It’s for this reason that I think before long, the metaphor will evolve toward “brains are like quantum/probabilistic computers.” Makes a lot more sense. Biomolecular systems in general seem to function this way — DNA and other molecules are very much, in a sense, computing. But you’re right, it’s not like fixed binary circuits and data.
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I would suggest considering that “computing” is more about the passing messages of information and processing that information, than it is about any specific type of device. I’d submit that nature is computing — although I’d also submit it’s a probabilistic computer, and that there is also a programmer. 
Are computers and magic the only possibilities? You know something a typical computer can’t do despite being Turing complete? Play a vinyl record. It can approximate it very closely, close enough for my purposes, to be sure. But we all know it isn’t quite the same. That’s not because record players are magic, it’s because they’re analog and the computer is digital.
I don’t entirely know what @scientist is getting at. But while neurons have all-or-nothing responses, they have analog inputs and variable rates of fire. Seeing as how computers and Turing machine in general have a set clock for state changes, I can see that very possibly one wouldn’t act quite like the other in a feedback control system, and that doesn’t mean I’m invoking Cartesian dualism.
If you want to understand brains, I think it might be worth considering that lots of animals have them, and for most controlling things is far more important than performing computer-like computations. Lampreys have brains, much simpler than ours but akin to them. They take in sensory input, control muscle movement, and learn basic things about their environment. I don’t think lampreys can do things like multiplying large numbers, although I would be delighted to see it demonstrated otherwise, and few people would bother using a universal computer as an analog for them.
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A couple decades of studying those inputs and processing. You really seem not get how exquisitely intricate (and also fundamentally different from man-made electronics) the body is. Its not that I think there is some physical law preventing it, its that I don’t think it is something that any hypothetical, future advancement could achieve. Obviously, this is my opinion rather than something testable.
There presumably lies a future full of technological advances far beyond anything I could possibly imagine. Likely a world where issues of scale and processing speed make descendants of computers that are effectively instantaneous and can interact with the world at nanoscale resolution.
To me, its not a question of processing, or chemistry, or physics. Its that your nervous system interacts with itself and the body through over 1013 interconnections. The output is a dynamic interaction of all of them. This interaction is dynamic on many simultaneous temporal and spatial scales and most of the function is a result of the physical nature of the material. As James put it, to me its just not a live hypothesis. I truly can’t conjure any path to even a thousandth of what it would take
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Stating that reality is analog and as such is fundamentally different than a digital computer may be true. But not all computers are digital, and quantum physics speaks to a quantized nature to reality — at least as far as we’ve yet been able to penetrate.
A digital computer can model an analog computer, and vice versa… 
As a nondualist, I don’t have to choose between being an idealist and being a materialist. 
But if we’re going to go with “everything is a computer when you really think about it” then “a brain is a computer” becomes a meaningless statement anyway.
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The all-or-none action potentials of neurons is for transmission, not processing. The actual processing is strictly analog. The signal is then digitized into a sequence of action potentials if it needs to be conveyed more than about a millimeter to prevent loss during transmission. The synapse then converts back to an analog system once the action potential reaches its destination
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That’s the Dark Place your brain goes at 2 AM when you can’t sleep. The place that causes us to visit therapists so that we can learn to avoid that place as much as we can.
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I knew the effects at the synapse and weighing of inputs, if you want to call it that, are analog. Is it possible for a neuron to have a response that doesn’t involve the all-or-none action potential, then, if it’s a shorter distance? I hadn’t heard that and it would be interesting to me.
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Yeah. Brains of very small organisms have no action potentials. And processing condensed into small spaces, for instance in your retina, is purely graded (analog) with no action potentials. In human vision the ganglion cell (4 cells deep) is the first to fire an action potential, because it is transmitting the signal into the brain. If photoreceptors had to convert to digital action potentials and then back we would lose temporal resolution for vision
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But that is the point. I agree with you that there will never be a point where someone takes a really long tape with a magnetic head and connects it to a bunch of vats of hormones, neurotransmitters and electricity and makes a body move around. That isn’t the question that “Is the brain a computer?” is asking though. It’s asking whether there is something special about the actual calculations that a brain performs such that only a brain can perform them.
It’s a little bit like if I ask are there things in space that don’t have a length? You can say to me, “Well the galaxy doesn’t have a length because I can’t see that we will ever build a big enough yard stick to measure it.” I agree with you there, but that doesn’t mean that a galaxy doesn’t have a length. Simply because we can’t build a potentially non-finite Turing machine to replicate a brain, doesn’t mean that the brain doesn’t have a morphism to such a thing. The way to prove that a brain isn’t a Turing machine is to demonstrate a calculation that a brain can do that a Turing machine in principle cannot. It doesn’t have anything to do with constructing a thing any more than know that the galaxy has a length has anything to do with building a yardstick.
You’re really focused on the calculations aspect. As has been pointed out by others, there’s SO much more the brain does other than calculations.
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I play CBC Radio As It Happens on my Internet radio, and plunge right after the summary. (Just the most recent one. The web page plays them all in reverse order.)
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The example that @Scientist gave was the ability to control a body through neurochemistry. I guess. To me that doesn’t seem all that relevant or impressive or fundamental to what a brain actually is, i.e. a thing that thinks. Computers control bodies through electronic interfaces all the time. What is your example of something a brain does, that isn’t a calculation and could never be expressed as a calculation or series of calculations?
The point was that so much of what the brain does are not calculations. This is why the brain isn’t a computer. Not that it makes a magically different kind of calculation
or as Otherbrother posted while I was writing this:
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This is where we most fundamentally disagree. Thinking (in the rational sense) is distantly secondary to what our brain is
Hopefully this story might illustrate:
After a mathematician collaborator of mine gave a talk in which he described a model of a single neuron that I made with over 200,000 partial differential equations, another mathematician asked me why the brain would be implementing 200k pde to produce an output well described by a 3 variable equation.
My response was that if I were to model the turbulent dynamics of the Mississippi river it would likely also be a dynamic systems model composed of a similarly large number of pdes, but that one would never conclude that the river must be making those calculations. With the brain, it processes information and so do our models/computers, so we often conclude they must be doing the same calculations. The brain doesn’t calculate those dynamics any more than the river; it IS those dynamic interactions
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As one quick and obvious example it physically produces important chemicals and hormones like dopamine, serotonin, etc.
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That is somehow a different kind of thing than controlling the body? Is a chemist (or better yet an automated machine) in a lab who synthesizes dopamine somehow more like a brain than something that does calculations?
You didn’t ask for an example of something it does that’s not related to “controlling the body.” You very specifically asked for an example that’s “not a calculation.” And I think that producing important physical excretions are clearly not the same thing as a calculation or series of calculations, whether or not calculations may be involved at some point in the process.
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