Can Turing Machines Understand Infinity?

• I
I had a conversation with someone once upon a time (it was quite a while back actually), and we came to the question of whether or not Turing machines could ever understand infinity. We agreed that we as humans are intimate with the extant and divisible infinities mainly through our sensory-perception, and more generally, we are able to grasp at analytic continuation. Although, computers work in modulo math and can only count up to the number of bits of information they can hold. Any opinions? Has there ever been any literature on this?

mfb
Mentor
What does "understand infinity" mean? Which output to which input corresponds to an understanding?

As Turing machines can in principle simulate a human brain: Do we "understand infinity"? If yes Turing machines can do so as well.

jedishrfu
Mentor
In my view to understand infinity means to know when to stop because looping forever would serve no purpose.

The problem with computing machines is that they can’t always know when to halt and thus are doomed to keep working a problem forever. It’s known as the halting problem and Turing proved that there is no general algorithm that can determine if a given program will halt or not.

https://en.m.wikipedia.org/wiki/Halting_problem

In any event, all halting problem answers are best summed up as 42, he said humorously. :-)

Janosh89
undecidable = irresolute?? Or does the latter have other connotations?

stevendaryl
Staff Emeritus
I had a conversation with someone once upon a time (it was quite a while back actually), and we came to the question of whether or not Turing machines could ever understand infinity. We agreed that we as humans are intimate with the extant and divisible infinities mainly through our sensory-perception, and more generally, we are able to grasp at analytic continuation. Although, computers work in modulo math and can only count up to the number of bits of information they can hold. Any opinions? Has there ever been any literature on this?

Well, you don't need infinite resources in order to reason about infinite objects. We can prove things about the natural numbers and the reals, etc., using finitely many axioms.

Gold Member
we as humans are intimate with the extant and divisible infinities mainly through our sensory-perception
Ah? How do our senses record infinities? When is the last time you saw, smelled, touched, tasted, or heard an inaccessible cardinal?

stevendaryl
Staff Emeritus
In my view to understand infinity means to know when to stop because looping forever would serve no purpose.

The problem with computing machines is that they can’t always know when to halt and thus are doomed to keep working a problem forever. It’s known as the halting problem and Turing proved that there is no general algorithm that can determine if a given program will halt or not.

But humans aren't any better. For problems like the Riemann hypothesis or Goldbach's conjecture, they may never know whether eventually they will prove them or refute them, or neither. They never know when to give up. Except when they get hungry or bored.

jedishrfu
Mentor
Ah? How do our senses record infinities? When is the last time you saw, smelled, touched, tasted, or heard an inaccessible cardinal?

We had a few in my yard. We could see them feeding but we couldn't touch as they were too skittish and flew away at the slightest sound.

jedishrfu
Mentor
But humans aren't any better. For problems like the Riemann hypothesis or Goldbach's conjecture, they may never know whether eventually they will prove them or refute them, or neither. They never know when to give up. Except when they get hungry or bored.

This is all true but sometimes we get lucky and discover Godel who's shown us that there are always undecidable statements in any system of logic and knowing that will give us pause. Also we can get tired or die from exhaustion and infinity becomes more infinite while others laugh at our folly.

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Gold Member
This is all true but sometimes we get lucky and discover Godel who's shown us that there are always undecidable statements in any system of logic.
First, far from "any" system of logic, but that is nit-picking. The more important point is that Gödel (Turing, Rosser, etc.) pointed out how to generate a few undecidables, and a few specific examples were pointed out (Cohen and Gödel, Matiyasevich et al, etc.), but unless a given problem fits into this list (modulo isomorphism or as a problem requiring the solution of an undecidable), or has been proven or disproven, or is part of a system in which all problems are decidable, there is no general procedure to tell if a given problem is undecidable. So we are back to square one, with machines and humans theoretically having the same problems.

jedishrfu
Mentor
First, far from "any" system of logic, but that is nit-picking. The more important point is that Gödel (Turing, Rosser, etc.) pointed out how to generate a few undecidables, and a few specific examples were pointed out (Cohen and Gödel, Matiyasevich et al, etc.), but unless a given problem fits into this list (modulo isomorphism or as a problem requiring the solution of an undecidable), or has been proven or disproven, or is part of a system in which all problems are decidable, there is no general procedure to tell if a given problem is undecidable. So we are back to square one, with machines and humans theoretically having the same problems.

You are back to square one, I am perfectly fine running in my hamster cage at work. :-)

Gold Member
I am perfectly fine running in my hamster cage at work.
Philip Dick wrote a short story "The Infinities" in which hamsters evolve into pure energy and zap a meanie human. Maybe you are on your way

jedishrfu
jedishrfu
Mentor
Philip Dick wrote a short story "The Infinities" in which hamsters evolve into pure energy and zap a meanie human. Maybe you are on your way

No, more likely to be attacked by the hamsters for maligning them and their wheel here.

PeroK