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(CW: self-plug, music/game geekery)

Some of you may be aware that I'm taking a few weeks' break in between jobs. This is one of the things I've been working on in the meantime! It's a case study about how Zoombinis (a favourite game of mine) does music.

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"The crucial aspect of the new 3-D forms has more to do with their unusual geometrical configuration than with the material itself, which suggests that similar strong, lightweight materials could be made from a variety of materials by creating similar geometric features.

“You could either use the real graphene material or use the geometry we discovered with other materials, like polymers or metals,” [department head] Buehler says, to gain similar advantages of strength combined with advantages in cost, processing methods, or other material properties (such as transparency or electrical conductivity)."

“You could either use the real graphene material or use the geometry we discovered with other materials, like polymers or metals,” [department head] Buehler says, to gain similar advantages of strength combined with advantages in cost, processing methods, or other material properties (such as transparency or electrical conductivity)."

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luuulz

> For the past five years, the easy notion of the American enemy has permeated the everyday lives of Russian people. A few years ago car stickers saying “Obama is a wanker” became fashionable and Russian establishments, from cafes to hairdressers’ salons, that carried the sign “we do not serve Obama here” were often promoted on state TV.

> A win for Clinton would have preserved these sentiments but Trump’s victory destroys them. Russian parliamentarians were applauding Trump, not realising that he actually undermines their political system, which will have to evaluate how it goes forward now that America is no longer the enemy.

> For the past five years, the easy notion of the American enemy has permeated the everyday lives of Russian people. A few years ago car stickers saying “Obama is a wanker” became fashionable and Russian establishments, from cafes to hairdressers’ salons, that carried the sign “we do not serve Obama here” were often promoted on state TV.

> A win for Clinton would have preserved these sentiments but Trump’s victory destroys them. Russian parliamentarians were applauding Trump, not realising that he actually undermines their political system, which will have to evaluate how it goes forward now that America is no longer the enemy.

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"Do not tell people what they did or did not suffer; it’s rude and always an asshole move. People can suffer things they don’t tell you about, and being told you didn’t suffer something you did feels like shit. Similarly, don’t tell someone they couldn’t possibly understand X experience because they are privileged; even if they’re not closeted, a lot of experiences are shared across marginalizations anyway. It’s probably wise to avoid speculating about the group membership of people you don’t know very well; there have been far too many awkward cases in which the privileged neurotypical turned out to be a mentally ill person. In general, whenever possible, stick to arguing about facts and evidence, instead of exploring why the person you’re arguing with believes the thing they believe; the latter often winds up condescending."

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[Another repost from my former Google Buzz feed, which I first posted on Aug 16, 2010.]

Formally, a mathematical proof consists of a sequence of mathematical statements and deductions (e.g. "If A, then B"), strung together in a logical fashion to create a conclusion. A simple example of this is a linear chain of deductions, such as "A~~> B -> C -> D -> E", to create the conclusion "A -> E". In practice, though, proofs tend to be more complicated than a linear chain, often acquiring a tree-like structure (or more generally, the structure of a directed acyclic graph), due to the need to branch into cases, or to reuse a hypothesis multiple times. Proof methods such as proof by contradiction, or proof by induction, can lead to even more intricate loops and reversals in a mathematical argument.~~

~~Unfortunately, not all proposed proofs of a statement in mathematics are actually correct, and so some effort needs to be put into verification of such a proposed proof. Broadly speaking, there are two ways that one can show that a proof can fail. Firstly, one can find a "local", "low-level" or "direct" objection to the proof, by showing that one of the steps (or perhaps a cluster of steps, see below) in the proof is invalid. For instance, if the implication C -> D is false, then the above proposed proof "A -> B -> C -> D -> E" of "A -> E" is invalid (though it is of course still conceivable that A -> E could be proven by some other route).~~

~~Sometimes, a low-level error cannot be localised to a single step, but rather to a cluster of steps. For instance, if one has a circular argument, in which a statement A is claimed using B as justification, and B is then claimed using A as justification, then it is possible for both implications A -> B and B -> A to be true, while the deduction that A and B are then both true remains invalid. (Note though that there are important and valid examples of near-circular arguments, such as proofs by induction, but this is not the topic of my discussion today.)~~

~~Another example of a low-level error that is not localisable to a single step arises from ambiguity. Suppose that one is claiming that A~~>B and B->C, and thus that A->C. If all terms are unambiguously well-defined, this is a valid deduction. But suppose that the expression B is ambiguous, and actually has at least two distinct interpretations, say B1 and B2. Suppose further that the A->B implication presumes the former interpretation B=B1, while the B->C implication presumes the latter interpretation B=B2, thus we actually have A->B1 and B2->C. In such a case we can no longer validly deduce that A->C (unless of course we can show in addition that B1->B2). In such a case, one cannot localise the error to either "A->B" or "B->C" until B is defined more unambiguously. This simple example illustrates the importance of getting key terms defined precisely in a mathematical argument.

