What are you counting?
Eliezer’s post How To Convince Me That 2 + 2 = 3 has an interesting consideration—if putting two sheep in a field, and putting two more sheep in a field, resulted in three sheep being in the field, would arithmetic hold that two plus two equals three?
I want to introduce another question. What exactly are you counting?
Imagine one sheep in one field, and another sheep in another. Now put them together. Do you now have two sheep?
“Of course!”
Ah, but is that -all- you have?
“What?”
Two sheep are more than twice as complex as a single sheep. It takes more than twice as many bits to describe two sheep than it takes to describe a single sheep, because, in addition to those two sheep, you now also have to describe their relationship to one another.
Or, to phrase it slightly differently, does 1+1=2?
Well, the answer is, it depends on what you’re counting.
If you’re counting the number of discrete sheep, 1+1=2. However, why is the number of discrete sheep meaningful?
If you’re a hunter counting, not herded sheep, but prey—two sheep is, roughly, twice as much meat as one sheep. 1+1=2. If you’re a herder, however, two sheep could be a lot more valuable than one—two sheep can turn into three sheep, if one is female and one is male. The value of two sheep can be more than twice the value of a single sheep. And if you’re a hypercomputer running Solomonoff Induction to try to describe sheep positional vectors, two sheep will have a different complexity than twice the complexity of a single sheep.
Which is not to say that one plus one does not equal two. It is, however, to say that one plus one may not be meaningful as a concept outside a very limited domain.
Would an alien intelligence have arrived at arithmetic? Depends on what it counts. Is arithmetic correct?
Well, does a set of two sheep contain only two sheep, or does it also contain their interactions? Depends on your problem domain; 1+1 might just equal 2+i.
In the strictest sense, “adding” sheep is a category error. Sheep are physical objects, you can put two sheep in a pen or imagine putting two sheep in a pen, but you aren’t “adding” them, that’s for numbers. Arithmetic is merely a map that can be fruitfully used to model (among many other things) certain aspects of sheep collection, separating sheep into groups, etc, under certain circumstances. When mathematical maps work especially well, they risk being confused with the territory, which is what I think is going on here. The “female sheep + male sheep” example should be thought of as an aspect of “putting sheep in pens for long periods of time” which addition does not model, not as an exception to “1+1=2”.
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.
On the other hand you don’t have to describe twice features shared by both sheep.
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 assume we agree that traditional math is extremely effective in predicting the physical world; I don’t see how this other idea of quantity remotely meets the same standard. Thus, even if an alien intelligence saw some other definition of quantity as more fundamental, it would probably agree with us on which concept is more likely to be shared by other species. (Just as we should be more surprised if an alien has a sense of humor than if it shares our mathematics.)
I’d also issue a warning against signaling cleverness via contrarian ontologies, if that’s what you’re doing.
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.
1 sheep in the field right now + 1 sheep in the field right now = 2 sheep in the field right now. If 1 + 1 != 2 then you’re overloading the “+” symbol, or you lost track of your units, or some other similar problem. Most of your post is undoubtedly true, but I don’t feel edified one bit.
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.
Best statement in the entire thread. Calling this hypothetical operation ‘counting’ or ‘+’ is needless obfuscation.
...and naively expect that their version of mathematics is a universal language, or that an AI with built-in intuitions about arithmetic hasn’t already been predisposed to think about things in human terms. Yes.
Although I’m less concerned ensuring the symbolic links involved are distinct, I will confess. “Yes, but that’s not addition” is missing the point; assuming no other linguistic barriers, somebody who counts sheep interactions as part of the equation is going to continually insist you’re leaving out something out when you insist that putting another sheep in the field only results in two sheep, and the person who insists that one sheep plus one sheep equals two sheep, period, is going to regard “sheep interactions” as utterly irrelevant to the equation.
In terms of units, 1 sheep + 1 sheep = 2 sheep + sheep interaction.
The additional sheep adds more to the field than another quantitative sheep.
This may just be my physical science education speaking, but adding two quantities of unit “sheep” and getting a quantity of unit “sheep” and a quantity of unit “sheep interaction” bothers me immensely. In every situation I’ve ever encountered where quantities were added, and I trusted whoever was doing the adding to have a very good understanding of when you get to describe something as “addition”, the resulting quantity had strictly the same dimensions as the originals. Whatever’s going on here, it’s probably not best described as addition (as Bundle so wonderfully explained).
That’s deliberate.
It’s not the abstract process of addition. It’s an entirely different way of counting reality; the abstract processes are necessarily different.
