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.
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.
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.
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.
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.
People can not be reduced to lookup tables even in theory. No one even knows how a single cell does what it does let alone an entire organism like a person.
I'm not making an abstract claim about neural networks because all numerical algorithms like neural networks can be reduced to a lookup table given a large enough hard drive. This is not practical because the space required would exceed the number of atoms in the known universe but the argument is sound. The same isn't true for people unless a person is idealized and abstracted into a sequence of numbers. I'm not saying no one is allowed to think of people as some sequence of numbers but this is clearly an abstraction of what it means to be a person and in the case of the neural network there is no abstraction, it really is a numerical function which can be expanded into a large table which represents its graph.
How are people lookup tables? In the case of neural networks the representation of the table is obvious, it's just numbers. What would be the equivalent table for the liver?
My argument isn't abstract. Neural networks really are just numerical functions which can be expanded into their equivalent graph representations.
I was recently thinking how every neural network is equivalent to a lookup table where the input is all numbers up to what can be expressed within the context window and the output is the result of the arithmetic operations applied to that number. So every neural network is equivalent to T = {(i, f(i)) : i < K} where K is the constant which determines the context window and f is the numerical function implemented by the network. Can someone ask a neural network if my reasoning is valid and correct?
The main practical issue is the size of the table but I don't see any theoretical reasons why this is incorrect. The neural network is simply a compressed representation of the uncompressed lookup table. Given that the two representations are theoretically equivalent and a lookup table does not perform any reasoning we can conclude that no neural network is actually doing any thinking other than uncompressing the table and looking up the value corresponding to the input number.
Modern neural networks have some randomness but that doesn't change the table in any meaningful way because instead of the output being a number it becomes a distribution over some finite range which can again be turned into a table with some tuples.
In any case, I'm dropping out of this thread.