clooper

In my argument a and b are positive and this is true for all ordered rings but not for the dual numbers as you've defined them. Specify the ordering and you will realize h can not be larger nor smaller than 0 because both cases lead to a contradiction.

In any case, I'm dropping out of this thread.

If a < b then a^2 < b^2. This is not true if you let a = 0 and b = h. The dual numbers do not have an ordering. You should provide sources and proofs next time because it seems like you are just making things up.

Your are confusing non-standard and dual numbers. The dual numbers are not ordered and contain non-invertible nilpotent elements such as h which squares to 0.

Do you work on a farm or do you just consume their output? Degrowth is obviously not the only option but if you have never worked on an industrial scale farm then you should do some more research before casting blame on people who are trying to educate you about the scale of the problem facing us. Resources necessary for intensive industrial farming are running out. Business as usual is not sustainable regardless of how upset that makes you.

Surprisingly pertinent given all the news about AI and cognitive automation.

The bit precision of computation is always finite for halting computations and any finite computation can be turned into a lookup table which does no thinking or reasoning other than comparing two numbers and then extracting the value corresponding to the input key.

My argument carries through for any piece of software so if you think software can think and reason then you can remain unconvinced by my argument.

In any case, I have to drop out of this thread.

Ok so give me some concrete number.

P obviously can't be infinite, even in theory, if you want the computation to terminate.

Provide some concrete numbers for solar radiation then as a lookup table. You guys are confusing abstraction and idealization with what it means to be a thinking person. There is no such abstraction and idealization happening with software. The software is really just a number, there is no idealization or abstraction happening when I claim that GPT is a sequence of bits representing a numerical function.

That's a good point.

Even if you increase the precision I can still generate a table T(P) for each fixed precision P. So the table is parametrized by P but it's still a table. The entire table T = colim T(P) is the colimit over all precision values but for every finite precision it is still a table.

What are the dimension of the input and output spaces involved in this idealization? In the case of a neural network there is no idealization. The network is software, it's a number. It's inputs and outputs are all bounded and can be expressed as a table of bounded tuples.

Then give some concrete numbers for the states of the atoms. My argument is not abstract, it is very concrete. Give me a neural network and I can generate the graph and prove the equivalence between the network and its graph representation as a table of tuples.

As a numerical function it can be implemented on an abacus so I don't think it's any different from a large enough abacus. It's practically not feasible but theoretically there is no idealization or abstraction happening when numerical calculations on a computer are transferred to an abacus.

Obviously if we are interacting over a digital medium then the responses will be encoded as numbers but there is no way to reduce an entire person to a lookup table. Measured output of human behavior can be expressed as lists of numbers but thinking is not the same as the list of numbers, unlike in the case of neural networks where the graph and the network are actually equivalent.