Professor Murkywaters is sought out by his students at lunch. Murky is eating a baloney sandwich. He eats quite a bit of baloney. The students come to his table in the cafeteria.
De: Professor Murkywaters can we join you? (as they sit down).
Murky: Sure, be my guests!
On: We overheard you talking to your graduate student, James, and we are full of questions! Especially since we were "talking" ''til after midnight last night (he blushes slightly).
Murky: Fire away! Umm this is good!
On: ok, you guys were discussing some abstract thing called a "modeling relation" as if it had something to do with the real world. That kind of threw me.
De: Me too. It seems like something so abstract and removed from what we actually do, but I heard you say it represents our whole way of perceiving and manipulating the real world.
On: Yeah, and when you said that EVERY representation of the real world somehow can be seen this way it seemed like you went to far.
Murky: Well, let's see if that's true or not. How do you guys "see" the world around you? Describe it to me. And then lets get fancier and tell me how scientists, both experimenters and theory nuts go about their business?
On: This is silly! I "see" with my eyes of course. Oh, you mean like view or observe or something don't you?
Murky: Yes, eventually you get some information about what is outside you into that cranium, right? There are lots of ways to do this. You have sense organs. You can extend those by inventing more precise ways of interacting with things, so-called measuring devices or meters. You can talk to people and listen to music and the radio. You watch TV. Finally, you use your "imagination". The end result of all these activities is a "something" in your mind. Lets give the something a name and call it a PERCEPT. At least some of these originate outside of us and then we can see them as part of some sort of external world.
De: Then experimental data eventually sits there right beside something we imagined? That sounds mushy again! Empirical facts are what establish our sense of reality. They must be objective. You can't afford to risk having them mix with the junk you produce up there! Oooops! Excuse me! Nothing personal intended.
Murky: That's where the modeling relation comes in. We need to be able to sort through all the "junk" we are processing up there. We do that by "guessing" at how to see more of the real world's reality that what little we get from senses or meters. Therefore we "construct" a parallel reality and see if it "works".
On: I kind of follow, but then I don't. How do we keep the wheat separate from the chaff?
Murky: We iterate to some extent. The modeling relation is an iterative process. It keeps on going like the little bunny with the drum.
De: Ok, HOW does it work?
Murky: Well, we are dealing with percepts, some good some bad. Some came from outside. Let's call the whole thing a "Natural System"since it hopefully came from the natural world outside, but is already converted into a percept via sense organs, meters, etc.
De: Where does that get us?
Murky: One thing is missing. We add a belief.
De: A BELIEF? More ^%&$# mush!
Murky: Well I can't prove this can you? But I bet you believe it. We all believe that when our percepts about the Natural System change it is due to a "cause" rather than random nonsense.
On: Well SOME of my classes.......
Murky: Come on now! How about you De?
De: Yes I see your point. We can agree to accept that. So you are saying that we assign causes to the changes in our percepts and that "explains" the fact that the Natural System is ever changing. But couldn't Tarot cards be seen as a cause?
Murky: Evidently some folks' modeling relations don't prevent that. But let's see how it might. We can't always manipulate the external world. We often must "observe". We can construct our own worlds though and try to use them to represent what is "really out there". Much of the time, but certainly not always, in science, especially hard science, we use mathematical objects to replace our percepts. Lets stick to this set of objects for this discussion, realizing that we are voluntarily limiting our universe of discourse.
On: Whoa! What do you mean by a mathematical "object"?
Murky: Well, things like sets, groups, topological spaces, categories, dynamic systems, etc. plus the relations between them and the relations which they satisfy. Is that not an arsenal for at least some of our percepts to be represented by?
On: I guess so.
De: Sure, what more could we need. OK I know! Mushy stuff too.
Murky: But it doesn't end there. There are the ACTUAL objects and then the symbols (or language) we use to represent them. If I say A is a "group" it still is the letter A and only now stands for a group because we agree to let it. Once we agree we can do something more. We can find ways to manipulate the symbols and see if we come up with real mathematical objects again. We call those manipulations propositions and if the lead us back into the world of mathematical objects, we say that they are "true" otherwise they are "false".
De: Wow! And I thought I understood math before. This is so much MORE than the writing and solving of equations. Let me see if I've got it. The only way we can see if propositions, which are manipulations of symbols standing for mathematical objects, are true or false is by passing back and forth between the world of propositions and the world of objects! Hey! WOW! THERE IS AN EMPIRICAL PART OF IT THEN! We have to, in a sense, sample the world of objects before we know a proposition is true or false!
On: That's kind of exciting! Math is like everything else then! But what does all this have to do with the Natural System?
Murky: Look at the structure. We have just described a "Formal System". It has basically the same structure as the Natural system. Objects in nature correspond to objects in math. Percepts correspond to the mathematical symbols. The relationships are the same!
De: Yeah, but where is causality in the Formal System?
Murky: That's the whole purpose of this exercise. In the FORMAL SYSTEM we get to play God! We get to "play" with propositions and pick the ones that seem to result in symbols being related in the same way that percepts are in the Natural System. We call the sequences of manipulations that work IMPLICATION and see it as our Formal System's counterpart to CAUSALITY.
On: So the Formal System becomes a MODEL for the Natural System! Hey, that's neat!
De: Hey big fella! You start to sound like one of us!
On: Well it makes sense! And again, I can see ways of doing this with things OTHER than math. So there! (pensive pause) Hey? How do we know how to match up the Formal System's precepts and causality with the Natural System's symbols and implications?
Murky: That's an art form! Modeling is an art form! The stuff we just talked about is linked through an encoding of the the Natural System into the Formal system and then, after the implication manipulations, a decoding to compare with the Natural System. If it all hangs together, the modeling relation has worked and we have a VALID model. But this is not a simple matter in itself and I have a class. Let's continue later. (he leaves with a goodbye)
On: Ummm...did you have lunch? ...oh yeah..later Prof.
De: Bye prof! No wanna eat together?