5 ms·

Notation And Thinking

85 points by Randy00 16y ago

gfodor 16y ago
The interesting thing to note is that most of this discussion around notation is being applied to mathematics. Trying to provide higher level notation (beyond ASCII) to programming apparently was given up on a long time ago. It's a shame.
- andrewl 16y ago
  There's always Fortress, which can be written in Unicode: http://projectfortress.sun.com/Projects/Community/ http://projectfortress.sun.com/Projects/Community/
  - derefr 16y ago
    Fortress code is still a linear stream of characters in a grammar that gets LR-parsed into an AST, though. Many other approaches are possible if you start with a different data structure—a digraph of tokens with node-identity, for example.
- jamii 16y ago
  Notation doesn't apply just to the symbols in which you write but the meanings and operations assigned to them and the way in which they can be combined. This applies as much to the expressiveness of language semantics as it does to the language syntax. Consider how things like first class functions change the way one approaches a problem and expresses the solution.
- Jach 16y ago
  It is a shame, but I'm not sure how we could fix that situation. I'd much prefer to type λ than 'lambda', but my keyboard doesn't have a λ key!
  - gfodor 16y ago
    No, but your iPad could.
  - pavel_lishin 16y ago
    Does it have a ⊄ key?
    - _delirium 16y ago
      No, but that's why I never type it directly. In TeX, one gets it via the ASCII notation \not\subset. This gives TeX-math the curious property that it's read in a different set of symbols than it's written in. There have been some attempts at that in pretty-printing programming languages, but none seem to have really caught on.
- maxwell 16y ago
  The problem with non-ASCII is that it's harder to read, because the pronunciation (even if only mental) is ambiguous. Imagine if you don't know how to pronounce Greek letters, for instance. I couldn't begin to pronounce APL, and thus would have to remember shapes. I think this is the reason everyone has an issue with Lisp's prefix notation for arithmetic. They read (+ 1 2 3) as "plus one two three" and it sounds weird. I wonder if a Lisp would sell if you only allowed alphanumeric function names, e.g. (sum 1 2 3) (Not to mention that "Lisp" is the worst name ever.) It's just a hunch, but I suspect the root cause for European imperialistic domination was that they had a more accessible writing notation. It could even be that the difficulty of Chinese writing actually produced "more Edisons", but likewise prevented diffusion. And if James Burke has taught me anything, it's that communication is way more important than inspiration or perspiration. It's exactly like Lisp. As it took Europe centuries to catch up with Chinese technology, it seems likely that it'll take Algol-style langs a century to catch up with Lisp concepts.
  - jessriedel 16y ago
    But your argument could be applied to mathematics, where powerful, non-ASCII symbols are very useful. What's different about programming?
    - maxwell 16y ago
      Nothing. I think non-ASCII symbols are a net loss in mathematics too. While they're powerful (taking succinctness to be equal to power), they come at the expense of accessibility. With great power comes great power consumption; there's a reason most people don't have jet engines under their hood or read mathematics: doing so is very expensive. I believe the cognitive cost outweighs the expressive benefit.
  - jeangenie 16y ago
    >> It's just a hunch, but I suspect the root cause for European imperialistic domination was that they had a more accessible writing notation. the greeks devised a fantastic writing system, including representations of vowels (which its predecessor lacked). aristotle was the first person to say words should be separated when written (before all letters were run together, leaving the reader to unscramble). your claim is a stretch, but it's interesting to note that reliable and easy ways to communicate and sustain our collective knowledge have become unexpectedly critical.
andrewl 16y ago
Numerals themselves are also notation, and I've always liked this from Alfred North Whitehead: "By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that ... a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility ... Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. [...] By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain. [...] It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle -- they are strictly limited in number, they require fresh horses, and must only be made at decisive moments." --from An Introduction to Mathematics, 1911
- scott_s 16y ago
  Amazing - that's the same argument people make regarding higher level abstractions in programming languages. For the first time, I've seen the connection between how we have been evolving expressing computations for computers and how we have evolved expressing computation for our own minds. Of course, I have known for a long time that math is a language unto itself, and that, say, the Arabic numeral system is superior in most ways to the Roman numeral system. But I never before saw that relationship on the same spectrum as programming languages.
- EricNV 16y ago
  We don't think of language as a technology, but it's actually the most important one we have.
thehotdon 16y ago
Notation is a solved problem. Math got it wrong, Lisp got it right.
dangrover 16y ago
For some reason, math notation has always felt less formal and rigorous to me than any programming language. It seems very "analog" and vague, whereas you can always run a program to get a result.
- ced 16y ago
