"And remember again - it's not that humans contain nothing of Validity.  You have concepts like 'or', and 'and'.  You have, in fact, more concepts than you need in order to make first-order logic complete, and some of them are redundant.  The human problem is not so much what we can never manage to derive in infinite time, as that we are too slow, and, even more than that, we tend to derive an awful lot of stuff that doesn't follow.  In dath ilan the Very Serious People used to complain a lot about how we were all being terrible at this, and I used to think of myself as being willing to pursue even riskier and wilder lines of reasoning than that, but now that I've read a book in Cheliax it really puts a lot of that into perspective."

"But I digress.  Humans contain shards, pieces, of the higher mathematical structure we call Validity, the content of necessity, the rules governing premises and conclusions, whose optimal answer is pinpointed by the completeness theorem.  Without these shards of Law, humans couldn't function at all.  These shards of Law within us are not manifested in a singular perfect engine that has been corrupted; rather, there are bits and pieces and shadows and correlates of Validity, glommed onto us by mistakes retained in the tiny spirals specifying the starting biology of our brains.  It's not that there's a perfect engine of Validity inside us that's corrupted.  It's not even that the parts of the perfect engine are distributed here and there inside us.  The human versions of logical concepts like our version of 'or' - often implying exclusive-orness, which isn't the logical version I showed you, but sometimes not being exclusive either - are more like weird shadows or correlates of pieces of Law.  Same goes for the human native version of 'Z implies Q', where in the human version it feels stranger to say that 'if I'm naked then I'm wearing a shirt' is clearly true since I'm not naked and I am wearing a shirt.  You can make logic out of the human pieces.  But at the same time, it's not a coincidence that you can make logic out of the human pieces, for all that those pieces were sloppily thrown together."

"Generation after generation, for millions of years, there were slight advantages in reproduction to the ancestors of humanity, who we call hominids, when they could do a better job of deducing unseen truths from the truths they already knew or guessed.  The human versions of 'and' and 'or' and 'implies' were built into us in order to do jobs including that job.  And because there is only a single complete structure of Validity in the realm of math - because there is a Law and a best Law and it's not even that hard a Law to find - all the bits and pieces of Validity that made their way into us, could have enough coherence and overlap in their messiness that a shadow of true Law could be born inside them."

"Validity is not the only principle with messy shards embedded into humans, in whose overlap and coherence the shadow of higher Law can be seen.  Another such principle is the one my people name Expected Utility, singled out as a unique answer by what we call coherence theorems.  If yesterday you trade two apples for twenty cherries, and then tomorrow you trade twenty cherries for one apple, you've gone around in a circle and ended up with fewer resources than you had when you started.  This is the bare start of a gesture in the direction of proofs that say, 'If you do a lot of deals with apples and cherries in ways that inconsistently value them relative to each other, you'll end up with strictly fewer of both apples and cherries than you could've had by doing different deals.'  If I'm rolling a die that must show either an even or an odd number, you'd be foolish to buy for eight silver pieces a gambling-ticket that pays seven silver pieces if the die shows an odd number, and six silver pieces if the die shows an even number.  This is the bare start of a gesture at the proof that, when you weight the probability of paths through time inside your mind, you should not weight a sub-possibility of a path more highly than you weight its whole."

"The principle of Expected Utility has indeed a sub-principle, which we call Probability, with rules and coherences of its own.  You may have noticed that a great number of conclusions that we need in everyday life do not follow with necessity from any facts that we are highly confident about; but there are also proofs about the best guesses you can extract from a state of uncertainty, and how you cannot do better than those without adding more data or more certainty into your premises."

"And remember again - it's not that humans contain nothing of Validity.  You have concepts like 'or', and 'and'.  You have, in fact, more concepts than you need in order to make first-order logic complete, and some of them are redundant.  The human problem is not so much what we can never manage to derive in infinite time, as that we are too slow, and, even more than that, we tend to derive an awful lot of stuff that doesn't follow.  In dath ilan the Very Serious People used to complain a lot about how we were all being terrible at this, and I used to think of myself as being willing to pursue even riskier and wilder lines of reasoning than that, but now that I've read a book in Cheliax it really puts a lot of that into perspective."

"But I digress.  Humans contain shards, pieces, of the higher mathematical structure we call Validity, the content of necessity, the rules governing premises and conclusions, whose optimal answer is pinpointed by the completeness theorem.  Without these shards of Law, humans couldn't function at all.  These shards of Law within us are not manifested in a centralized single engine whose voice we sometimes ignore; rather, there are bits and pieces and shadows and correlates of Validity, glommed onto us by mistakes retained in the tiny spirals specifying the starting biology of our brains.  It's not that there's a perfect engine of Validity inside us that's corrupted.  It's not even that the parts of the perfect engine are distributed here and there inside us.  The human versions of logical concepts like our version of 'or' - often implying exclusive-orness, which isn't the logical version I showed you, but sometimes not being exclusive either - are more like weird shadows or correlates of pieces of Law.  Same goes for the human native version of 'Z implies Q', where in the human version it feels stranger to say that 'if I'm naked then I'm wearing a shirt' is clearly true since I'm not naked and I am wearing a shirt.  You can make logic out of the human pieces.  But at the same time, it's not a coincidence that you can make logic out of the human pieces, for all that those pieces were sloppily thrown together."

