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Abd comments on Logical Pinpointing - Less Wrong
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I love this inquiry.
Numbers do not appear in reality, other than "mental reality." 2+2=4 does not appear outside of the mind. Here is why:
To know that I have two objects, I must apply a process to my perception of reality. I must recognize the objects as distinct, I must categorize them as "the same" in some way. And then I apply another process, "counting." That is applied to my collected identifications, not to reality itself, which can just as easily be seen as unitary, or sliced up in a practically infinite number of ways.
Number, then, is a product of brain activity, and the observed properties of numbers are properties of brain process. Some examples.
I put two apples in a bowl. I put two more apples in the bowl. How many apples are now in the bowl?
We may easily say "four," because most of the time this prediction holds. However, it's a mixing bowl, used as a blender, and what I have now is a bowl of applesauce. How many apples are in the bowl? I can't count them! I put four apples in, and none come out! Or some smaller number than four. Or a greater number (If I add some earth, air, fire, and water, and wait a little while....)
Apples are complex objects. How about it's two deuterium molecules? (Two deuterons each, with two electrons, electronically bound.) How about the bowl is very small, confining the molecules, reducing their freedom of movement, and their relative momentum is, transiently, close to zero?
How many deuterons? Initially, four, but ... it's been calculated that after a couple of femtoseconds, there are none, there is one excited atom of Beryllium-8, which promptly decays into two helium nuclei and a lot of energy. In theory. It's only been calculated, it's not been proven, it merely is a possible explanation for certain observed phenomena. Heh!
The point here: the identity of an object, the definition of "one," is arbitrary, a tool, a device for organizing our experience of reality. What if it's two red apples and two green apples? They don't taste the same and they don't look the same, at least not entirely the same. What we are counting is the identified object, "apple." Not what exists in reality. Reality exists, not "apples," except in our experience, largely as a product of language.
The properties of numbers, so universally recognized, follow from the tools we evolved for predicting behavior, they are certainly not absolutes in themselves.
Hah! "Certainly." That, with "believe" is a word that sets off alarms.
It's almost a tautology. What we have is an iterated identification. There are two objects that are named "apple," they are identical in identification, but separate and distinct. This appears in time. I'm counting my identifications. The universality of 1+1 = 2 is a product of a single brain design. For an elephant, the same "problem" might be "food plus food equals food."
Basically, you're saying that for an elephant, apples behave like clouds, because the elephant has a concept of apple that is like our concept of cloud. (I hope real elephants aren't this dumb). I like this a lot, it clarifies what I felt was missing from the cloud analogy.
Having it explicitly stated is helpful. It leads to the insight that at bottom, outside of directly useful concepts and into pure ontology/epistemology, there are no isolated individual integers. There is only relative magnitude on a broad continuum. This makes approaching QM much simpler.
Mmmm. This is all projected onto elephants, but maybe something like what you say. I was just pointing to a possible alternate processing mode. An elephant might well recognize quantity, but probably not through counting, which requires language. Quantity might be recognized directly, by visual comparison, for example. Bigger pile/smaller pile. More attraction vs. less attraction, therefore movement toward bigger pile. Or smell.
Would you argue, then, that aliens or AIs might not discover the fact that 1 + 1 = 2, or even consider it a fact at all?
Okay, I don' t have to speculate or argue. I'm an alien, and I don't consider it a "fact," unless fact is defined to include the consequences of language. I.e, as an alien, I can see your process, and, within your process, I see that "1 + 1 = 2" is generally useful to your survival. That I'll accept as a fact. However, if you believe that 1 + 1 = 2 is a "fact," such that 1 + 1 <> 2 is necessarily "false," I think you might be unnecessarily limited, harming long-term survival.
It's also useful to my survival, normally. Sometimes not. Sometimes 1 + 1 = 1, or 1 + 1 = 0, work better. I'm not kidding.
The AI worth thinking about is one which is greater than human, so that a human can recognize the limitation of fixed arithmetic indicates to me that a super-human AI would be able to do that or more.
It seems to me like there are at least two possible definitions of "reality". One is "the set of all true statements" (such as "the sky is blue", "salt dissolves in water", and "2 + 2 = 4"), and the other is "the set of all 'aspects of the world'"—things like "in world W, there is an electron at point p and time t", whose truth could, in principle, be varied independently of all other "aspects of the world".
The choice of definition seems more or less arbitrary.