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Originally Posted by Joe Durnavich
Language is indeed rich with metaphor. But as for what words mean, we typically learn that from contexts of use.
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Whereas that doesn't apply to models and metaphors... how?
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Would you say that they all collapse down to a singular entity—a “model”--that somehow packages all the senses into a singular something?
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No one who knows what models are thinks their purpose is to package "all the senses into a particular something". To understand what models are, one should restrict to their actual purpose.
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Can a person understand something only by forming a model of it?
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No one has claimed that models are the "only" way to do anything-- merely that they are an excellent way to do many things that define our intelligence. I think the abiilty to make models is a crucial element of what one might call advanced intelligence.
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If so, how does he come to understand his model?
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Here's how. He notices the way his mind thinks, that it seeks unifying characteristics of various seemingly separate things. Presumably this is an instinctive element of intelligence. Then he models those unifying characteristics as a single entity of some kind, like "mother" or "toy". When this unification is applied to the very process of seeking unification, the concept of a "model" emerges. So yes, the process of modeling is what is used to define the meaning of the term-- it's all about noticing what our intelligence is doing.
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The difference I was thinking of was “creating a simpler reality” versus “dealing with reality in a simpler way” or maybe “discovering the simpler aspects of reality.”
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Well that's your problem right there-- no one who uses models thinks that it involves "creating a simpler reality". It merely involves creating a
model, that is what is being "created". The model needs to be simpler than the reality, yes, but it does not need to be mistaken as a "type of reality". You are just adding that part yourself.
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Circles and ellipses don't have to exist in the mind or elsewhere.
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Circles and ellipses, in the precise mathematical meanings of the words, do exist in the mind, and they don't exist anywhere else. That's just a fact, and it means that they are both
models.
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We may be shown (imperfect) drawings in math textbooks that we learn to call circles, then we learn to relate other objects like hula-hoops to them.
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Exactly, but a key step in that process is to notice things about circles, to find their unifying characteristics. That may be done vaguely, as a child might do, or precisely, as a mathematician might. All that is different is the sophistication of the model. If I asked you to draw me a circle, do you think you would call to mind objects that other people have told you are circles, and draw one of them? No, you would simply draw what you carry in your mind as your model of a circle. That's also how you know the size of what you draw doesn't matter-- you have figured out that the size of a circle does not affect its circularity, and thus you include in your model the concept that size doesn't matter.
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If talk of circles refers to anything, it is not to “creations of our minds”, but to physical objects like drawings (or rings or hula-hoops, etc.) that serve as a public referents for circles.
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The mathematicians who have proven all kinds of theorems about circles, entirely using mental reasoning, should be interested to know that. Last I checked, none of them ever proved anything by "referring" to a hula hoop-- they refer to mental constructs like
definitions and
axioms, which are, of course, the building blocks of models.
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Later in life, we learn stricter techniques for dealing with the objects we encounter as circular, but all without there being some sort of imaginary perfect circle that serves as a referent for the term.
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That is where you are quite wrong. The "imaginary perfect circle" is a crucial "referent", and mathematicians use that referent
all the time. Those who don't use that referent have a much vaguer concept of what a circle is-- they are simply using a cruder model.
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What I am trying to do here, in case I am not doing it so well, is to look at what a teacher might do to teach a child about circles and ellipses and to identify what exists in those contexts and what takes place.
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Yes, that's fine-- a teacher may indeed teach a child that way. And later, when the child's mind is more sophisticated, it will be time for the child to recognize that what he was doing all along was learning a model. Eventually, the child will understand how to master the models, and will begin to learn mathematics. Or they won't-- the general mathematical abilities of our youths is in many quarters in dismal shape. Maybe it's because they deny the existence and usefulness of models.