For the CS topics you listed, e.g., programming languages, that field has for decades just cried out for some good math but gotten nearly none! Clearly compiling and executing a program are necessarily mathematically something, understood or not, powerful or not. For progress, we need to understand the subject mathematically. E.g., given a programming language statement, what are its mathematical properties so that given a sequence of two such statements, what are the mathematical properties? Can't see any? Okay: Change programming languages until can. Then with the properties, what can we conclude about the program? Anything useful, say, about correctness, running time, storage utilization, approaching limitations? Given some properties, can we have some code transformations with some known properties? Are some such useful? If not, can we define some useful programming languages that admit such transformations?
There are some simple, illustrative, pregnant cases that just leap off the screen: Write a collection of routines for sort by surrogate, i.e., finding a permutation, maybe honoring sort 'stability', applying a permutation to an array, chasing through the fact that each permutation is a product of disjoint cycles, etc. Then with such routines, will see some properties that are close to algebraic, e.g., where some combinations of the routines are the identity transformation, where one routine is the algebraic inverse of another, where we might have commutativity and/or associativity, or are equivalent to other combinations, etc. Can we preserve sort 'stability'? So, we see that we should have an 'algebra' of those routines and be able to derive known properties. Then we see that due to 'boundary' limitations, etc., our algebra is only roughly correct. So, maybe we should fix that.
This is old stuff. It has leaped off the screen for decades. We want some known properties we can use to derive new, useful properties. Instead all we get is programming languages that compete in some absurd 'beauty contest' or that 'encourage' programming 'styles' seen as 'helpful'. Sicko.
E.g., it has been common now in server farms to be concerned about reliability. So, we set up systems so that when system A gets too busy or fails, then system B takes over, etc. However, what we've seen, including at some of the most important farms, for decades, up to this year, is that these intuitive approaches far too easily go "splat" in the mud: They are 'unstable', have problems that propagate, and are surprisingly UNreliable.
So, we need some new math of 'reliability' for such server farms, math that can write us some guarantees. That math will start with the real problems, have definitions, theorems, and proofs, and, if successful, some solid tools for building reliable server farms.
All across computing, CS needs to give us progress, and for that they need to proceed essentially only mathematically.
It's the math. The CS is the math. Server farm reliability is just the application of the math and is not the CS. Ignoring math, CS is "A little people. A silly people."! Until CS gets serious about the math, it is writing checks its methodology can't cash.
To get serious about math, CS first has to learn LP well!
The would you believe probability? How about stochastic processes? Any stochastic processes in server farms and networks? Any uses of the axiomatic derivation of the Poisson process? Any roles for one of the strongest inequalities in math, the martingale inequality and the associated convergence theorem? Measure preserving transformations as in ergodic theory -- I'll answer that one, "Yes!". The renewal theorem and loads at server farms? How 'bout a little in probability?
Not really, as your raving all over the place and still not making a lot of sense, argument-wise.
I guess I'll just agree to disagree, I feel bad for making your write so much.
Just a few of parting shots:
>Clearly compiling and executing a program are necessarily mathematically something, understood or not, powerful or not. For progress, we need to understand the subject mathematically.
If you think that there has been no progress in these areas, I'm sorry, but you really don't know what you are talking about. Could that progress have been faster, had the maths behind it been formalized? Maybe so, yet I don't see that many mathematicians working on branch prediction on deep pipelining processor architectures, and plenty of other things that lowly CS-ers have worked out. Want to see more maths in all of CS? Maybe you need to bring them to it, in the form of useful tools that can deal with fast moving targets.
>what are the mathematical properties? Can't see any? Okay: Change programming languages until can.
I regret to say that that's not as easy as it sounds. There are issues of tractability, and more importantly, there's this not-quite-cottage industry of software development, which needs good-enough languages to use _today_.
> this is old stuff. It has leaped off the screen for decades. We want some known properties we can use to derive new, useful properties.
You mean like Agda, Epigram, and countless other formal systems for program specification, derivation and correctness proving?
> Instead all we get is programming languages that compete in some absurd 'beauty contest' or that 'encourage' programming 'styles' seen as 'helpful'. Sicko.
This coming from a mathematician is strange to hear. Elegance and expressiveness are important when programming. It makes for succinctness and less bugs, and improves programmer productivity (which down here in the real world, means $$$).
>How about stochastic processes? Any stochastic processes in server farms and networks?
How about them? Plenty of very CS-oriented research based on stochastic processes. I should know, I did my doctoral research within a 45 person team on networks and network modelling, where most of the big guys were mathematicians. These people (not me, I do very applied stuff in a very bastardized field) breathe and live Markov processes and queuing systems. I don't see what you're complaining about here.
>E.g., it has been common now in server farms to be concerned about reliability. So, we set up systems so that when system A gets too busy or fails, then system B takes over, etc. However, what we've seen, including at some of the most important farms, for decades, up to this year, is that these intuitive approaches far too easily go "splat" in the mud: They are 'unstable', have problems that propagate, and are surprisingly UNreliable.
