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> Math is beyond computation. We are talking about the same thing, but I would actually put this the other way around. Computation and computability is "the f

by gf000 8d ago

> Math is beyond computation. We are talking about the same thing, but I would actually put this the other way around. Computation and computability is "the final frontier". Math is a "subset" of that. Doesn't matter if we choose ZFC or in the future discover some "better" subset of core axioms, we will always hit limits where BB will trivially skip over whatever we could prove (let alone Gödel's theorems). > given fixed input of the statement with no other information sources Also, this is just trivially avoidable, so not sure if we really should be concerned about this limitation. An LLM in a loop where it can write on a tape can be Turing complete, ergo it can compute anything computable and is "bigger" than math at that point.

magicalist 8d ago
> Computation and computability is "the final frontier". Math is a "subset" of that. Maybe I'm misunderstanding you point, but I don't know how widely this would be held as true. Are you defining "math" as _only_ what can be proven under some particular formal system?
- gf000 8d ago
  Well, I only know how to define computability in terms of Turing machines. For math I don't have a fix definition, but it's surely a bit more specific than that (e.g. I wouldn't consider the computation that prints a 0 at the same place for infinity math) - but of course I do see the circularity in my argument: a Turing machine is a mathematical object in and of itself. Though being able to talk about something doesn't necessarily change which is "bigger". As for the other direction, this gets a bit more into the philosophy behind math itself. Constructive math's territory is "easy" - but I am on the opinion that if humans (or any intelligent physical entity) are at most Turing-complete [1], then any non-constructive math "steps" or thoughts must also be at most computable. Well, unfortunately I can't prove whether math done by transcendent entities are also computable, though. In any case, I am no mathematician, so whatever I think regarding this topic may not have much relevance to anyone, only done CS course with quite a bit of math, but that's obviously not the same. [1] I believe religion is an escape hatch here from an argument perspective
  - streetfighter64 8d ago
    > if humans (or any intelligent physical entity) are at most Turing-complete This is a bit of a strange assumption to make. I do agree that a human, if it had infinite memory, would be an universal machine, i.e. capable of computing any given Turing machine [0]. But would that be the limits of its capabilities? It's far from certain. You'll get into the philosophy of free will (funnily enough, a sort of inverted Turing test), i.e. for a given human with infinite memory, is there a Turing machine that exactly replicates the behavior of that human? Is our behavior governed entirely by rules? Would that imply that a human themselves is a kind of Chinese room [1]? > any non-constructive math "steps" or thoughts must also be at most computable. What does it mean for a "thought" to be computable? Compare to Gödel's incompleteness theorem. Clearly the act of stating the thought, or writing down the theorem, is computable. But proving it to be true or false may very well be impossible. [0] https://en.wikipedia.org/wiki/Universal_Turing_machine https://en.wikipedia.org/wiki/Universal_Turing_machine [1] https://en.wikipedia.org/wiki/Chinese_room https://en.wikipedia.org/wiki/Chinese_room
    - qarl 8d ago
      > But would that be the limits of its capabilities? It's far from certain. Do you agree that humans are physical systems? My understanding is that any physical system can be evaluated to any degree of accuracy by a computer, no?
      - streetfighter64 8d ago
        > any physical system can be evaluated to any degree of accuracy by a computer That's an interesting hypothesis, but I don't know why you'd assume it to be true at face value. It's a bit unclear how you would even define "evaluated", given that we don't yet have a mathematical model of all of physics as we know it. [0] And then consider unknown unknowns. > Do you agree that humans are physical systems? Do you consider humans _with infinite memory_ as physical systems? Do you consider computers _with infinite memory_ as physical systems? [0] https://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Model https://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Mo...
streetfighter64 8d ago
> Computation and computability is "the final frontier". Math is a "subset" of that. In what sense? BB(n) is a prime example of an object that can be mathematically defined, yet is not computable. Or see BBB(n) for an "even more" uncomputable function. [0] > An LLM in a loop where it can write on a tape can be Turing complete What does this mean? A given LLM, like a given C program, can't really be Turing complete or not in a meaningful sense. The C programming language, or the concept of LLMs in general can be said to be Turning complete or not. Do you mean to state that LLMs in general are not Turing complete, but being "in a loop" somehow makes a difference? > it can compute anything computable and is "bigger" than math at that point Again, in what sense is it "bigger" than math? Lots of things are Turing complete, I wouldn't classify lambda calculus as "bigger" than math. [0] https://wiki.bbchallenge.org/wiki/Beeping_Busy_Beaver https://wiki.bbchallenge.org/wiki/Beeping_Busy_Beaver
rsrsrs86 7d ago
You need to do some studying _without_ chat gpt if you like math.
- gf000 6d ago
  Care to give some explanation and correction then?

