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> Imagine if software engineers were actual engineers instead of glorified script kiddies. Maybe you shouldn't need to be an engineer to write a document with

by dewey 6y ago

> Imagine if software engineers were actual engineers instead of glorified script kiddies. Maybe you shouldn't need to be an engineer to write a document with some formulas inside?

jimktrains2 6y ago
You don't. Maybe to edit such a complex program that you're using to convert your equations to some other pretty format you should.
konjin 6y ago
1 + 1 = 2 Aren't formulas easy to insert in plain text? The problem with maths notation is that it was invented by sighted people for sighted people as a short hand for very complex ideas, which at the time weren't fully understood. If you want to type equations use s-expressions, the clarity you get from saying what you mean is astonishing. (define integral (lambda (function) ;; Implementation of Risch algorithm left ;; as an exercise to the motivated reader. )) (define definite-integral (lambda (lower-bound upper-bound function) (let ((anti-derivative (integral function))) (- (anti-derivative upper-bound) (anti-derivative lower-bound))))) Hey look, you don't need the dummy variable explicitly any more. It's almost like it's a relic from a time before people understood what function application actually was. If you want the implicit mess (and incredible power) of higher maths, be prepared to deal with the the mess of typography. Which is why you need TeX, or worse.
- Tainnor 6y ago
  You're skimping on some of the complexity by equating a definition with a particular way of computing it, but that's completely inadequate for mathematics, as there are many things we can't compute (either in theory or in practice). In particular, you definition of integral assumes that integrable functions always have an antiderivative, which is wrong.
  - JadeNB 6y ago
    > In particular, you definition of integral assumes that integrable functions always have an antiderivative, which is wrong. I was objecting at the same time as you were, but I don't think this is the right objection. It's true that not every integrable function has an elementary anti-derivative, but every integrable function f does have an anti-derivative F, at least in the sense that F is almost everywhere differentiable, and the derivative is almost everywhere equal to f. (And, of course, if f is continuous, then F is everywhere differentiable, and its derivative everywhere equals f.)
    - Tainnor 6y ago
      If you say "at least in the sense that F is almost everywhere differentiable", you're already redefining "anti-derivative" to some extent, I feel. But this is just arguing over semantics. Even then, I think your claim that every integrable function has a "generalized antiderivative" is also only true for the Riemann integral. The Dirichlet function is Lebesgue integrable, but it doesn't have an antiderivative even in this weaker sense.[^1] And mathematicians generally prefer the Lebesgue integral. I think the more important insight here is that integration fundamentally isn't defined through the anti-derivative, and that the two notions are actually related is a deep theorem, rather than just a definition. And the fact that non-elementary antiderivatives exist is interesting in theory, but in practice you can't use them directly for anything. In particular, in practical situations you will often use numerical methods to integrate a function which will not be based on any notion of anti-derivative at all. [^1] Edit: I think I was wrong here. If you take the function identically zero, then its derivative is identically zero and as such equal to the Dirichlet function almost everywhere. So this is not a counterexample. I still think it's weird to call than an "antiderivative" though.
      - JadeNB 6y ago
        > If you say "at least in the sense that F is almost everywhere differentiable", you're already redefining "anti-derivative" to some extent, I feel. But this is just arguing over semantics. It is, but let's! Before we re-define the anti-derivative, we'd have to define it. A sensible definition is: a function F is an anti-derivative of a function f if F is everywhere differentiable, and if F' = f everywhere. By this definition, not every integrable function has an anti-derivative. On the other hand, we could also just choose to define—not re-define!