Re: Set Theory: Should You Believe


herbzet@xxxxxxx wrote:

I think of logic as a branch of philosophy, no?

I always take it that philosophy has no subject matter but can be
'applied' (better - is an exploration of) to any subject--matter. So,
no, I do not think of logic as a branch of philosophy. If we are in
doubt about that, then remember that the philosopher can come along and
undermine the claim that logic is a branch of philosophy by presenting
a foundation of all logics:
If logic is a subject matter in philosophy, then the subject matter
itself performs the activity of philosophy. In that case, logic could
not examine itself. And indeed, logic does not examine itself: logic is
examined through the independent activity of philosophy. Philosophy is
not a topic with its own subject matter, but an activity.

Infinity in mathematics is challenged by 'finitism' and
constructivism'.
Numbers are constructed, not found. So there is no 'infinite number'.
Infinity, like zero, has no limit. So as a number is defined and
identified by a limit, or by some act (otherwise we could not
distinguish one number from another), then as infinity and xero have no
limit, they are not identifiable as numbers. It is the act of making a
number that identifies a number, and not merely the appearance. The
appearance is called a numeral.


1) It is unclear whether you are speaking for yourself, or
if you are assuming the voice of those whose opinions
you disagree with, or both. Perhaps you wrote in some haste.

2) You are aware, perhaps, that the term "limit" has a precise,
technical meaning in mathematics. It is widely used in
this technical sense, and in particular it has been used
extensively in the construction of some classes of numbers.


I am sure the meaning of 'limit' is technical, but I can state it as an
axiom of metaphysics if you like, that every technical term can be
reduced to originary, conceivable, and simple behaviours of everyday
attitudes and objects. Especially for a term like 'limit' which has
many uses in mathematics, it is helpful I think to find its originary
use and meaning, rather than imagine that 'its' use is justified in a
multiple of cases without having an idea of what 'it' might be. What
might the 'originary' use of limit be? As an 'argument' approaches a
point, let us call the point a limit, or we can call the limit the last
move of the argument. We can reduce that further to 'that which is
required to be identified'. That probably reduces again to 'that which
is identifiable'. In this case, for example, each of the elements in a
sequence is identified without stipulating a particular end-point.
But a sequence of numbers demands an end-point or limit for each of its
elements, or it would not be a sequence - composed of identifiable
elements. It follows that to stipulate a sequence without a limit
undermines the concept of sequence. We might then conclude that there
is no infinite sequence.

In discussion _of_ mathematics, it might be well to find
a synonym to use in its place, if you are not using the
term in this technical sense, in order to keep confusion to
a minimum. Or, alternatively, to make clear in what sense,
exactly, you are using the term.

It is difficult to comment on the substance of your post, since
I am unclear on what it is that you are asserting. Perhaps you
could further explicate? I enjoy reading your writing.

--
hz

.

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