  For some reason, math notation has always felt less formal and rigorous to me than any programming language. Think of how liberating it would be if you could just write pseudo-code, and the computer would figure out what you meant, instead of complaining about a missing semi-colon. That's how mathematics is typically written: for other mathematicians to read. But the underlying meaning conveyed is usually very, very precise, more so than in a typical program (which may have bugs in spite of being "formal and rigorous"). you can always run a program to get a result That's the distinction between procedural and declarative information. Most lower animals, and most programming languages are firmly in the procedural camp. Perhaps the biggest breakthrough in Homo Sapiens was the ability to do declarative reasoning, which is much more flexible and powerful. There are a few baby steps in declarative programming languages, like Prolog and Mathematica, but the revolution has yet to happen IMO.
  - maxwell 16y ago
    Think of how liberating it would be if you could just write pseudo-code, and the computer would figure out what you meant, instead of complaining about a missing semi-colon. I've noticed Lisp, Python, Ruby, JavaScript, etc. hackers don't tend to talk about "pseudo-code", and I believe it's that they think in their chosen notation, rather than mentally compiling it. I know I do. I think in JavaScript, not "pseudo-code". Not that I wouldn't opt for more elegant notation (e.g. CoffeeScript).
  - dangrover 16y ago
    Well, if by pseudo-code you mean super-terse language fitted for the problem at hand, then yes, that's much better than having to remember semicolons and stuff. But not if you mean not having to fully specify the details of new ideas you're expressing in your chosen language/notation (e.g. the mythical "powerpoint compiler" that turns ideas into code). If your program doesn't compile, or has a fatal flaw in its logic -- whether it's written declaratively or procedurally -- you have to fix it. You don't get to just say "Oh, this is an exercise for the reader, fuck you." I guess my problem really is with teaching more than the actual subject matter. Math, at its best, is a formal system just like anything else. You build understanding through axioms and proving things and you can always test an idea by trying it out. At least, so I hear. I've never actually had it taught that way, and have never been able to make myself think of it that way. In my studies, mathematicians usually prove things more by fiat and intimidation and hand-waving, not by actually working through a system and its rules. It's hard to actually trace reasoning down the entire "stack" like you would when working in code. The only GOOD math-y books I've read are Sipser's Theory of Computation and GEB. The tone of these is "Here's how shit is, and here's why Y makes sense if you accept X", not "Here's the problems you'll be doing on the test, and you have to practice them so you can solve them with pen and paper in less than 2 minutes each" When teaching introductory programming to newbies, part of the goal is to get people to think of it as a consistent formal system that you can do anything you want within to reach your goals. If something doesn't compile or you find a bug, you work through and test your thinking at every way to see what the problem is. But math is usually taught as a set of heuristics for solving pre-defined types of problems. You have to just take it on faith half the time. I wish I could kind of have some kind of epiphany and suddenly understand math, but it's never going to happen.
kragen 16y ago
It's disappointing that this whole article ends without ever mentioning Ken Iverson's famous Turing Award Lecture, which is entitled "Notation as a tool of thought": http://www.jsoftware.com/papers/tot.htm http://www.jsoftware.com/papers/tot.htm Perhaps the most interesting modern controversy about notation for programs is the one about point-free style in, for example, Haskell; advocates claim that it makes refactoring ("calculating") much easier, while opponents claim that it makes programs hard to understand. They may both be right. A fun paper on this from 1994 is "An Introduction to the Bird-Meertens Formalism", http://www.comlab.ox.ac.uk/people/jeremy.gibbons/publications/nzfpdc-squiggol.ps.gz http://www.comlab.ox.ac.uk/people/jeremy.gibbons/publication... --- it shows a step-by-step transformation from a one-line program that takes cubic time, but is obviously correct, to a one-line program that takes linear time. It occurs to me that 1994 is rather a long time ago now, though. What's the current status of this line of work?
amichail 16y ago
It's amazing how many people don't know proper asymptotic notation and use big O for everything. It doesn't help that research papers use wrong asymptotic notation.
- scott_s 16y ago
  I suspect the problem is not that most people do not know the proper notation, but that they are unfamiliar with the concepts behind big omega and big theta.
splat 16y ago
Tangentially related but interesting nonetheless: http://en.wikipedia.org/wiki/Abuse_of_notation http://en.wikipedia.org/wiki/Abuse_of_notation
jessriedel 16y ago
> The power of the [Dirac bra-ket] notation is that can be a symbol, expression, or even words that describe the state values. No, if this were true, then it wouldn't make any difference what the combination of braces, brackets, parentheses, and pipes were. All that would matter is that you could label a vector by an arbitrary word. That's hardly the importance of the notation. Rather, the asymmetry of the notation for a vector |x> and it's mirror image the dual vector <y| emphasizes the canonical duality isomorphism on inner product spaces. The "hanging" pipe on a vector remind one that one must attach a dual vector (possibly sandwiching operators) in order to form a scalar. As the author later notes, there are other identities which become trivial when using the notation. Incidentally, it's interesting that Einstein notation (which the author only briefly mentions) is basically a more powerful--though less suggestive--version of bra-ket notation. Basically, you can represent Dirac vectors in Einstein notation as variables with a superscript letter and Dirac dual vectors as variables with a subscript letter. Operators then always have one superscript and one subscript. The power of Einstein notation is that it allows high-dimensional tensors than Dirac. The GR textbook by Wald has a good discussion of this, and Penrose write poetically about it (and his preferred graphical notation) in "The Road to Reality".