—an anti-derivative of f to be a function F that is almost everywhere differentiable, and such that F' = f almost everywhere. This definition is more complicated, but also more inclusive; and it handles everything the old definition could. In this respect it is, and it's no accident, exactly like the Lebesgue integral vis a vis the Riemann integral. Lebesgue's integral has a more complicated definition than Riemann's, and we could call it a re-definition; but, since it handles everything that Riemann's does (with the same answer), we could say in retrospect that Lebesgue's was the correct definition, and Riemann's was just the special case we happened to discover first. > I think the more important insight here is that integration fundamentally isn't defined through the anti-derivative, and that the two notions are actually related is a deep theorem, rather than just a definition. Certainly I agree with this! > And the fact that non-elementary antiderivatives exist is interesting in theory, but in practice you can't use them directly for anything. In particular, in practical situations you will often use numerical methods to integrate a function which will not be based on any notion of anti-derivative at all. Here again I'd argue over semantics, though I'd concede it's much more a matter of personal preference than my argument above, which I think has mathematical weight behind it. Namely, I'd argue that the numerical integration is doing something directly with the non-elementary anti-derivative, namely, evaluating it at a point—just like we call reading off the value of, say, the sine of an angle from our calculator doing something directly with the sine, even though what we're really doing is summing sufficiently many terms in a Taylor-series approximation. > [^1] Edit: I think I was wrong here. If you take the function identically zero, then its derivative is identically zero and as such equal to the Dirichlet function almost everywhere. So this is not a counterexample. I still think it's weird to call than an "antiderivative" though. I agree that it's not a counterexample for the reason you say, and there's no arguing with perceptions of something being weird; it certainly is counter to intuition built out of Riemann integrals. And yet, if we didn't steel ourselves to handle this weirdness, we'd have to say that it didn't have an anti-derivative at all; and why artificially restrict our theorems to match our intuition, rather than expanding our intuition to meet our theorems?
- JadeNB 6y ago
  > Hey look, you don't need the dummy variable explicitly any more. It's almost like it's a relic from a time before people understood what function application actually was. If you actually perform definite integration, as a limit of Riemann (or Lebesgue, if you like) sums, or even try to approximate it numerically, you're going to have some dummy variable cropping back up again. As your comment indicates, when Risch's theorem proves that there is no elementary anti-derivative, you're going to be out of luck.
HarryHirsch 6y ago
Back in the day we had engineers writing documents from markup. We called them typesetters. It was a qualified job, you don't expect the task to be something Joe Random could to with 6 weeks of training.
- SilasX 6y ago
  At risk of gatekeeping, it doesn’t sound like typesetters merit the title of engineer.
  - formerly_proven 6y ago
    That's because it wasn't.
- dewey 6y ago
  Back in the days operating a computer also needed weeks of training and was a qualified job. That doesn't mean it always has to stay that way.
- throw0101a 6y ago
  > Back in the day we had engineers writing documents from markup. Back in the day secretaries were creating electronic documents with troff and nroff: > The first version of Unix was developed on a PDP-7 which was sitting around Bell Labs. In 1971 the developers wanted to get a PDP-11 for further work on the operating system. In order to justify the cost for this system, they proposed that they would implement a document-formatting system for the Bell Labs patents department[1]. This first formatting program was a reimplementation of McIllroy's roff, written by Joe F. Ossanna. * https://en.wikipedia.org/wiki/Troff https://en.wikipedia.org/wiki/Troff
  - rmbeard 6y ago
    OP, here. It's not that long ago, I've used troff myself. Or maybe it was that long ago, but it just seems like yesterday.
falseprofit 6y ago
We're not talking about the difficulty of writing a document using LaTeX, which I think we can agree is more or less proportional to the degree of control one has over the finished product. What's at stake here is the difficulty of editing the source code of LaTeX, which would be necessary in order to make the resulting pdf files more compatible with screen readers for the visually impaired.

