5 ms·

How much math is knowable? [video]

115 points by diaphanous 2y ago

munchler 2y ago
This is an excellent talk by Scott Aaronson, although it's not something that can be digested as quickly as the typical HN post. The summary slide at the end is perhaps a good introduction to the scope of the topic: * In math, we are finite beings trying to apprehend the infinite * The Busy Beaver function actually quantifies that (!) * Even the finite can exceed the scope of the cosmos. That's where we need physics and complexity theory * Quantum computers look like they're already slightly expanding the scope of what mathematical statements we can know * Can we know even more than that? Depends what the ultimate laws of physics are
- jerf 2y ago
  "although it's not something that can be digested as quickly as the typical HN post" I really want a view of HN that is something like the upvotes divided by the number of comments (although, not necessarily linear). Those aren't necessarily the "best" by every metric, not saying this is the "best" view or anything, but it would be an interesting one.
  - gsf_emergency 2y ago
    Pretty sure dang (or other mods) has that view as a sidebar or something, maybe you can request that as an exclusive feature for top karmies. It would, as a bonus, already cover the edge cases you haven't thought about..
  - nh23423fefe 2y ago
    true. lots of comments is an indicator of low quality
- analog31 2y ago
  >>> In math, we are finite beings trying to apprehend the infinite Watched the entire lecture. Fascinating. An amusing aside is that this is the first time I've encountered that statement, where it wasn't just an empty assertion, but actually had some meat on the bones.
deleted 2y ago
[deleted]
calf 2y ago
What precisely is meant by simulating the laws of physics? There's a common understanding pitfall there, in that a lot of lay people quickly end up talking about "is the universe a computer" or "infinite Turing machines have nothing to do with the real world" sort of confusing arguments. I liked how this talk addressed the topic yet have questions about this talk's overarching argument. It's kind of an argument yet more like a philosophical position about CS--I know at least one paper that critiques the philosophical commitments made in something like the Church Turing thesis, etc., that paper called Aaronson the strong American style perspective or something, I forget, at any rate the issue is how transparent these philosophical premises are which those of us from computer science may take for granted. I also liked-disliked those one slide proofs. I couldn't follow them, I know they are right (and are great exercises for a student to go over offline) and that they are in service of showing that modern understanding can do the results in one slide and so forth, but yet there's an accepted style of academic presentation that kind of misleadingly "performs" in a way that is cognitively impossible for the audience to actually follow unless they already know it, technical talks are kind of a backwards lie in that way. There was a recent popular book by a French mathematician who said as much, that 99.9% of the time we can hardly understand mathematics talks and books and it's kind of an open secret how professionals deal with this phenomenon. Edit: I took undergrad CS theory but never "got" the exposure/fascination with Busy Beaver, but after this talk it became clearer to me that the reason for the Busy Beaver function is that it's a meta-function, it is a function about Turing machines, which is how the BBF gets the special properties/results that it does. Which immediately reminds me of Chaitin's constant which also tries to encode/talk about the properties of Turing machines, e.g. Kolmogorov style complexity which was not explicitly mentioned in the talk.
- tromp 2y ago
  > it is a function about Turing machines, which is how the BBF gets the special properties/results that it does. The properties of the Busy Beaver function carry over to any other computational model you could base it on. E.g. the lambda calculus based BBλ[1] shares most of its properties. > Which immediately reminds me of Chaitin's constant which also tries to encode/talk about the properties of Turing machines, Chaitin's work on Algorithmic Information Theory (aka Kolmogorov Complexity) was actually focussed on custom Lisp languages of his own design rather than Turing Machines. One could be confused though by some of his articles on LISP program-size complexity like [2] where he calls his LISP interpreter a UTM (Universal Turing Machine). [1] https://oeis.org/A33347 https://oeis.org/A33347 [2] https://www.jucs.org/jucs_2_5/the_limits_of_mathematics/Chaitin_G_J.html https://www.jucs.org/jucs_2_5/the_limits_of_mathematics/Chai...
meroes 2y ago
Hoping to watch the video soon. What's still a mystery to me is how Turing Machines (infinite, abstract, non-physical) are so useful for us finite beings. Similar could be said for mathematical infinity.
- feoren 2y ago
  The fact that Turing machines are abstract and infinite is a simplification. Turing machines are an attempt to create the simplest possible model of "computation", without compromising on what "computation" means. It's a way to prove statements about the nature of computation, that's all. A model of computation that is finite, concrete, and/or physical is less useful because it contains unnecessary details that are secondary to the nature of computation. If you're not trying to prove fundamental things about computation, you probably never need to care about Turing machines. Most cases where infinity is invoked are done to simplify things. Forget whether infinity exists in the real world or not: ask only whether it's useful. After all, does the number 17 really exist? All models are wrong; some models are useful.
  - meroes 2y ago
    It’s a simplification that all kinds of infinites are implicated by infinite models? To understand a finite computation, we (some amount of scientists and mathematicians) must understand at least ZFC? Why not pick 10^180 or something as the highest number/states then? Yes even natural numbers are infinite, so I don’t quite buy at face value it’s a simplification when we could cutoff these infinities because no computation will run 10^180 for attoseconds or have that many states. And I would think all the mysteries of ZFC for example being ousted would be simpler. I think for your simplification it may have been clearer to say abstraction.
    - feoren 2y ago
      > Why not pick 10^180 or something as the highest number/states then? Because then we have to find all sorts of limits to all of our proofs, and qualify everything, and waste time proving that our results are under this limit? If we prove something about an algorithm parameterized by some N, and we rely on a Turing Machine in our proof, then we'd also have to figure out which N hits our limit of 10^180 states on the tape, so we can qualify the limits of our proof. This could be very difficult, since the translation to a Turing machine model can be arbitrarily complex. Why bother? It doesn't help anyone. > To understand a finite computation, we (some amount of scientists and mathematicians) must understand at least ZFC? Turing machines are for proving things about the nature of computation itself. You seem to be imagining that you need to understand Turing machines before you'll be allowed to make statements about some specific computation. That's simply not true. If Turing machines help you prove something, use them; if not, ignore them. Putting an arbitrary limit on them only makes them worse at proving things. I'm not really sure why you're so focused on ZFC. On one hand, the vast majority of math proofs (and therefore CS proofs that are based on them) assume ZFC, usually without even bothering to mention it. On the other hand, the axiom of choice seems completely irrelevant for Turing machines. Although the tape is infinite, it's almost always implicitly assumed that the state is finite at every step (the rest of the tape is 0). You'd have to initialize the tape with some infinite pattern to get around this, and that's definitely out of the ordinary. It's probably better to think of the state as "arbitrarily large" rather than infinite. I suspect you could get away with assuming Peano arithmetic for most CS proofs anyone cares about. But also, yes, some number of scientists and mathematicians are expected to understand the basic axioms of math if they're going to prove mathematical theorems.