by gf000 8d ago

magicalist 8d ago

> Computation and computability is "the final frontier". Math is a "subset" of that. Maybe I'm misunderstanding you point, but I don't know how widely this would be held as true. Are you defining "math" as _only_ what can be proven under some particular formal system?

gf000 8d ago
Well, I only know how to define computability in terms of Turing machines. For math I don't have a fix definition, but it's surely a bit more specific than that (e.g. I wouldn't consider the computation that prints a 0 at the same place for infinity math) - but of course I do see the circularity in my argument: a Turing machine is a mathematical object in and of itself. Though being able to talk about something doesn't necessarily change which is "bigger". As for the other direction, this gets a bit more into the philosophy behind math itself. Constructive math's territory is "easy" - but I am on the opinion that if humans (or any intelligent physical entity) are at most Turing-complete [1], then any non-constructive math "steps" or thoughts must also be at most computable. Well, unfortunately I can't prove whether math done by transcendent entities are also computable, though. In any case, I am no mathematician, so whatever I think regarding this topic may not have much relevance to anyone, only done CS course with quite a bit of math, but that's obviously not the same. [1] I believe religion is an escape hatch here from an argument perspective
- streetfighter64 8d ago
  > if humans (or any intelligent physical entity) are at most Turing-complete This is a bit of a strange assumption to make. I do agree that a human, if it had infinite memory, would be an universal machine, i.e. capable of computing any given Turing machine [0]. But would that be the limits of its capabilities? It's far from certain. You'll get into the philosophy of free will (funnily enough, a sort of inverted Turing test), i.e. for a given human with infinite memory, is there a Turing machine that exactly replicates the behavior of that human? Is our behavior governed entirely by rules? Would that imply that a human themselves is a kind of Chinese room [1]? > any non-constructive math "steps" or thoughts must also be at most computable. What does it mean for a "thought" to be computable? Compare to Gödel's incompleteness theorem. Clearly the act of stating the thought, or writing down the theorem, is computable. But proving it to be true or false may very well be impossible. [0] https://en.wikipedia.org/wiki/Universal_Turing_machine https://en.wikipedia.org/wiki/Universal_Turing_machine [1] https://en.wikipedia.org/wiki/Chinese_room https://en.wikipedia.org/wiki/Chinese_room
  - qarl 8d ago
    > But would that be the limits of its capabilities? It's far from certain. Do you agree that humans are physical systems? My understanding is that any physical system can be evaluated to any degree of accuracy by a computer, no?
    - streetfighter64 8d ago
      > any physical system can be evaluated to any degree of accuracy by a computer That's an interesting hypothesis, but I don't know why you'd assume it to be true at face value. It's a bit unclear how you would even define "evaluated", given that we don't yet have a mathematical model of all of physics as we know it. [0] And then consider unknown unknowns. > Do you agree that humans are physical systems? Do you consider humans _with infinite memory_ as physical systems? Do you consider computers _with infinite memory_ as physical systems? [0] https://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Model https://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Mo...

streetfighter64 8d ago

> Computation and computability is "the final frontier". Math is a "subset" of that. In what sense? BB(n) is a prime example of an object that can be mathematically defined, yet is not computable. Or see BBB(n) for an "even more" uncomputable function. [0] > An LLM in a loop where it can write on a tape can be Turing complete What does this mean? A given LLM, like a given C program, can't really be Turing complete or not in a meaningful sense. The C programming language, or the concept of LLMs in general can be said to be Turning complete or not. Do you mean to state that LLMs in general are not Turing complete, but being "in a loop" somehow makes a difference? > it can compute anything computable and is "bigger" than math at that point Again, in what sense is it "bigger" than math? Lots of things are Turing complete, I wouldn't classify lambda calculus as "bigger" than math. [0] https://wiki.bbchallenge.org/wiki/Beeping_Busy_Beaver https://wiki.bbchallenge.org/wiki/Beeping_Busy_Beaver

rsrsrs86 7d ago

You need to do some studying _without_ chat gpt if you like math.

gf000 6d ago
Care to give some explanation and correction then?