00:00:19.000 Aside from the physical limitation imposed by science,
00:00:23.000 Peter is limited in its memory and process and speed, for example.
00:00:27.000 Are there theoretical limits imposed by, say, mathematics?
00:00:36.000 The quantum physicist and philosopher David Deutsch
00:00:43.000 Imagine a super powerful computer of the future.
00:00:46.000 Imagine further that this computer is to be used as a virtual reality
00:00:57.000 At the present point in time, virtual reality is still very experimental.
00:01:02.000 Around a decade or so ago, it was all the rage,
00:01:05.000 and it seemed that soon we would be able to don a helmet
00:01:08.000 and be suddenly transported to a computer-generated world,
00:01:14.000 Because the graphics would be so clear, the sound so crisp,
00:01:18.000 and other sensations so vivid as to completely fool our senses
00:01:22.000 into believing that we really were somewhere else.
00:01:25.000 This would apparently make holidaying far cheaper.
00:01:28.000 Come on, boffons. I want my virtual trip to Argentina.
00:01:31.000 Of course, the reality of virtual reality hasn't yet lived up to the promise.
00:01:47.000 virtual reality machines could be effectively unlimited in their ability to render
00:01:54.000 There is a distinction to be made here between what is physically possible
00:02:01.000 Something which is physically possible is something that is permitted to occur by the laws of physics.
00:02:06.000 That is, something which does not require you to say
00:02:09.000 use up all of the available energy in the universe,
00:02:18.000 On the other hand, something which is logically possible,
00:02:21.000 like say, running faster than light, is certainly logically possible
00:02:28.000 Now, all physically possible environments can be programmed into a virtual reality generator.
00:02:33.000 So, for example, we could program a sufficiently powerful computer
00:02:37.000 to simulate what it would be like to have a civilization on Mars,
00:02:43.000 These physically possible scenarios would be straightforward to dial into a
00:02:47.000 supercomputer acting as a virtual reality machine.
00:02:50.000 They don't, after all, transgress any laws of physics.
00:02:54.000 But what about all logically possible environments?
00:03:01.000 This argument was first put forward by Geon Cantor,
00:03:08.000 and their proofs about the limits of mathematics and computation.
00:03:11.000 The argument used is known as a diagonal argument,
00:03:18.000 Imagine you have a computer from the far distant future.
00:03:21.000 It's as powerful as you like, with almost unlimited memory
00:03:30.000 It can render environments that are finite in terms of the space and time
00:03:37.000 A representation of infinite space by computer is a fairly straightforward process,
00:03:42.000 And if you play some computer games, you will know what I mean.
00:03:45.000 You can walk away from the action and just keep going off into the blue yonder.
00:03:49.000 One neat only program is simulation with what is effectively a loop of space.
00:04:00.000 What are the limitations in rendering virtual reality environments,
00:04:04.000 even given the most powerful supercomputer of the future?
00:04:07.000 Are there environments that it cannot even in principle hope to create?
00:04:13.000 Imagine that the number of logically possible environments is infinite,
00:04:20.000 You imagine that our supercomputer can render an infinite set of logically possible environments.
00:04:25.000 You imagine further that these possible environments are written down in a list,
00:04:52.000 You get the picture, add in for an item, covering all the possible environments
00:05:02.000 Environment one, environment two, environment three, and so on,
00:05:05.000 of all the possible environments that our virtual reality simulator can generate.
00:05:13.000 there are still environments and infinity of them actually that the computer cannot render.
00:05:24.000 For the first five minutes, it's different to environment one.
00:05:35.000 And for the second five minutes, it's different to environment two.
00:05:45.000 And for the third five minutes, you guessed it's different to environment three.
00:05:56.000 Now because it's different to environment one for the first five minutes,
00:06:02.000 Because it's different to environment two, it cannot be environment two.
00:06:07.000 For similar reasons, it cannot be environment three, four, or five, or any environment now list.
00:06:15.000 And yet we created our list by saying that it not only contained the set of all
00:06:21.000 environments that could be rendered by the computer, but also that the list was infinite.
00:06:26.000 Despite the list being infinite, it cannot contain everything logically possible.
00:06:36.000 This diagonal argument establishes that there are limits to computation,
00:06:41.000 that there are certain things a computer cannot do.
00:06:44.000 In this case, in terms of creating virtual reality environments,
00:06:48.000 there's certain environments that it can't actually recreate even if they're logically possible.
00:06:53.000 But more importantly, diagonal arguments in general illustrate limitations of mathematics and computation more generally.
00:07:00.000 It also suggests that certain kinds of infinity are bigger than others.
00:07:05.000 In this case, the infinite set of all environments that can be rendered by a supercomputer,
00:07:10.000 you're certainly smaller than the infinite set of all environments that it can render plus the ones it cannot.
00:07:16.000 I hope you've enjoyed this topcast episode, which is part of the series on the nature of mathematics and computation.
00:07:28.000 Try downloading some of the others from the website, and I'll see you again soon.