"Generation after generation, for millions of years, there were slight advantages in reproduction to the ancestors of humanity, who we call hominids, when they could do a better job of deducing unseen truths from the truths they already knew or guessed.  The human versions of 'and' and 'or' and 'implies' were built into us in order to do jobs including that job.  And because there is only a single complete structure of Validity in the realm of math - because there is a Law and a best Law and it's not even that hard a Law to find - all the bits and pieces of Validity that made their way into us, could have enough coherence and overlap in their messiness that a shadow of true Law could be born inside them."

"Validity is not the only principle with messy shards embedded into humans, in whose overlap and coherence the shadow of higher Law can be seen.  Another such principle is the one my people name Expected Utility, singled out as a unique answer by what we call coherence theorems.  If yesterday you trade two apples for twenty cherries, and then tomorrow you trade twenty cherries for one apple, you've gone around in a circle and ended up with fewer resources than you had when you started.  This is the bare start of a gesture in the direction of proofs that say, 'If you do a lot of deals with apples and cherries in ways that inconsistently value them relative to each other, you'll end up with strictly fewer of both apples and cherries than you could've had by doing different deals.'  If I'm rolling a die that must show either an even or an odd number, you'd be foolish to buy for eight silver pieces a gambling-ticket that pays seven silver pieces if the die shows an odd number, and six silver pieces if the die shows an even number.  This is the bare start of a gesture at the proof that, when you weight the probability of paths through time inside your mind, you should not weight a sub-possibility of a path more highly than you weight its whole."

"The principle of Expected Utility has indeed a sub-principle, which we call Probability, with rules and coherences of its own.  You may have noticed that a great number of conclusions that we need in everyday life do not follow with necessity from any facts that we are highly confident about; but there are also proofs about the best guesses you can extract from a state of uncertainty, and how you cannot do better than those without adding more data or more certainty into your premises."

"And remember again - it's not that humans contain nothing of Validity.  You have concepts like 'or', and 'and'.  You have, in fact, more concepts than you need in order to make first-order logic complete, and some of them are redundant.  The human problem is not so much what we can never manage to derive in infinite time, as that we are too slow, and, even more than that, we tend to derive an awful lot of stuff that doesn't follow.  In dath ilan the Very Serious People used to complain a lot about how we were all being terrible at this, and I used to think of myself as being willing to pursue even riskier and wilder lines of reasoning than that, but now that I've read a book in Cheliax it really puts a lot of that into perspective."

"But I digress.  Humans contain shards, pieces, of the higher mathematical structure we call Validity, the content of necessity, the rules governing premises and conclusions, whose optimal answer is pinpointed by the completeness theorem.  Without these shards of Law, humans couldn't function at all.  These shards of Law within us are not manifested in a centralized single engine whose voice we sometimes ignore; rather, there are bits and pieces and shadows and correlates of Validity, glommed onto us by mistakes retained in the tiny spirals specifying the starting biology of our brains.  It's not that there's a perfect engine of Validity inside us that's corrupted.  It's not even that the parts of the perfect engine are distributed here and there inside us.  The human versions of logical concepts like our version of 'or' - often implying exclusive-orness, which isn't the logical version I showed you, but sometimes not being exclusive either - are more like weird shadows or correlates of pieces of Law.  Same goes for the human native version of 'Z implies Q'.  In the human version it feels stranger to say that 'if I'm naked, that implies I'm wearing a shirt' is extremely true about me because I'm not naked and I am wearing a shirt.  You can make logic out of the human pieces."

"But for all that the human pieces were sloppily thrown together, it's no coincidence that you can make the universal logic out of them.  Generation after generation, for millions of years, there were slight advantages in reproduction to the ancestors of humanity, who we call hominids, when they could do a better job of deducing unseen truths from the truths they already knew or guessed.  The human versions of 'and' and 'or' and 'implies' were built into us in order to do jobs including that job.  And because there is only a single complete structure of Validity in the realm of math - because there is a Law and a best Law and it's not even that hard a Law to find - all the bits and pieces of Validity that made their way into us, could have enough coherence and overlap in their messiness that a shadow of true Law could be born inside them."

