“Can we really say that world-states and mathematical formalism are totally independent? I would argue otherwise.” I understand that we can formulate propositions that involve both real world states and mathematical formulations, but there generally needs to be some kind of bridge, ie. “let’s assume that the moon is size X with mass Y around the earth with size A and mass B in a perfect circle”.
The notions of truth involved in basic real world statements and mathematical expressions appear to be quite different. The truth of real world statements seems to be defined in a kind of objective manner in that the world corresponds to this state or it doesn’t (even if we don’t know what the state is). Mathematical statements seem to be constructed as they can only ever said to be true within a particular system or not.
“In other words, our computational abilities are limited in this way precisely because of the liar’s paradox”—the Liar paradox and the proof of the halting problem both use self-reference, but that really isn’t grounds to say that the halting problem is due to the Liar paradox.
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.)
“Can we really say that world-states and mathematical formalism are totally independent? I would argue otherwise.” I understand that we can formulate propositions that involve both real world states and mathematical formulations, but there generally needs to be some kind of bridge, ie. “let’s assume that the moon is size X with mass Y around the earth with size A and mass B in a perfect circle”.
The notions of truth involved in basic real world statements and mathematical expressions appear to be quite different. The truth of real world statements seems to be defined in a kind of objective manner in that the world corresponds to this state or it doesn’t (even if we don’t know what the state is). Mathematical statements seem to be constructed as they can only ever said to be true within a particular system or not.
“In other words, our computational abilities are limited in this way precisely because of the liar’s paradox”—the Liar paradox and the proof of the halting problem both use self-reference, but that really isn’t grounds to say that the halting problem is due to the Liar paradox.
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.