Why the alphabet of the digital information is composed of 2 elements? please don't offer an answer about "it's an electronic thing" or something like that, just keep reading.

I don't understand why we use a dictionary, a lexicon, with just 2 elements to express the entire digital world.

My point is about the fact that information is entropy, things that are predictable or just don't vary over time don't carry any information, so if you want information you want entropy, and if you want entropy why you stick with just 2 elements for your alphabet ? It's like the minimum before making the thing completely "flat" and meaningless like an array full of 0 or Z, an array of 0 and 1 it's just the next step after that .

But the main reason why I don't accept the answers with "electronics in them" is that this kind of studies were based on math and theoretical stuff, Boole wasn't even a regular student, He was a self taught mathematician, and chances are that he was trying to solve or study a practical problem without even caring about electronic devices given the times he was in .

If you also consider the information as a bingo where you can call numbers from 1 to 100, and you can call 1 number for each session, you have 1% of probability to call any number in the range, and this probability doesn't change between sessions . But this doesn't grant you any entropy, it's just probability .

For example if you consider a single char a number in the Bingo and the fact that every char is contributing to a word with its positional value, the fact that the hello word is composed over time by

h
he
hel
hell
hello


with each char having the same probability this consideration, on its own, doesn't give you any entropy.

Things change when you introduce a dictionary of a given language and you associate it with the pool of chars you are fishing from. If you have that dictionary the word hello becomes predictable, and thus with less entropy and more order, by the time you write hel or h or he depending on the context and semantics .

So I guess the answer is that we use binary because no one speaks binary and there is no dictionary of words in binary and the binary is not sensitive to the context and thus retains more entropy ? Because the context plays a big role in lowering the entropy and let things appear more ordered then what they really are, and entropy can be also taught as just something "random" with less order because you don't know what to expect because there is no semantic. And I don't think that the positional value can be considered as a giveaway for semantic, you have the same probability of picking either 0 or 1 in your next cell of information, positional value doesn't change anything.

I'm probably terribly confused and confusing, but why we use binary ?
You are really asking three different questions:


Why are computers based on binary?
Why did Boole use a binary system?
What does all of this have to do with entropy?


The answer to the first question, as you suspect, relies on implementation details; see for example this question. In other circumstances, such as wired or wireless communication, information is transmitted in larger chunks, what is known as modulation.

The answer to the second question is simpler: Boole was formulating the laws of thought, that is logic, in which each statement is either true or false. That's your binary distinction.

Finally, I can't really follow your third question. According to Shannon's source coding theorem, you can optimally compress information using any channel. Text is usually stored as text since it's easier to process that way, but we can compress it to reduce its size to roughly the entropy. In some sense, text is stored using an alphabet of size roughly 64, including lower and uppercase letters, digits and punctuation marks. Other data is stored in bytes, which is an alphabet of size 256. All of this has nothing to do with entropy, rather it's ease of processing. These are just some thoughts, since I can't really understand this question.