But doesn't Shannon rely on symbols and alphabets? What if the information is presymbolic?

Smolensky's pdp paper modeled presymbolic processing with the harmonium, the first restricted boltzman machine.

No it doesn't rely on symbols.

You have a piece of copper and circuitry between your house and your ISP. Back in the day, there was no digital processing. It's just copper, amplifiers, filters, and mechanical switches. Why did dial-up internet never exceed (within a factor of 2) 30 kbps? Because that's the bandwidth of the channel. It's not that modem designers missed a more complicated, neuronal, way to encode the information.

It's really amazing that information and bandwidth can be as fundamental as temperature and mass. There's nothing symbolic about it in that lens. Symbols only come in because we know how to do important computations in the digital domain.

Analog television, FM radio, telegraph lines, telephone lines, smoke signals, the human vocal tract, all of them have bandwidths.

I'm pretty sure the way analog degrees are freedom are mapped onto symbols is very important. In principle, if you have infinite SNR over a limited bandwidth, you can have infinite rate of data transfer - e.g. if you can have infinitely fine voltage resolution (in reality limited by the thermal noise floor, but you can always increase transmit power). So in that sense the mapping between information and bandwidth depends on SNR.

From the wiki page on QAM: "Arbitrarily high spectral efficiencies can be achieved with QAM by setting a suitable constellation size, limited only by the noise level and linearity of the communications channel." [https://en.wikipedia.org/wiki/Quadrature_amplitude_modulatio...]

Unfortunately I used bandwidth informally (how most people use it), to mean something measured in bits per second, not Hertz.

You're absolutely right that SNR plays a role. But I don't see why you need to map to symbols.

Calculating Shannon entropy does rely on the symbolic alphabet and its probabilities of occurrance (see Wikipedia on Shannon entropy); however, I don't know how bandwidth is calculated. What makes you think it is as fundamental as temperature? One thing I know is that thermodynamic entropy has never been fully commensurated with information entropy -- unless someone has a ref otherwise.

Dialup internet did exceed 30kbps -- and that's because the bandwidth of the channel (the copper wire) was not the limiter. That's why DSL works and can in theory reach 1 gbps (https://en.m.wikipedia.org/wiki/Digital_subscriber_line)

I believe it is the channel plus the receiving mechanism plus the sending mechanism plus the encoding mechanism that determine the bandwidth. I am not asserting this, but that is my understanding.

Dial up internet used an analog transfer domain to encode the information. DSL does not, and is an entirely different technology (and is not 'dial-up').

56kbps modems and the like relied on digital telephone lines. I'm not sure where the 30kbps number comes from though - earlier pre-digital modems did go faster than that. Although you could argue those were 'pre-internet' as well...

Wait, but that's my whole point -- it's the same channel, but higher bandwidth.

If you're defining the 'channel' as 'local copper pair', then, sure?

But the other end of the local copper pair switched and became digital, which changes the channel in my eyes at least. There was then a series of bandwidth increases in the digital realm, giving significantly more efficient and effective use of the channel.

It's not the same channel, because DSL requires equipment fairly close to the subscriber. You can't transmit DSL on telephone utility poles for miles, like you can dialup.