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Symbolic math is not the same thing as formalized math, and doesn't carry with it the same guarantees of correctness. That's the price one pays for having black

Symbolic math is not the same thing as formalized math, and doesn't carry with it the same guarantees of correctness. That's the price one pays for having black box functions spit out answers to hard problems in a reasonable time, they won't spit out proofs of correctness alongside their answer. Any symbolic system should be treated with care, but they can of course be extremely useful. My typical use case is to have it compute very challenging symbolic expressions for me. I then treat it as a "very plausible hypothesis" which I then prove.

monochr 12y ago
That's only true for closed source systems. You can easily debug open source ones, look at their insides and so on. This whole article is really just telling people to use the scientific method in computing too: if you don't know how to reproduce a result it's not science. Fast computation which a human can't possibly do is something that's new from the last 30 years and most scientists still don't know how to deal with it. It is very sad to see that statisticians have switched to R, biologists to python, computer scientists to the gnu tool chain, yet physics and maths seem to have been colonized by mathematica when you have a whole set of open source tools which are superior in every way: pari pg, gnu arbitrary precision libraries, axiom, maxima and sage is you want everything under one roof with a unified interface.
- tel 12y ago
  > "Easily debug" One of the major driving forces in the design of computer proof systems is that they (a) output proofs along with assertions that are (b) trivially verifiable. Indeed, verification is a major philosophical and practical force behind the theory of any type theory. Ultimately, this is a major component of the design of a proof checker because while lots of its pieces can be complex and scary and potentially buggy, the proof verifier must always be tiny and easy and impeccable because it bears the entire burden of safety. A similar argument cannot really be made for CAS, I think. Even open source ones. (There's a catchy name for this principle, but I forget it. "Somebody's Something")
- fddr 12y ago
  As a former physicist that used Mathematica heavily not that long ago, I have to disagree. It would be great if there were open source alternatives that were "superior in every way" but that is just not the case. For some use cases at least, Mathematica is clearly better than anything else I have tried. I feel like something like ipython notebooks with the right combination of libraries might eventually get there, but that is unfortunately still years away.
  - monochr 12y ago
    As another physicist who used mathematica what you mean is that you were too lazy to think about what assumptions were made implicitly in the calculations you were performing and liked mathematica because it did them automagically for you. This is shown no where better in the paper than when they try to calculate Integrate[Exp[-pt](Sinh[t])ˆ3, {t, 0, Infinity}]. In a good cas, such as maxima, you will get no result and a ton of errors. Which is what you should get without specifying what sort of variable p is, is it a matrix, polynomial, group of some sort? That it's implicitly assumed to be a real number that might turn out to be complex under some circumstances isn't a feature, it's a bug.
    - fddr 12y ago
      I am genuinely surprise at how confidently you (quite wrongly) diagnosed my problem. What I did mean, was that for my use cases, I found Mathematica was superior to the alternatives. You are welcome to think this was because I was lazy or misguided, but it was definitely not because I liked not having to specify domains for my variables (and I do remember having to write Assuming[p>0 && x \in Reals, ...] and the like often, it definitely does not assume everything is a positive real). One thing I used Mathematica a lot for was numerical integration of ODEs (that had been derived using the CAS part of Mathematica and had some pretty nasty coefficients). NDSolve in my experience was just better than competition. You can definitely get nonsense out of it, but with a modicum of care it works incredibly well.
- conistonwater 12y ago
