be too, but so many are instead fascinated by the 10 year, 2 year, 1 week problems around the fashion of a programming tool. No wonder everyone is always talking about the rapid pace of learning in high technology.
ksk 13y ago
No, I too am talking about a hierarchy of internally consistent knowledge. Have you implemented a compiler or a interpreter for a popular language? I could be wrong but I strongly suspect you haven't. If you have ever implemented a compiler you understand the language at a fundamental level and when stumped with a bug or a highly technical puzzle or what have you - you get that "aha" moment where its like "Ofcource ! How else could it be. This language feature must be using this construct because of X or Y reason and the compiler has to work this way because this part here doesnt allow for it to work any other way, etc ,etc and the optimizer has xyz amount of scratch registers so abc condition could never happen, and on and on."
- cousin_it 13y ago
  My comment was talking about a stronger kind of consistency, maybe "inevitability" would be a better word for it. You could make many different tiny changes to the rules of C and still end up with a workable programming language. (Many people have done that, there are tons of languages derived from C.) You could say that the rules of C are "random" - after learning the first n rules, you cannot use logic to predict the n+1st. So learning the rules of C is more about memorization than logic. Math is different: when you learn a piece of math, you know exactly how it follows logically from the pieces you learned before, and you know it couldn't be any other way. That's why it feels more reasonable to me to inquire about math, rather than the details of C. Lawrence Kesteloot said it well at http://www.teamten.com/lawrence/writings/what_interests_me.html http://www.teamten.com/lawrence/writings/what_interests_me.h...: On a planet far away they have universities. In those universities they have many of the same subjects we teach here, such as math, biology, philosophy, history, computer science, psychology, and literature. But although the subjects of the classes are the same, the contents are different. They teach a history class, but their history is different from ours. So are their biology and literature. I’m not interested in subjects where their content is different from ours. In some classes, though, the content is the same. They’re teaching the same physics class (assuming they’re at the same level of understanding we’re at) and the same math classes (with a different base). In computer science most of what they teach is different, but their theory of computation is surely the same. Their biology is different, but their teaching of evolution is the same. Their philosophy is vastly different (think how different Eastern and Western philosophies are on earth), but their philosophy of science is probably very similar. Even if this point of view doesn't make sense to some people, I subscribe to it because it's very inspiring to me. YMMV.
  - ksk 13y ago
    >You could say that the rules of C are "random" - after learning the first n rules, you cannot use logic to predict the n+1st. You're mis-applying things here. The language is not something to be theorized by itself. Like any (edit:spell) other body of knowledge, it has its own set of initial axioms and the 'C standard' which defines the language is simply a collection of rules other meta-knowledge which are based on those. In that sense it is internally consistent. Thats what I meant.
alexk7 13y ago
Standard mathematical notation is not inevitable and is not unique. It is on par with most programming languages for being quite random. Both programmers and mathematicians use an imperfect tool (programming language, mathematical notation) to achieve goals that may, or may not, be eternal.
- cousin_it 13y ago
  That's a fair point, and math notation is famously more ambiguous than programming languages. But it seems to me that the concepts denoted by math notation (e.g. numbers, arithmetical operations, functions...) are inevitable and unique, while the concepts denoted by strings in a programming language are much fuzzier and more arbitrary (except for languages that have a denotational semantics, which are few and far between).
  - agentultra 13y ago
    People often speak of the axioms in maths when they describe how, "elegant," and, "consistent," it is. I'm not a mathematician but I've read Gödel's "proof" and know enough that there are no guarantees. It matters what system of axioms you use and requires the reader to recognize the limits of those axioms in defining the system of logic they they imply. Or some such. Point being that there is no system that is complete. There are large bodies of mathematics these days that have specialized their notations and axioms to their domains. It is incredibly difficult to maintain a masterful knowledge of all of their idiosyncrasies and notations.
kazagistar 13y ago
Mathematicians tend to miss one important ideal; a major consumer of your code needs to be other programmers who read it. Hence why mathematicians often end up writing code that, while more terse, is a jumbled mess of single letter variables and layered abstractions.
blt 13y ago
Scheme has a bit of flavor of the "inevitable" to me...

