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I encountered below statement by Alan M. Turing here:

"The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false."

I am not a native English speaker. Could anyone explain it in plain English?

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    $\begingroup$ perhaps, it's better suited for philosophy portal rather to hard science like CS $\endgroup$ – Bulat Apr 19 '19 at 9:43
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    $\begingroup$ @Bulat I was going to say the same -- and redirect to English Language Learners -- but then I realised that there is some CS-related content that can be explained in an answer, which probably wouldn't be picked up on, in other parts of Stack Exchange. $\endgroup$ – David Richerby Apr 19 '19 at 9:45
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    $\begingroup$ A good example is iteration of the transformation z := z² + c, where z and c are complex numbers. What happens if I take any starting point on the plane z and iterate, will the number go to infinity or not? An ordinary fellow would say, yeah, this will give you two regions or maybe a few more where the value goes to zero and for the rest it goes to infinity. Relatively unsurprising. Then Mandelbrot comes along and actually plots the regions on the the plane defined by this simple "machine". As the result comes out of the dotmatrix printer, this simple "machine" proves itself ... weird. $\endgroup$ – David Tonhofer Apr 19 '19 at 16:56
  • $\begingroup$ Facebook and other social media are a great example of this... A lot of the consequences of their algorithms are not something that was expected by the creators (or anyone really). $\endgroup$ – aslum Apr 19 '19 at 20:52
  • $\begingroup$ A rather quirky individual once referred to this using a fire metaphor: "The bigger you build your bonfire of knowledge, the more darkness is revealed to your startled eye" $\endgroup$ – JacobIRR Apr 19 '19 at 22:05
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Mathematicians and philosophers often assume that machines (and here, he probably means "computers") cannot surprise us. This is because they assume that once we learn some fact, we immediately understand every consequence of this fact. This is often a useful assumption, but it's easy to forget that it's false.

He's saying that systems with simple, finite descriptions (e.g., Turing machines) can exhibit very complicated behaviour and that this surprises some people. We can easily understand the concept of Turing machines but then we realise that they have complicated consequences, such as the undecidability of the halting problem and so on. The technical term here is that "knowledge is not closed under deduction". That is, we can know some fact $A$, but not know $B$, even though $A$ implies $B$.

Honestly, though, I'm not sure that Turing's argument is very good. Perhaps I have the benefit of writing nearly 70 years after Turing, and my understanding is that the typical mathematician knows much more about mathematical logic than they did in Turing's time. But it seems to me that mathematicians are mostly quite familiar with the idea of simple systems having complex behaviour. For example, every mathematician knows the definition of a group, which consists of just four simple axioms. But nobody – today or then – would think, "Aha. I know the four axioms, therefore I know every fact about groups." Similarly, Peano's axioms give a very short description of the natural numbers but nobody who reads them thinks "Right, I know every theorem about the natural numbers, now. Let's move on to something else."

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    $\begingroup$ Historically, the early 20th century had a strong academic belief in "solving" mathematics. E.g., Hilbert's program, and Whitehead+Russel's Principia Mathematica. Godel's work resolved that quest negatively, but I imagine it took some time for academia to fully embrace this notion; even fully acknowledging the correctness of Godel, people would still remember the grand ideas of Hilbert. I think Turing writing only two decades after Godel would be addressing his audience with this context in mind. $\endgroup$ – BurnsBA Apr 19 '19 at 13:13
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    $\begingroup$ I would question whether most mathematicians know "much more about mathematical logic" than Turing did. But it is obvious that almost all contemporary humans have vastly more practical experience of what machines (and particularly computers) can do than he did. $\endgroup$ – alephzero Apr 19 '19 at 18:48
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    $\begingroup$ @alephzero That's not what I said! I said that the average mathematician today knows more about mathematical logic than the average mathematician during Turing's time. $\endgroup$ – David Richerby Apr 19 '19 at 19:28
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    $\begingroup$ Your argument seems to be not that Turing's argument isn't good, but that it is unnecessary or directed at a strawman. I strongly suspect Turing had real people make arguments like this to him, so I don't think he's making a strawman out of nothing. As Discrete lizard states in a comment, Turing is only saying that a particular argument against machines surprising us is bad. Your answer just says that that this argument is bad has become even more obvious over time. That said, people (though usually not experts) still make arguments in this vein today. $\endgroup$ – Derek Elkins left SE Apr 20 '19 at 0:11
  • $\begingroup$ It is the absence of epistemic closure. $\endgroup$ – Dan D. Apr 20 '19 at 1:00
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Just an example - given chess rules, anyone should immediately figure the best strategy to play chess.

