phyllo, which I am sure that you are mistaken, i’m not sure why. What do you mean when you say “apply” the original abstraction? What does that mean and why would you do that?
gib -
Why are we bringing causality into this?
Depends what you mean. I have never designed (engineered) a car and wouldn’t know where to start. I think I know what car is, but i think there are many people who know better what a car is than i do. We all know what numbers are well enough to use them to balance our checkbooks, but when we’re dealing with ontology, we need to know more than that. Mathematical realists (if you know what I mean) probably live rich, full lives. They just don’t make sense as philosophers.
So we don’t get it confused with referentiality.
Yes, that’s more or less what I was talking about.
Well, again, “cause” is one of those philosophically loaded words. The reason i question this is that what so often happens here is that people tend to run away with a term. Or an idea. I think that’s what phyllo has done.
An abstraction might very well become the standard by which reality is judged, but nothing I have said here implies that. Where I tend to be strict is when I will say that my willingness to call a coffee can cylindrical is probably dependent on just how close the shape of the can comes to a mathematical cylinder, but that doesn’t mean the coffee tastes any better.
people around here tend to get a bit overwrought over nothing.
An original abstraction build on the various grounds for/ of successive abstractions. The original ground may be a structural product of the degree of intent/causual proportion of designation of prior abstractions.
By ‘apply’, I mean you test whether something falls into an abstracted category. You see something and ask : Is that a ‘dog’? Is that a ‘cylinder’?
If you are looking at one of the dogs from which you derived the abstraction ‘dog’, you will answer - Yes that is a ‘dog’.
If you are looking at one of the cylinders from which you derived the abstraction ‘cylinder’ … I think you have to answer - Yes that is a ‘cylinder’. Otherwise, the abstraction has been altered to the point that the original referent is no longer recognizable.
Well, my contention is that if you say “yes, that’s a cylinder” then you are mistaken as a matter of fact, so let’s take the dog example.
If it really is one of the very dogs from which you derived the category, then there is no problem. All you are saying is that the label you have applied is apt. There could hardly be a problem with this - it’s just a label.
And if it is one of the original cylinders from which you derived the ‘cylinder’ category, then it works differently?
If you say “it is an X”, then you are speaking of the abstract category and saying that “it falls within the category X”.
If you say, “it is the X” or “it is X”, then you are speaking of that one specific item alone, not a category.
“It is a dog” and “it is a cylinder” are both correct even though the item isn’t the exact categorical definition of the abstract category. You have said that it falls within the category, not that it IS that category.
“It is Dog” would be saying that “it” is the very definition of the category “dog”. And the cap letter is to indicate that it is the very categorical definition, as in “God”.
It’s also grammatically incorrect.
The problem with the cylinder is that it is a second-level category. We see cylinders in the world (or, if you like, cylindrically shaped objects), and we draw out the first level abstraction: the Cylinder (not the capital ‘C’). This is like “Dog” with the capital ‘D’.
What geometrists denote when they utter “cylinder” is a second level abstraction of cylinders. When they say that the cylinders they are talking about don’t really exist in the world, what they mean is that they exist only as first level abstractions that we have made in our minds. They are talking about the first-level category. They are talking about the properties and relations of this (these?) first level abstraction of cylinders, what must be axiomatically or deductively true (or approximately true) of any real-world cylindrically shaped object that falls into the first level category of Cylinders.
At this level of philosophizing, I wouldn’t say this is a dog. I would say this is not a dog.
Please explain.
Possibly what is far more psychologically interesting is how is it a child can immediately identify THIS as a cat?
phyllo -
No. There is no empirical analog for the cylinder at all. Mathematical forms, geometric entities have no examples in the real world. There are cylindrical objects - approximations - it’s the reverse of the abstraction from empirical objects.
I would explain it differently than gib does. As I have said, mathematics is a special case. Mathematical objects are not abstracted from the real world at all. That’s what confused Plato and so many after him.
Perhaps using the example of a square-shaped acre will illuminate. An acre is not measured according to land surface, but according to a horizontal projection above the land itself. For this reason, a mountain acre has a larger surface area than a flatland acre. This projection is technically “to infinity”. There is no “square” on the land itself.
Now, one of geometry’s first uses was in land surveying, and the first knowledge of geometry must surely have been empirically derived. But mathematics has come along some since that day. And just as you can “do math” without understanding the theory of numbers, you can “do geometry”, to an extent, without knowing the theories and axioms behind it. But even with Euclid, we can see the beginnings of the theoretical underpinnings of geometry - in fact, that’s why he is still famous. He fully realized that the objects we see in real life were at best approximations of the forms he studied - Euclid didn’t think that a mathematical cylinder was abstracted from the real world, but was a “pure” abstraction.
Perhaps some study of Euclid is all that is required to understand my point. Now, Plato didn’t have the benefit of Euclid’s work, but was unfortunately more of an influence on the lay person than Euclid, and is, sadly, more of an influence to this day. Ignoring Euclid is an example of the perils of getting stuck on Plato.
That’s the short answer, phyllo.
It is interesting that you mention Euclid. You use a Euclidean geometric abstraction of circles. That gives another clue that there is something amiss with the abstraction. Why should the axioms of Euclidean geometry be the correct way to abstract circles? Why not Non-Euclidean geometry -hyperbolic or elliptic geometry? Modern physics says that space-time is hyperbolic.
Use different axioms and you get different properties of the abstracted object.
But philosophers are deciding on one particular set of axioms- why? Justified by what? It can’t be physical reality, because the circle drawn on paper and the cylinders fabricated out of aluminum are now disconnected from the abstraction and cannot act a verification of the abstraction.
It was just an example, phyllo. No one here is deciding which axioms to use - I am simply pointing out that mathematics is based upon axioms and not upon empirical observation. It is my very point that this is not based upon empirical reality. It seems we have agreed.
A cylindrically shaped piece of steel basically fits your image of what a cylinder is. You could just call it a cylinder. A furry thing with four legs and one ear basically fits your image of what a dog is. You could just call it a dog. Both dog and cylinder are categories we create to group things together in certain ways.
Not sure who I’m agreeing and disagreeing with here, by the way. Just chiming in.
‘your image’= the abstraction
I don’t see a reason to think about a cylinder any differently than about a dog.
Me neither. There is no ideal cylinder and there is no ideal dog. Both are equally abstract, though one is described mathematically in a simple way. I guess that makes it more clinical or something, or more abstract, but not in any fundamental philosophical sense that I can see. Also, I think human beings invented the cylindrical shape, then we make them or see similar shapes in natural things. Whereas I don’t think we invent ‘dog’ in the abstract, in the absence of a bunch of similar beings we call ‘dogs’.
The difference it that there is only an ideal cylinder but no ideal dog.