The story is supposed to be the icing, not the cake.
If I read an elaborate story that describes something that is apparently mathematically absurd (big integer + 1 = a small decimal) then I expect the post to go on to illustrate or explain some counter-intuitive truth or at least a speculative mathematical theory.
Problem is, everything collapses with one contradiction. So rigorously, there is nothing more to tell.
Now you can conceive of some sort of world in which the truths of mathematics are empirical, contingent and changeable, for instance, so that one contradiction is not much of a biggie. That would be quite fun, but, alas, I don’t have time for much fiction nowadays. Maybe someone else could try?
There are systems (relevant logic, for example) which do not collapse under one contradiction—to some extent, the fragility of classical logic is due to very strong assumptions that were built into it about how powerful math will turn out to be before Godel’s incompleteness and other undecidability results were discovered (and, at least in pop math like you and I are familiar with, they’re still not really fully digested).
Charlie Stross has also written post-arithmetic-consistency sf: “Dark Integers”. I’m a little surprised that Ted Chiang’s story didn’t contain any attempts to build devices to exploit the inconsistency.
The story is supposed to be the icing, not the cake.
If I read an elaborate story that describes something that is apparently mathematically absurd (big integer + 1 = a small decimal) then I expect the post to go on to illustrate or explain some counter-intuitive truth or at least a speculative mathematical theory.
Problem is, everything collapses with one contradiction. So rigorously, there is nothing more to tell.
Now you can conceive of some sort of world in which the truths of mathematics are empirical, contingent and changeable, for instance, so that one contradiction is not much of a biggie. That would be quite fun, but, alas, I don’t have time for much fiction nowadays. Maybe someone else could try?
There are systems (relevant logic, for example) which do not collapse under one contradiction—to some extent, the fragility of classical logic is due to very strong assumptions that were built into it about how powerful math will turn out to be before Godel’s incompleteness and other undecidability results were discovered (and, at least in pop math like you and I are familiar with, they’re still not really fully digested).
Charlie Stross has also written post-arithmetic-consistency sf: “Dark Integers”. I’m a little surprised that Ted Chiang’s story didn’t contain any attempts to build devices to exploit the inconsistency.
It’s a Greg Egan story actually.
You’re right, I apologize.
And it’s a sequel to “Luminous”.