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Luke_A_Somers comments on Consistent extrapolated beliefs about math? - Less Wrong
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Marbles exist.
I believe we can apply (at least certain formulations of) integers as elements in models about marbles such as questions about how many marbles fulfill some sharp-line criterion (such as being in a particular urn).
If a formulation of Integers is inconsistent, then that formulation is not a good model for marbles.
If a formulation of integers does not allow you to say things about marbles that you want to be able to say, then that formulation is not powerful enough.
Generally there's a belief that there an infinitive amount of integers but we don't believe that there something like an infinitive amount of marbles.
Okay. If I want to know something about marbles and someone begins invoking infinities, that raises a yellow flag for me.
If I want to know about the asymptotic behavior of an algorithm, then no such flag is raised.
I think you meant to say "We don't believe that there are infinite different amounts of marbles", since the amount of marbles in an infinite amount of marbles isn't even described by an integer.
So if I want to describe filling a vase with a marble being cut in half when a n unsplit would fall off I have to come up with a more powerfull formulation of integers that allows fractions? That can't be it because fractions are no longer a model of integers.
No, you just broke your model by doing something not covered by it.
But it covers only what it is able to say. Thus any attempt to be more expressive breaks it.
edit: actually the theory works just fine. It isn't even broken but it is a different theory. If I would had said that this was a theory of "amounts" this would have been clearly progress that should be welcomed. But what if in my pretheoretic sense I equivocate "integers" and "amounts" (as could be assumed if I can't fraction). Thus when wanting a better theory it's ambigous whether I want or don't want it to cover that kind of scenario.
Exactly. If it acts like integers, then use integers. The above example, you tried to use integers despite the underlying phenomenon not acting like integers. That broke it.
The question is more about your beliefs about marbles, and what would happen if you tried to extrapolate those beliefs all the way. My argument is that some parts of Platonism wouldn't survive such extrapolation.