On Philosophy

June 4, 2007

The Speed of Thought

Filed under: Mind — Peter @ 12:00 am

I claim that there is a theoretical upper bound on how fast a system can think (processes information, have conscious experiences, whatever) that is proportional to its size. Meaning that the larger a system is the slower it will think, assuming that it is thinking as fast as it possibly can. Now some people familiar with computers and their construction may be nodding their heads at this point (making a processor larger generally requires decreasing its clock speed). If you are one of those people you might just want to skip to the final paragraph.

So why is there an upper bound on the speed of thought? Well there is an upper bound on how fast information can pass from one part of the system to another, that being the speed of light. So consider the system below. The ellipse labeled A represents the system in some initial state and B represents the state of the system after a single step in its thought process incorperating the initial state A has been carried out.

Obviously in the above set up this single thought step cannot occur faster than d (the diameter of the system) divided by the speed of light. Of course this is a bit of an idealization. It is possible that the thinking done by this system is reflected in only some part of what I have labeled B, with some parts of B being completely unaffected by the state at A. Certainly this is possible, but consider what it means to say that the whole system is a thinking system, it means that every part of the system (or at least most of them) are involved in thinking. So suppose that this part of B remains unconnected in its processing from the rest of the system, but interacts only with itself. Thus this part could think faster then d / c. However it would then be proper to say that it was this part of the system that is the thinking system, and not the system that we initially thought was. And of course this smaller part will have some diameter, d2, and it will thus have its own upper bound of d2 / c. Or, on the other hand, it may be that the state of this smaller part of B is later has an influence on the whole state of the system. But in that case the time it takes for the whole system to complete a single step in its thought process is greater than d / c. So our upper bound holds (upper bound in regards to thinking speed; d / c is a lower bound when it comes to the time elapsed for a single thought step).

This is also an idealization in another way. In most thinking systems of the biological kind it doesn’t look like the whole brain processes the previous state of the whole brain. Rather the processing starts at one location in the brain and then moves along to a different location, until eventually the whole brain (or most of it) has become involved in thinking. Such a situation is illustrated below:

In this illustration k is the average distance from one active area to another, d is the average diameter of each active area, and e is the diameter of the entire system. I won’t go into the mathematical details, but a little thought will show that the time it takes for the entire system to have a thought is greater than d / c, our upper bound. (This is because it takes extra time for information to travel from one active area to another since they are spatially separated.)

Now exactly what a single thought step corresponds to is hard to say. Does a conscious experience take one thought step to occur or more than one? But exactly what they correspond to doesn’t really matter since we know that whatever they are they take longer and longer to occur the larger the system is. Which means that a system which thinks at some particular speed will think at half that speed when it doubles in size, since its thought steps are taking twice as long, and the speed at which it thinks is proportional to how long it takes for a thought step to occur.

Of course this doesn’t have very many practical applications. But it does have a rhetorical application. Specifically against those who say that the universe has a mind of its own, or is the mind of god. Now the observable universe has a diameter of approximately 100 billion light years (note: the whole universe is probably larger in size, but how much larger is hard to say). Which means that if the universe were a mind a single thought step would take approximately 100 billion years, if the universe was thinking as fast as it possibly could. Of course the universe was smaller at the beginning, but even when it was one second old it was at least 8 light years in diameter, making the time it would take to complete a single thought step 8 years.* So if the universe is the mind of god then god is a really slow thinker.

* Technical note: actually as long as the universe was expanding faster than the speed of light it would actually be impossible for it to complete a single thought step. At this point in the universe’s history at best local regions of the universe could be thinking.

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5 Comments

  1. Also, those like me who say that the universe ought to have a mind of its own, if materialism is true, and that therefore it is probably false. Of course we are also fantastically slow thinkers… I’m thinking of the rapid reactions of insects and small mammals… But nonetheless we pride ourselves on the quality of our thoughts (their breadth and depth). We can after all destroy all of those animals and ourselves if we want. A God is a God for all that it might have done a lot of amazing thinking a long time ago, and be now only thinking a few inconceivably amazing things, as we potter about trying not to destroy everything by mistake!

    Comment by Enigman — June 4, 2007 @ 8:09 am

  2. If materialism is true it implies that the universe almost certainly does not have a mind, because the parts of the universe are not causally connected in the right kind of way to pass along and process information.

    Comment by Peter — June 4, 2007 @ 8:26 am

  3. Yes, sorry, I meant, not materialism, but just the thought that there are no souls, that minds arise from matter, etc. (my terminology is really sloppy). But why not causally connected in the right way, though? E.g. neutrinos and such must have wavefunctions that are spread out over huge distances. And some photons too (not to mention dark matter, or gravitational waves)… As for processing, could that not piggy-back upon local sapients, in a sort of modular way? Or in some other way?

    Comment by Enigman — June 4, 2007 @ 8:44 am

  4. Sure photons go everywhere, but the processes that create them are essentially random, except in the case of humanity, which produces structured radio waves. And nothing in the universe recieves and processes photons in such a way as to think using them, photons are simply converted to heat upon impact.

    Comment by Peter — June 4, 2007 @ 8:49 am

  5. Related point: there is also a theoretical upper limit on the information that can be contained within a certain volume of space containing a certain amount of energy/matter – the so called Beckenstein bound. And there is also a limit of an object to perform computation in terms of operations per second (“amount of thinking”) , that is equal – for a kilogram of matter – to 2PI/Planck’s constant ~ 5 * 10^50 ops. Just in case you haven’t heard of these things. :)

    Oh, and the light-speed limitation of communication betwen varius parts of a mind you mentioned also presents a problem for designers of future artificial inteligences. Some day AIs will probably grow so vast that they’ll span several light years across… how can we assure, that all their parts will act in a coherent, beneficial ways!? Well, this is one of the subproblems of Friendly AI (http://www.singinst.org/)… that actualy isn’t so pressing right now. :)

    Oh well, back to work – studying for exams…

    Comment by nicnevem — June 4, 2007 @ 1:07 pm


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