The other way to find an error in a proof is to obtain a "high level" or "global" objection, showing that the proof, if valid, would necessarily imply a further consequence that is either known or strongly suspected to be false. The most well-known (and strongest) example of this is the counterexample. If one possesses a counterexample to the claim A->E, then one instantly knows that the chain of deduction "A->B->C->D->E" must be invalid, even if one cannot immediately pinpoint where the precise error is at the local level. Thus we see that global errors can be viewed as "non-constructive" guarantees that a local error must exist somewhere.

A bit more subtly, one can argue using the structure of the proof itself. If a claim such as A->E could be proven by a chain A->B->C->D->E, then this might mean that a parallel claim A'->E' could then also be proven by a parallel chain A'->B'->C'->D'->E' of logical reasoning. But if one also possesses a counterexample to A'->E', then this implies that there is a flaw somewhere in this parallel chain, and hence (presumably) also in the original chain. Other examples of this type include proofs of some conclusion that mysteriously never use in any essential way a crucial hypothesis (e.g. proofs of the non-existence of non-trivial integer solutions to a^n+b^n=c^n that mysteriously never use the hypothesis that n is strictly greater than 2, or which could be trivially adapted to cover the n=2 case).

While global errors are less constructive than local errors, and thus less satisfying as a "smoking gun", they tend to be significantly more robust. A local error can often be patched or worked around, especially if the proof is designed in a fault-tolerant fashion (e.g. if the proof proceeds by factoring a difficult problem into several strictly easier pieces, which are in turn factored into even simpler pieces, and so forth). But a global error tends to invalidate not only the proposed proof as it stands, but also all reasonable perturbations of that proof. For instance, a counterexample to A->E will automatically defeat any attempts to patch the invalid argument A->B->C->D->E, whereas the more local objection that C does not imply D could conceivably be worked around.

(There is a mathematical joke in which a mathematician is giving a lecture expounding on a recent difficult result that he has just claimed to prove. At the end of the lecture, another mathematician stands up and asserts that she has found a counterexample to the claimed result. The speaker then rebuts, "This does not matter; I have two proofs of this result!". Here one sees quite clearly the distinction of impact between a global error and a local one.)

It is also a lot quicker to find a global error than a local error, at least if the paper adheres to established standards of mathematical writing.

To find a local error in an N-page paper, one basically has to read a significant fraction of that paper line-by-line, whereas to find a global error it is often sufficient to skim the paper to extract the large--scale structure. This can sometimes lead to an awkward stage in the verification process when a global error has been found, but the local error predicted by the global error has not yet been located. Nevertheless, global errors are often the most serious errors of all.

It is generally good practice to try to structure a proof to be fault tolerant with respect to local errors, so that if, say, a key step in the proof of Lemma 17 fails, then the paper does not collapse completely, but contains at least some salvageable results of independent interest, or shows a reduction of the main problem to a simpler one. Global errors, by contrast, cannot really be defended against by a good choice of proof structure; instead, they require a good choice of proof strategy that anticipates global pitfalls and confronts them directly.

One last closing remark: as error-testing is the complementary exercise to proof-building, it is not surprising that the standards of rigour for the two activities are dual to each other. When one is building a proof, one is expected to adhere to the highest standards of rigour that are practical, since a single error could well collapse the entire effort. But when one is testing an argument for errors or other objections, then it is perfectly acceptable to use heuristics, hand-waving, intuition, or other non-rigorous means to locate and describe errors. This may mean that some objections to proofs are not watertight, but instead indicate that either the proof is invalid, or some accepted piece of mathematical intuition is in fact inaccurate. In some cases, it is the latter possibility that is the truth, in which case the result is deemed "paradoxical", yet true. Such objections, even if they do not invalidate the paper, are often very important for improving one's intuition about the subject.

Formally, a mathematical proof consists of a sequence of mathematical statements and deductions (e.g. "If A, then B"), strung together in a logical fashion to create a conclusion. A simple example of this is a linear chain of deductions, such as "A

The other way to find an error in a proof is to obtain a "high level" or "global" objection, showing that the proof, if valid, would necessarily imply a further consequence that is either known or strongly suspected to be false. The most well-known (and strongest) example of this is the counterexample. If one possesses a counterexample to the claim A->E, then one instantly knows that the chain of deduction "A->B->C->D->E" must be invalid, even if one cannot immediately pinpoint where the precise error is at the local level. Thus we see that global errors can be viewed as "non-constructive" guarantees that a local error must exist somewhere.

