I can only blame the usual slopiness of mathematicians,
That’s “sloppiness” :P.
I don’t consider these errors to be of the kind that damages credibility. That may be self-serving, though, since I make them all the time. But then again, I am a mathematician.
I’m curious, why are mathematicians sloppier than others?
I think it’s because we’re mainly focused on getting ideas right—most of the time, writing out the equation is merely a confirmation of what we allready know to be true. So often, a mathmo will write out a series of equations where the beginning will be true, the middle completely wrong, and the conclusion correct.
As for general linguistic sloppiness, that probably derives from the feeling that “hey my math is good, so don’t mess me about my words”.
If that’s true, I’ve wasted a significant chunk of my life reviewing my writings for errors. :-(
I’ve done that too—I’m just not very good at catching them. And it’s only a waste if you have a typo-tolerant audience.
I think it’s because we’re mainly focused on getting ideas right—most of the time, writing out the equation is merely a confirmation of what we allready know to be true. So often, a mathmo will write out a series of equations where the beginning will be true, the middle completely wrong, and the conclusion correct.
I wonder why that doesn’t work in cryptography. There are several well-known examples of “security proofs” (proof of security of a crypto scheme under the assumption that some computational problem is hard) by respected researchers that turn out many years after publication to contain errors that render the conclusions invalid.
Or does this happen just as often in mathematics, except that mathematicians don’t care so much because their errors don’t usually have much real-world impact?
Or does this happen just as often in mathematics, except that mathematicians don’t care so much because their errors don’t usually have much real-world impact?
The strongest theorems are those that have multiple proofs, or where the idea of the proof is easy to grasp (think Godel’s incompleteness theorem). Proofs that depend on every detail of a long tedious calculation, and only on that, are rare.
Proof that err by using implicit lemmas, or assuming results they can’t assume, are much more common, and mathematicians know this and are much more on guard for those errors.
The strongest theorems are those that have multiple proofs, or where the idea of the proof is easy to grasp (think Godel’s incompleteness theorem). Proofs that depend on every detail of a long tedious calculation, and only on that, are rare.
But those kinds of proofs are not rare in cryptography. Which suggests that there’s a selection effect going on in mathematics, where mathematicians choose which problems to work on partly by how likely the solutions to those problems will involve “strong” theorems with multiple proofs and perhaps be easily accessed by their intuitions.
Now what happens when the problem picks you, instead of you picking the problem? That is the situation we’re in, I think, so sloppiness is a worse problem than you might expect.
Both math and crypto contain errors. Are they the result of sloppiness? the kind of sloppiness Stuart Armstrong attributes to mathematicians?
I don’t know much about crypto. LF is said to be repeatedly wrong (in crypto? in another field?). That must constitute a kind of sloppiness. Is it correlated with other kinds of sloppiness?
i see two kinds of sloppiness I see attributed in this thread to mathematicians: (1) that detectable by copyediting; (2) focusing on the hard parts and trusting the easy parts to take care of themselves. (2) can lead to (1). There’s a third kind of sloppiness common in senior mathematicians: they supply the proof, but refuse to give the statement. Much of the difference is probably material that mathematicians include that people in CS simply omit. (is crypto published in conferences?)
Both math and crypto contain errors. Are they the result of sloppiness? the kind of sloppiness Stuart Armstrong attributes to mathematicians?
According to Stuart, in math there are often errors where “beginning will be true, the middle completely wrong, and the conclusion correct”. I was saying that this kind of error doesn’t seem to occur often in crypto, and trying to figure out why, with no bragging intended. Do you have another hypothesis, besides the one I gave?
LF is said to be repeatedly wrong (in crypto? in another field?).
My working hypothesis is that math and crypto are very similar, this kind of error occurs frequently, and you just don’t notice. What little crypto I know could be called complexity theory. I’ve read very little and heard it mainly orally. I’ve experienced this kind of error, certainly in oral complexity theory and I think in oral crypto. Of course, there’s a difference when people are trying to reconstruct proofs that are stamped by authority.
I thought it possible you were talking about the person LF.
Yes, mathematicians produce errors that stand for decades, but they aren’t errors that are detectable to copy-editing. Surely the errors you mention in crypto weren’t caused by substituting pz for px! How could it have stood so long with such an error? Also, such a random error is unlikely to help the proof. If you are approximating something and you want different sources of error to cancel, then a minus sign could make all the difference in the world, but people know that this is the key in the proof and pay more attention. Also, if flipping a sign makes it true, it’s probably false, not just false, but false with a big enough error that you can check it in small examples.
“Beware of bugs in the above code; I have only proved it correct, not tried it.″ - Knuth
I’ve heard stories (from my math professors in college) of grad students who spent multiple years writing about certain entities, which have all sorts of very interesting properties. However, they were having difficulties actually constructing one. Eventually it was demonstrated that there aren’t any, and they had been proving the interesting things one could do if one had an element of the empty set.
Only yesterday I read this digression in Girard’s The Blind Spot:
Once in a while I would like to indulge into an anecdote concerning the genesis of the proof. The criterion was found by the end of 1985; then I remained more than six months making circles around the “splitting tensor”. One nice day of August 1986, I woke up in a camping of Siena and I had got the proof: I therefore sat down and wrote a manuscript of 10 pages. One month later, I was copying this with a typewriter, and I discovered that one of my lemmas was wrong: no importance, I made another lemma! This illustrates the fact, neglected by the formalist ideology, that a proof is not putting side by side logical rules, it is a global perception: since I had found the concept of empire, I had my theorem and the faulty lemma was no more than a misprint.
(1) Not all errors damage credibility (in my eyes) to a significant degree.