The folks over at Google would like a word or two with you on this.
> Until CS gets serious about the math, it is writing checks its methodology can't cash.
Maybe, and yet it seems to be doing not at all bad, as far as I can see. Would more and better maths be useful? Surely so. Would relegating CS to "footnotes" make any sense? We surely disagree there.
There's a whole continuum in CS, from the highly abstract and "math-y" to very applied stuff that has a very faint, if any, relation with what mathematicians normally worry about, and yet is critical for actual applications. I just find it odd that you don't see that fact.
There is a trend towards more maths in plenty of CS topics that were once just "practical stuff", so maybe the day will come when maths departments will branch out into real-world CS, and CS students will need more math, as you'd like. Given that we've been doing "maths" for 3K years, and CS for less than 100, you might need to wait a bit, though.
Anyway, thanks for the discussion, and have a nice weekend.
You are starting with an axiom that I'm talking nonsense and then bending everything I write in that direction instead of reading what I write. E.g.,
"I regret to say that that's not as easy as it sounds. There are issues of tractability, and more importantly, there's this not-quite-cottage industry of software development, which needs good-enough languages to use _today_."
I didn't say it was "easy", and clearly my point is progress and not current software development. Sure, for my project, I'm typing in code in Visual Basic .NET. For writing code today, all things considered for my context, that is about the best option.
But what I'm typing in is hardly different from what anyone might have typed in all the way back to Algol.
Here's the point: People type in all that code, and then what? Can some software go over it, report properties, do some transformations with known, useful properties? Not very much. Such things won't be better for Visual Basic than they were for Algol or anything between.
So, they type in the code, and then all they have is just that code. They can desk check it, test it, try it, revise it, etc., but it's all still digging the Panama Canal with a teaspoon one canal at a time. That is, there's no real automation and no significant exploitation of mathematical properties that could lead to automation.
Or, return to most of the rest of engineering where we have specifications of resistors, capacitors, inductors, transistors, fans, copper wire, steel sheets, aluminum bars, etc. with ohms, farads, density, tensile strength, electrical conductivity, etc. Okay, now compare with some DLL or its source code: What engineering properties do we have. Essentially none. It's like building with iron before we knew about tensile strength.
Sure, with some such research progress, eventually programmers would benefit. Obviously the intention is real progress in software productivity. We're not going to get such productivity by more of the same that got us C, C++, Java, Python, Visual Basic .NET, C#, etc.
Also for another of your points, I'm not talking about anything like branch prediction or deep pipelining. That's basically what to do with with the hardware to execute an existing instruction set.
Also the point is not math or not. The point is progress. Math will be necessary but not sufficient. That is, just stuffing in some math won't yield more than the old trick of putting two extra, unused transistors in a radio to claim nine transistors instead of seven.
Your point that there's a lot of good CS to do without 'mathematizing' the field is not promising for research or significant progress.
> Sure, for my project, I'm typing in code in Visual Basic .NET. For writing code today, all things considered for my context, that is about the best option.
Maybe you should try something other than VB.NET, then. There are ways of constructing correct code, if you're patient enough, that is. If you're using the CLR, why not use F# instead of VB? That'd be a big step towards being able to prove some things about your code. Or if you really want to take out the big guns, go for Coq, Isabelle, Agda, Epigram, etc.
> Can some software go over it, report properties, do some transformations with known, useful properties?
Yes, in many cases. There's a whole lot of work in static code analysis, and refactoring tools. It's not perfect (the halting problem being non-decidable and all that) but there have been _significant_ improvements since the Algol60 days, and that's not counting functional languages and type-theoretic approaches.
> Or, return to most of the rest of engineering
That goes a bit off topic, but at the stage we are in, for most practical purposes, "software engineering" is not really engineering. Except maybe when done by NASA, but then again, that's not a practical approach either.
> Also for another of your points, I'm not talking about anything like branch prediction or deep pipelining. That's basically what to do with with the hardware to execute an existing instruction set.
Oh, but you were talking about compilers. Modern optimizing compilers have to take those things into account, among many other things.
> Clearly compiling and executing a program are necessarily mathematically something, understood or not, powerful or not. For progress, we need to understand the subject mathematically.
(from your previous post)
My point was that very significant progress has been made in many fields, even if not necessarily formalized.
> Also the point is not math or not.
Sorry, but when you state that CS should be a footnote in a math book, you are kinda making the point that everything in CS (even in those sub-domains that are eminently practical) should be math-based to do anything meaningful. This is demonstrably not true.
> The point is progress.
And I've agreed with you on this. More and better maths can help advance CS. But we knew that already.
> Your point that there's a lot of good CS to do without 'mathematizing' the field is not promising for research or significant progress.
