I define intelligence as the capacity of one entity to create complex internal representations that can be used to predict about it's environment the likely outcome from a particular configuration or action, based on the regularities of the environment.

Entity can be organic life form or (potentially) a natural or artificial inorganic intelligence.

The "can be used" instead of "uses" also covers a case of "abstract only" entity that constructs purely theoretical abstract model (think mathematics) and does not use it for any adaptation purposes.

The capacity should be synergistic in the sense that we should choose the optimal scale so that we don't consider parts of an intelligence or a group of intelligent beings that act independently. Sub-components should have significant lower capacity even reduced to the scale of sub-component. Supra-groups should have limited synergistic capacities.

There is an interesting problem regarding entities that would just develop a kind of mathematical knowledge based on various sets of theorems. You could really use such entity as an adviser when you want to improve your model about the world. However, there are many potential sets of axioms that can define various mathematics. You still need a "practical" intelligence that learns from environment in order to chose the right set of axioms and the useful set of theorems from the infinite possible propositions that can be derived from the axioms.

Such "abstract only" entities would not store internally any representations of the environment regularities. We can understand their representations as shortcuts for the regularities that would be otherwise harder to observe by trial and error by a "practical intelligence" (even using a formal system).

How can we understand the nature of such abstract knowledge? How is that they are useful at all in understanding environment?

There are simple mathematical relations that can be intuitively derived from the environment. For example the result of many things increases linear with the quantity. A squared land that is 3 times bigger on the edge than another squared land will produce 9 times the production. Actually, the production from the bigger land might be slightly bigger then 9 times, because maybe it becomes more efficient to work it.

Having such mathematical apparatus can be very useful in constructing a bridge for example. You just need to be able to calculate the resulting resistance of the bridge based on limited size measurements and observed scaling laws. It's not a proof that it will hold that much, there are also non-linear and non-polynomial laws. Therefore, there is always something inferred from the environment that talks about how the environment will behave.

Then what the mathematics is about? I guess mathematics can "compress" processes that can be simulated in the real environment. Let's imagine I don't have a formal mathematical system so I create a system based on rocks, to know how many horses I must receive for my cows. If a happen to consider that 1 horse worth 2 cows, I could check the deal without inventing numbers. When I send a cow to my neighbor, I add a stone in a heap. When I receive a horse, I remove 2 stones from the heap. The change is fair only if I remove all the stones based on received horses.

I could make it simpler if I would know some math, but I did not had to. I just had to create a kind of "isomorphism" between cows, horses and rocks. I had to add one additional axiom, that repeating the operation in any order does not make a difference. Actually this might not be entirely true practically, the last few cows worth much more than the first cows if this is your source of living. But classical math is a good approximation of reality in many cases.

We are forced to admit that mathematics only provides symbolic chains of equivalences for the moving rocks processes. We already know that we can simulate any calculable function based on very simple Turing machines, that can be emulated even with rocks. The equivalence with reality is a simple leap of faith. We used to thought that the acceleration is linear with force and inverse proportionally with mass. This was Newton, but Einstein added that at big speeds it is a little different. How sure are we that an even better approximation is not possible?

We can only model reality based on simple patterns that can be re-presentable mechanically, for example with rocks. We can even prove theorems with such systems, just that they are inherently limited to their equivalence assumptions.

Dear reader, please leave a message if you exist! ;) Also, please share this article if you find it interesting. Thank you.

Entity can be organic life form or (potentially) a natural or artificial inorganic intelligence.

The "can be used" instead of "uses" also covers a case of "abstract only" entity that constructs purely theoretical abstract model (think mathematics) and does not use it for any adaptation purposes.

The capacity should be synergistic in the sense that we should choose the optimal scale so that we don't consider parts of an intelligence or a group of intelligent beings that act independently. Sub-components should have significant lower capacity even reduced to the scale of sub-component. Supra-groups should have limited synergistic capacities.

There is an interesting problem regarding entities that would just develop a kind of mathematical knowledge based on various sets of theorems. You could really use such entity as an adviser when you want to improve your model about the world. However, there are many potential sets of axioms that can define various mathematics. You still need a "practical" intelligence that learns from environment in order to chose the right set of axioms and the useful set of theorems from the infinite possible propositions that can be derived from the axioms.

Such "abstract only" entities would not store internally any representations of the environment regularities. We can understand their representations as shortcuts for the regularities that would be otherwise harder to observe by trial and error by a "practical intelligence" (even using a formal system).

How can we understand the nature of such abstract knowledge? How is that they are useful at all in understanding environment?

There are simple mathematical relations that can be intuitively derived from the environment. For example the result of many things increases linear with the quantity. A squared land that is 3 times bigger on the edge than another squared land will produce 9 times the production. Actually, the production from the bigger land might be slightly bigger then 9 times, because maybe it becomes more efficient to work it.

Having such mathematical apparatus can be very useful in constructing a bridge for example. You just need to be able to calculate the resulting resistance of the bridge based on limited size measurements and observed scaling laws. It's not a proof that it will hold that much, there are also non-linear and non-polynomial laws. Therefore, there is always something inferred from the environment that talks about how the environment will behave.

Then what the mathematics is about? I guess mathematics can "compress" processes that can be simulated in the real environment. Let's imagine I don't have a formal mathematical system so I create a system based on rocks, to know how many horses I must receive for my cows. If a happen to consider that 1 horse worth 2 cows, I could check the deal without inventing numbers. When I send a cow to my neighbor, I add a stone in a heap. When I receive a horse, I remove 2 stones from the heap. The change is fair only if I remove all the stones based on received horses.

I could make it simpler if I would know some math, but I did not had to. I just had to create a kind of "isomorphism" between cows, horses and rocks. I had to add one additional axiom, that repeating the operation in any order does not make a difference. Actually this might not be entirely true practically, the last few cows worth much more than the first cows if this is your source of living. But classical math is a good approximation of reality in many cases.

*A formal system is more like a very compressed external memory system.*We are forced to admit that mathematics only provides symbolic chains of equivalences for the moving rocks processes. We already know that we can simulate any calculable function based on very simple Turing machines, that can be emulated even with rocks. The equivalence with reality is a simple leap of faith. We used to thought that the acceleration is linear with force and inverse proportionally with mass. This was Newton, but Einstein added that at big speeds it is a little different. How sure are we that an even better approximation is not possible?

We can only model reality based on simple patterns that can be re-presentable mechanically, for example with rocks. We can even prove theorems with such systems, just that they are inherently limited to their equivalence assumptions.

**What if the Universe is just slightly more complex than we can symbolize with a bunch of rocks? What if the Universe is a little more complex than what we can compress in our limited brains + formal systems?****Intelligent life forms****If there are regularities in the outside environments and there are actions that have predicted outcome, a life form might learn chains of regularities and their probabilities. Probably simple Bayesian proportions might give an intuition and can derive our entire logic and our formal systems.**

Dear reader, please leave a message if you exist! ;) Also, please share this article if you find it interesting. Thank you.

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