  > That's only true for closed source systems. What? This is a software bug. Open source software has bugs just as closed source software has bugs. The average user of sage is not going to go hunting for a bug in sage's source, whether it's available or not, which is just the same as with Mathematica. Examining an unfamiliar code base will be of no help when you want to know why a certain integral was evaluated incorrectly. > open source tools which are superior in every way They are not superior. At best they are equivalent. What you list is a number of disparate tools that may or may not cooperate together well. What Mathematica provides by way of competition is a set of reasonably polished packages, all in one place. Mathematica also makes it possible to quickly write a one-liner that solves a problem and lets you move along. This experience is simply not there for tools like sage. Sage's plotting (matplotlib, IIRC) is just not comparable.
  - monochr 12y ago
    The average user of sage is a phd in a field that uses maths extensively. I really don't see how you can think that someone like that is dumb enough to not be able to debug a python function call to the point where something went wrong. It is literally one days worth of work to learn how to use the python debugging tools and it's largely trivial compared to the everyday work you do.
    - conistonwater 12y ago
      I think that's not right at all. Sage is a fairly massive project, most of which is implemented in separate libraries (not necessarily in python) that are then bound together under sage. To say that you need to know how to debug a python function call is a great understatement. The function call is not the issue. The issue is that it is a large unfamiliar code base, and the bug, as in this case, would not be an incorrect None somewhere, but will be a mathematical bug somewhere, like an incorrectly written formula. Chasing that is far far more difficult. Bearing in mind also that the average phd mathematician has at best cursory knowledge of programming and software engineering, enough to get on with mathematics, expecting them to dive into sage and fix things is just unrealistic. > it's largely trivial compared to the everyday work you do I think this is also a very unrealistic expectation of users, however skilled they may be as mathematicians.
- tzs 12y ago
  > That's only true for closed source systems. You can easily debug open source ones, look at their insides and so on. Who is "you"? In fact, very few people can debug a modern symbolic mathematics system. They are fiendishly complex.
- dalke 12y ago
  "if you don't know how to reproduce a result it's not science." Astronomy is one of many sciences which can never reproduce its results in the way you're talking about. Do you not consider astronomy to be a science? In any case, this paper was looking for counter-examples. If found, then they would be all the evidence that's needed. The method to compute the counter-examples is nice, but irrelevant.
  - monochr 12y ago
    You have astronomy as scinece confused with astronomy as historical records. I can't recreate an eclipse in 500 CE but I can calculate when another like it will happen in the same place it did.
    - dalke 12y ago
      The dynamic instability of the solar system puts a limit on that predictability. In any case, tell me about how you can predict gamma-ray bursts. When will the next detection of extrasolar neutrinos occur? How do we recreate the Big Bang? I am not confused. I regard historical sciences like astronomy, geology, paleontology and archaeology equally part of science even though there isn't the high level of reproducibility of, say, most chemistry. I also regard nuclear bomb physics to be a science, even though by law it's impossible to reproduce those tests.
      - monochr 12y ago
        So get better telescopes for the tolerance you want. We have no problems making predictions over thousands of years with current technology when it comes to stellar mechanics that agree extremely well with the historical record. Yours seems to be a very medieval mindset. Just because we are ignorant of the initial conditions of a system doesn't mean we are ignorant of the equations by which it evolves. And the example of chemistry is just bizarre. If we couldn't reproduce the same reaction down to the atom time and time again our silicone based infrastructure would have filed a very long time ago. Similarly for nuclear bomb tests, if they were truly irreproducible then things like [1] should happen a lot more often than not. [1] https://en.wikipedia.org/wiki/Castle_Bravo https://en.wikipedia.org/wiki/Castle_Bravo

