Continued...

I suppose it depends on the meaning of "understand", but when used to imply an intellectual understanding, as I mean the term, I'd say that word is more or less synonymous with modeling.
That is an interesting model of "toy" that you are describing. Yet, it is simply another model to help understand that activity. That is the core inconsistency in your position-- you are saying "here's my model-- see, it doesn't involve modeling". But it does.

But what is "it" that works here? Your model. We can agree that the purpose of models is to convey some intellectual advantage, unification, or simplification-- that does not make them not models. Everything you are talking about here is just how you model things. Those models are not unique, nor do they invalidate other models, nor do they invalidate the concept of a model, even as it applies to what you are doing.
To which I say:

Yes, your model involves some smearing, it tries to encompass more things, which has both advantages and disadvantages depending on the desired result, but it is silly to think that makes it not a model. In many cases, the things you encompass merely serve to weaken the power of the model, as would be true in almost all cases as applied to physics but certainly less so when applied to the models of psychology or sociology.
Yes-- that's the whole point of a model.

You misunderstand my point, I said the noticing the desire to unify is part of understanding the modeling process (modeling the modeling, the issue you brought up), I did not say, nor mean, that that is itself part of the modeling process. The self-referential nature of modeling modeling is not a contradiction, it is a key element of intelligence that remains an enigma.
But you are missing how Bobby knows those are circles even though they've never specifically been identified as such. It's simply because he has created a model of circles, and uses the model to recognize those similarities in reality.

No, he says, "yes, that is obviously true." The circle in the mind has always been there, it is the way he developed the ability to identify circles. Otherwise, all he could do is recall items that have been associated with that label.

This can be shown by an example. If I tell you that 2, 3, 5, 7, 11, 13, and 17 are the first seven "prime numbers", you could then identify those numbers as prime if you saw them somewhere else. But what would you need to do to identify 19 as prime? You would need to notice some characteristic of these numbers, in this case a mathematical property. Once you notice that property, you have created a model for what a "prime number" is (in mathematics, we can skip this and refer directly to the definition, but with reality, we never see the "rule book"). To the extent that this model works when applied to other numbers, you will say you "understand" prime numbers. But without that model of what primeness means, you would never have any hope of finding other prime numbers. It is just the same with circles, their mathematical property is just a little simpler to notice than the property that makes a number prime. No model, no capacity to generalize, no understanding.

That would be the incredibly clunky way our minds would have to work if we could not make models for these things. Fortunately, we can. That is basically the difference between "google science" and inteligence-- google can only categorize and differentiate, never model, never understand.
No, "apples" are in the mind too, insofar as a-p-p-l-e is part of a language that exists in our minds. The word represents something we find in a tree (or something that is represented by the word "tree)", but when something stops being an apple, or a tree, is a completely arbitrary element of semantics, invented by us. Reality has no idea what an "apple" is, and can make a continuous distribution of a collection of atoms from what we call DNA, to a seed, to a green apple, ato red apple, to a decayed worm-eaten apple, to some worm excrement. The rest is all us, all our models of what we decide is the important attribute worth calling an apple.
Correct-- there is, and that is the important one for the intellectual activity of understanding. We use it all the time-- little progress has ever been made without it, despite your complaints.
To my ear, that just says a circle is a type of model. You are simply modeling what a circle model can entail, but your model for conceptualizing a circle is far from unique, and as I said, has advantages and disadvantages. It might be useful for understanding the sociology of the role of circles in human culture, but it would clearly be terribly overblown and useless for developing mathematical theorems about circles, and that's why it is not used for same.
The way you are choosing to model it, perhaps. The "reality" is none of those things-- it could also be viewed as just a bunch of atoms obeying laws of physics-- except that too is just another model for what is happening. It's models all the way down-- get used to it!
You can sure split a hair, but I see no important difference in those statements. Do you?

So the theorems I can prove about circles are unique to my own physiology? Thank goodness models don't work that way! This is supposed to convince me you have a "better way" to think about circles?
I'll bet that is not at all how you learned to draw circles. I'll bet you did it the same way everyone else did-- my forming a model of what a circle should look like, then trying to create one yourself. You didn't need to show the result to anyone, you could just look at it yourself, and compare it to your concept of circle (perhaps informed by a reference circle that you could see did conform to your mental model). When it didn't look like the referent, you modified and tried again, training your hand using the referent which in turn was informed by the model. After awhile, you could do the comparison with your own idea of a circle, your own model.

This is quite easy to prove. Have you ever drawn a circle, using that so-called muscle memory you are describing, and then looked at it and erased part and redrawn it better? Did you do that because of some muscle memory in the process of drawing it? No. Did you do that because you referred to something you were told was a circle and noticed a problem? No. I know perfectly well what you did-- you compared your result to exactly that inner concept of a circle that you claim you don't have. How else would you be led to redraw it?

You see, our inner model of a circle is the whole point of a circle, that's the whole reason we are talking about circles here and not the coastline of France. You probably have a vague model of the coastline of France, and no muscle memory in drawing it, yet you could do your best at drawing what you remember. You will be referring to your inner model of the coastline of France. But we are talking about circles because they are so much easier to model, so much easier to unify, than the coastline of France. Your teacher insisted on you being able to model circles, but not the coastline of France, expressly because the former is easy to model and the latter quite difficult. It is a perfect example of the power of models, and when they are most useful.
No, that would be the case if you were a trained pigeon with no intelligence. In fact, what you did was to unify that lack of correction into a concept of "size doesn't matter". That then becomes part of the model. As you become more sophisticated mathematically, you can even solidify this into a mathematical concept called "scale invariance". Your insistence that such modeling skills are irrelevant is quite pointless, as it denies essentially all of mathematics and how we use it, in favor of dog training.
Certainly not, that would be the last thing I would suggest! Instead, they need to understand what mathematics is, and how to apply it for their benefit, in terms of the application of intelligence not dog training. Are people in charge of curricula thinking like you do? No offense, but that would explain a lot.

__________________
Logic is the grammar of truth.

Meaning and absolute certainty are incompatible, and profound meaning and absolute certainty are profoundly incompatible.

The only thing intelligence is capable of is recognizing itself.