85 points by Randy00 16y ago

The interesting thing to note is that most of this discussion around notation is being applied to mathematics. Trying to provide higher level notation (beyond ASCII) to programming apparently was given up on a long time ago. It's a shame.

andrewl 16y ago
There's always Fortress, which can be written in Unicode: http://projectfortress.sun.com/Projects/Community/ http://projectfortress.sun.com/Projects/Community/
- derefr 16y ago
  Fortress code is still a linear stream of characters in a grammar that gets LR-parsed into an AST, though. Many other approaches are possible if you start with a different data structure—a digraph of tokens with node-identity, for example.
jamii 16y ago
Notation doesn't apply just to the symbols in which you write but the meanings and operations assigned to them and the way in which they can be combined. This applies as much to the expressiveness of language semantics as it does to the language syntax. Consider how things like first class functions change the way one approaches a problem and expresses the solution.
Jach 16y ago
It is a shame, but I'm not sure how we could fix that situation. I'd much prefer to type λ than 'lambda', but my keyboard doesn't have a λ key!
- gfodor 16y ago
  No, but your iPad could.
- pavel_lishin 16y ago
  Does it have a ⊄ key?
  - _delirium 16y ago
    No, but that's why I never type it directly. In TeX, one gets it via the ASCII notation \not\subset. This gives TeX-math the curious property that it's read in a different set of symbols than it's written in. There have been some attempts at that in pretty-printing programming languages, but none seem to have really caught on.
maxwell 16y ago
The problem with non-ASCII is that it's harder to read, because the pronunciation (even if only mental) is ambiguous. Imagine if you don't know how to pronounce Greek letters, for instance. I couldn't begin to pronounce APL, and thus would have to remember shapes. I think this is the reason everyone has an issue with Lisp's prefix notation for arithmetic. They read (+ 1 2 3) as "plus one two three" and it sounds weird. I wonder if a Lisp would sell if you only allowed alphanumeric function names, e.g. (sum 1 2 3) (Not to mention that "Lisp" is the worst name ever.) It's just a hunch, but I suspect the root cause for European imperialistic domination was that they had a more accessible writing notation. It could even be that the difficulty of Chinese writing actually produced "more Edisons", but likewise prevented diffusion. And if James Burke has taught me anything, it's that communication is way more important than inspiration or perspiration. It's exactly like Lisp. As it took Europe centuries to catch up with Chinese technology, it seems likely that it'll take Algol-style langs a century to catch up with Lisp concepts.
- jessriedel 16y ago
  But your argument could be applied to mathematics, where powerful, non-ASCII symbols are very useful. What's different about programming?
  - maxwell 16y ago
    Nothing. I think non-ASCII symbols are a net loss in mathematics too. While they're powerful (taking succinctness to be equal to power), they come at the expense of accessibility. With great power comes great power consumption; there's a reason most people don't have jet engines under their hood or read mathematics: doing so is very expensive. I believe the cognitive cost outweighs the expressive benefit.
- jeangenie 16y ago
  >> It's just a hunch, but I suspect the root cause for European imperialistic domination was that they had a more accessible writing notation. the greeks devised a fantastic writing system, including representations of vowels (which its predecessor lacked). aristotle was the first person to say words should be separated when written (before all letters were run together, leaving the reader to unscramble). your claim is a stretch, but it's interesting to note that reliable and easy ways to communicate and sustain our collective knowledge have become unexpectedly critical.

