Calculus 201...
Estudia: Hey Phi! I’ve been doing some thinking and some reading about concepts since we last got together. It is pretty complex, but I think I’m catching on.
Philo: Good, what have you learned?
Estudia: Well, one thing I read was really fascinating. I was reading this study in Scientific American about these psychologists who were working with children. They would do things like make a model of a real home and hide a toy in it, say, behind a chair. They would show a kid the model with a little toy hidden in the same place and then send the kid into the real room to find the toy. Two year olds couldn’t do it, but three year olds could. It took some maturity to learn to associate a model or a picture with the real thing. Then something else they did that was cool. They had the kid explore a real home with normal kid size objects like a chair, table and plant. They told the kids that they could miniaturize the room by turning on a machine and leaving the room along for a while. They then brought the kid back into the same room but with all the objects replaced by tiny ones, like a foot high chair or table. If the kid was very young they would try to sit in the chair just like it was a regular one. They knew what it was for, it had the right shape, but they couldn’t seem to understand that it existed with a different scale and wouldn’t function like a regular chair. This got me thinking about concepts and how we must learn to recognize objects as being of the same type but with just different measurements. Really interesting.
Philo: I haven’t seen that, but I bet you are on to something there. I wonder if the scientists doing the study even have an understanding that they are observing the young minds grasping new concepts. We form concepts by differentiation of groups of things that can be distinguished from each other by some means of measurement. It’s implicit at first, like the kids, struggling to use the mini-chair like the real one because they don’t yet understand the difference in sizes. They hold the concept of chair in their minds as a shape and a function but not as a continuum of sizes. They must “get it” by three or so if such behavior no longer occurs after a certain maturity has developed.
Estudia: So, what does this say about all concepts?
Philo: There has to be some unit of measure common to the objects that will form the concept. You can use shape to differentiate tables from chairs but not from metal objects or blue things. No way to measure the difference between a chair and a blue object. Like apples and oranges, you could distinguish but you could form the concept of fruit and talk about sugar content, origin (is it the edible reproductive body of a seed plant) or whatever. Ayn Rand used the concept of “Conceptual Common Denominator” or CCD to explain how we form concepts. Things with a CCD, like shape, can be grouped together. Say chairs all with that “h” shape. Then when we omit the measurements of all the individual chairs we are left with the concept of “chair”.
Estudia: So, we have to use measurement to differentiate objects into groups. Say, all those things with “h” shape and used for sitting. Then we integrate all those objects with a common unit of measurement like shape into the unit concept of “chair” by ignoring the difference in size measurements.
Philo: Exactly and Ayn Rand’s formal definition is: “A concept is a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted.”
Estudia: So what is the purpose of all this theory?
Philo: Well, Ms. Rand covered all the main kinds of concepts like motion, relationship (up, down, right left, etc.) and materials. She showed how the concepts are derived in the same manner and fit her definition. As a philosopher she had to explain concepts so that we would know the rules to follow when using our minds to understand reality. But, more importantly, she had to have a way to explain or validate conceptual knowledge. By understanding our mental process we can now defend our mental processes from the mystics who say that our concepts are just some form of revelation. Or if a skeptic claims that our concepts are just social inventions, now you can defend yourself against such ignorance.
Estudia: So with this theory of concepts we can even validate reason I suppose?
Philo: Very good. We have to know that our concepts are valid tools of cognition and if we do know it, we can save ourselves from the irrationality that surrounds us. We now know that concepts are based on facts of reality, and these concepts do refer to facts of reality.
Estudia: I can appreciate that. The way you explained it, I can see that a concept refers to facts that are processed by our human minds. The concept includes all the things it refers to and doesn’t omit or change the properties of those things. Everything about all chairs is included in the concept of “chair” we are just going to not focus on all the different sizes, colors, designs and construction materials, etc. that can be used. Cool math for sure. Listen, I’ve got to run, but next time you have got to explain more about higher levels of concepts like “fruits” or “animals” and not just things like tables and chairs.