I contend that there has been significant progress in many CS areas without 'mathematizing' them. That is a fact. I also stated, in my previous post, that I agree that maths could help improve this progress. I think my problem with your position is that you're talking in absolutes in topics where those absolutes clearly don't hold.
"Sorry, but when you state that CS should be a footnote in a math book, you are kinda making the point that everything in CS (even in those sub-domains that are eminently practical) should be math-based to do anything meaningful. This is demonstrably not true."
I'm exaggerating, but, still there is a reasonable point here. CS is about some 'science', and call that the mathematical part where we have some solid material worth being called 'science', For the rest, call that 'computer practice' or some such.
The upside of my view is that for some serious progress we're going to have to use some serious methodology. So, I'm proposing if not mathematical physics envy then applied math envy. Applied math didn't take on how to design the display lights in a scientific pocket calculator although without the lights the thing wouldn't work.
My view is not the most extreme: Last year I communicated with a CS prof whose position was that CS is looking for the 'fundamentals of computation'. Hmm .... It sounds like he believes that the P versus NP question should be right at the top of the list, and I don't. I'll settle for anything that is solid and a contribution, even if small, to the 'science' and not just to current practice.
LIkely many people here know the current state of programming language research much better than I do. If that field has gotten nicely mathematical with some solid material, great. The progress from Fortran, Cobol, Algol, Basic, PL/I, Pascal, C, C++, etc. was pragmatic and important and of enormous value to the economy but not much progress in a 'science' of programming languages, and that progress, with poor methodology, has slowed as we might have expected.
Maybe I'm saying that, in 1920, if we wanted a really good airplane, then maybe we should set aside the wood, linen, and glue and go do some aerodynamiocs calculations, discover Reynolds number, and discover that those really thin wings were a mistake. Or, observational astronomy was just a lot of curiosity until Newton came along and made progress in understanding the universe. The practical chemists had discovered a LOT, but by applying quantum mechanics they made HUGE progress.
If we are going to make the huge progress we want in computing, then history suggests that we can't be just pragmatic and that "theory is under-rated". We can't expect that chemistry will do much to help CS, but the obvious tool is math, to turn CS, the science, part into some applied math.
For the CS topics you listed, e.g., programming languages, that field has for decades just cried out for some good math but gotten nearly none! Clearly compiling and executing a program are necessarily mathematically something, understood or not, powerful or not. For progress, we need to understand the subject mathematically. E.g., given a programming language statement, what are its mathematical properties so that given a sequence of two such statements, what are the mathematical properties? Can't see any? Okay: Change programming languages until can. Then with the properties, what can we conclude about the program? Anything useful, say, about correctness, running time, storage utilization, approaching limitations? Given some properties, can we have some code transformations with some known properties? Are some such useful? If not, can we define some useful programming languages that admit such transformations?
There are some simple, illustrative, pregnant cases that just leap off the screen: Write a collection of routines for sort by surrogate, i.e., finding a permutation, maybe honoring sort 'stability', applying a permutation to an array, chasing through the fact that each permutation is a product of disjoint cycles, etc. Then with such routines, will see some properties that are close to algebraic, e.g., where some combinations of the routines are the identity transformation, where one routine is the algebraic inverse of another, where we might have commutativity and/or associativity, or are equivalent to other combinations, etc. Can we preserve sort 'stability'? So, we see that we should have an 'algebra' of those routines and be able to derive known properties. Then we see that due to 'boundary' limitations, etc., our algebra is only roughly correct. So, maybe we should fix that.
This is old stuff. It has leaped off the screen for decades. We want some known properties we can use to derive new, useful properties. Instead all we get is programming languages that compete in some absurd 'beauty contest' or that 'encourage' programming 'styles' seen as 'helpful'. Sicko.
E.g., it has been common now in server farms to be concerned about reliability. So, we set up systems so that when system A gets too busy or fails, then system B takes over, etc. However, what we've seen, including at some of the most important farms, for decades, up to this year, is that these intuitive approaches far too easily go "splat" in the mud: They are 'unstable', have problems that propagate, and are surprisingly UNreliable.
So, we need some new math of 'reliability' for such server farms, math that can write us some guarantees. That math will start with the real problems, have definitions, theorems, and proofs, and, if successful, some solid tools for building reliable server farms.
All across computing, CS needs to give us progress, and for that they need to proceed essentially only mathematically.
It's the math. The CS is the math. Server farm reliability is just the application of the math and is not the CS. Ignoring math, CS is "A little people. A silly people."! Until CS gets serious about the math, it is writing checks its methodology can't cash.
To get serious about math, CS first has to learn LP well!
The would you believe probability? How about stochastic processes? Any stochastic processes in server farms and networks? Any uses of the axiomatic derivation of the Poisson process? Any roles for one of the strongest inequalities in math, the martingale inequality and the associated convergence theorem? Measure preserving transformations as in ergodic theory -- I'll answer that one, "Yes!". The renewal theorem and loads at server farms? How 'bout a little in probability?
Understanding now?