"Validity is not the only principle with messy shards embedded into humans, in whose overlap and coherence the shadow of higher Law can be seen.  Another such principle is the one my people name Expected Utility, singled out as a unique answer by what we call coherence theorems.  If yesterday you trade two apples for twenty cherries, and then tomorrow you trade twenty cherries for one apple, you've gone around in a circle and ended up with fewer resources than you had when you started.  This is the bare start of a gesture in the direction of proofs that say, 'If you do a lot of deals with apples and cherries in ways that inconsistently value them relative to each other, you'll end up with strictly fewer of both apples and cherries than you could've had by doing different deals.'  If I'm rolling a die that must show either an even or an odd number, you'd be foolish to buy for eight silver pieces a gambling-ticket that pays seven silver pieces if the die shows an odd number, and six silver pieces if the die shows an even number.  This is the bare start of a gesture at the proof that, when you weight the probability of paths through time inside your mind, you should not weight a sub-possibility of a path more highly than you weight its whole."

"The principle of Expected Utility has indeed a sub-principle, which we call Probability, with rules and coherences of its own.  You may have noticed that a great number of conclusions that we need in everyday life do not follow with necessity from any facts that we are highly confident about; but there are also proofs about the best guesses you can extract from a state of uncertainty, and how you cannot do better than those without adding more data or more certainty into your premises."

"And remember again - it's not that humans contain nothing of Validity.  You have concepts like 'or', and 'and'.  You have, in fact, more concepts than you need in order to make first-order logic complete, and some of them are redundant.  The human problem is not so much what we can never manage to derive in infinite time, as that we are too slow, and, even more than that, we tend to derive an awful lot of stuff that doesn't follow.  In dath ilan the Very Serious People used to complain a lot about how we were all being terrible at this, and I used to think of myself as being willing to pursue even riskier and wilder lines of reasoning than that, but now that I've read a book in Cheliax it really puts a lot of that into perspective."

"But I digress.  Humans contain shards, pieces, of the higher mathematical structure we call Validity, the content of necessity, the rules governing premises and conclusions, whose optimal answer is pinpointed by the completeness theorem.  Without these shards of Law, humans couldn't function at all.  These shards of Law within us are not manifested in a centralized single engine whose voice we sometimes ignore; rather, there are bits and pieces and shadows and correlates of Validity, glommed onto us by mistakes retained in the tiny spirals specifying the starting biology of our brains.  It's not that there's a perfect engine of Validity inside us that's corrupted.  It's not even that the parts of the perfect engine are distributed here and there inside us.  The human versions of logical concepts like our version of 'or' - often implying exclusive-orness, which isn't the logical version I showed you, but sometimes not being exclusive either - are more like weird shadows or correlates of pieces of Law.  Same goes for the human native version of 'Z implies Q'.  In the human version it feels stranger to say that 'if I'm naked, that implies I'm wearing a shirt' is extremely true about me because I'm not naked and I am wearing a shirt.  You can make logic out of the human pieces."

"But for all that the human pieces were sloppily thrown together, it's no coincidence that you can make a valid logic out of them.  Generation after generation, for millions of years, there were slight advantages in reproduction to the ancestors of humanity, who we call hominids, when they could do a better job of deducing unseen truths from the truths they already knew or guessed.  The human versions of 'and' and 'or' and 'implies' were built into us in order to do jobs including that job.  And because there is only a single complete structure of Validity in the realm of math - because there is a Law and a best Law and it's not that hard a Law to find - all the bits and pieces of Validity that made their way into us, could have enough coherence and overlap in their messiness that a shadow of true Law could be born inside them."

"Validity is not the only principle with messy shards embedded into humans, in whose overlap and coherence the shadow of higher Law can be seen.  Another such principle is the one my people name Expected Utility, singled out as a unique answer by what we call coherence theorems.  If yesterday you trade two apples for twenty cherries, and then tomorrow you trade twenty cherries for one apple, you've gone around in a circle and ended up with fewer resources than you had when you started.  This is the bare start of a gesture in the direction of proofs that say, 'If you do a lot of deals with apples and cherries in ways that inconsistently value them relative to each other, you'll end up with strictly fewer of both apples and cherries than you could've had by doing different deals.'  If I'm rolling a die that must show either an even or an odd number, you'd be foolish to buy for eight silver pieces a gambling-ticket that pays seven silver pieces if the die shows an odd number, and six silver pieces if the die shows an even number.  This is the bare start of a gesture at the proof that, when you weight the probability of paths through time inside your mind, you should not weight a sub-possibility of a path more highly than you weight its whole."

"The principle of Expected Utility has indeed a sub-principle, which we call Probability, with rules and coherences of its own.  You may have noticed that a great number of conclusions that we need in everyday life do not follow with necessity from any facts that we are highly confident about; but there are also proofs about the best guesses you can extract from a state of uncertainty, and how you cannot do better than those without adding more data or more certainty into your premises."