Given that any stock is a function over time as well, there should theoretically exist a neural net that can approximate the stock price for the future? This reasoning is obviously wrong, what is my exact error of thought though?
The library of babel contain every possible finite description of every possible function.
Wait long enough, and one leaky bag will emit the number "42". If you get the initial conditions just right (and quantum nondeterminism isn't really a thing), then you'll also get a good approximation to the stock market.
...now if you don't explicitly represent U as a parameter, you'll have it implicit in the function. So your "neural network" contains the entire state of the freakin universe (!!).
Ergo, contingent on your stance on theologic immanence vs. transcendence, what you'd call "neural network approximation of the stock's price function" is probably quite close to what other call... God (!).
(Now, if relativity as we know it is right, you might get aways with a "smaller slice of U" - lear about "light cone". And to phrase this in karelp's explanation context: you'd need to know U to know which of the practically infinitely many such neural networks to pick. The core of (artificial) intelligence is not neural networks in themselves, it's learning, the NN is a quite boring computational structure, but you can implement tractable learning strategies for it, both in code, and in living cells as evolution has shown...)
There isn't one, you're just overestimating the value of existence-propositions.
In practice, knowing that something exists is not a very useful result - it is often more useful to know that something does NOT exist (e.g. solution to the halting problem).
Train a model today so that it is overfitting for a given stock. It would predict everything very accurately upto today. The ONLY way to make sure that the results are completely isolated from the market is to not make the result available to ANYONE (how do you know that an isolated human observer is not leaking data with some unknown phenomena... say quantum entanglement with particles in other people's brains, for example). So, the ONLY way to test the models is back-testing.
You can extend that to saying that for any given point in the future (say, this is a reference point), there will be an overtrained model which will backtest perfectly i.e. the theoritical model that works at any time during the past to predict the exact stock price in the future upto the point of reference.
It is just that the neural network would have to compute a model of the entire world or even the universe on an atomic scale. It would be computationally unfeasible but not theoretically impossible.
It is theoretically possible that the universe we live is already being computed on a neural network in some other external universe.
In particular, the way the stockmarkets are distributed the function of time is likely relativistic and every participant is acting under incomplete information even in the infinite limit.
Also, you have to be cautious what any function in this context really means, as I imagine it means differentiable functions (after somebody mentioned the Ackermann function, which is not anywhere differentiable).
There isn't one for the future, unless said future is somehow predetermined?
Is it, given enough input data?
Does this discussion then distill down to philosophy?
Do living beings have agency, or are they simply very complex NNs?
How one answers that speaks to consciousness as much as it does the prospect of a predictive stock price model.
Implying, there is a wealth of thought devoted to inteligence! That fact is actually proving the conjecture in a nicely constructive way by itself, that we are thoughtful indeed, if only you believe this axiomatically like. The quintessential theorem was distiled by Descartes, of course, wherefore he is remembered.
So I was just surprised by their use of language as it seems to imply parent thought we would be closer to or there already with our developments of AI tech.
The value of a dice is also a function over time. Can we learn this function with a neural net? No, because our features don't include the nitty gritty details of each throw so it's essentially random.