It’s very close to addition, though, and may reflect reality better than addition. Could you work with a mathematical system in which new units come apparently out of nowhere?
If you’re not talking about addition, then why in the name of Odin’s glorious beard did you use phrases like “+”, “addition”, and “counting” in reference to whatever this mathematical operation is? I hope you can imagine why that would be horribly confusing to us. You’d have to specify what this operation is before I contemplate the relation between its input units and output units.
I believe that was the point the article was attempting to get across. To my impression, OrphanWilde seems to be attempting to convey a concept he does not yet fully understand for which there have not yet been any formalizations and/or for which no words or accurate english/human-linguistic description exists.
My own interpretation tends towards a “feeling” of the following being an approximate description of this operation: “model the first element, model the second element, model the joining of these two elements, model the two elements as a whole of ‘firstandsecondelement’”
To me, this seems clearly nonequal to “first element” + “second element”, but I’d also agree that not mentioning this crucial distinction is confusing.
Reread my post. I didn’t use them in reference to that mathematical operation, except in the end, where the problem domain would be different (and hence the operators could conceivably mean something different). I in fact said that “Which is not to say that one plus one does not equal two. It is, however, to say that one plus one may not be meaningful as a concept outside a very limited domain.”
I -did- do this in my response to you, because the confusion was in a sense important; you can’t outright deny the existence of sheep interactions, you can only point out that this isn’t addition. Which allowed me to make this point: “It’s very close to addition… and may reflect reality better than addition.”
I’m not attempting to define this operation, only present its conceivable existence. There are two points to this post: First, that any defined subset of mathematics is not universal. (That is, mathematics is not in fact a universal language, any more than “Language” is a universal language.) Second, that any defined subset of mathematics is a nonideal representation of reality, and that it would frankly be surprising if an advanced intelligence chose to use the same mathematics we chose through our biased processes.
I think the trick is that there’s an intermediate layer of map, in between arithmetic and physical objects, which you aren’t seeing: finite set cardinality. You aren’t “adding” sheep, you’re taking the union of two disjoint sets of sheep. Everything between arithmetic and cardinality is provable mathematics, but cardinality maps much more closely to the operation you’re actually performing on sheep. In this example, you can’t get from addition on number of sheep to addition on economic value, because economic-value(set-of-sheep) is a function which does not have sigma-additivity (which is a fancy way of saying that the value of a set of sheep is not always equal to the sum of the values of the subsets).
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.
What do you mean when you use the term ‘counting’ or for that matter ‘+’? You seem to be using these ideas in different ways at different times, which is a language problem not a real problem: like trying to use one word that means alberzle and bargulum when you may need to use different words.
I have yet to find a concept that does not fit the description ‘may not be meaningful outside a limited domain.’ It seems like there is a generalizable mistake among some very intelligent people that makes them want something to be more useful than it is. Yes, addition is only useful for putting together “individual” (however you want to define that, often pragmatically) objects into “groups.” You can’t add x+y and get a delicious pie as a result. But any piece of knowledge is only useful within the domain of its application: why would you think otherwise?
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.
How is this a useful insight?
“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.
I’m not sure what you mean by “an AI which isn’t constrained by the same biases inherent to the human perspective”; I know what I mean when I say that but it might not be what you mean.
If by “realize that arithmetic in itself is derived from a bias inherent to the human perspective” you mean “realize that an alien might say that 1 + 1 = 1.3″ then I don’t see how that would help you build anything.
1.3 may be a more useful answer than 2.
I responded elsewhere with this:
“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.”
Imagine you’re an alien, for a moment, whose mathematics don’t have any of the trigonometric functions—no sine, no cosine, no tangent. Whenever they’re called for in their mathematics, they do a fourier transform of an infinite series of aperiodic waves, although they would never understand them -as- aperiodic waves, but as simple equations. This is an equally valid way of representing the trigonometric functions—but there would be a lot of very intractable mathematical problems.
Before you call that ridiculous, we didn’t have set theory until the 19th century; it permitted the solution of a lot of mathematical problems we had, until then, been struggling with. Set theory overcame a lot of the problems arithmetic had struggled with. New mathematical models have arisen since then, such as category theory.
It’s useful, therefore, to recognize arithmetic as a model, and one we may have a bias for, in consideration that another model might be more useful. More specifically, it’s useful, when building AI for example, not to build into it a requisite bias for a particular model, if your goal is to permit it to solve problems which we have thus found far intractable; you may be building into it the very structural problems which have made it intractable for us.