115 points by diaphanous 2y ago

This is an excellent talk by Scott Aaronson, although it's not something that can be digested as quickly as the typical HN post. The summary slide at the end is perhaps a good introduction to the scope of the topic: * In math, we are finite beings trying to apprehend the infinite * The Busy Beaver function actually quantifies that (!) * Even the finite can exceed the scope of the cosmos. That's where we need physics and complexity theory * Quantum computers look like they're already slightly expanding the scope of what mathematical statements we can know * Can we know even more than that? Depends what the ultimate laws of physics are

jerf 2y ago
"although it's not something that can be digested as quickly as the typical HN post" I really want a view of HN that is something like the upvotes divided by the number of comments (although, not necessarily linear). Those aren't necessarily the "best" by every metric, not saying this is the "best" view or anything, but it would be an interesting one.
- gsf_emergency 2y ago
  Pretty sure dang (or other mods) has that view as a sidebar or something, maybe you can request that as an exclusive feature for top karmies. It would, as a bonus, already cover the edge cases you haven't thought about..
- nh23423fefe 2y ago
  true. lots of comments is an indicator of low quality
analog31 2y ago
>>> In math, we are finite beings trying to apprehend the infinite Watched the entire lecture. Fascinating. An amusing aside is that this is the first time I've encountered that statement, where it wasn't just an empty assertion, but actually had some meat on the bones.

deleted 2y ago

[deleted]

calf 2y ago

What precisely is meant by simulating the laws of physics? There's a common understanding pitfall there, in that a lot of lay people quickly end up talking about "is the universe a computer" or "infinite Turing machines have nothing to do with the real world" sort of confusing arguments. I liked how this talk addressed the topic yet have questions about this talk's overarching argument. It's kind of an argument yet more like a philosophical position about CS--I know at least one paper that critiques the philosophical commitments made in something like the Church Turing thesis, etc., that paper called Aaronson the strong American style perspective or something, I forget, at any rate the issue is how transparent these philosophical premises are which those of us from computer science may take for granted. I also liked-disliked those one slide proofs. I couldn't follow them, I know they are right (and are great exercises for a student to go over offline) and that they are in service of showing that modern understanding can do the results in one slide and so forth, but yet there's an accepted style of academic presentation that kind of misleadingly "performs" in a way that is cognitively impossible for the audience to actually follow unless they already know it, technical talks are kind of a backwards lie in that way. There was a recent popular book by a French mathematician who said as much, that 99.9% of the time we can hardly understand mathematics talks and books and it's kind of an open secret how professionals deal with this phenomenon. Edit: I took undergrad CS theory but never "got" the exposure/fascination with Busy Beaver, but after this talk it became clearer to me that the reason for the Busy Beaver function is that it's a meta-function, it is a function about Turing machines, which is how the BBF gets the special properties/results that it does. Which immediately reminds me of Chaitin's constant which also tries to encode/talk about the properties of Turing machines, e.g. Kolmogorov style complexity which was not explicitly mentioned in the talk.

