I think “what should be done” is generally different question that “what kind of mindsets there are” and I would prefer to disentangle them.
My claims about mindsets roughly are
there is important and meaningful distinction between “security mindset” and “mathematical mindset” (as is between 1-10^(-16) and 1)
also between “mathematical mindset” and e.g. “physics mindset”
the security mindset may be actually closer to some sort of scientific mindset
the way of reasoning common in maths is fragile in some sense
As I understand it (correct me if I’m wrong), your main claim roughly is “we should have a deep understanding how these systems works at all”.
I don’t think there is much disagreement on that.
But please note that Scott’s post in several places makes explicit distinction between the kind of understanding achieved in mathematics, and in science. The understanding we have how rockets work is pretty much on the physics side of this—e.g. we know we can disregard gravitational waves, radiation pressure, and violations of CP symmetry.
To me, this seems different from mathematics, where it would be somewhat strange to say something like “we basically understand what functions and derivatives are … you can just disregard cases like the Weierstrass function”.
(comment to mods: I would actually enjoy a setting allowing me to not see the karma system at all, the feedback it is giving me is “write things which people would upvote” vs. “write things which are most useful—were I’m unsure, see some flaws,...”. )
I agree. When I think about the “mathematician mindset” I think largely about the overwhelming interest in the presence or absence, in some space of interest, of “pathological” entities like the Weierstrass function. The truth or falsehood of “for all / there exists” statements tend to turn on these pathologies or their absence.
How does this relate to optimization? Optimization can make pathological entities more relevant, if
(1) they happen to be optimal solutions, or
(2) an algorithm that ignores them will be, for that reason, insecure / exploitable.
But this is not a general argument about optimization, it’s a contingent claim that is only true for some problems of interest, and in a way that depends on the details of those problems.
And one can make a separate argument that, when conditions like 1-2 do not hold, a focus on pathological cases is unhelpful: if a statement “fails in practice but works in theory” (say by holding except on a set of sufficiently small measure as to always be dominated by other contributions to a decision problem, or only for decisions that would be ruled out anyway for some other reason, or over the finite range relevant for some calculation but not in the long or short limit), optimization will exploit its “effective truth” whether or not you have noticed it. And statements about “effective truth” tend to be mathematically pretty uninteresting; try getting an audience of mathematicians to care about a derivation that rocket engineers can afford to ignore gravitational waves, for example.
I think “what should be done” is generally different question that “what kind of mindsets there are” and I would prefer to disentangle them.
My claims about mindsets roughly are
there is important and meaningful distinction between “security mindset” and “mathematical mindset” (as is between 1-10^(-16) and 1)
also between “mathematical mindset” and e.g. “physics mindset”
the security mindset may be actually closer to some sort of scientific mindset
the way of reasoning common in maths is fragile in some sense
As I understand it (correct me if I’m wrong), your main claim roughly is “we should have a deep understanding how these systems works at all”.
I don’t think there is much disagreement on that.
But please note that Scott’s post in several places makes explicit distinction between the kind of understanding achieved in mathematics, and in science. The understanding we have how rockets work is pretty much on the physics side of this—e.g. we know we can disregard gravitational waves, radiation pressure, and violations of CP symmetry.
To me, this seems different from mathematics, where it would be somewhat strange to say something like “we basically understand what functions and derivatives are … you can just disregard cases like the Weierstrass function”.
(comment to mods: I would actually enjoy a setting allowing me to not see the karma system at all, the feedback it is giving me is “write things which people would upvote” vs. “write things which are most useful—were I’m unsure, see some flaws,...”. )
I agree. When I think about the “mathematician mindset” I think largely about the overwhelming interest in the presence or absence, in some space of interest, of “pathological” entities like the Weierstrass function. The truth or falsehood of “for all / there exists” statements tend to turn on these pathologies or their absence.
How does this relate to optimization? Optimization can make pathological entities more relevant, if
(1) they happen to be optimal solutions, or
(2) an algorithm that ignores them will be, for that reason, insecure / exploitable.
But this is not a general argument about optimization, it’s a contingent claim that is only true for some problems of interest, and in a way that depends on the details of those problems.
And one can make a separate argument that, when conditions like 1-2 do not hold, a focus on pathological cases is unhelpful: if a statement “fails in practice but works in theory” (say by holding except on a set of sufficiently small measure as to always be dominated by other contributions to a decision problem, or only for decisions that would be ruled out anyway for some other reason, or over the finite range relevant for some calculation but not in the long or short limit), optimization will exploit its “effective truth” whether or not you have noticed it. And statements about “effective truth” tend to be mathematically pretty uninteresting; try getting an audience of mathematicians to care about a derivation that rocket engineers can afford to ignore gravitational waves, for example.