What are you counting?

In the strictest sense, adding anything except abstract numbers is a category error.

My point has less to do with the map not correlating to the territory, as that the map is not necessarily as intuitive, as a map, as we might expect; arithmetic may be a lot more tightly correlated to a distinctly human way of perceiving the world than naive expectation would lead to be the case.

I would argue that this is a form of "encoding" and "compressing" of the data, and that in a strictly atomic sense would produce a fuzzy description unless both sheep actually shared their atomic components. I use the word "atomic" here to refer to whatever form of most elementary description tickles your fancy, be it strings or wave functions or quarks or bosons or gluons or what-have-yous - the most elemental and fundamental pieces with which the two sheep are built, whatever that may truly be.

Arguably, and I'm really not sure about this second hypothesis, the Total Message Length might turn out to be even greater than before if you attempt to produce any sort of "accurate" description of both sheep by way of "shared features".

Very true. One alien intelligence might count 1.3, instead of 2, or 2.08.

I don't agree with you that traditional math is extremely effective in predicting the physical world. It is in fact very bad at predicting anything that isn't an extremely isolated situation. It's -theoretically- very good at predicting the physical world, but by the same token, you can -theoretically- use fourier transforms of aperiodic waves instead of trigonometric functions.

And mathematical systems exist without -any- concept of quantity. Set theory and category theory, for example, both exist without an internal representation of quantity.

To say, based on that we've had success with our particular brand of mathematics, that we should expect similar mathematical systems in use by aliens - why, this is to ignore entirely the implications of the UTM Theorem. There are an infinite number of analogous mathematical systems with analogous predictive powers.

If 1 + 1 != 2 then you're overloading the "+" symbol, or you lost track of your units, or some other similar problem.

I think the idea was that perhaps for some alien intelligence the "+" symbol could be useless or even meaningless, and something else would be in the place of "the most simple abstract computational operation". Then the aliens could naively expect that every intelligence in the universe must know this very basic operation.

In terms of units, 1 sheep + 1 sheep = 2 sheep + sheep interaction.

The additional sheep adds more to the field than another quantitative sheep.

The intermediate map resolves the map-territory problems with arithmetic, but doesn't really address the problem here, which is more than that the map of arithmetic doesn't correspond perfectly with the territory, but that there are -other valid maps- of that same territory which can evaluate the same real-world operation (adding a sheep to a field) on the basis of a different abstract operation.

"What are you counting" is the operative question here; arithmetic only makes sense if you count additive properties.

I am using counting to refer to any process by which a number is assigned as a symbol for a property.

I use + in both its concrete - this field now contains two sheep - and its abstract - this set now contains two sheep - meanings. I hope the context makes it clear when I am using each meaning, and why; because the lack of clarity is, in fact, important. See my response to the first comment, in which I deliberately used its concrete meaning in response to somebody using the abstract meaning. Both of us are in fact correct, and the confusion itself is meaningful.

Because it -is- a linguistic problem - and because linguistic problems can, in fact, be real problems.

Viliam Bur encapsulated what I was trying to establish pretty well: "I think the idea was that perhaps for some alien intelligence the "+" symbol could be useless or even meaningless, and something else would be in the place of "the most simple abstract computational operation". Then the aliens could naively expect that every intelligence in the universe must know this very basic operation."

But more than that - if your basic operations are different, it's possible to come to very different conclusions.

One of my biggest revelations in mathematics was in statistics, when, after the class (including me) worked unsuccessfully for a couple of hours to integrate an equation, the instructor (who I'm sure was laughing at us) walked up to the board, converted into a different coordinate system, and integrated the now very easily integrated equation in about thirty seconds.

If your basic operations are different, you might be able to come to conclusions you otherwise were unable to come to.

• I have yet to find a concept that does not fit the description 'may not be meaningful outside a limited domain.'

And in response to this I'll ask: How many people accept this about the fundamental descriptors they use in their mathematics? How many people operate on the assumption that mathematics are a universal language, or that the universe runs on mathematics (which I generally interpret to mean -their- mathematics)?

• You can't add x+y and get a delicious pie as a result.

And yet most recipes follow a basically arithmetic formula: Add 5 units of meat, add 375 units of heat over 10 units of time.

The point, although it's a long way around to coming, is that arithmetic may be a fundamentally -human- way of evaluating the universe. It goes without saying that it's not the ideal model in many scenarios. And for those considering how to build AI, particularly those interested in solving intractable problems, it may be worth letting it come to its own model.

"Useful" is dependent upon an ends to that use.

If your goal is to design an AI which isn't constrained by the same biases inherent to the human perspective, it could be useful to realize that arithmetic in itself is derived from a bias inherent to the human perspective, for example.