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I wish there was a mathematical theory for what 'understanding' is. For instance, I too suspect that whatever we call a function of a system, something that the

I wish there was a mathematical theory for what 'understanding' is. For instance, I too suspect that whatever we call a function of a system, something that the system does, can not simply occur by random parts coming together. To be clear, I don't think a layer of sands has the function of filtering water. It does filter water, but it does it in the virtue of static property, in the first of the geometry of the system not the mechanics. There is something about systems that do something that I can't name, having limited knowledge on the subject. Could anyone give me a hint on research done on this?

RodgerTheGreat 14y ago
Er- you seem to be describing the concept of an "emergent property" of a system, but beyond that I don't really understand what you're asking.
pirateking 14y ago
I believe the phenomenon you are referring to is emergence[1][2]. Most of the research in this area I have come across is related to education and epistemology. I would recommend reading works of Seymour Papert and Jean Piaget. I think the mathematical theory you are looking for to explain "what understanding is", is actually the philosophical motivation for the field of mathematics itself. At least it is why I am really interested in studying it these days. [1] http://en.wikipedia.org/wiki/Emergence http://en.wikipedia.org/wiki/Emergence [2] http://plato.stanford.edu/entries/properties-emergent/ http://plato.stanford.edu/entries/properties-emergent/
- cristianpascu 14y ago
  Emergence sounds right. But I think there's something more. The properties of a car are emergent on the properties of its part. But the function of the car is not necessarily so. There is intent, something that a pile of rocks does not have, although it has emergent properties.
  - pirateking 14y ago
    It might be easier if you warp your viewpoint across multiple frames of reference. The intent of the car (to travel) is a human mental construct, which can be independent of its physical properties. There are the physical properties of parts, driving higher level mechanical systems, all working together to create an even higher level model with its own emergent properties (speed, handling, fuel economy, etc.). The ability to discern the intent of a car is an emergent property of our culture. We grow up seeing wheels in motion all around us, houses have garages, roads and parking lots exist. A car can just as easily be viewed as a mobile shelter or a pizza transport machine. Similarly, a pile of rocks may actually have intent in some cultures. A pair of sticks can map to eating utensil after all.
    - cristianpascu 14y ago
      Then my question is how can function occur in the natural world without intent. Natural selection explains why that function is kept once it occurs. Think of it like this: a space of N dimensions, for N properties, with an infinite number of values for each. Small variations (genetic mutations) of each of these property values, will move the position of the system in this N dimensions space. Initially, most of the parameters are null. But, there's a clear trajectory, or a set of possible trajectories from the initial position to the position where, by a certain combination of parameters, a function occurs. When there's intent, it's very easy to see the path from A to B. But when there's no intent, and not foreknowledge of the possibility of a particular function (a special position in the N space), finding that particular spot by random variations sound enormously improbable. Does this low probability have a name or something?
gatlin 14y ago
As another commenter said, this sounds like the basis of math (or CS) itself. Understanding is emergent. Inference and learning are, to me, the pillars of it and we can do those. Understanding is the ability to internalize some idea and make correct inferences about it. But then what is an idea? And down the rabbit hole I go ...

RodgerTheGreat 14y ago

Er- you seem to be describing the concept of an "emergent property" of a system, but beyond that I don't really understand what you're asking.

pirateking 14y ago

I believe the phenomenon you are referring to is emergence[1][2]. Most of the research in this area I have come across is related to education and epistemology. I would recommend reading works of Seymour Papert and Jean Piaget. I think the mathematical theory you are looking for to explain "what understanding is", is actually the philosophical motivation for the field of mathematics itself. At least it is why I am really interested in studying it these days. [1] http://en.wikipedia.org/wiki/Emergence http://en.wikipedia.org/wiki/Emergence [2] http://plato.stanford.edu/entries/properties-emergent/ http://plato.stanford.edu/entries/properties-emergent/

cristianpascu 14y ago
Emergence sounds right. But I think there's something more. The properties of a car are emergent on the properties of its part. But the function of the car is not necessarily so. There is intent, something that a pile of rocks does not have, although it has emergent properties.
- pirateking 14y ago
  It might be easier if you warp your viewpoint across multiple frames of reference. The intent of the car (to travel) is a human mental construct, which can be independent of its physical properties. There are the physical properties of parts, driving higher level mechanical systems, all working together to create an even higher level model with its own emergent properties (speed, handling, fuel economy, etc.). The ability to discern the intent of a car is an emergent property of our culture. We grow up seeing wheels in motion all around us, houses have garages, roads and parking lots exist. A car can just as easily be viewed as a mobile shelter or a pizza transport machine. Similarly, a pile of rocks may actually have intent in some cultures. A pair of sticks can map to eating utensil after all.
  - cristianpascu 14y ago
    Then my question is how can function occur in the natural world without intent. Natural selection explains why that function is kept once it occurs. Think of it like this: a space of N dimensions, for N properties, with an infinite number of values for each. Small variations (genetic mutations) of each of these property values, will move the position of the system in this N dimensions space. Initially, most of the parameters are null. But, there's a clear trajectory, or a set of possible trajectories from the initial position to the position where, by a certain combination of parameters, a function occurs. When there's intent, it's very easy to see the path from A to B. But when there's no intent, and not foreknowledge of the possibility of a particular function (a special position in the N space), finding that particular spot by random variations sound enormously improbable. Does this low probability have a name or something?

gatlin 14y ago

As another commenter said, this sounds like the basis of math (or CS) itself. Understanding is emergent. Inference and learning are, to me, the pillars of it and we can do those. Understanding is the ability to internalize some idea and make correct inferences about it. But then what is an idea? And down the rabbit hole I go ...