Re: SUO: "Abstract" and "dimensionality" - Re: SUMOaxiomatization
Mike et al.,
That article about abstract objects in the Stanford Encyclopedia is
useful, but the author, like most 20th century analytic philosophers,
is unfortunately ignorant of the history of his own subject. He cites
Frege as the chief authority for the fundamental distinctions, even
though Frege was almost as confused as his buddies.
In the passage below, I have copied the excerpt from the Stanford
Encyc. article that discusses Frege's contribution. I certainly agree
that Frege was a very smart guy. But like most people who quote him,
Frege was ignorant of the long history of logic, especially the major
contributions by the medieval scholastics.
Frege preached against "psychologism" in logic, but he constantly
fell back into the old muddled ways of thought in his own terminology,
with his repeated use of words like "Gedanke" (thought) and "Urteil"
(judgment). Peirce used the scholastic term "propositio" (proposition),
which is sufficiently neutral to serve as the "mind-independent"
content without presupposing a necessary dependence on any kind of mind.
Even though the scholastics were devoted Christians (and one of the
chief authors, Peter of Spain, later became Pope John XXI), they were
never so naive as to claim support from the "mind of God" in order to
develop logic.
Most of the following quotation discusses ways out of Frege's muddle,
which the medieval scholastics and, of course, Peirce avoided with
their more sophisicated theories of semiotics.
The author also mentions Bolzano and Brentano as having made claims
that were "similar" to Frege's. That also distorts history, since both
of them were very familiar with the scholastic writings, and unlike
Frege, they did not succumb to the muddled terminology that they were
arguing against.
John Sowa
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Source: http://plato.stanford.edu/entries/abstract-objects/
One signal event in this development is Frege’s insistence that the
objectivity and a priori of the truths of mathematics entail that
numbers are neither material beings nor ideas in the mind. If numbers
were material things (or properties of material things), the laws of
arithmetic would have the status of empirical generalizations. If
numbers were ideas in the mind, then the same difficulty would arise,
as would countless others. (Whose mind contains the number 17? Is
there one 17 in your mind and another in mine? In that case, the
appearance of a common mathematical subject matter is an illusion.)
In The Foundations of Arithmetic (1884), Frege concludes that numbers
are neither external ‘concrete’ things nor mental entities of any sort.
Later, in his essay "The Thought" (Frege 1918), he claims the same
status for the items he calls thoughts -- the senses of declarative
sentences -- and also, by implication, for their constituents, the
senses of subsentential expressions. Frege does not say that senses
are "abstract". He says that they belong to a "third realm" distinct
both from the sensible external world and from the internal world of
consciousness. Similar claims had been made by Bolzano (1837), and
later by Brentano (1874) and his pupils, including Meinong and Husserl.
The common theme in these developments is the felt need in semantics
and psychology as well as in mathematics for a class of objective (i.e.,
non-mental) supersensible entities. As this new "realism" was absorbed
into English speaking philosophy, the traditional term "abstract" was
enlisted to apply to the denizens of this "third realm".
Frege’s way of drawing the distinction is an instance of what Lewis
(1986) calls the Way of Negation. Abstract objects are defined as those
that lack certain features possessed by paradigmatic concrete things.
Nearly every explicit characterization in the literature has this
feature. There are, however, several significant difficulties with this
approach, at least in its most familiar implementations.
According to Frege’s explicit account, the items in the "third realm"
are non-mental and non-sensible. But it is unclear what it means to call
an object mental or mind-dependent; and to the extent that the notion is
intelligible, it is quite unclear whether abstract objects in general
satisfy the condition. It is commonly supposed, for example, that the
game of chess is an abstract entity (Dummett 1973). But there is
certainly a sense in which the game would not have existed were it not
for the mental activity of human beings. So at least one sort of
mind-dependence would appear to be compatible with abstractness.
Moreover, it has sometimes been maintained that the paradigmatic
abstract entities -- mathematical objects, universals -- exist only as
ideas in the mind of God. The view may be outlandish; but is it a view
according to which abstract entities do not exist? Or is it rather a
view
ccording to which certain abstract entities are also mind-dependent?
Insofar as the latter interpretation is not straightforwardly
contradictory, the definition of "abstract" should not require
mind-independence.
Perhaps more importantly, Frege’s identification of the abstract with
the realm of non-sensible non-mental things entails that unobservable
physical objects such as quarks and electrons should be classified as
abstract entities. But this is at odds with standard usage, and almost
certainly with Frege’s intention.