A bit more subtly, one can argue using the structure of the proof itself. If a claim such as A->E could be proven by a chain A->B->C->D->E, then this might mean that a parallel claim A'->E' could then also be proven by a parallel chain A'->B'->C'->D'->E' of logical reasoning. But if one also possesses a counterexample to A'->E', then this implies that there is a flaw somewhere in this parallel chain, and hence (presumably) also in the original chain. Other examples of this type include proofs of some conclusion that mysteriously never use in any essential way a crucial hypothesis (e.g. proofs of the non-existence of non-trivial integer solutions to a^n+b^n=c^n that mysteriously never use the hypothesis that n is strictly greater than 2, or which could be trivially adapted to cover the n=2 case).

While global errors are less constructive than local errors, and thus less satisfying as a "smoking gun", they tend to be significantly more robust. A local error can often be patched or worked around, especially if the proof is designed in a fault-tolerant fashion (e.g. if the proof proceeds by factoring a difficult problem into several strictly easier pieces, which are in turn factored into even simpler pieces, and so forth). But a global error tends to invalidate not only the proposed proof as it stands, but also all reasonable perturbations of that proof. For instance, a counterexample to A->E will automatically defeat any attempts to patch the invalid argument A->B->C->D->E, whereas the more local objection that C does not imply D could conceivably be worked around.

(There is a mathematical joke in which a mathematician is giving a lecture expounding on a recent difficult result that he has just claimed to prove. At the end of the lecture, another mathematician stands up and asserts that she has found a counterexample to the claimed result. The speaker then rebuts, "This does not matter; I have two proofs of this result!". Here one sees quite clearly the distinction of impact between a global error and a local one.)

It is also a lot quicker to find a global error than a local error, at least if the paper adheres to established standards of mathematical writing.

To find a local error in an N-page paper, one basically has to read a significant fraction of that paper line-by-line, whereas to find a global error it is often sufficient to skim the paper to extract the large--scale structure. This can sometimes lead to an awkward stage in the verification process when a global error has been found, but the local error predicted by the global error has not yet been located. Nevertheless, global errors are often the most serious errors of all.

It is generally good practice to try to structure a proof to be fault tolerant with respect to local errors, so that if, say, a key step in the proof of Lemma 17 fails, then the paper does not collapse completely, but contains at least some salvageable results of independent interest, or shows a reduction of the main problem to a simpler one. Global errors, by contrast, cannot really be defended against by a good choice of proof structure; instead, they require a good choice of proof strategy that anticipates global pitfalls and confronts them directly.

One last closing remark: as error-testing is the complementary exercise to proof-building, it is not surprising that the standards of rigour for the two activities are dual to each other. When one is building a proof, one is expected to adhere to the highest standards of rigour that are practical, since a single error could well collapse the entire effort. But when one is testing an argument for errors or other objections, then it is perfectly acceptable to use heuristics, hand-waving, intuition, or other non-rigorous means to locate and describe errors. This may mean that some objections to proofs are not watertight, but instead indicate that either the proof is invalid, or some accepted piece of mathematical intuition is in fact inaccurate. In some cases, it is the latter possibility that is the truth, in which case the result is deemed "paradoxical", yet true. Such objections, even if they do not invalidate the paper, are often very important for improving one's intuition about the subject.

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"As someone who used to define herself vis-a-vis her work ethic, to become unable to act upon that work ethic is nearly intolerable. My deep fear is that I'm secretly slothful and am using chronic illness to disguise the sick rot of laziness within myself. Surely I can rouse myself from this bed and bring myself to my desk?"

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This is bloody beautiful.

"If you're trying to pass the class, then pass it with minimum effort. Anything else is wasted motion.

If you're trying to ace the class, then ace it with minimum effort. Anything else is wasted motion...

The slackers fail to deploy their full strength because they realize that the quality line is not their preference curve. The tryers deploy their full strength at the wrong target, in attempts to go as far right as possible, wasting energy on a fight that is not theirs. So take the third path: remember what you're fighting for."

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Risk, reward, and rules.

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"If you keep doing it– if you stare unflinchingly at the void– eventually you will come to a sense of acceptance, of peace, even of relief. You will not be able to comprehend the scale of human suffering, of wild animal suffering, of death. But you do not have to pretend it does not exist. It will not destroy you. And it is always easier to look your problems square in the face than to pretend that they don’t exist at all."

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