Re:Making math more Lego-like
by klodolph on Saturday March 04, 2017 @03:30AM
(#13789974)
> Why does Pi and Sigma provide a better interface to these operations then apply and reduction notation of LISP?
I can answer that. You have hit the nail on the head here, that pi and sigma are in fact much better when you're doing math.
Unlike most programming languages, mathematical notation is not limited to using characters in one font and size on a series of lines, one after the other. Math notation is 2D with whatever fonts you please. So we take advantage of that. When you see a big symbol like a sigma or pi, you know that it's a higher-order operator like 'reduce' in Lisp. There's a whole family of symbols that you can put there, not just sigma and pi, but integrals, tensor products, coproducts, unions, intersections, exterior products, etc.
Or in short, that's just how mathematicians write certain types of higher-order math, by using a big font. They're all vaguely similar to "reduce" in the sense that they apply the inner expression to each element of a set and produce a single output. However, they don't translate to "reduce" in any concrete way. You can't rewrite summations in math as "reduce" in Lisp, because while they produce a single output, there's not one particular way in which that happens. For example, summations and integrals make existential claims, while unions and intersections do not.
I think that's the biggest difference between programming and math here—much math is nonconstructive, many of these operators are not total, and therefore it doesn't make sense to think of them as functions, and therefore it doesn't make sense to write them as functions—because functions are total.
To rephrase that, in math, sum() is not a function, so we don't write it using function notation. This to me is clearer—we avoid getting confused by using the same notation for two different things (functions and non-functions).
I can answer that. You have hit the nail on the head here, that pi and sigma are in fact much better when you're doing math.
Unlike most programming languages, mathematical notation is not limited to using characters in one font and size on a series of lines, one after the other. Math notation is 2D with whatever fonts you please. So we take advantage of that. When you see a big symbol like a sigma or pi, you know that it's a higher-order operator like 'reduce' in Lisp. There's a whole family of symbols that you can put there, not just sigma and pi, but integrals, tensor products, coproducts, unions, intersections, exterior products, etc.
Or in short, that's just how mathematicians write certain types of higher-order math, by using a big font. They're all vaguely similar to "reduce" in the sense that they apply the inner expression to each element of a set and produce a single output. However, they don't translate to "reduce" in any concrete way. You can't rewrite summations in math as "reduce" in Lisp, because while they produce a single output, there's not one particular way in which that happens. For example, summations and integrals make existential claims, while unions and intersections do not.
I think that's the biggest difference between programming and math here—much math is nonconstructive, many of these operators are not total, and therefore it doesn't make sense to think of them as functions, and therefore it doesn't make sense to write them as functions—because functions are total.
To rephrase that, in math, sum() is not a function, so we don't write it using function notation. This to me is clearer—we avoid getting confused by using the same notation for two different things (functions and non-functions).