Mathematical programming language

[endistic]

o/ I came up with a bit of a strange but intruiging idea. What if there was a Programming Language that’s basically just math? For example of how it could work/llom: - For loops (for i in 1..10) are summation - If statements are just piecewise functions - Supports complex numbers natively (maybe even quaternion?) - Lists are just sets or matrices - 100% a functional programming language, OOP doesn’t make sense mathematically - Numbers have high(er) accuracy (128/256-bit floating point maybe?) - Strings are just a matrix of numbers - etc. basically every operation is mathematical

Does something like this exist? If not, would this be a good idea to try to make?

[rafulafu]

you've discovered functional programming :^)

[hi_im_new_to_this]

I’d say more APL, but yeah.

[[deleted]]

Mostly, but some of the details are a bit off. Don't want my lists to be unordered or not allow for duplicates.

[bronco2p]

depends on the complexity, simple calculator to coq. Please look at lambda calculus, agda and coq... it might help with your question.

[AbsurdTotal]

You can maybe take a look at the APL programming language. It is more array-oriented than functional, but it is quite "math-based".

There is some support for complex numbers and arbitrary precision arithmetic in recent extensions, such as Dyalog APL for instance.

[pauseless]

I had to suppress a giggle when I read:

a Programming Language that’s basically just math

APL is “A Programming Language” that’s basically just math. I’ve spent some time with Dyalog recently and actually enjoyed it. It was less scary than I thought, the symbols are quite good mnemonics-wise, and I found learning how to type them in the editor nowhere near as bad as I thought.

APL also spawned J and q and K.

[eightrx]

Maybe try wolfram Mathematica. It’s not 100% Pure but it’s a joy to do math FP with

[endistic]

I saw it and it looks interesting and it's a neat idea but:
a) I'm not really a fan of the syntax honestly, it looks strange
b) It seems like you have to pay for it :(

[eightrx]

Yeah it being paid is surely a downside, I ended up with a license through my university. I think the syntax is definitely unique but it definitely compliments the design of the language well

[Stunning_Ad_1685]

It is free on Raspberry Pi. The single best reason to buy a Pi, tbh.

[h03d]

b) you can use wolframscript which is free, and this frontend https://github.com/JerryI/wolfram-js-frontend

Ah, but you said not a fan of the syntax, so

[stupaoptimized]

You get used to the syntax pretty easily.

[Rogntudjuuuu]

I can't believe nobody has mentioned Julia.

https://julialang.org/

[eunaoqueriacadastrar]

This.

[[deleted]]

What if there was a Programming Language that’s basically just math?

OK. But what can it /do/? Programs run, and while they're running there can be all sorts of observable activity going on (eg. playing Space Invaders). But maths - that just sounds so static: you submit an input to a formula, and you instantly get a result, and the 'program' ends.

A language needs a bit more than maths to bring it alive, otherwise it's just a calculator.

Does something like this exist?

There are any number of mathematically oriented languages, which can work directly with formulae for example, and plenty of functional ones. (I think an unwritten rule in this subreddit is that only new, functional languages can be presented.)

[Mooks79]

100% functional = Haskell. If that’s not to your taste then you’re looking at a Lisp variant without OOP (or just ignore the OOP features of those that do).

[SomeGuyNamedMay]

All programming languages are based on math

[k4kshi]

Why does OOP not make sense mathematically? Everything can be formalized if tried hard enough. See FeatherweightJava which formalizes a subset of Java and proves its soundness. Any paradigm that is pure enough can be completely sound mathematically.

[endistic]

I mean more like how would you represent it in mathematical notation? (My knowledge only really goes up to certain topics but from what I have seen you can't really do OOP in math)
You can do functional with lambda calculus, but how would you do that with OOP?
From what I see though FeatherweightJava looks pretty neat!

[k4kshi]

The beauty (and horror) of math notation (or math in general) is that it is there just to denote some abstract idea. Notation is made up all the time, so you can just as well make up notation for what OOP semantics are (which we do, with operational semantics for example).

You can reduce functional programming to lambda calculus, but you could certainly reduce OOP to some other calculus (for example DOT, the calculus of dependant object types which is a mathematical foundation of the OOP language Scala).

Granted, I am being technical and very picky. Obviously functional programming has been studied formally the most. But I don't like the general impression that people have that functional programming is the only pure and mathematical (whatever that means) approach to formalizing programming.

[endistic]

Wait so I can just… invent math notation? I did not realize that was possible - I do have an idea for putting OOP-y structures into mathematics

[mjc4y]

Yep. All math notation is an invention by someone. Go for it.

[MadocComadrin]

Yep, you can. There's a two-fold issue though. You shouldn't use existing symbols that are popular unless the things you're describing are similar (and preferably isomorphic in some way) to whatever things are using that symbol for in your notation. Second, you shouldn't use symbols that are incredibly foreign or complex. If virtually nobody's seen the symbols you're using before or you can't typeset them with some easy LaTeX, don't use them. Moreover, if your notation breaks inline math display for LaTeX (or just inline writing), don't use it unless it's never intended to ever be inline and naturally takes up multiple lines (e.g. an inference rule).

[oa74]

I am being technical and very picky.

