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The same argument would say that it's impossible to write mathematics papers without proof assistants (Coq, etc). Obviously that is bogus, mathematicians are n

The same argument would say that it's impossible to write mathematics papers without proof assistants (Coq, etc). Obviously that is bogus, mathematicians are not 'deluded' about their ability to check their work themselves. It's just a skill.

cwzwarich 11y ago
In the analogy between programming and theorem-proving, programs correspond to proofs. When mathematicians informally check a proof, they are usually judging whether a complete proof could be produced, not whether the presented proof is a complete correct proof. Programmers are not given the same affordance.
asQuirreL 11y ago
I feel like the analogy would be "writing mathematical proofs without writing down any of the propositions" (And indeed, the Curry-Howard correspondence lends credence to this). Writing down types for programs is not just useful because the computer can check them, but also because it commits what is in one's mind. What's more, type-checkers offer a lot more help in checking types than Coq does at proving mathematical propositions (There is a reason, after all, that one is an assistant, whereas the other is a checker), so, relatively speaking, writing a proof without Coq is not as hard as typing a program without a type-checker.
seliopou 11y ago
Not the case. The same argument does not apply because of the nature of mathematical, and to a certain extent scientific. All three--mathematics, science, and computing (programming)--are similar in that their primary currency is knowledge that can be concisely described and embodied in instruments (tools), either of which can then be used by others without too much of a concern for where that knowledge or instrument came from as long as there's a good instruction manual nearby. The knowledge and instruments are used to create more knowledge, which is embodied in more instruments, and the cycle continues on and on etc. The difference between computing and mathematics is the form that the knowledge and the instruments take. In mathematics, the concepts you're dealing with (or inventing, since I'm not a mathematical realist) are very clean and they stack well. You also have virtually no instruments to speak of. As a mathematician, it's not too common that you reduce your theories to practice. That's something that's left to the lesser sciences, such as computing. In computing, the relationship between between knowledge and instruments is that of identity. At the very least, the two are conflated in casual discussion. More concretely, the "knowledge" of computing is typically a model of some problem domain and the the instruments are software, which in turn manipulate the physical world (that's what your JavaScript is doing) in order to perform some task or measurement in service of solving some problem in your domain. So in mathematics, you have no correspondence between knowledge and instruments to do deal with. In computing, that's all you're dealing with except people are only really interested in the knowledge as its embodiment in the instruments. Static typing ensures that some sliver of that knowledge is explicit and checked. It doesn't mean it corresponds to the thing in your brain, but it does ensure that whatever's written down is embodied in your code. Mathematicians have no such constraints to deal with. They would if the proofs that they wrote were formal proofs, but they're not. Most, if not all, mathematicians know this. Most programmers, though, don't.
- mpweiher 11y ago
  > Static typing ensures that some sliver of that knowledge is explicit and checked. No. Testing ensures that the knowledge is checked. As in: run experiments. Empirical evidence. Static typing only checks one set of assumptions against another set of assumptions. Don't get me wrong, this is better than nothing. But the only real check is actually running the code, getting empirical evidence. In every other engineering discipline, math is seen as a tool, simulations are cost saving measure because running experiments for everything is too expensive. Only in the discipline where gathering empirical evidence is actually generally cheaper than trying to prove things (with mostly dubious value) is there this odd notion that using math is superior to running the experiment. If flying planes or running wind tunnel tests were as quick and cheap as running computer simulations, there would be no more computer simulations in aviation. Compare Knuth: "Beware of bugs in the above code; I have only proved it correct, not tried it."
  - seliopou 11y ago
    First, > some sliver Second, I for the most part agree with you in the ways that are relevant to the point you're trying to make. However, the knowledge/tool distinction is a vague (in that they may fall on a spectrum) and ambiguous (both words tend to be moving targets), which is why I chose the term instrument instead. Mathematics is a tool only in the sense that it is knowledge that is applied towards some benefit, e.g., efficiency. Also, > Compare Knuth: "Beware of bugs in the above code; I have only proved it correct, not tried it." This is again "proof" in the day-to-day mathematical sense of a convincing argument, not in the formal sense for writing down an expression per line such that each line can be logically deduced from some preceding lines through the the use of axioms. A type checker is literally a search procedure through the space of derivation trees to find a formal proof that this term has this type.
    - isolate 11y ago
      Type checking is a conservative analysis, and the type correctness of a program is undecidable. But you can still rely on the knowledge generated by your type checker without having to run your program. Among other things, this allows you to optimize your program automatically.

