3 ms·

Not that big of a mountain, for a thing to exist it has to be implementable, and only computable things are implementable or else they would not “fit” into a un

Not that big of a mountain, for a thing to exist it has to be implementable, and only computable things are implementable or else they would not “fit” into a universe.

User23 3y ago
And yet we know that most* functions aren't computable. Are we saying most of mathematics doesn't exist? If that's true then how can we talk about it? I'm asking these questions rhetorically, but they're serious questions that need to be answered (or at least attempted) to maintain even a pretense of intellectual coherency. You certainly can make the constructivist argument that only the rationals exist and real numbers and everything that builds on them is some kind of fever dream, but personally I've never seen any even remotely compelling exposition of that position. Maybe that's just a "me problem" though? I really don't know I find that this kind of metaphysic pushes up to and sometimes past my cognitive ability.
- fellowmartian 3y ago
  Constructivism seems obviously* true to me, and these Aleph arithmetics do in fact seem like a fever dream, but I’m not qualified enough to provide the exposition you’re asking for. To me real numbers exist, but only as approximation functions bounded by available memory and compute, and not as members of an uncountably infinite set in some platonic meta reality. Check out the works and interviews of Joscha Bach if you haven’t already, he’s influenced my thinking on this quite a bit. *obviously not in the dismissive sense, but in the sense as “we hold these truths to be self-evident”
  - meroes 3y ago
    We don’t even get to “the reals” before this issue crops up though. Irrational numbers were rigorously studied before infinity and the reals, as far back as 500 BC. I’m not a computer scientist so I may have gap here, but demonstrating two quantities are incommensurable (showing no unit makes up two quantities m and n times, m and n being integers), does not seem like something possible to approximate empirically or computationally in many cases. The precision required may be one step beyond your current capacity. Constructivism may be right. But I don’t have a good argument for why finished computation or empirical approximation (there’s always limits to measurement) is the be all end all. Unless we take them to be the final adjudicators, why shouldn’t there be incommensurable quantities? We need very strong arguments they provide the final say, but we know they have limits, their capacity/memory.

Not that big of a mountain, for a thing to exist it has to be implementable, and only computable things are implementable or else they would not “fit” into a universe.

User23 3y ago

And yet we know that most* functions aren't computable. Are we saying most of mathematics doesn't exist? If that's true then how can we talk about it? I'm asking these questions rhetorically, but they're serious questions that need to be answered (or at least attempted) to maintain even a pretense of intellectual coherency. You certainly can make the constructivist argument that only the rationals exist and real numbers and everything that builds on them is some kind of fever dream, but personally I've never seen any even remotely compelling exposition of that position. Maybe that's just a "me problem" though? I really don't know I find that this kind of metaphysic pushes up to and sometimes past my cognitive ability.

fellowmartian 3y ago
Constructivism seems obviously* true to me, and these Aleph arithmetics do in fact seem like a fever dream, but I’m not qualified enough to provide the exposition you’re asking for. To me real numbers exist, but only as approximation functions bounded by available memory and compute, and not as members of an uncountably infinite set in some platonic meta reality. Check out the works and interviews of Joscha Bach if you haven’t already, he’s influenced my thinking on this quite a bit. *obviously not in the dismissive sense, but in the sense as “we hold these truths to be self-evident”
- meroes 3y ago
  We don’t even get to “the reals” before this issue crops up though. Irrational numbers were rigorously studied before infinity and the reals, as far back as 500 BC. I’m not a computer scientist so I may have gap here, but demonstrating two quantities are incommensurable (showing no unit makes up two quantities m and n times, m and n being integers), does not seem like something possible to approximate empirically or computationally in many cases. The precision required may be one step beyond your current capacity. Constructivism may be right. But I don’t have a good argument for why finished computation or empirical approximation (there’s always limits to measurement) is the be all end all. Unless we take them to be the final adjudicators, why shouldn’t there be incommensurable quantities? We need very strong arguments they provide the final say, but we know they have limits, their capacity/memory.