(2) Nonetheless, even if an error doesn’t damage your credibility, avoiding it shows that you care about not wasting your readers’ time.
To expand on point (1), I’m inclined to be pretty forgiving if
the error was in a routine computation;
the computation is almost as easy to do myself as to verify; and
the computation serves only as an example, not as a formally necessary part of the description of the ideas.
In some fields of mathematics, papers are almost entirely prose, with very little computation (in the sense of manipulating formulas). In these fields, the proofs are communicated using words, not equations, though the words have very precise definitions.
That’s “sloppiness” :P.
I don’t consider these errors to be of the kind that damages credibility. That may be self-serving, though, since I make them all the time. But then again, I am a mathematician.
No, “slopiness” works too—mathematicians have more slope in their heads ;-P
I’m curious, why are mathematicians sloppier than others?
If that’s true, I’ve wasted a significant chunk of my life reviewing my writings for errors. :-(
I think it’s because we’re mainly focused on getting ideas right—most of the time, writing out the equation is merely a confirmation of what we allready know to be true. So often, a mathmo will write out a series of equations where the beginning will be true, the middle completely wrong, and the conclusion correct.
As for general linguistic sloppiness, that probably derives from the feeling that “hey my math is good, so don’t mess me about my words”.
I’ve done that too—I’m just not very good at catching them. And it’s only a waste if you have a typo-tolerant audience.
I wonder why that doesn’t work in cryptography. There are several well-known examples of “security proofs” (proof of security of a crypto scheme under the assumption that some computational problem is hard) by respected researchers that turn out many years after publication to contain errors that render the conclusions invalid.
Or does this happen just as often in mathematics, except that mathematicians don’t care so much because their errors don’t usually have much real-world impact?
The strongest theorems are those that have multiple proofs, or where the idea of the proof is easy to grasp (think Godel’s incompleteness theorem). Proofs that depend on every detail of a long tedious calculation, and only on that, are rare.
Proof that err by using implicit lemmas, or assuming results they can’t assume, are much more common, and mathematicians know this and are much more on guard for those errors.
But those kinds of proofs are not rare in cryptography. Which suggests that there’s a selection effect going on in mathematics, where mathematicians choose which problems to work on partly by how likely the solutions to those problems will involve “strong” theorems with multiple proofs and perhaps be easily accessed by their intuitions.
Now what happens when the problem picks you, instead of you picking the problem? That is the situation we’re in, I think, so sloppiness is a worse problem than you might expect.
This reads to me like macho bragging.
Both math and crypto contain errors. Are they the result of sloppiness? the kind of sloppiness Stuart Armstrong attributes to mathematicians?
I don’t know much about crypto. LF is said to be repeatedly wrong (in crypto? in another field?). That must constitute a kind of sloppiness. Is it correlated with other kinds of sloppiness?
i see two kinds of sloppiness I see attributed in this thread to mathematicians: (1) that detectable by copyediting; (2) focusing on the hard parts and trusting the easy parts to take care of themselves. (2) can lead to (1). There’s a third kind of sloppiness common in senior mathematicians: they supply the proof, but refuse to give the statement. Much of the difference is probably material that mathematicians include that people in CS simply omit. (is crypto published in conferences?)
According to Stuart, in math there are often errors where “beginning will be true, the middle completely wrong, and the conclusion correct”. I was saying that this kind of error doesn’t seem to occur often in crypto, and trying to figure out why, with no bragging intended. Do you have another hypothesis, besides the one I gave?
What is LF?
My working hypothesis is that math and crypto are very similar, this kind of error occurs frequently, and you just don’t notice. What little crypto I know could be called complexity theory. I’ve read very little and heard it mainly orally. I’ve experienced this kind of error, certainly in oral complexity theory and I think in oral crypto. Of course, there’s a difference when people are trying to reconstruct proofs that are stamped by authority.
I thought it possible you were talking about the person LF.
Yes, mathematicians produce errors that stand for decades, but they aren’t errors that are detectable to copy-editing. Surely the errors you mention in crypto weren’t caused by substituting pz for px! How could it have stood so long with such an error? Also, such a random error is unlikely to help the proof. If you are approximating something and you want different sources of error to cancel, then a minus sign could make all the difference in the world, but people know that this is the key in the proof and pay more attention. Also, if flipping a sign makes it true, it’s probably false, not just false, but false with a big enough error that you can check it in small examples.
“Beware of bugs in the above code; I have only proved it correct, not tried it.″ - Knuth
I’ve heard stories (from my math professors in college) of grad students who spent multiple years writing about certain entities, which have all sorts of very interesting properties. However, they were having difficulties actually constructing one. Eventually it was demonstrated that there aren’t any, and they had been proving the interesting things one could do if one had an element of the empty set.
http://en.wikipedia.org/wiki/Principle_of_explosion
Mathematicians do make errors. Sometimes they brush them aside as trivial (like Girard in Nesov’s example), but sometimes they care a lot.
Only yesterday I read this digression in Girard’s The Blind Spot:
(1) Not all errors damage credibility (in my eyes) to a significant degree.
(2) Nonetheless, even if an error doesn’t damage your credibility, avoiding it shows that you care about not wasting your readers’ time.
To expand on point (1), I’m inclined to be pretty forgiving if
the error was in a routine computation;
the computation is almost as easy to do myself as to verify; and
the computation serves only as an example, not as a formally necessary part of the description of the ideas.
In some fields of mathematics, papers are almost entirely prose, with very little computation (in the sense of manipulating formulas). In these fields, the proofs are communicated using words, not equations, though the words have very precise definitions.
Reviewing your writing costs you time, but it saves a mental hiccup on the part of a lot of readers. Multiply?