Of course, it doesn't work. Even people aren't equal, and computers may outperform us due to their better abilities to make conclusions from the facts.

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    $\begingroup$ Not sure that's a good example. People do readily come up with chess strategies, as soon as they properly grasp the rules, and though these strategies are obviously flawed and useless against more experienced players and modern engines, they would have been good enough against early computer chess engines. $\endgroup$ – leftaroundabout Apr 20 '19 at 21:58
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    $\begingroup$ My point exactly that not only people are different, but computers are different too, so stupid computers of Turing era doesn't mean that they always will be stupid. You may need to know, though, that Turing died long before computers started playing chess. $\endgroup$ – Bulat Apr 21 '19 at 4:54
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    $\begingroup$ I think this is a good example, and captures the essence of Turing's paragraph. $\endgroup$ – copper.hat Apr 22 '19 at 0:40
  • $\begingroup$ @leftaroundabout So ..., is chess a draw when optimally played or a win by white, or by black? More to the point: A relatively recent discovery that extremely long endgames are possibly lead to a revision of the 50-move-draw rules - such a discovery would count as a "surprise" in the sence of the quote $\endgroup$ – Hagen von Eitzen Apr 22 '19 at 10:09
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This is the idea of emergence, which is when complex behavior results from the interaction of relatively simple rules. There are lots of examples of this in nature, as that link points out. Insect colonies, bird flocks, schools of fish, and of course, consciousness. In a flock of birds or school of fish, each individual in the swarm is only making decisions based on the others immediately surrounding them, but when you put a bunch of those individuals together all following those rules, you start to see more coordinated behavior than you'd expect without a higher level plan. If you go on Youtube and watch demonstrations of robot swarms, you see that they all avoid hitting each other and work in unison. Surprisingly this doesn't need to be accomplished by having a single central computer coordinate the behavior of each individual robot but can instead be done using swarm robotics where, like the insects or the birds or the fish, each robot is making local decisions which leads to emergent coordination.

Another interesting demonstration of emergent behavior is Conway's Game of Life. The rules for the game are extremely simple, but can lead to very fascinating results

A tempting argument against the ability of computers to gain human-intelligence is to say that since they can only do precisely what they're programmed to do, they must only exhibit the intelligence that we program them with. If this were true, then we would also not expect the relatively simple behavior of neurons to give rise to human intelligence. Yet as far as we can tell, this IS the case and consciousness is an emergent property of neural processing. I'm sure Turing would have loved to see what's become possible today with the use of artificial neural networks

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    $\begingroup$ Thanks for mentioning the emergence. You add some optimism to my pessimism about A.I through computation. $\endgroup$ – smwikipedia Apr 20 '19 at 13:04
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People might assume that if I write a program, and I understand the algorithm completely, and there are no bugs, then I should know what the output of that program would be, and that it should not surprise me.

Turing says (and I agree) that this is not the case: The output can be surprising. The solution to a travelling salesman problem can be surprising. The best way to build a full adder can be surprising. The best move in a chess game can be surprising.

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  • $\begingroup$ This does explain why computers could be surprising which is the first half of the quote, but you do not address the part of the quote that explains why a particular argument that machines cannot surprise is fallacious. $\endgroup$ – Discrete lizard♦ Apr 19 '19 at 15:08

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