Translating Mathematics into Code: Examples in Java, Python, Haskell and Racket • Hacker News Board

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

#31

Post by Klockan » Tue, Feb 10, 2015, 8:07 AM UTC

tel wrote: post #9

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…

Java has no functions and is the language which gets harped on the most, so this is obviously not the main issue.

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

#32

Post by asuidyasiud » Tue, Feb 10, 2015, 8:48 AM UTC

orbifold wrote: post #25

asuidyasiud wrote: post #22

Earlier quoted context omitted.

There is nothing destructive about the map M: A x A -> A.

In fact there is, take A to be the space of differential forms on some manifold, and M the wedge product. Given two differential forms eta, mu, there is no way to recover eta and mu from M(eta,mu). Even simpler take two arbitrary numbers multiply them, unless they were prime you have no way of telling from the product what the two original numbers were. In other words multiplication destroys information. In the case…

No, applying a map does not mutate or consume the parameters, and therefore no information is destroyed. Function application in mathematics is referentially transparent.

To make this point perfectly clear: Whenever you encounter an expression such as "f(x)", you may freely re-use the expression "x" in a different place. This is a matter independent from the category you choose to work with -- for example, the expression "psi \otimes psi" makes sense in the monoidal category of Hilbert spaces.

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

#33

Post by Klockan » Tue, Feb 10, 2015, 9:01 AM UTC

kyllo wrote: post #29

brudgers wrote: post #12

This: public Node(SortedSet left, T element, SortedSet right) { this.left = left ; this.right = right ; this.element = element ; Isn't really mathematics. Nodes have a label. Any record attached to the node is business logic. It doesn't matter if the tuple is (first, last, address, department) or (value left, right), these are not part of the mathematics. Sure (value, left, right) looks like math, but it's just a con…

The article seems to be not-so-subtly hinting that Java is an anti-math programming language, and I agree. The language encourages each developer to reinvent and reimplement key concepts in terms of "classes" that are much more succinctly described in languages that embrace abstract algebra. The example of using inheritance and dynamic dispatch to represent a union type, is a perfect example.

Union types is not a key concept in abstract algebra, I bet that most mathematicians have never even heard of it. I'd like to hear about its use in abstract algebra, how would you define the union type of the simplest of simple objects without breaking its operation: a group with a group?

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

#34

Post by orbifold » Tue, Feb 10, 2015, 9:14 AM UTC

asuidyasiud wrote: post #32

orbifold wrote: post #25

Earlier quoted context omitted.

In fact there is, take A to be the space of differential forms on some manifold, and M the wedge product. Given two differential forms eta, mu, there is no way to recover eta and mu from M(eta,mu). Even simpler take two arbitrary numbers multiply them, unless they were prime you have no way of telling from the product what the two original numbers were. In other words multiplication destroys information. In the case…

No, applying a map does not mutate or consume the parameters, and therefore no information is destroyed. Function application in mathematics is referentially transparent. To make this point perfectly clear: Whenever you encounter an expression such as "f(x)", you may freely re-use the expression "x" in a different place. This is a matter independent from the category you choose to work with -- for example, the expres…

This largely depends on your viewpoint, the map M : A * A \to A was meant to be a morphism in an arbitrary monoidal category. An element like x, would be interpreted as a morphism x : I -> A (in general the objects A in a monoidal category, or any category for that matter have no notion of elements). The expression f(x) is interpreted as the composition f . x, so no you are not free to use x a second time, once you formed that composition. This is swept under the rug in most cases because monoidal categories like Set are cartesian and allow you to freely copy morphisms like x : I -> A, via the Delta map Delta : A -> A * A. Explicitly you have Delta(x) = Delta . x = x * x. In general it might not be possible to even determine if two morphisms x : I -> A and y : I -> A are the same.

In some cases you might be able to assume from the start that you have a certain number of equal morphisms x : I -> A "in reserve", for example in linear logic you have the operator ! ("of course") which gives you an arbitrary number of identical morphisms to play with.

In any case this distinction is quite subtle and I understand, why you might assume that I'm simply misunderstanding things. In particular I should emphasize that almost no programming languages work this way, although with some effort you would be able to recast typical cryptographic / numerical code in this language.

It is also really easy that to see for example in the case of addition that indeed information is destroyed, clearly the map

(a,b) -> (a+b,a-b)

has an inverse if the characteristic isn't 2, the subsequent projection to the first factor destroys information. Categories in which that is possible have maps d_A : A -> I.

What tel is pointing out above, is that Delta and d together form a comonoid.

While psi \otimes psi certainly makes sense, the map psi \to psi \otimes psi is not linear and therefore not a morphism (Physicists call this the no cloning theorem for some reason).

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

#35

Post by Klockan » Tue, Feb 10, 2015, 9:22 AM UTC

I wont believe in functional programming as long as not even mathematicians prefer it.

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

#36

Post by nova » Tue, Feb 10, 2015, 9:54 AM UTC

orbifold wrote: post #34

asuidyasiud wrote: post #32

Earlier quoted context omitted.

