| | Cal wrote, The abstraction "two" only refers to itself.
Wow! You said it. I didn't. The next thing you'll be telling me is that a ruler can measure itself; a scale, weigh itself; or a clock, time itself. Of course, there are self-referential concepts. The concept "concept" refers to itself, but only indirectly, because it refers to all concepts including itself. But a concept cannot refer only to itself, any more than ruler can measure its own length. A concept or an abstraction must, at the very least, refer to something other than itself. Then, you say,
Maybe the idea of a Platonic archetype is not so strange. From Wikipedia:
Platonism is considered to be, in mathematics departments the world over, the predominant philosophy of mathematics, especially regarding the foundations of mathematics.
Which shows the folly of consensus, even in the special sciences. Or are you now telling me that you believe in Plato's heaven?
The difference with the abstraction "red" is that the latter can't be defined without reference to the physical world (like the response of our visual system to light of certain wavelengths or something like that), while "two" can be defined without any reference to the physical world.
On the contrary, grasping the idea of a particular quantity requires an awareness of separate and distinct existents, from which one abstracts the idea. The abstraction retains the quantitative relationship, but omits the kind of existents from which the abstraction is derived. The quantity or number thus abstracted can then be applied to any existential units, but must apply to some in order to be meaningful. As you acknowledged in your previous reply, there is no such thing as pure two. Although one can reason mathematically without specifying the units, existential units are nevertheless implied in order for the number 'two' to make sense. It must be two somethings--two Objectivists, two Platonists, etc.
If you find it difficult to separate the abstract notion of "two" from its application in the real world, what are in your opinion the real world existents to which "+" refers? Or the number i? An integral? A 2735258-dimensional sphere?
The "+" sign refers to the concept of addition--of adding one thing to another. So "2 + 2 = 4" refers to the number of units that you get if you add two units to two other units. You're not seriously suggesting that the concept of addition has no existential referents, are you?
As for the number i, it's not so much a number as an operator like the "+" sign. As Ronald Pisaturo puts it,
The "imaginary number" i is an operator that must be applied to a unit not once but twice in order to yield a unit that is the negative of the original unit--i.e., i(squared) x 1 units = -1 units. When the operator is applied to a unit only once, it yields a unit that is not reducible to the original unit or its negative. Thus the operator i applies in cases where there are kinds of units that bear such kinds of relations. One such case is units of length in a two-dimensional plane, where the operator i is analogous to a right-angle rotation. For example, if the base unit is a mile east, then i operated on that unit is a mile north; i operated on a mile north (which is like operating twice on a mile east) is a mile west, which negates a mile east; i operated on a mile west is a mile south, which negates a mile north; and i operated on a mile south is a mile east again. Each of these four types of units is negated by the type of unit produced with two successive operations of i.
Just as the negative operator allowed us to express two kinds of units--positive and negative--in terms of one kind of unit, the i operator now lets us express four kinds of units in terms of one kind of unit. E.g., if 1 mile east is our base unit, then 3 miles north can be expressed as 3i miles east. Just as with the negative operator, the i operator achieves the result of restoring the condition of the uniform unit. (Ronald Pisaturo, "Mathematics in One Lesson--Conclusion," The Intellectual Activist, October 1998)
Mathematical operators or concepts of method have meaning insofar as they pertain to real relationships; otherwise they are not valid concepts. Such concepts are, of course, abstractions from abstractions, but the higher level abstractions ultimately depend upon and presuppose the lower level ones.
One can apply the same sort of analysis to your other two examples--the integral and a 2735258-dimensional sphere in order to determine if they bear any legitimate relationship to reality. Clearly, the integral does. And just as clearly, the 2735258-dimensional sphere does not. Of course, one can arrive at invalid abstractions, like the concept of God--which is a pure consciousness, without anything that is conscious.
Come to think of it, God is sort of like your view of number, isn't it--a pure abstraction that is off somewhere in Plato's heaven? So I shouldn't be too surprised if, in all seriousness, you should trot out the notion of a God or supernatural spirit. That is, after all, where your epistemology is leading you!
- Bill
(Edited by William Dwyer
on 2/04, 6:44pm)
(Edited by William Dwyer
on 2/04, 11:52pm)
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