andrewl 16y ago

Numerals themselves are also notation, and I've always liked this from Alfred North Whitehead: "By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that ... a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility ... Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. [...] By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain. [...] It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle -- they are strictly limited in number, they require fresh horses, and must only be made at decisive moments." --from An Introduction to Mathematics, 1911

scott_s 16y ago
Amazing - that's the same argument people make regarding higher level abstractions in programming languages. For the first time, I've seen the connection between how we have been evolving expressing computations for computers and how we have evolved expressing computation for our own minds. Of course, I have known for a long time that math is a language unto itself, and that, say, the Arabic numeral system is superior in most ways to the Roman numeral system. But I never before saw that relationship on the same spectrum as programming languages.
EricNV 16y ago
We don't think of language as a technology, but it's actually the most important one we have.

thehotdon 16y ago

Notation is a solved problem. Math got it wrong, Lisp got it right.

dangrover 16y ago

For some reason, math notation has always felt less formal and rigorous to me than any programming language. It seems very "analog" and vague, whereas you can always run a program to get a result.

ced 16y ago
For some reason, math notation has always felt less formal and rigorous to me than any programming language. Think of how liberating it would be if you could just write pseudo-code, and the computer would figure out what you meant, instead of complaining about a missing semi-colon. That's how mathematics is typically written: for other mathematicians to read. But the underlying meaning conveyed is usually very, very precise, more so than in a typical program (which may have bugs in spite of being "formal and rigorous"). you can always run a program to get a result That's the distinction between procedural and declarative information. Most lower animals, and most programming languages are firmly in the procedural camp. Perhaps the biggest breakthrough in Homo Sapiens was the ability to do declarative reasoning, which is much more flexible and powerful. There are a few baby steps in declarative programming languages, like Prolog and Mathematica, but the revolution has yet to happen IMO.
- maxwell 16y ago
  Think of how liberating it would be if you could just write pseudo-code, and the computer would figure out what you meant, instead of complaining about a missing semi-colon. I've noticed Lisp, Python, Ruby, JavaScript, etc. hackers don't tend to talk about "pseudo-code", and I believe it's that they think in their chosen notation, rather than mentally compiling it. I know I do. I think in JavaScript, not "pseudo-code". Not that I wouldn't opt for more elegant notation (e.g. CoffeeScript).
- dangrover 16y ago
  Well, if by pseudo-code you mean super-terse language fitted for the problem at hand, then yes, that's much better than having to remember semicolons and stuff. But not if you mean not having to fully specify the details of new ideas you're expressing in your chosen language/notation (e.g. the mythical "powerpoint compiler" that turns ideas into code). If your program doesn't compile, or has a fatal flaw in its logic -- whether it's written declaratively or procedurally -- you have to fix it. You don't get to just say "Oh, this is an exercise for the reader, fuck you." I guess my problem really is with teaching more than the actual subject matter. Math, at its best, is a formal system just like anything else. You build understanding through axioms and proving things and you can always test an idea by trying it out. At least, so I hear. I've never actually had it taught that way, and have never been able to make myself think of it that way. In my studies, mathematicians usually prove things more by fiat and intimidation and hand-waving, not by actually working through a system and its rules. It's hard to actually trace reasoning down the entire "stack" like you would when working in code. The only GOOD math-y books I've read are Sipser's Theory of Computation and GEB. The tone of these is "Here's how shit is, and here's why Y makes sense if you accept X", not "Here's the problems you'll be doing on the test, and you have to practice them so you can solve them with pen and paper in less than 2 minutes each" When teaching introductory programming to newbies, part of the goal is to get people to think of it as a consistent formal system that you can do anything you want within to reach your goals. If something doesn't compile or you find a bug, you work through and test your thinking at every way to see what the problem is. But math is usually taught as a set of heuristics for solving pre-defined types of problems. You have to just take it on faith half the time. I wish I could kind of have some kind of epiphany and suddenly understand math, but it's never going to happen.