No, applying a map does not mutate or consume the parameters, and therefore no information is destroyed. Function application in mathematics is referentially transparent. To make this point perfectly clear: Whenever you encounter an expression such as "f(x)", you may freely re-use the expression "x" in a different place. This is a matter independent from the category you choose to work with -- for example, the expres…

This largely depends on your viewpoint, the map M : A * A \to A was meant to be a morphism in an arbitrary monoidal category. An element like x, would be interpreted as a morphism x : I -> A (in general the objects A in a monoidal category, or any category for that matter have no notion of elements). The expression f(x) is interpreted as the composition f . x, so no you are not free to use x a second time, once you f…

To check if I understand the argument: I'm guessing you're talking about the internal logic or language of some category, in contrast to the external one used to talk about that category.

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

#37

Post by asuidyasiud » Tue, Feb 10, 2015, 9:57 AM UTC

orbifold wrote: post #34

asuidyasiud wrote: post #32

Earlier quoted context omitted.

No, applying a map does not mutate or consume the parameters, and therefore no information is destroyed. Function application in mathematics is referentially transparent. To make this point perfectly clear: Whenever you encounter an expression such as "f(x)", you may freely re-use the expression "x" in a different place. This is a matter independent from the category you choose to work with -- for example, the expres…

This largely depends on your viewpoint, the map M : A * A \to A was meant to be a morphism in an arbitrary monoidal category. An element like x, would be interpreted as a morphism x : I -> A (in general the objects A in a monoidal category, or any category for that matter have no notion of elements). The expression f(x) is interpreted as the composition f . x, so no you are not free to use x a second time, once you f…

I understand where you are come from, but you are conflating different meta-levels (external vs. internal language/logic): You are absolutely free to use any formal expression "x" a second time, and - crucially - it will have the same mathematical meaning if you choose to do so, as can be seen by pondering over the tautology "P(x) P(x)". This is not a property of the category that you chose to work with, but rather of the formal language that we use when engaging in mathematics, as you have demonstrated yourself when you had permitted that "psi \otimes \psi" made sense.

The non-existence of a morphism "psi -> psi \otimes psi" and the notion of "destroyed information" that you are discussing in the rest of your post is independent from all of this. If you wish I can elaborate on the "no cloning theorem".

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

#38

Post by orbifold » Tue, Feb 10, 2015, 10:37 AM UTC

asuidyasiud wrote: post #37

orbifold wrote: post #34

Earlier quoted context omitted.

This largely depends on your viewpoint, the map M : A * A \to A was meant to be a morphism in an arbitrary monoidal category. An element like x, would be interpreted as a morphism x : I -> A (in general the objects A in a monoidal category, or any category for that matter have no notion of elements). The expression f(x) is interpreted as the composition f . x, so no you are not free to use x a second time, once you f…

I understand where you are come from, but you are conflating different meta-levels (external vs. internal language/logic): You are absolutely free to use any formal expression "x" a second time, and - crucially - it will have the same mathematical meaning if you choose to do so, as can be seen by pondering over the tautology "P(x) P(x)". This is not a property of the category that you chose to work with, but rather o…

Well ok, then this was a simple misunderstanding, or miscommunication. My knowledge is largely based on reading of http://ncatlab.org/nlab/show/internal+logic and several other pages on that wiki, plus graduate training in math and physics. Based on your last reply I'm confident you have a similar background. Logic and category theory are not my specialty and I probably haven't expressed myself as clearly as I would in professional communication.

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

#39

Post by throwaway283719 » Tue, Feb 10, 2015, 10:41 AM UTC

Klockan wrote: post #33

kyllo wrote: post #29

Earlier quoted context omitted.

The article seems to be not-so-subtly hinting that Java is an anti-math programming language, and I agree. The language encourages each developer to reinvent and reimplement key concepts in terms of "classes" that are much more succinctly described in languages that embrace abstract algebra. The example of using inheritance and dynamic dispatch to represent a union type, is a perfect example.

Union types is not a key concept in abstract algebra, I bet that most mathematicians have never even heard of it. I'd like to hear about its use in abstract algebra, how would you define the union type of the simplest of simple objects without breaking its operation: a group with a group?

That would be a groupoid - http://en.wikipedia.org/wiki/Groupoid

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

#40

Post by orbifold » Tue, Feb 10, 2015, 10:47 AM UTC

nova wrote: post #36

orbifold wrote: post #34

Earlier quoted context omitted.

This largely depends on your viewpoint, the map M : A * A \to A was meant to be a morphism in an arbitrary monoidal category. An element like x, would be interpreted as a morphism x : I -> A (in general the objects A in a monoidal category, or any category for that matter have no notion of elements). The expression f(x) is interpreted as the composition f . x, so no you are not free to use x a second time, once you f…

To check if I understand the argument: I'm guessing you're talking about the internal logic or language of some category, in contrast to the external one used to talk about that category.

Right I think that was the point of misunderstanding between me and asuidyasiud. If you think of x as an assumption in linear logic for example, then there is a difference if you assume x or x^2 and so on. Since in programming languages such x : I -> A are values of type A, this then ties in with duplication and destruction I was going on about above. I tried to express the sentiment that in many ways it is actually natural to think of those kinds of logic as more fundamental, because it makes the duplication of values (unnatural for things like resources, spaces of state, etc.) explicit or even impossible.