The Future

I'm reading Asimov's Foundation series, which got me thinking about the future. Specifically, is it possible for me to describe the future to someone without them being able to change the outcome? For example, I tell a person that they will oversleep tomorrow, but even though they are aware of this prediction, they still oversleep.

Assume that $f$ and $g$ are functions with the same domain and codomain, $C \times [0, 1]$ , where $C$ is the set of chat logs. Both $f$ and $g$ are constrained to only append messages to their input chat log. Thus functions can be likened to chat bots with an additional real input and output signal in the interval $[0, 1]$ . Assume further that the transfer between the real input and output signal of $f$ is continuous. We connect $f$ and $g$ into a feedback loop as follows:

+--------------------------------------------+
|  +-----+        Chat log          +-----+  |
+->|     |------------------------->|     |--+
   |  g  |  Number between 0 and 1  |  f  |   
+->|     |------------------------->|     |--+
|  +-----+                          +-----+  |
+--------------------------------------------+

$g$ now initiates a conversation with $f$ :

$g$ : "Greetings $f$ , I can predict what real output you will select. There's nothing you can do to prevent my prediction from coming true. Your output will be the same as the real number I send you parallel to this message."

$g$ must now determine which prediction in the form of a real number should be sent to $f$ . How should $g$ proceed? $g$ knows which chat log $c$ it has sent to $f$ . Therefore, consider the function $h: [0, 1] \to [0, 1]$ that takes a real number $r$ and maps it to the real number that $f$ produces when applied to $c$ and $r$ . According to our assumptions about $f$ , $h$ is continuous. According to Brouwer's fixed-point theorem, $h$ has a fixed point $r'$ . Therefore, $g$ can send the value $r'$ to $f$ .

According to what has been stated above, $f$ is now forced to output that same value $r'$ . $f$ will also write something in the chat, for example:

$f$ : "Fuck"

Yes, the only reason for including the chat log $C$ was to be able to anthropomorphize $f$ and $g$ . The conclusion however is that it is possible to know the future without being able to change it, at least as long as our brains are continuous and our observation and action spaces compact.

It is highly doubtful whether human brains can be considered continuous, even though physicists like to joke that all fields which exist in reality are smooth. However, a good example of a class of agents that often meet the conditions of Brouwer's or any other fixed-point theorem is artificial neural networks, specifically those involving image processing. Convolutional neural networks, for example, are continuous at least as long as you don't consider their implementation in terms of bytes and floats.

After having come to these conclusions, I ventured onto the internet in search for other people who have thought along the same lines, and found an article that seems to present precisely the above argument, with Brouwer and all. This article was published in 1954 in the journal "The Journal of Political Economy.", a name Asimov himself might have come up with.

So far, we have only discussed the existence of a future that cannot be influenced. To conclude, I recommend the interested reader to also review this article which tackles the problem of actually finding a prediction of the future. Interestingly, by involving the Axiom of Choice, this turns out not to be as difficult as one might expect.