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Striking parallels between mathematics and software engineering

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Re: Striking parallels between mathematics and software engineering

#21

Post by alicez »

sdenton4 wrote: post #15
sedachv wrote: post #4

This is why one of the most valuable uses of time for mathematicians and especially software engineers is to study the historical development of techniques and technologies. Understanding how matrices are a product of the quest for finding solutions to systems of linear equations gives you a much better idea for when and how to apply matrix techniques. Most math textbooks limit themselves to "The determinant is defin…

To be a bit contrarian, linear algebra is waaaay more useful than just as an outgrowth of solving systems of equations. My own point of view is that linear algebra is by far the most successful part of mathematics: 'Most' questions you can come up with have satisfactory answers. This is in contrast to, say, number theory, where there's a bunch of nice elementary results and a lot of interesting questions that seem ni…

Linear algebra is one of my favorite subjects. Its design is beautifully simple, yet extremely powerful. Half of modern machine learning (and all of Matlab) is built on matrix algebra. And the existence of fast software for numeric linear algebra makes it practically applicable.

The link to graph theory is beautiful, too. Entries in an matrix can represent edge weights, and taking a random walk on a graph can be represented as a matrix-vector multiplication, and the stationary distribution is the singular vector. How cool is that? When you start to link together abstractions from different fields of mathematics and science, you get these fantastic insights that are just mind bogglingly awesome. This is what makes all the pain of wading through an ocean of symbols and equations worthwhile, imho.

Re: Striking parallels between mathematics and software engineering

#22

Post by placebo »

I've had a similar realisation regarding the parallels between software design and scientific theories. In science you have observed phenomena, and you try to find the best (preferably simplest) theory/abstraction that will both explain them and also be able to explain phenomena that you haven't observed yet. In software engineering, you have a set of current requirements and you try to find the best (and hopefully simplest) architecture/abstraction that will meet the current requirements but also be able to naturally accommodate requirements that you haven't foreseen. Both science and software engineering try to devise abstractions that try to convey some "higher truth" than the given data, and both invariably need to be reassembled into a higher abstraction when reality doesn't agree with them, but the test of a good scientific theory as well as a good software architecture is how long can the abstraction hold until something doesn't fit the model.

Re: Striking parallels between mathematics and software engineering

#23

Post by sdenton4 »

luddypants wrote: post #17
sedachv wrote: post #4

This is why one of the most valuable uses of time for mathematicians and especially software engineers is to study the historical development of techniques and technologies. Understanding how matrices are a product of the quest for finding solutions to systems of linear equations gives you a much better idea for when and how to apply matrix techniques. Most math textbooks limit themselves to "The determinant is defin…

Can you suggest any good reads which incorporate the history & motivation of linear algebra? I have had some experience with the common text books, but never put much effort into incorporating it into the way I think, precisely because it did not seem worth my time to just memorize methods without much context.

'Proofs and refutations' is a fantastic book about the process of mathematical exploration, whose primary worked example ('What is a polytope? And what does Euler's Formula mean?') ends up being a non-obvious translation of geometry into linear algebra. It's a really wonderful read, and will give a bit of a sense of how mathematicians think about linear algebra, in a way that most textbooks don't.

Re: Striking parallels between mathematics and software engineering

#24

Post by sedachv »

sdenton4 wrote: post #15
sedachv wrote: post #4

This is why one of the most valuable uses of time for mathematicians and especially software engineers is to study the historical development of techniques and technologies. Understanding how matrices are a product of the quest for finding solutions to systems of linear equations gives you a much better idea for when and how to apply matrix techniques. Most math textbooks limit themselves to "The determinant is defin…

To be a bit contrarian, linear algebra is waaaay more useful than just as an outgrowth of solving systems of equations. My own point of view is that linear algebra is by far the most successful part of mathematics: 'Most' questions you can come up with have satisfactory answers. This is in contrast to, say, number theory, where there's a bunch of nice elementary results and a lot of interesting questions that seem ni…

> My own point of view is that linear algebra is by far the most successful part of mathematics: 'Most' questions you can come up with have satisfactory answers. This is in contrast to, say, number theory, where there's a bunch of nice elementary results and a lot of interesting questions that seem nigh impossible to solve.

I view that as "if you can find a way to express your problem in terms of linear equations, there are well-known techniques for finding solutions."

Re: Striking parallels between mathematics and software engineering

#25

Post by [deleted] »

placebo wrote: post #22

I've had a similar realisation regarding the parallels between software design and scientific theories. In science you have observed phenomena, and you try to find the best (preferably simplest) theory/abstraction that will both explain them and also be able to explain phenomena that you haven't observed yet. In software engineering, you have a set of current requirements and you try to find the best (and hopefully s…

[deleted]

Re: Striking parallels between mathematics and software engineering

#27

Post by tel »

CurtMonash wrote: post #19
eliben wrote: post #8

Interesting. While I agree with the general premise (a lot of mathematics is a human-made construction), the concrete examples she provided feel somewhat forced. Especially the comparison to OOP - I personally don't see how it adds to understanding the various subdivisions defined by abstract algebra.

That's because the inheritance in abstract algebra is so bloody obvious. At least when I studied math, it was the first subject in which the template was: Let's define something, and consider what we can figure out about it. Now let's add another property, and see what we can figure out. Now let's add yet another property, and see what happens. Etc. One winds up familiar with the whole progression group --> abelian g…

This sort of subtyping tree available in abstract algebra isn't the same as an inheritance tree. Inheritance merely layers constructors.

What you really want to consider is interface subtyping and layering. Actually, more than just interfaces you want to also carry along laws. For instance, a group is a monoid with an inverse operation appended (interface concatenation) cut down by the fact that the inverse operation must be "nice" (law concatenation). If you only append the interfaces you end up with free structures.

Re: Striking parallels between mathematics and software engineering

#28

Post by swatow »

I don't think it is ever tenable to claim that mathematics is purely a human construction. The Formalist viewpoint is that mathematics is just the manipulation of symbols, according to certain rules. But Formalists accept that these rules regarding manipulation of symbols are themselves mathematical. That is, the Formalists are actually Platonists with regard to metamathematics. The Platonist says "1 + 1 = 2 is a meaningful, true statement" while the formalist says "that P is a correct proof that 1 + 1 = 2, is meaningful, true statement".

Re: Striking parallels between mathematics and software engineering

#29

Post by daniel-levin »

As someone who grew up writing code, and is now studying mathematics at a tertiary institution, I was quite surprised to read that parallels between mathematics and software engineering are 'surprising'.

On the contrary, mathematics has formed the basis (no pun intended) of so much software engineering. Take for example the very concept of a function/subroutine/method/ - this comes straight from the world of mathematics (albeit with minor modifications to make it convenient).

The algorithms that do all the heavy lifting in order to facilitate this web browsing experience are all grounded in mathematics - memory management in your kernel, database {everything}, even HTML layout management! The whole of complexity and asymptotic analysis is actually just mathematics.

Many of the pioneers in computer science originated as mathematicians. Alan Turing, John McCarthy and Donald Knuth for example.

It may not seem like it on a daily basis writing CRUD apps in an OO language, but software engineering is inextricably linked to mathematics. Such results are the furthest from surprising!

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