cwzwarich 11y ago

In the analogy between programming and theorem-proving, programs correspond to proofs. When mathematicians informally check a proof, they are usually judging whether a complete proof could be produced, not whether the presented proof is a complete correct proof. Programmers are not given the same affordance.

asQuirreL 11y ago

I feel like the analogy would be "writing mathematical proofs without writing down any of the propositions" (And indeed, the Curry-Howard correspondence lends credence to this). Writing down types for programs is not just useful because the computer can check them, but also because it commits what is in one's mind. What's more, type-checkers offer a lot more help in checking types than Coq does at proving mathematical propositions (There is a reason, after all, that one is an assistant, whereas the other is a checker), so, relatively speaking, writing a proof without Coq is not as hard as typing a program without a type-checker.

seliopou 11y ago

Not the case. The same argument does not apply because of the nature of mathematical, and to a certain extent scientific. All three--mathematics, science, and computing (programming)--are similar in that their primary currency is knowledge that can be concisely described and embodied in instruments (tools), either of which can then be used by others without too much of a concern for where that knowledge or instrument came from as long as there's a good instruction manual nearby. The knowledge and instruments are used to create more knowledge, which is embodied in more instruments, and the cycle continues on and on etc. The difference between computing and mathematics is the form that the knowledge and the instruments take. In mathematics, the concepts you're dealing with (or inventing, since I'm not a mathematical realist) are very clean and they stack well. You also have virtually no instruments to speak of. As a mathematician, it's not too common that you reduce your theories to practice. That's something that's left to the lesser sciences, such as computing. In computing, the relationship between between knowledge and instruments is that of identity. At the very least, the two are conflated in casual discussion. More concretely, the "knowledge" of computing is typically a model of some problem domain and the the instruments are software, which in turn manipulate the physical world (that's what your JavaScript is doing) in order to perform some task or measurement in service of solving some problem in your domain. So in mathematics, you have no correspondence between knowledge and instruments to do deal with. In computing, that's all you're dealing with except people are only really interested in the knowledge as its embodiment in the instruments. Static typing ensures that some sliver of that knowledge is explicit and checked. It doesn't mean it corresponds to the thing in your brain, but it does ensure that whatever's written down is embodied in your code. Mathematicians have no such constraints to deal with. They would if the proofs that they wrote were formal proofs, but they're not. Most, if not all, mathematicians know this. Most programmers, though, don't.

mpweiher 11y ago
> Static typing ensures that some sliver of that knowledge is explicit and checked. No. Testing ensures that the knowledge is checked. As in: run experiments. Empirical evidence. Static typing only checks one set of assumptions against another set of assumptions. Don't get me wrong, this is better than nothing. But the only real check is actually running the code, getting empirical evidence. In every other engineering discipline, math is seen as a tool, simulations are cost saving measure because running experiments for everything is too expensive. Only in the discipline where gathering empirical evidence is actually generally cheaper than trying to prove things (with mostly dubious value) is there this odd notion that using math is superior to running the experiment. If flying planes or running wind tunnel tests were as quick and cheap as running computer simulations, there would be no more computer simulations in aviation. Compare Knuth: "Beware of bugs in the above code; I have only proved it correct, not tried it."
- seliopou 11y ago
  First, > some sliver Second, I for the most part agree with you in the ways that are relevant to the point you're trying to make. However, the knowledge/tool distinction is a vague (in that they may fall on a spectrum) and ambiguous (both words tend to be moving targets), which is why I chose the term instrument instead. Mathematics is a tool only in the sense that it is knowledge that is applied towards some benefit, e.g., efficiency. Also, > Compare Knuth: "Beware of bugs in the above code; I have only proved it correct, not tried it." This is again "proof" in the day-to-day mathematical sense of a convincing argument, not in the formal sense for writing down an expression per line such that each line can be logically deduced from some preceding lines through the the use of axioms. A type checker is literally a search procedure through the space of derivation trees to find a formal proof that this term has this type.
  - isolate 11y ago
    Type checking is a conservative analysis, and the type correctness of a program is undecidable. But you can still rely on the knowledge generated by your type checker without having to run your program. Among other things, this allows you to optimize your program automatically.