by dewey 6y ago

> Imagine if software engineers were actual engineers instead of glorified script kiddies. Maybe you shouldn't need to be an engineer to write a document with some formulas inside?

jimktrains2 6y ago

You don't. Maybe to edit such a complex program that you're using to convert your equations to some other pretty format you should.

konjin 6y ago

1 + 1 = 2 Aren't formulas easy to insert in plain text? The problem with maths notation is that it was invented by sighted people for sighted people as a short hand for very complex ideas, which at the time weren't fully understood. If you want to type equations use s-expressions, the clarity you get from saying what you mean is astonishing. (define integral (lambda (function) ;; Implementation of Risch algorithm left ;; as an exercise to the motivated reader. )) (define definite-integral (lambda (lower-bound upper-bound function) (let ((anti-derivative (integral function))) (- (anti-derivative upper-bound) (anti-derivative lower-bound))))) Hey look, you don't need the dummy variable explicitly any more. It's almost like it's a relic from a time before people understood what function application actually was. If you want the implicit mess (and incredible power) of higher maths, be prepared to deal with the the mess of typography. Which is why you need TeX, or worse.

Tainnor 6y ago
You're skimping on some of the complexity by equating a definition with a particular way of computing it, but that's completely inadequate for mathematics, as there are many things we can't compute (either in theory or in practice). In particular, you definition of integral assumes that integrable functions always have an antiderivative, which is wrong.
- JadeNB 6y ago
  > In particular, you definition of integral assumes that integrable functions always have an antiderivative, which is wrong. I was objecting at the same time as you were, but I don't think this is the right objection. It's true that not every integrable function has an elementary anti-derivative, but every integrable function f does have an anti-derivative F, at least in the sense that F is almost everywhere differentiable, and the derivative is almost everywhere equal to f. (And, of course, if f is continuous, then F is everywhere differentiable, and its derivative everywhere equals f.)
  - Tainnor 6y ago
    If you say "at least in the sense that F is almost everywhere differentiable", you're already redefining "anti-derivative" to some extent, I feel. But this is just arguing over semantics. Even then, I think your claim that every integrable function has a "generalized antiderivative" is also only true for the Riemann integral. The Dirichlet function is Lebesgue integrable, but it doesn't have an antiderivative even in this weaker sense.[^1] And mathematicians generally prefer the Lebesgue integral. I think the more important insight here is that integration fundamentally isn't defined through the anti-derivative, and that the two notions are actually related is a deep theorem, rather than just a definition. And the fact that non-elementary antiderivatives exist is interesting in theory, but in practice you can't use them directly for anything. In particular, in practical situations you will often use numerical methods to integrate a function which will not be based on any notion of anti-derivative at all. [^1] Edit: I think I was wrong here. If you take the function identically zero, then its derivative is identically zero and as such equal to the Dirichlet function almost everywhere. So this is not a counterexample. I still think it's weird to call than an "antiderivative" though.
    - JadeNB 6y ago
      > If you say "at least in the sense that F is almost everywhere differentiable", you're already redefining "anti-derivative" to some extent, I feel. But this is just arguing over semantics. It is, but let's! Before we re-define the anti-derivative, we'd have to define it. A sensible definition is: a function F is an anti-derivative of a function f if F is everywhere differentiable, and if F' = f everywhere. By this definition, not every integrable function has an anti-derivative. On the other hand, we could also just choose to define—not re-define!—an anti-derivative of f to be a function F that is almost everywhere differentiable, and such that F' = f almost everywhere. This definition is more complicated, but also more inclusive; and it handles everything the old definition could. In this respect it is, and it's no accident, exactly like the Lebesgue integral vis a vis the Riemann integral. Lebesgue's integral has a more complicated definition than Riemann's, and we could call it a re-definition; but, since it handles everything that Riemann's does (with the same answer), we could say in retrospect that Lebesgue's was the correct definition, and Riemann's was just the special case we happened to discover first. > I think the more important insight here is that integration fundamentally isn't defined through the anti-derivative, and that the two notions are actually related is a deep theorem, rather than just a definition. Certainly I agree with this! > And the fact that non-elementary antiderivatives exist is interesting in theory, but in practice you can't use them directly for anything. In particular, in practical situations you will often use numerical methods to integrate a function which will not be based on any notion of anti-derivative at all. Here again I'd argue over semantics, though I'd concede it's much more a matter of personal preference than my argument above, which I think has mathematical weight behind it. Namely, I'd argue that the numerical integration is doing something directly with the non-elementary anti-derivative, namely, evaluating it at a point—just like we call reading off the value of, say, the sine of an angle from our calculator doing something directly with the sine, even though what we're really doing is summing sufficiently many terms in a Taylor-series approximation. > [^1] Edit: I think I was wrong here. If you take the function identically zero, then its derivative is identically zero and as such equal to the Dirichlet function almost everywhere. So this is not a counterexample. I still think it's weird to call than an "antiderivative" though. I agree that it's not a counterexample for the reason you say, and there's no arguing with perceptions of something being weird; it certainly is counter to intuition built out of Riemann integrals. And yet, if we didn't steel ourselves to handle this weirdness, we'd have to say that it didn't have an anti-derivative at all; and why artificially restrict our theorems to match our intuition, rather than expanding our intuition to meet our theorems?
JadeNB 6y ago
> Hey look, you don't need the dummy variable explicitly any more. It's almost like it's a relic from a time before people understood what function application actually was. If you actually perform definite integration, as a limit of Riemann (or Lebesgue, if you like) sums, or even try to approximate it numerically, you're going to have some dummy variable cropping back up again. As your comment indicates, when Risch's theorem proves that there is no elementary anti-derivative, you're going to be out of luck.

HarryHirsch 6y ago

Back in the day we had engineers writing documents from markup. We called them typesetters. It was a qualified job, you don't expect the task to be something Joe Random could to with 6 weeks of training.

SilasX 6y ago
At risk of gatekeeping, it doesn’t sound like typesetters merit the title of engineer.
- formerly_proven 6y ago
  That's because it wasn't.
dewey 6y ago
Back in the days operating a computer also needed weeks of training and was a qualified job. That doesn't mean it always has to stay that way.
throw0101a 6y ago
> Back in the day we had engineers writing documents from markup. Back in the day secretaries were creating electronic documents with troff and nroff: > The first version of Unix was developed on a PDP-7 which was sitting around Bell Labs. In 1971 the developers wanted to get a PDP-11 for further work on the operating system. In order to justify the cost for this system, they proposed that they would implement a document-formatting system for the Bell Labs patents department[1]. This first formatting program was a reimplementation of McIllroy's roff, written by Joe F. Ossanna. * https://en.wikipedia.org/wiki/Troff https://en.wikipedia.org/wiki/Troff
- rmbeard 6y ago
  OP, here. It's not that long ago, I've used troff myself. Or maybe it was that long ago, but it just seems like yesterday.

falseprofit 6y ago

We're not talking about the difficulty of writing a document using LaTeX, which I think we can agree is more or less proportional to the degree of control one has over the finished product. What's at stake here is the difficulty of editing the source code of LaTeX, which would be necessary in order to make the resulting pdf files more compatible with screen readers for the visually impaired.