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Exactly. This is basically just a for loop in a symbolic notation. Math would be an entirely more enjoyable subject if it were communicated in more practical t

Exactly. This is basically just a for loop in a symbolic notation. Math would be an entirely more enjoyable subject if it were communicated in more practical terms that can be executed on a CPU.

gmfawcett 10y ago
Well, not really a for-loop, it's the definition of a set. There are so many interesting objects in math that cannot be executed on a computer. You could define the set of permutations of the naturals, for example, which is an infinitely large set; there's no algorithm you can execute which could produce such a set on your computer. Math would be severely limited if it could only deal with things that were computable on your workstation.
- khedoros1 10y ago
  You could define the function to generate arbitrary sections of the set. It's not like you're limited to actually generating the entire set, in order to work with it in a practical sense.
- matchagaucho 10y ago
  True... in my haste I mistook set for sigma (my other pet peeve where symbols make CS translation unnecessarily difficult).
  - pron 10y ago
    Sigma isn't a for loop, either. You can sum over infinite sequences. A for loop -- or any computation for that matter -- implies a certain dynamic process, with a certain computational complexity. A mathematical expression has no dynamics; it just describes an object. So Σ 1/(2^n) doesn't mean "sum up all 2^-n" -- i.e., it is not a process -- but simply, "the number equal to the sum of all 2^-n". Of course, summation of infinite sequences requires its own definition using limits. You can have math that is entirely computational -- it's called constructive math -- but the result is a very different math. For example, in constructive math, all functions over the reals are continuous, and not all subsets of finite sets are finite.
    - matchagaucho 10y ago
      But... come on. Sigma is basically a for-loop if you're translating Math symbolism into code. The edge case of summing infinite sequences has no practical application in CS.
      - pron 10y ago
        > The edge case of summing infinite sequences has no practical application in CS. The number 2 is \Sigma_{n=0}^{\infty}1/2^n. So you're "using" sigma every time you use the number 2, which may be practical in some programs. The fact that you can't write a program that sums an infinite sequence using an infinite number of computational steps doesn't mean uncomputable objects have no practical application. In fact, the formalism Lamport talked about in his lecture makes common use of them, and you do, too, every time you use floating point arithmetics. When you use floating point numbers, it's very convenient to think of them not as the very complicated objects that they are, but as a real number (an uncomputable object) with some error term; in fact, that's how floating point is thought of in the design of many numerical algorithms. In other words, objects that are directly representable on a computer, are often conveniently thought of as approximations of uncomputable objects. If you can't write what the non-computable object is in a language designed to assist in reasoning about how algorithms work -- which is the subject of Lamport's talk -- you're making life much harder for yourself. Another problem of thinking of summation as a for-loop is that it makes you think of the definition as an algorithm, which it isn't. For example 4 * 5 = \Sigma_{i=1}^{5}4, but both of them are just different representations of the number 20. In a program it may make a big difference if you're writing 20, 4 * 5, or `for(i in 0..4) sum+=4`. In mathematical notation, all three are the same. It's not like one uses a cheap multiplication operation and the other an expensive for-loop.
    - MarkMMullin 10y ago
      Sigma and power aren't for loops, but they do represent the melding of iteration and operation - the key is your definition of the sigma above - silly computers are always trying to 'do' things, and often math can simplify problems too big to 'do' by fiddling with representation - so I do agree with you, I just think you're being cruel to the for loop example :-)

Exactly. This is basically just a for loop in a symbolic notation. Math would be an entirely more enjoyable subject if it were communicated in more practical terms that can be executed on a CPU.

gmfawcett 10y ago

Well, not really a for-loop, it's the definition of a set. There are so many interesting objects in math that cannot be executed on a computer. You could define the set of permutations of the naturals, for example, which is an infinitely large set; there's no algorithm you can execute which could produce such a set on your computer. Math would be severely limited if it could only deal with things that were computable on your workstation.

khedoros1 10y ago
You could define the function to generate arbitrary sections of the set. It's not like you're limited to actually generating the entire set, in order to work with it in a practical sense.
matchagaucho 10y ago
True... in my haste I mistook set for sigma (my other pet peeve where symbols make CS translation unnecessarily difficult).
- pron 10y ago
  Sigma isn't a for loop, either. You can sum over infinite sequences. A for loop -- or any computation for that matter -- implies a certain dynamic process, with a certain computational complexity. A mathematical expression has no dynamics; it just describes an object. So Σ 1/(2^n) doesn't mean "sum up all 2^-n" -- i.e., it is not a process -- but simply, "the number equal to the sum of all 2^-n". Of course, summation of infinite sequences requires its own definition using limits. You can have math that is entirely computational -- it's called constructive math -- but the result is a very different math. For example, in constructive math, all functions over the reals are continuous, and not all subsets of finite sets are finite.
  - matchagaucho 10y ago
    But... come on. Sigma is basically a for-loop if you're translating Math symbolism into code. The edge case of summing infinite sequences has no practical application in CS.
    - pron 10y ago
      > The edge case of summing infinite sequences has no practical application in CS. The number 2 is \Sigma_{n=0}^{\infty}1/2^n. So you're "using" sigma every time you use the number 2, which may be practical in some programs. The fact that you can't write a program that sums an infinite sequence using an infinite number of computational steps doesn't mean uncomputable objects have no practical application. In fact, the formalism Lamport talked about in his lecture makes common use of them, and you do, too, every time you use floating point arithmetics. When you use floating point numbers, it's very convenient to think of them not as the very complicated objects that they are, but as a real number (an uncomputable object) with some error term; in fact, that's how floating point is thought of in the design of many numerical algorithms. In other words, objects that are directly representable on a computer, are often conveniently thought of as approximations of uncomputable objects. If you can't write what the non-computable object is in a language designed to assist in reasoning about how algorithms work -- which is the subject of Lamport's talk -- you're making life much harder for yourself. Another problem of thinking of summation as a for-loop is that it makes you think of the definition as an algorithm, which it isn't. For example 4 * 5 = \Sigma_{i=1}^{5}4, but both of them are just different representations of the number 20. In a program it may make a big difference if you're writing 20, 4 * 5, or `for(i in 0..4) sum+=4`. In mathematical notation, all three are the same. It's not like one uses a cheap multiplication operation and the other an expensive for-loop.
  - MarkMMullin 10y ago
    Sigma and power aren't for loops, but they do represent the melding of iteration and operation - the key is your definition of the sigma above - silly computers are always trying to 'do' things, and often math can simplify problems too big to 'do' by fiddling with representation - so I do agree with you, I just think you're being cruel to the for loop example :-)