To paraphrase the proof of the halting problem in dangerously abbreviated common English: We can program a machine such that ‘Machine X halts iff Machine X never halts.’ Does the liar’s paradox contain any properties of interest that are not also contained in this proof? A reasonably complex formal language should be able to (and indeed, does) capture this paradox just as easily as a translation to Spanish or Mandarin.
The sequence, How to Convince Me That 2+2=3 may be germane to this discussion as well, for its discussion of belief-in-arithmetic as an empirical process. It is true enough that you can create mathematical models that do not map easily on to our material experiences, i.e. certain non-Euclidian geometries. But at the same time, it’s trivially obvious that our experiences are structured in predictable ways. When we prove things about the limits of what structure is, in otherwise abstract mathematical ways, we have also produced useful predictions about the behavior of the material world.
(Incidentally, quotes are a bit more readable if you use the “>” notation, demonstrated under the ‘show help’ menu.)
To paraphrase the proof of the halting problem in dangerously abbreviated common English: We can program a machine such that ‘Machine X halts iff Machine X never halts.’ Does the liar’s paradox contain any properties of interest that are not also contained in this proof? A reasonably complex formal language should be able to (and indeed, does) capture this paradox just as easily as a translation to Spanish or Mandarin.
The sequence, How to Convince Me That 2+2=3 may be germane to this discussion as well, for its discussion of belief-in-arithmetic as an empirical process. It is true enough that you can create mathematical models that do not map easily on to our material experiences, i.e. certain non-Euclidian geometries. But at the same time, it’s trivially obvious that our experiences are structured in predictable ways. When we prove things about the limits of what structure is, in otherwise abstract mathematical ways, we have also produced useful predictions about the behavior of the material world.
(Incidentally, quotes are a bit more readable if you use the “>” notation, demonstrated under the ‘show help’ menu.)
“We can program a machine such that ‘Machine X halts iff Machine X never halts.’”—good point. I can see that now.