The people who, when shown something like the surreal numbers, exclaim "Wow! I can't believe numbers work like that!"
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Replying to @CurlOfGradient
What kind of evidence should we use to know which structures are real?
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Replying to @ObjectOfObjects @CurlOfGradient
If by "real" you mean "corresponding to something observable," then none of these systems are real.
#Ultrafinitism (Or perhaps more specifically,#Actualism)2 replies 0 retweets 1 like -
Replying to @davidmanheim @ObjectOfObjects
Even if our universe were finite, the real numbers are a perfectly coherent system that one can make true or false statements about. I don’t get ultrafinitists
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Replying to @CurlOfGradient @ObjectOfObjects
It's not that it's an incoherent system, it's that there is nothing that it maps to in reality. And making the mistake of thinking that it is meaningful in practice leads to lots of problems. So we should be careful not to claim that the "real" numbers are more than a toy.
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Replying to @davidmanheim @ObjectOfObjects
As a physicist I use the real numbers to describe reality all the time. You can't do physics without them.
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Replying to @CurlOfGradient @ObjectOfObjects
Yes, you use a convenient mapping of states to the real number system. But accepted physics implies there literally cannot be evidence that there are arbitrary precision numbers for position/velocity, much less infinite information density, as the real numbers would imply.
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Replying to @davidmanheim @ObjectOfObjects
Just because our measurements of reality can't be real numbers doesn't mean the state of the universe can't involve them. And even if we discover the universe is a complicated cellular automaton, current physics is so accurate we'll go on using it for most purposes anyway.
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So there *is* something that the real numbers map to in reality; our current model of physics, which is absurdly accurate and useful even if it turns out to be wrong. I don't see the "but it's not the base level" argument as having a point for the "reality" of real numbers.
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Replying to @CurlOfGradient @ObjectOfObjects
You're arguing that because X allows Y, where Y is known to be a simplified model that doesn't capture everything, therefore X is real? Yes, "real" numbers are conceptually useful and consistent. That doesn't imply they exist in the territory, it means they help with mapmaking.
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No numbers exist in the territory. If the "base" level of reality is fully described by integer mathematics, that doesn't mean integers are "real" and all the rest of the number systems are "not real". I think we have very different definitions of "real" when it comes to math.
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Replying to @CurlOfGradient @ObjectOfObjects
Yes, I think that is a large part of the problem. But, to get back to why I made the claim originally, I think that eliminating the assumption that infinities exist solves lots more problems than it creates.
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Replying to @davidmanheim @ObjectOfObjects
I'm confused. What do you mean by assuming infinities "exist"?
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