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Parallels Between Math and Software Engineering

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Re: Parallels Between Math and Software Engineering

#11

Post by Yomammas_Lemma »

alexlarsson wrote: post #9
vezzy-fnord wrote: post #7

Earlier quoted context omitted.

The perception of FP as being somehow more "math-y" is nothing but a bias. There is no intrinsic magical property of "mathiness", it's just that the operational semantics of functional languages are much more well-defined. Logic programming is itself firmly based in axiomatic semantics and has a similar a priori system of reasoning to it, though distinct from FP. Imperative languages can be modeled well on Hoare logi…

Also, many purely mathematical models of computing like turing machines are definitely not functional.

Well, nobody is proposing that Turing Machines be the foundation of mathematics, whereas the sort of type theory used in FP is being taken pretty seriously as a foundation.

Re: Parallels Between Math and Software Engineering

#15

Post by mafribe »

vezzy-fnord wrote: post #7
agumonkey wrote: post #6

Not trying to evangelize but FP was an great hint for that. Seeing 'tangible' (that I can create, see, step through) incarnations of groups, monoids, transitive relations etc gave an operational grounding to abstract algebra. Something needed for some of us before see the abstraction behind the notation, and understanding it.

The perception of FP as being somehow more "math-y" is nothing but a bias. There is no intrinsic magical property of "mathiness", it's just that the operational semantics of functional languages are much more well-defined. Logic programming is itself firmly based in axiomatic semantics and has a similar a priori system of reasoning to it, though distinct from FP. Imperative languages can be modeled well on Hoare logi…

I agree that FP isn't more mathematical than other forms of programming.

     the operational semantics of functional languages are much more well-defined.
No. The operational semantics of any language that has a compiler or interpreter is as well-defined as possible. What you may mean is that the meta-theory of (pure) FP is better understood than the meta-theory of other paradigms, typically because FP has simpler meta-theoretic properties.

     Logic programming is itself firmly based in axiomatic semantics
This is a non-standard use of the term "axiomatic semantics" which is usually referring to Hoare-logic with pre- and post-conditions. Prolog has (simplifying a bit) a proof-theoretic / proof-search / resolution-unification-based semantics.

   [Hoare-logics for imperative languages] is comparatively understudied, though.
That used to be the case, but the last decade has seen extreme progress in this matter, and Hoare logics for imperative languages are routinely used these days.

Re: Parallels Between Math and Software Engineering

#16

Post by adam930 »

matheweis wrote: post #3

I've had a similar thought as the author, and often wondered - could we develop alternative systems for intermediate-to-advanced mathematical concepts that would make it easier to parse?

Yes, in fact there already is such a formalism: http://faculty.luther.edu/~macdonal/

From the first page:

"Geometric algebra and its extension to geometric calculus unify, simplify, and generalize vast areas of mathematics that involve geometric ideas, including linear algebra, multivariable calculus, real analysis, complex analysis, and euclidean, noneuclidean, and projective geometry. They provide a unified mathematical language for physics (classical and quantum mechanics, electrodynamics, relativity), the geometrical aspects of computer science (e.g., graphics, robotics, computer vision), and engineering."

Re: Parallels Between Math and Software Engineering

#17

Post by thinkpad20 »

a3_nm wrote: post #10

It is also interesting that there are many parallels between software engineering and the design of mathematical proofs (or theoretical CS proofs, which I am more familiar with). In theoretical CS, people talk of catching and fixing "bugs" in proofs, namely, mistakes that make the proof fail but can hopefully be fixed while sticking to essentially the same idea. One can "refactor" proofs, in superficial ways (e.g., r…

If you are writing programs in Coq or another dependently typed programming language, these parallels between math and programming are not just incidental; they are one and the same. Theorems are stated as functions, whose type signatures reflect the theorem being proven. Even the gap that you mention at the end of your post is bridged, since the type checker will verify that your proof is correct (modulo the trust in the metatheory of the language itself, and its implementation). Things like refactoring, choosing more or less specific statements of theorems (e.g. more or less abstraction in a library), interfaces, etc, all become exactly analogous to the same kinds of problems faced in software engineering. It's really quite remarkable.

(Disclaimer: I'm hardly an expert in these languages; I just dabble.)

Re: Parallels Between Math and Software Engineering

#19

Post by euske »

The other day I realized that a man-made law is also a bit like mathematics or computer software. It is carefully designed and constructed. Ideally, it is intended to work like a machine with as little room for human discretion as possible. And just like mathematics, adding an another "axiom" to the law has far, far-reaching consequences.

Re: Parallels Between Math and Software Engineering

#20

Post by mshron »

Well said. Advanced math is mostly about working with properties higher up the chain of abstraction, and then seeing what happens when you bring the insights learned up there back down to more concrete examples.

From an OO point of view, the real numbers inherit almost every useful trait: they're a field, they have a topology, they have a measure. Studying the parent classes, so to speak, gives you abstract algebra, topology, and analysis, respectively.

Once you get the basics of each, you can study how they interact. Then, once that stuff is clear, they can be recombined in beautiful ways to give you new objects to study.

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