kragen 16y ago

It's disappointing that this whole article ends without ever mentioning Ken Iverson's famous Turing Award Lecture, which is entitled "Notation as a tool of thought": http://www.jsoftware.com/papers/tot.htm http://www.jsoftware.com/papers/tot.htm Perhaps the most interesting modern controversy about notation for programs is the one about point-free style in, for example, Haskell; advocates claim that it makes refactoring ("calculating") much easier, while opponents claim that it makes programs hard to understand. They may both be right. A fun paper on this from 1994 is "An Introduction to the Bird-Meertens Formalism", http://www.comlab.ox.ac.uk/people/jeremy.gibbons/publications/nzfpdc-squiggol.ps.gz http://www.comlab.ox.ac.uk/people/jeremy.gibbons/publication... --- it shows a step-by-step transformation from a one-line program that takes cubic time, but is obviously correct, to a one-line program that takes linear time. It occurs to me that 1994 is rather a long time ago now, though. What's the current status of this line of work?

amichail 16y ago

It's amazing how many people don't know proper asymptotic notation and use big O for everything. It doesn't help that research papers use wrong asymptotic notation.

scott_s 16y ago
I suspect the problem is not that most people do not know the proper notation, but that they are unfamiliar with the concepts behind big omega and big theta.

splat 16y ago

Tangentially related but interesting nonetheless: http://en.wikipedia.org/wiki/Abuse_of_notation http://en.wikipedia.org/wiki/Abuse_of_notation

jessriedel 16y ago

> The power of the [Dirac bra-ket] notation is that can be a symbol, expression, or even words that describe the state values. No, if this were true, then it wouldn't make any difference what the combination of braces, brackets, parentheses, and pipes were. All that would matter is that you could label a vector by an arbitrary word. That's hardly the importance of the notation. Rather, the asymmetry of the notation for a vector |x> and it's mirror image the dual vector <y| emphasizes the canonical duality isomorphism on inner product spaces. The "hanging" pipe on a vector remind one that one must attach a dual vector (possibly sandwiching operators) in order to form a scalar. As the author later notes, there are other identities which become trivial when using the notation. Incidentally, it's interesting that Einstein notation (which the author only briefly mentions) is basically a more powerful--though less suggestive--version of bra-ket notation. Basically, you can represent Dirac vectors in Einstein notation as variables with a superscript letter and Dirac dual vectors as variables with a subscript letter. Operators then always have one superscript and one subscript. The power of Einstein notation is that it allows high-dimensional tensors than Dirac. The GR textbook by Wald has a good discussion of this, and Penrose write poetically about it (and his preferred graphical notation) in "The Road to Reality".