Philo: Promise. I’ll see you whenever.
Philo: Good, what have you learned?
Estudia: Well, one thing I read was really fascinating. I was reading this study in Scientific American about these psychologists who were working with children. They would do things like make a model of a real home and hide a toy in it, say, behind a chair. They would show a kid the model with a little toy hidden in the same place and then send the kid into the real room to find the toy. Two year olds couldn’t do it, but three year olds could. It took some maturity to learn to associate a model or a picture with the real thing. Then something else they did that was cool. They had the kid explore a real home with normal kid size objects like a chair, table and plant. They told the kids that they could miniaturize the room by turning on a machine and leaving the room along for a while. They then brought the kid back into the same room but with all the objects replaced by tiny ones, like a foot high chair or table. If the kid was very young they would try to sit in the chair just like it was a regular one. They knew what it was for, it had the right shape, but they couldn’t seem to understand that it existed with a different scale and wouldn’t function like a regular chair. This got me thinking about concepts and how we must learn to recognize objects as being of the same type but with just different measurements. Really interesting.
Philo: I haven’t seen that, but I bet you are on to something there. I wonder if the scientists doing the study even have an understanding that they are observing the young minds grasping new concepts. We form concepts by differentiation of groups of things that can be distinguished from each other by some means of measurement. It’s implicit at first, like the kids, struggling to use the mini-chair like the real one because they don’t yet understand the difference in sizes. They hold the concept of chair in their minds as a shape and a function but not as a continuum of sizes. They must “get it” by three or so if such behavior no longer occurs after a certain maturity has developed.
Estudia: So, what does this say about all concepts?
Philo: There has to be some unit of measure common to the objects that will form the concept. You can use shape to differentiate tables from chairs but not from metal objects or blue things. No way to measure the difference between a chair and a blue object. Like apples and oranges, you could distinguish but you could form the concept of fruit and talk about sugar content, origin (is it the edible reproductive body of a seed plant) or whatever. Ayn Rand used the concept of “Conceptual Common Denominator” or CCD to explain how we form concepts. Things with a CCD, like shape, can be grouped together. Say chairs all with that “h” shape. Then when we omit the measurements of all the individual chairs we are left with the concept of “chair”.
Estudia: So, we have to use measurement to differentiate objects into groups. Say, all those things with “h” shape and used for sitting. Then we integrate all those objects with a common unit of measurement like shape into the unit concept of “chair” by ignoring the difference in size measurements.
Philo: Exactly and Ayn Rand’s formal definition is: “A concept is a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted.”
Estudia: So what is the purpose of all this theory?
Philo: Well, Ms. Rand covered all the main kinds of concepts like motion, relationship (up, down, right left, etc.) and materials. She showed how the concepts are derived in the same manner and fit her definition. As a philosopher she had to explain concepts so that we would know the rules to follow when using our minds to understand reality. But, more importantly, she had to have a way to explain or validate conceptual knowledge. By understanding our mental process we can now defend our mental processes from the mystics who say that our concepts are just some form of revelation. Or if a skeptic claims that our concepts are just social inventions, now you can defend yourself against such ignorance.
Estudia: So with this theory of concepts we can even validate reason I suppose?
Philo: Very good. We have to know that our concepts are valid tools of cognition and if we do know it, we can save ourselves from the irrationality that surrounds us. We now know that concepts are based on facts of reality, and these concepts do refer to facts of reality.
Estudia: I can appreciate that. The way you explained it, I can see that a concept refers to facts that are processed by our human minds. The concept includes all the things it refers to and doesn’t omit or change the properties of those things. Everything about all chairs is included in the concept of “chair” we are just going to not focus on all the different sizes, colors, designs and construction materials, etc. that can be used. Cool math for sure. Listen, I’ve got to run, but next time you have got to explain more about higher levels of concepts like “fruits” or “animals” and not just things like tables and chairs.
Philo: Promise. I’ll see you whenever.
posted by Rp | 10:11 PM
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