tel 13y ago

A thousand times this. Mathematicians are spoilt by maths riches of eternal problems. CS ought to be too, but so many are instead fascinated by the 10 year, 2 year, 1 week problems around the fashion of a programming tool. No wonder everyone is always talking about the rapid pace of learning in high technology.

ksk 13y ago

No, I too am talking about a hierarchy of internally consistent knowledge. Have you implemented a compiler or a interpreter for a popular language? I could be wrong but I strongly suspect you haven't. If you have ever implemented a compiler you understand the language at a fundamental level and when stumped with a bug or a highly technical puzzle or what have you - you get that "aha" moment where its like "Ofcource ! How else could it be. This language feature must be using this construct because of X or Y reason and the compiler has to work this way because this part here doesnt allow for it to work any other way, etc ,etc and the optimizer has xyz amount of scratch registers so abc condition could never happen, and on and on."

cousin_it 13y ago
My comment was talking about a stronger kind of consistency, maybe "inevitability" would be a better word for it. You could make many different tiny changes to the rules of C and still end up with a workable programming language. (Many people have done that, there are tons of languages derived from C.) You could say that the rules of C are "random" - after learning the first n rules, you cannot use logic to predict the n+1st. So learning the rules of C is more about memorization than logic. Math is different: when you learn a piece of math, you know exactly how it follows logically from the pieces you learned before, and you know it couldn't be any other way. That's why it feels more reasonable to me to inquire about math, rather than the details of C. Lawrence Kesteloot said it well at http://www.teamten.com/lawrence/writings/what_interests_me.html http://www.teamten.com/lawrence/writings/what_interests_me.h...: On a planet far away they have universities. In those universities they have many of the same subjects we teach here, such as math, biology, philosophy, history, computer science, psychology, and literature. But although the subjects of the classes are the same, the contents are different. They teach a history class, but their history is different from ours. So are their biology and literature. I’m not interested in subjects where their content is different from ours. In some classes, though, the content is the same. They’re teaching the same physics class (assuming they’re at the same level of understanding we’re at) and the same math classes (with a different base). In computer science most of what they teach is different, but their theory of computation is surely the same. Their biology is different, but their teaching of evolution is the same. Their philosophy is vastly different (think how different Eastern and Western philosophies are on earth), but their philosophy of science is probably very similar. Even if this point of view doesn't make sense to some people, I subscribe to it because it's very inspiring to me. YMMV.
- ksk 13y ago
  >You could say that the rules of C are "random" - after learning the first n rules, you cannot use logic to predict the n+1st. You're mis-applying things here. The language is not something to be theorized by itself. Like any (edit:spell) other body of knowledge, it has its own set of initial axioms and the 'C standard' which defines the language is simply a collection of rules other meta-knowledge which are based on those. In that sense it is internally consistent. Thats what I meant.

alexk7 13y ago

Standard mathematical notation is not inevitable and is not unique. It is on par with most programming languages for being quite random. Both programmers and mathematicians use an imperfect tool (programming language, mathematical notation) to achieve goals that may, or may not, be eternal.

cousin_it 13y ago
That's a fair point, and math notation is famously more ambiguous than programming languages. But it seems to me that the concepts denoted by math notation (e.g. numbers, arithmetical operations, functions...) are inevitable and unique, while the concepts denoted by strings in a programming language are much fuzzier and more arbitrary (except for languages that have a denotational semantics, which are few and far between).
- agentultra 13y ago
  People often speak of the axioms in maths when they describe how, "elegant," and, "consistent," it is. I'm not a mathematician but I've read Gödel's "proof" and know enough that there are no guarantees. It matters what system of axioms you use and requires the reader to recognize the limits of those axioms in defining the system of logic they they imply. Or some such. Point being that there is no system that is complete. There are large bodies of mathematics these days that have specialized their notations and axioms to their domains. It is incredibly difficult to maintain a masterful knowledge of all of their idiosyncrasies and notations.

kazagistar 13y ago

Mathematicians tend to miss one important ideal; a major consumer of your code needs to be other programmers who read it. Hence why mathematicians often end up writing code that, while more terse, is a jumbled mess of single letter variables and layered abstractions.

blt 13y ago

Scheme has a bit of flavor of the "inevitable" to me...