monochr 12y ago

That's only true for closed source systems. You can easily debug open source ones, look at their insides and so on. This whole article is really just telling people to use the scientific method in computing too: if you don't know how to reproduce a result it's not science. Fast computation which a human can't possibly do is something that's new from the last 30 years and most scientists still don't know how to deal with it. It is very sad to see that statisticians have switched to R, biologists to python, computer scientists to the gnu tool chain, yet physics and maths seem to have been colonized by mathematica when you have a whole set of open source tools which are superior in every way: pari pg, gnu arbitrary precision libraries, axiom, maxima and sage is you want everything under one roof with a unified interface.

tel 12y ago
> "Easily debug" One of the major driving forces in the design of computer proof systems is that they (a) output proofs along with assertions that are (b) trivially verifiable. Indeed, verification is a major philosophical and practical force behind the theory of any type theory. Ultimately, this is a major component of the design of a proof checker because while lots of its pieces can be complex and scary and potentially buggy, the proof verifier must always be tiny and easy and impeccable because it bears the entire burden of safety. A similar argument cannot really be made for CAS, I think. Even open source ones. (There's a catchy name for this principle, but I forget it. "Somebody's Something")
fddr 12y ago
As a former physicist that used Mathematica heavily not that long ago, I have to disagree. It would be great if there were open source alternatives that were "superior in every way" but that is just not the case. For some use cases at least, Mathematica is clearly better than anything else I have tried. I feel like something like ipython notebooks with the right combination of libraries might eventually get there, but that is unfortunately still years away.
- monochr 12y ago
  As another physicist who used mathematica what you mean is that you were too lazy to think about what assumptions were made implicitly in the calculations you were performing and liked mathematica because it did them automagically for you. This is shown no where better in the paper than when they try to calculate Integrate[Exp[-pt](Sinh[t])ˆ3, {t, 0, Infinity}]. In a good cas, such as maxima, you will get no result and a ton of errors. Which is what you should get without specifying what sort of variable p is, is it a matrix, polynomial, group of some sort? That it's implicitly assumed to be a real number that might turn out to be complex under some circumstances isn't a feature, it's a bug.
  - fddr 12y ago
    I am genuinely surprise at how confidently you (quite wrongly) diagnosed my problem. What I did mean, was that for my use cases, I found Mathematica was superior to the alternatives. You are welcome to think this was because I was lazy or misguided, but it was definitely not because I liked not having to specify domains for my variables (and I do remember having to write Assuming[p>0 && x \in Reals, ...] and the like often, it definitely does not assume everything is a positive real). One thing I used Mathematica a lot for was numerical integration of ODEs (that had been derived using the CAS part of Mathematica and had some pretty nasty coefficients). NDSolve in my experience was just better than competition. You can definitely get nonsense out of it, but with a modicum of care it works incredibly well.
conistonwater 12y ago
> That's only true for closed source systems. What? This is a software bug. Open source software has bugs just as closed source software has bugs. The average user of sage is not going to go hunting for a bug in sage's source, whether it's available or not, which is just the same as with Mathematica. Examining an unfamiliar code base will be of no help when you want to know why a certain integral was evaluated incorrectly. > open source tools which are superior in every way They are not superior. At best they are equivalent. What you list is a number of disparate tools that may or may not cooperate together well. What Mathematica provides by way of competition is a set of reasonably polished packages, all in one place. Mathematica also makes it possible to quickly write a one-liner that solves a problem and lets you move along. This experience is simply not there for tools like sage. Sage's plotting (matplotlib, IIRC) is just not comparable.
- monochr 12y ago
  The average user of sage is a phd in a field that uses maths extensively. I really don't see how you can think that someone like that is dumb enough to not be able to debug a python function call to the point where something went wrong. It is literally one days worth of work to learn how to use the python debugging tools and it's largely trivial compared to the everyday work you do.
  - conistonwater 12y ago
    I think that's not right at all. Sage is a fairly massive project, most of which is implemented in separate libraries (not necessarily in python) that are then bound together under sage. To say that you need to know how to debug a python function call is a great understatement. The function call is not the issue. The issue is that it is a large unfamiliar code base, and the bug, as in this case, would not be an incorrect None somewhere, but will be a mathematical bug somewhere, like an incorrectly written formula. Chasing that is far far more difficult. Bearing in mind also that the average phd mathematician has at best cursory knowledge of programming and software engineering, enough to get on with mathematics, expecting them to dive into sage and fix things is just unrealistic. > it's largely trivial compared to the everyday work you do I think this is also a very unrealistic expectation of users, however skilled they may be as mathematicians.
tzs 12y ago
> That's only true for closed source systems. You can easily debug open source ones, look at their insides and so on. Who is "you"? In fact, very few people can debug a modern symbolic mathematics system. They are fiendishly complex.
dalke 12y ago
"if you don't know how to reproduce a result it's not science." Astronomy is one of many sciences which can never reproduce its results in the way you're talking about. Do you not consider astronomy to be a science? In any case, this paper was looking for counter-examples. If found, then they would be all the evidence that's needed. The method to compute the counter-examples is nice, but irrelevant.
- monochr 12y ago
  You have astronomy as scinece confused with astronomy as historical records. I can't recreate an eclipse in 500 CE but I can calculate when another like it will happen in the same place it did.
  - dalke 12y ago
    The dynamic instability of the solar system puts a limit on that predictability. In any case, tell me about how you can predict gamma-ray bursts. When will the next detection of extrasolar neutrinos occur? How do we recreate the Big Bang? I am not confused. I regard historical sciences like astronomy, geology, paleontology and archaeology equally part of science even though there isn't the high level of reproducibility of, say, most chemistry. I also regard nuclear bomb physics to be a science, even though by law it's impossible to reproduce those tests.
    - monochr 12y ago
      So get better telescopes for the tolerance you want. We have no problems making predictions over thousands of years with current technology when it comes to stellar mechanics that agree extremely well with the historical record. Yours seems to be a very medieval mindset. Just because we are ignorant of the initial conditions of a system doesn't mean we are ignorant of the equations by which it evolves. And the example of chemistry is just bizarre. If we couldn't reproduce the same reaction down to the atom time and time again our silicone based infrastructure would have filed a very long time ago. Similarly for nuclear bomb tests, if they were truly irreproducible then things like [1] should happen a lot more often than not. [1] https://en.wikipedia.org/wiki/Castle_Bravo https://en.wikipedia.org/wiki/Castle_Bravo