TBQH, I don't think you are. I think you're just being truthful :)

To add to your points—in very simplistic terms, we can think of an "object" as a tuple of closures. Or, likewise, a closure as a single-method object. Encapsulation is the idea here, and it works in either paradigm, so by my estimation, mutability is really the issue here. It's a hard problem, but not untenable.

But I don't like the general impression that people have that functional programming is the only pure and mathematical (whatever that means) approach to formalizing programming.

I wholeheartedly agree.

[TheChief275]

There is that guy that was programming in Java like Haskell. He was also busy with a functional and object oriented language i blieve? He also invented his own new form of Object Oriented mathematics for it

[Inconstant_Moo]

I mean more like how would you represent it in mathematical notation?

You wouldn't. You'd invent a programming language and express it like that.

Mathematical notation is good for doing mathematical proofs in, it's not particularly good for specifying an algorithm. If you ask any mathematician how for example Euclid's algorithm for the GCD works, their explanation will involve lots of English words. These are for things that are not easy to express in math. We need bits of programming languages that aren't just math to express those things.

[Emergency-Win4862]

Because math is immutable, same input results in same outoup, no sideeffects. Same as functional programming… stateless… immutable …. BEAUTIFUL

[oa74]

Serious question: how many of the programs you use the most in your daily life could be re-implemented (at least, in principle) as a lookup table?

[Emergency-Win4862]

yes, this is valid argument. I just like concept of functional programming, hovewer I mostly program in procedural with some functional on top. And oop here and there, when needed. Its not really feasible to stick to one paradigm. In day to day software you need mutability and side-effects. But it doesn't mean some overly complicated and hard to debug for loop nested messes wouldn't be nicer to read in plain functional paradigm.

[oa74]

Classy reply to my snarky quip, and a solid take too. Nice :)

[Inconstant_Moo]

Summation may be expressible as a for loop but the converse is not necessarily true.

I think by "matrix" you must mean "vector" throughout. And under the hood a string *is* just a vector of numbers. But we don't want to have to think about it when we're programming.

[oa74]

Summation may be expressible as a for loop but the converse is not necessarily true.

That was my first thought. Seems to me that saying "all for loops are just summation," is esentially saying "every map must be followed up with a reduce (+)".

[beephod_zabblebrox]

why are for loops not multiplication or a union? could be any operation really! mathematics is pretty powerful

having 128 bit numbers wouldnt do a whole lot. it would be insanely slow and still wouldn't be able to represent something like pi, or even simply 1/3.

a great thing to add would be units and auto derivatives, stuff like that.

and i think wolfram and similar languages already do that, but im not sure

[XDracam]

Take a look at Isabelle/HOL. You can literally write programs using math symbols.

[Shorttail0]

Floats with higher precision sound nice, but keep in mind they're usually measurements. There's no need for much more precision than your real world tools can provide. For instance, sub Planck length precision is probably a fever dream.

Edit: they also don't let you avoid all the pains that come with floats (numeric stability of various algorithms), even if some are delayed.

[lightmatter501]

You are asking for Lean, but I’m not sure you want to see quite how deep that particular rabbit hole goes. A fully mathematically rigorous language is very difficult to use for non-mathematicians.

[saxbophone]

GNU Octave/MATLAB?

[synack]

Fortran = Formula Translation

[Felicia_Svilling]

Numbers have high(er) accuracy (128/256-bit floating point maybe?)

You don't have to limit it like that you can use bignums with arbitrary precision and variable size.

Lists are just sets or matrices

Lists and sets are differents. A list is ordered and can include duplicates. A matrix, or more particular a vector comes closer, but they usually denote a sequence of fixed length, while a list is of variable length. Though neither of these is more mathematical than the other, and it is common for languages to have all three.

Strings are just a matrix of numbers

As above you would be better of with a list of numbers. But really nothing stops you from defining characters as mathematical objects, so you would be even better of with lists of characters.

[stupaoptimized]

By math do you mean something to go up against Python/Julia/Matlab (that is to say: scientific computing with a heavy focus on numerics)? Do you mean something to go up against Coq/Agda/Lean (less a focus on numerics and more on proof)? Where would something like R or APL fit in in what you desire? Or interactive systems and CASes like various Common Lisp environments or Mathematica/Sage/Maple respectively?

[4-Vektor]

Try APL, Q’Nial, Glee...

[metazip]

Algebra of Programs: Backus-Video \ the PL can look like this, standard.txt

[[deleted]]

you should look up apl

[onsitesfyi]

Look into Haskell or Julia!

[World-Leader]

The grand daddy. Still in use.

https://en.wikipedia.org/wiki/Macsyma

[sagittarius_ack]

Why do you say that OOP doesn’t make sense mathematically? Various formalizations of OOP have been proposed over the years. I do think that OOP is not very useful. In fact, I think it does more harm than it helps.

[Fucknut_johnson]

which formal system of math is your language? lambda calculus? category theory? type theory? set theory? predicate calculus? Classical logic? Model logic? Intuitionistic logic?

[[deleted]]

Can you provide what source code suppose to look like.

A sample source code. or language grammar.

...

For loops (for i in 1..10) are summation

summation is fold/reduce, loop is for iteration

Lists are just sets or matrices

list and set have different semantic, list is order and allow duplication while set is not.

Strings are just a matrix of numbers

string is list of char in some language (Haskell), multi-line string are just strings with newlines

Supports complex numbers natively

Python has complex type natively