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.

Calculus 101...

Philo: Ok, you asked what do concepts really refer to, and I’m going to try to explain it to you. I mentioned that both the mystics and the skeptics claim that concepts are not concretes but are separated from them. The mystics think they must be true only in some supernatural realm while the skeptics claim that concepts are only social whim. But neither of these approaches explains concepts or grounds them in reality. What we need, to do this, is mathematics.

Estudia: Math! I can count and I can measure but I’m not so hot with higher math.

Philo: Not to worry, we only need to think about what measurement is.

Estudia: Oh, that’s easy. Measurement is the identification of a quantitative relationship determined by the application of a standard unit. The meter being the distance between two scratch marks on a metal bar or the kilogram being a platinum/iridium cylinder kept in...

Philo: Alright already, you’ve got it! Measurement involves two concretes, the thing being measured and the thing used to measure it with, i.e. the standard or something traceable back to the standard.

Estudia: Right, I know that. The unit must be appropriate to what is being measured. You use the foot to count off distances and hands to measure horses.

Philo: And seconds for time. Every unit of measurement is a particular one of the things being measured. The standard unit is a particular specific instance of something, be it time, or weight or length. Whatever we choose for the standard will be something that is within our range of perception, and then we can subdivide or multiply that unit to compare it to whatever it is that we are measuring.

Estudia: Ah, that is interesting because I never thought it that way. When we measure things we expand our consciousness beyond what we can perceive directly. We can relate whatever we are measuring back to the unit. So if a nanometer is really small, and it is as there are about 80,000 of them needed to cross a typical human head hair, we can’t know that directly with our senses but we can relate it back to that bar of metal with two scratch marks on it.

Philo: When we measure, we use standard units which are things that we can directly grasp. All things are related to what we can perceive.

Estudia: So what does this have to do with concepts?

Philo: Well, this is the terrific thing that Ayn Rand discovered. She saw that there is a connection between conceptualization and measurement. Measurement is a process that uses things that humans can perceive as a base, and then takes those concretes and extends them to endless things outside of our ability to perceive thus making the whole universe accessible to us. Things are related by a quantitative means as there is a mathematical relationship between the concretes being considered.

Estudia: Oh, yea, let me say it. A concept integrates a lot of similar concretes by taking advantage of the fact that they are all related by some quantitative characteristic. When I form a concept, I retain the characteristic, but I omit the measurements. For example, I can see that I may have understood the concept of weight by noting that things take different amounts of effort to lift them but everything takes some effort. I understood that some how as a child.

Philo: Very good. Your mind formed the concept “weight” by noting that weight must exist in some quantity, but it also may exist in any quantity. You later learned that weight is a property that can be quantitatively related to a unit of weight. You learned what a pound or a kilogram felt like. Still the concept of weight was only related to that unit. The weight of any concrete could be of any quantity from very small to massive. The weight of any concrete could then be identified. All weights are commensurable, meaning that they are related quantitatively to the same unit.

Estudia: Ok, I can see that for concepts like weight, or length, or time, but what about real concretes like tables and chairs and pencils, coins and things like that? I don’t see that there is a unit chair or a standard pencil.

Philo: No, a concept like chair differs slightly in that chairs in general have certain shapes, usually an “h” shape with a seat and maybe a back and legs. These are characteristics that can have all sorts of measurements from the doll-house to the side of a mountain. The concept chair omits all the possible measurements that can be involved from the length and width of the seat, the height of the back, angle of the legs, the color, covering, hardness, temperature, conductivity, and much more. The concept “chair” also includes all past, present and future chairs no matter how they may vary. In our mind we can include all possible variations and can expand the concept at any time to include new designs like chairs without backs or legs or whatever, but they we may want to include some forms in different concepts like “stools”, “benches, or whatever.

Estudia: So, we have to grasp similarities between things in order to form a concept. If we note some characteristic that is similar but differs in mathematical quantity only then it can form the bases for a concept.

Philo: Which may or may not be useful. You may need to form a more limiting concept like “stools” out of certain classes of “chairs” because it is useful. Specialist are good at forming concepts to include narrower and narrower groups of things out of a large class that most of us would not need or care about.

Estudia: Our minds must do this automatically for us? I mean, I don’t think a child has any notion of measurement or of what measurements to exclude when forming a concept. They just observe similarities and characteristics and can form a concept and then they learn that a word is associated with that concept and they can use it to think about or identify concretes.

Philo: Yes, but really what goes on first is a process of abstraction. Our minds separate or abstract characteristics or attributes from their measurements. We don’t have to know that a form of measurement is implicit in the formation of a concept. We simply observe similarities. Chairs all have a different shape and function than pencils or tables or rocks. The abstracted characteristics of all the chairs we have observed is retained and used to form the concept “chair” and this view of existents allows us to integrate all things with similar characteristics into a single concept.

Estudia: But...

Philo: No, buts, for now. We have talked enough. Think about abstraction and measurement omission until we meet again. This is an important subject and requires a little more elucidation so I want to take it slowly. I’ll meet you here next time. For now it’s, good bye, au revoir, and auf Wiedersehen!