We know it doesn't. There's no metaphorical linear tape in the brain; it's a network of neurons. But a Turing machine can simulate a network of neurons (see Machine Learning), just as a brain can model a Turing machine (see Programmer). There are currently things a brain can think about that a Turing machine cannot, but the question is whether that will continue to be true despite the steady advance of (computer) science.
The article above has a good summary of the problems with the idea of neural networks as simulations of biological neural networks:
These appeals to biology are problematic, because most connectionist networks are actually not so biologically plausible (Bechtel and Abrahamsen 2002: 341–343; Bermúdez 2010: 237–239; Clark 2014: 87–89; Harnish 2002: 359–362). For example, real neurons are much more heterogeneous than the interchangeable nodes that figure in typical connectionist networks. It is far from clear how, if at all, properties of the interchangeable nodes map onto properties of real neurons. Especially problematic from a biological perspective is the backpropagation algorithm. The algorithm requires that weights between nodes can vary between excitatory and inhibitory, yet actual synapses cannot so vary (Crick and Asanuma 1986). Moreover, the algorithm assumes target outputs supplied exogenously by modelers who know the desired answer. In that sense, learning is supervised. Very little learning in actual biological systems involves anything resembling supervised training.
https://plato.stanford.edu/entries/computational-mind/#ArgFo...
Such as? Are these things forbidden by theory, or just simply beyond current engineering practice?
You say this as though we understand what "thinking" is. We do not.
If that includes "have a conscious experience," then you're the one who's going to have to descend into hand wavy nonsense to explain how that's possible. Unless you've solved the hard problem of consciousness.
If there is nothing inherently remarkable beyond scale then no hand waving needed.
The remarkable aspect is that of conscious awareness. You (presumably) have an experience of the world, in a way that a computer does not. Paraphrasing Nagel, "there is something it is like to be you."
Most people don't think that this is true of an executing computer program, for example - it executes whatever its instructions are, and even a self-modifying program, as you described, doesn't change that.
There is no known way to write a computer program which has conscious awareness, and no plausible reason that scale should affect this. If you scale up a computer program, or a computational neural network, there's no reason to believe that it wouldn't just be a very big machine, blindly executing its instructions with no conscious awareness.
The proposed explanations that do exist are all nothing but handwaving at this point, hence my original comment. The burden of explanation here is on those who claim that the brain is nothing but a computing device, since our current models of computing devices can't explain consciousness.
Primates, which are very similar to us, do not have a comparable subjective experience. We know that because we can communicate with them and they don't have that much to say.
There are also brains in other animals that share a lot of similarities and are larger than ours.
Yet no beings (that's were aware of) have a subjective experience as rich as humans or can self-referentially communicate about that experience.
But turn up the clock speed on my gaming rig far enough and I get to debate the meaning of existence with it?
You have to ignore so much that is obvious to come to a reductive, materialistic conclusion like that.
[Edit]
Not sure if it's worth adding this but I do think we can build a computer that passes the Turing test. I also believe (sooner rather than later) we'll have a Siri-like AI that will provide enough companionship that a relationship can be formed with it.
We could even teach that AI to discuss subjective experience in a believable way.
I'm saying that your hypothetical gaming rig because it can simulate any type of classical computation including running a simulation of your brain and thus would be sufficient given a programmer.