tromp 2y ago
> it is a function about Turing machines, which is how the BBF gets the special properties/results that it does. The properties of the Busy Beaver function carry over to any other computational model you could base it on. E.g. the lambda calculus based BBλ[1] shares most of its properties. > Which immediately reminds me of Chaitin's constant which also tries to encode/talk about the properties of Turing machines, Chaitin's work on Algorithmic Information Theory (aka Kolmogorov Complexity) was actually focussed on custom Lisp languages of his own design rather than Turing Machines. One could be confused though by some of his articles on LISP program-size complexity like [2] where he calls his LISP interpreter a UTM (Universal Turing Machine). [1] https://oeis.org/A33347 https://oeis.org/A33347 [2] https://www.jucs.org/jucs_2_5/the_limits_of_mathematics/Chaitin_G_J.html https://www.jucs.org/jucs_2_5/the_limits_of_mathematics/Chai...

meroes 2y ago

Hoping to watch the video soon. What's still a mystery to me is how Turing Machines (infinite, abstract, non-physical) are so useful for us finite beings. Similar could be said for mathematical infinity.

feoren 2y ago
The fact that Turing machines are abstract and infinite is a simplification. Turing machines are an attempt to create the simplest possible model of "computation", without compromising on what "computation" means. It's a way to prove statements about the nature of computation, that's all. A model of computation that is finite, concrete, and/or physical is less useful because it contains unnecessary details that are secondary to the nature of computation. If you're not trying to prove fundamental things about computation, you probably never need to care about Turing machines. Most cases where infinity is invoked are done to simplify things. Forget whether infinity exists in the real world or not: ask only whether it's useful. After all, does the number 17 really exist? All models are wrong; some models are useful.
- meroes 2y ago
  It’s a simplification that all kinds of infinites are implicated by infinite models? To understand a finite computation, we (some amount of scientists and mathematicians) must understand at least ZFC? Why not pick 10^180 or something as the highest number/states then? Yes even natural numbers are infinite, so I don’t quite buy at face value it’s a simplification when we could cutoff these infinities because no computation will run 10^180 for attoseconds or have that many states. And I would think all the mysteries of ZFC for example being ousted would be simpler. I think for your simplification it may have been clearer to say abstraction.
  - feoren 2y ago
    > Why not pick 10^180 or something as the highest number/states then? Because then we have to find all sorts of limits to all of our proofs, and qualify everything, and waste time proving that our results are under this limit? If we prove something about an algorithm parameterized by some N, and we rely on a Turing Machine in our proof, then we'd also have to figure out which N hits our limit of 10^180 states on the tape, so we can qualify the limits of our proof. This could be very difficult, since the translation to a Turing machine model can be arbitrarily complex. Why bother? It doesn't help anyone. > To understand a finite computation, we (some amount of scientists and mathematicians) must understand at least ZFC? Turing machines are for proving things about the nature of computation itself. You seem to be imagining that you need to understand Turing machines before you'll be allowed to make statements about some specific computation. That's simply not true. If Turing machines help you prove something, use them; if not, ignore them. Putting an arbitrary limit on them only makes them worse at proving things. I'm not really sure why you're so focused on ZFC. On one hand, the vast majority of math proofs (and therefore CS proofs that are based on them) assume ZFC, usually without even bothering to mention it. On the other hand, the axiom of choice seems completely irrelevant for Turing machines. Although the tape is infinite, it's almost always implicitly assumed that the state is finite at every step (the rest of the tape is 0). You'd have to initialize the tape with some infinite pattern to get around this, and that's definitely out of the ordinary. It's probably better to think of the state as "arbitrarily large" rather than infinite. I suspect you could get away with assuming Peano arithmetic for most CS proofs anyone cares about. But also, yes, some number of scientists and mathematicians are expected to understand the basic axioms of math if they're going to prove mathematical theorems.