Translating Mathematics into Code: Examples in Java, Python, Haskell and Racket

I'm aware the author is a professor of CS, but this is a statement I always see people originally trained as programmers claim, and as a mathematician and a programmer I respectfully disagree.

Mathematics does what you define it to do, and if you can communicate your idea to another person it doesn't matter whether you "mutated" a variable or not. Likewise, you could say that there's no such thing as mutation in programs because the value of a variable in a language that supports mutation depends implicitly on time; you just don't express that in the source code. It's a matter of perspective, and more perspectives are often better than fewer. In math if you want to index a variable to have "pretend mutation" to avoid confusion you can, but there's no rule about it. If you feel the best way to communicate that idea is with mutation, then that doesn't stop it from being mathematics.

Re: Translating Mathematics into Code: Examples in Java, Python, Haskell and Racket

#5

Post by meric » Tue, Feb 10, 2015, 1:05 AM UTC

j2kun wrote: post #4

> Mathematics has no side effects. I'm aware the author is a professor of CS, but this is a statement I always see people originally trained as programmers claim, and as a mathematician and a programmer I respectfully disagree. Mathematics does what you define it to do, and if you can communicate your idea to another person it doesn't matter whether you "mutated" a variable or not. Likewise, you could say that there'…

>> the value of a variable in a language that supports mutation depends implicitly on time

Mind blown.

Re: Translating Mathematics into Code: Examples in Java, Python, Haskell and Racket

#6

Post by the_af » Tue, Feb 10, 2015, 1:28 AM UTC

j2kun wrote: post #4

> Mathematics has no side effects. I'm aware the author is a professor of CS, but this is a statement I always see people originally trained as programmers claim, and as a mathematician and a programmer I respectfully disagree. Mathematics does what you define it to do, and if you can communicate your idea to another person it doesn't matter whether you "mutated" a variable or not. Likewise, you could say that there'…

> Likewise, you could say that there's no such thing as mutation in programs because the value of a variable in a language that supports mutation depends implicitly on time; you just don't express that in the source code.

But this is precisely the problem with mutation and programming languages:

    ...some code...
    y = f(x)
    ...more code...
    z = g(x)

Is x in the two lines the same? Often it's not. This is what's surprising to a mathematician. It would help if the "time" was made explicit, i.e. if you had x(t1) and x(t2), in which case it's quite obvious you need to take a closer look. But in mainstream imperative languages there is no support for this, which is very much a problem.

Re: Translating Mathematics into Code: Examples in Java, Python, Haskell and Racket

#7

Post by platz » Tue, Feb 10, 2015, 1:30 AM UTC

meric wrote: post #5

j2kun wrote: post #4

> Mathematics has no side effects. I'm aware the author is a professor of CS, but this is a statement I always see people originally trained as programmers claim, and as a mathematician and a programmer I respectfully disagree. Mathematics does what you define it to do, and if you can communicate your idea to another person it doesn't matter whether you "mutated" a variable or not. Likewise, you could say that there'…

>> the value of a variable in a language that supports mutation depends implicitly on time Mind blown.

Welcome to FRP

Re: Translating Mathematics into Code: Examples in Java, Python, Haskell and Racket

#8

Post by [deleted] » Tue, Feb 10, 2015, 1:59 AM UTC

[deleted]

Re: Translating Mathematics into Code: Examples in Java, Python, Haskell and Racket

#9

Post by tel » Tue, Feb 10, 2015, 2:15 AM UTC

j2kun wrote: post #4

> Mathematics has no side effects. I'm aware the author is a professor of CS, but this is a statement I always see people originally trained as programmers claim, and as a mathematician and a programmer I respectfully disagree. Mathematics does what you define it to do, and if you can communicate your idea to another person it doesn't matter whether you "mutated" a variable or not. Likewise, you could say that there'…

I don't think anyone, mathematicians or functional programmers, have a problem with mutation [0]. They have a problem with names suddenly changing without being too explicit about that fact.

In particular, mathematics certainly doesn't prohibit you from using whatever kind of notation you like to explain your concept, but it has suggested a certain primacy of the notion of a (pure) function. The reasons are both ones of mathematical "elegance" (whatever that means) and pedagogy. They're a fantastically important fundamental building block.

Mathematicians would never claim that pure functions are the only thing around, but by and large they'd assume that's what you were talking about given no other indication. Furthermore if you introduced something new (say something with a notion of mutation) then you'd merely be obliged to explain exactly what that new thing is with sufficient formality so that the reader can be assured that its behavior is completely reasonable.

All of this holds identically for functional programming. The only additional challenge is that FPers are accidental intuitionists so you can't merely name the behavior of a new kind of notation but must actually exhibit an interpreter of that new notation into more common notation. This is significantly more heavy-handed than most mathematicians would do.

[0] Or any other side effects. The whole argument is parametric in your choice of side effect.

Re: Translating Mathematics into Code: Examples in Java, Python, Haskell and Racket

#10

Post by j2kun » Tue, Feb 10, 2015, 3:00 AM UTC

the_af wrote: post #6

j2kun wrote: post #4

> Mathematics has no side effects. I'm aware the author is a professor of CS, but this is a statement I always see people originally trained as programmers claim, and as a mathematician and a programmer I respectfully disagree. Mathematics does what you define it to do, and if you can communicate your idea to another person it doesn't matter whether you "mutated" a variable or not. Likewise, you could say that there'…

> Likewise, you could say that there's no such thing as mutation in programs because the value of a variable in a language that supports mutation depends implicitly on time; you just don't express that in the source code. But this is precisely the problem with mutation and programming languages: ...some code... y = f(x) ...more code... z = g(x) Is x in the two lines the same? Often it's not. This is what's surprising…

I think what's surprising to a mathematician is if the fact that x can change is not stated prior to witnessing it. So you might say that immutability is a reasonable default for mathematics, but again not necessary as the author claims.

While I think programmers might find it surprising, in mathematical proofs context is often the primary tool one uses to figure out what the hell is going on locally in some expression. For example, indices like time and the size of a problem (in combinatorics) are often dropped because the reader understands that the context is asymptotic.