"When it comes to algebra over continuous quantites," Keltham says, gesturing at the tactics written between the steps of the equations, "we have rules like being allowed to multiply both sides by the same quantity, or divide both sides by the same quantity so long as it isn't zero.  If you imagine building a mind to reason inside a universe that was full of hidden order that could be described by algebra - if it was an observer surrounded by, like, piles of fruit containing twice as many cherries as apples, that sort of thing, it was just how that world worked - then you could imagine building that mind with rules like, 'If I believe an equation, I should also believe that equation with both sides multiplied by the same quantity' or 'If I believe an equation, I can believe that equation with both sides divided by the same quantity, so long as I already believe that quantity isn't zero.'  I say this to introduce a new topic: the concept of hidden order within the rules of reasoning themselves.  There are hidden patterns and deep explanations to be found in this subject matter, as, in my world, there was a reason why snowflakes had sixfold symmetry."

"As a very simple example, the rule 'You can divide by nonzero quantities' can be seen as a pure special case of 'You can multiply by any quantity.'  To say you can divide both sides by 2 is the same as saying you can multiply both sides by 1/2.  The reason you can't divide both sides by zero is that zero is the only continuous quantity which lacks an inverse.  Once you see things from that angle, in fact, you might say that it's a simpler viewpoint to say that there's just one rule to use there, about valid inference in algebra: the rule that you can multiply both sides by any quantity.  Say just that, and you don't need that darned rule with the extra complication about 'Oh well you can divide by anything unless it might be zero.'  You just have the rule that you can multiply by anything, and the rule that everything except zero has an inverse."

"But meanwhile, back in the real world, we deal more with the equivalent of triangles and red things than the equivalent of numbers and addition.  I mean, this world has both, but still, let's go back to shapes and colors and sizes.  What sort of truth-preserving rules analogous to 'you can multiply both sides by any quantity' in algebra, might we use to combine beliefs like these?"

Z.  All triangular things are red.
H.  All red things are large.

"When it comes to algebra over continuous quantites," Keltham says, gesturing at the tactics written between the steps of the equations, "we have rules like being allowed to multiply both sides by the same quantity, or divide both sides by the same quantity so long as it isn't zero.  If you imagine building a mind to reason inside a universe that was full of hidden order that could be described by algebra - if it was an observer surrounded by, like, piles of fruit containing twice as many cherries as apples, that sort of thing, it was just how that world worked - then you could imagine building that mind with rules like, 'If I believe an equation, I should also believe that equation with both sides multiplied by the same quantity' or 'If I believe an equation, I can believe that equation with both sides divided by the same quantity, so long as I already believe that quantity isn't zero.'  I say this to introduce a new topic: the concept of hidden order within the rules of reasoning themselves.  There are hidden patterns and deep explanations to be found in this subject matter, as, in my world, there was a reason why snowflakes had sixfold symmetry."

"As a very simple example, the rule 'You can divide by nonzero quantities' can be seen as a pure special case of 'You can multiply by any quantity.'  To say you can divide both sides by 2 is the same as saying you can multiply both sides by 1/2.  The reason you can't divide both sides by zero is that zero is the only continuous quantity which lacks an inverse.  Once you see things from that angle, in fact, you might say that it's a simpler viewpoint to say that there's just one rule to use there, about valid inference in algebra: the rule that you can multiply both sides by any quantity.  Say just that, and you don't need that darned rule with the extra complication about 'Oh well you can divide by anything unless it might be zero.'  You just have the rule that you can multiply by anything, and the rule that everything except zero has an inverse.  You could also add the rule about division, nothing invalid would happen to you if you did, but it would be redundant; the mind you were constructing could reach the same conclusions either way.  Through perceiving hidden order in the rules of reasoning, you would be able to simplify the mind's thought processes and arrive to the same ends - though it might also take longer to reason that way, it might take extra steps if you eliminated the extra rule."

"But meanwhile, back in the real world, we deal more with the equivalent of triangles and red things than the equivalent of numbers and addition.  I mean, this world has both, but still, let's go back to shapes and colors and sizes.  What sort of truth-preserving rules analogous to 'you can multiply both sides by any quantity' in algebra, might we use to combine beliefs like these?"

Z.  All triangular things are red.
H.  All red things are large.

"When it comes to algebra over continuous quantities," Keltham says, gesturing at the tactics written between the steps of the equations, "we have rules like being allowed to multiply both sides by the same quantity, or divide both sides by the same quantity so long as it isn't zero.  If you imagine building a mind to reason inside a universe that was full of hidden order that could be described by algebra - if it was an observer surrounded by, like, piles of fruit containing twice as many cherries as apples, that sort of thing, it was just how that world worked - then you could imagine building that mind with rules like, 'If I believe an equation, I should also believe that equation with both sides multiplied by the same quantity' or 'If I believe an equation, I can believe that equation with both sides divided by the same quantity, so long as I already believe that quantity isn't zero.'  I say this to introduce a new topic: the concept of hidden order within the rules of reasoning themselves.  There are hidden patterns and deep explanations to be found in this subject matter, as, in my world, there was a reason why snowflakes had sixfold symmetry."

"As a very simple example, the rule 'You can divide by nonzero quantities' can be seen as a pure special case of 'You can multiply by any quantity.'  To say you can divide both sides by 2 is the same as saying you can multiply both sides by 1/2.  The reason you can't divide both sides by zero is that zero is the only continuous quantity which lacks an inverse.  Once you see things from that angle, in fact, you might say that it's a simpler viewpoint to say that there's just one rule to use there, about valid inference in algebra: the rule that you can multiply both sides by any quantity.  Say just that, and you don't need that darned rule with the extra complication about 'Oh well you can divide by anything unless it might be zero.'  You just have the rule that you can multiply by anything, and the rule that everything except zero has an inverse.  You could also add the rule about division, nothing invalid would happen to you if you did, but it would be redundant; the mind you were constructing could reach the same conclusions either way.  Through perceiving hidden order in the rules of reasoning, you would be able to simplify the mind's thought processes and arrive to the same ends - though it might also take longer to reason that way, it might take extra steps if you eliminated the extra rule."

"But meanwhile, back in the real world, we deal more with the equivalent of triangles and red things than the equivalent of numbers and addition.  I mean, this world has both, but still, let's go back to shapes and colors and sizes.  What sort of truth-preserving rules analogous to 'you can multiply both sides by any quantity' in algebra, might we use to combine beliefs like these?"

Z.  All triangular things are red.
H.  All red things are large.