On Thu, 23 May 2013 14:46:08 +0000, benj wrote:
>
> Ah! Mathematics isn't "invented" or made up in someone's mind, it is
> "discovered"!
Now you are getting it.
The branch of mathematics that is formally known as "analysis" is
pure discovery as every practitioner will readily concede.
Other "branches" of mathematics, such as algebra, some will claim
are acts of human creativity. But if the process of creation is
examined carefully, it is realized that such "creation" is tantamount
to an increasing generalization based on pre-existing, or pre-discovered,
forms. Function "spaces" or the theory of "groups" are just two examples
of this pseudo-creativity (which is actually discovery).
> Just where is this place where square roots of minus one
> are laying about on the dirt waiting to be discovered?
You actually answer this question yourself:
>
> NO numbers are real
> things. "Two" isn't a real thing it's an abstraction of the set of all
> couples.
>
This is *immanence* and it *is* an actual reality.
Given a set of premises there is only one conclusion that is
permitted based upon the indwelling logical necessity.
Furthermore, *any* intelligent entity, not just the human,
could discover this conclusion because the necessity
is beyond any such intelligence. In this sense the
conclusion is very, very real.
>
> So you say math is a "continuum" and totally self-consistent over that
> "continuum"? An interesting premise. But since you know everything there
> is to know, could you please tell us lesser beings if parallel lines meet
> at some point or do they never meet. Both things seem to be in the
> "continuum" and yet are contradictory.
>
The basis for mathematics is the number "continuum" which is the
idea of an unbroken dimensional extent. We cannot model the world
without it.
Although the continuum is, well, continuous, is has been "broken"
or disaggregated into a set of distinct entities which we formally
call the complex numbers. The real numbers, although distinct, together
create an unbroken "continuum" which may seem to be thoroughly impossible
and even paradoxical but that's the inhuman beast known as mathematics.
That's is what I meant by the "continuum" of complex numbers and it does
possess a definite structure and properties, all of which have been
discovered by human investigation.
In such a continuum, "parallel lines" would actually be equations of
a certain form. (We don't talk about "lines" in the modern world.
We only speak of equations. "Lines" were the feeble concepts of an
ancient civilization.)
These equations do not intersect in certain spaces (or point sets),
but we can choose to "complete" or "extend" the space to include
points at infinity. (Some call this the projective plane.) When
this is accomplished, all the equations now have simultaneous solutions
(i.e. all parallel lines will meet). There is no contradiction. The
issue is that some spaces are more "complete" than others.
There are many reasons for wanting to "extend" the space, but we
would be getting far off from our original track to consider these
reasons.
We would also be getting off track to consider another fundamental
number system: the integers. Although the integers are a subset
of the continuum, they also can be considered separately, but when
we do so, serious problems will arise. The ancient Diophantine
equations are an example of "mixing" particle concepts, or integers,
with the continuum.
Again, unfortunately, this is a vast topic that cannot be adequately
covered here.
>
> "Two" isn't a real thing it's an abstraction of the set of all
> couples. The number two is just a symbol for that abstraction. Your
> transference of those abstractions to reality is fantasy of the highest
> order.
>
Here we find the gist of the disagreement.
The "two-ness" of an aggregate is an attribute which can be
discovered by *any* intelligence. Indeed, there is some evidence
that even non-human animals can discern the number of objects
in certain collections, i.e. can count. So obviously, the idea
of a counting number, that is, "two-ness" or "three-ness" and
so on, does not depend on our human existence. The "number"
is actually there for any intelligence to discover.
We humans can discover more than lower animals, but there is
certainly a limit to our intelligence. This means that there
exist relations which we cannot discover, in the same way that
a lower animal cannot discover, for example, a quadratic
relationship or the number pi.
So, to conclude, there is no need to transfer the abstraction
to reality in some act of human creation. The abstraction
is already there. Just ask your pet dog or canary. (Don't
bother with your goldfish; he doesn't know.)
>
> Is pi real? Are circles real? Do circles even exist? If circles don't
> exist, how in hell can pi be anything but a figment of some drunken
> Italian's imagination? Pi it NOT "present" in anything! Pi is an
> abstraction and mental construct. To think it exists is mathematical
> ignorance of the highest order.
>
Circles are immanent virtually everywhere.
When things move to where they are not, such as when when we spill
a drop of liquid onto the floor, or when that drop diffuses into
the surrounding space, a circle (or sphere) is formed.
When I swing a bolas above my head, voila, there is a circle.
In the same way that we discover "three-ness" we, or any other
intelligent being, will discover the circle, and hence pi.
If pi is a mental construct then it must pre-exist in the brain,
in the same way that grammar or parental instincts pre-exist.
We all should catch a glimpse of pi whether or not we ever consider
a circle. But this never happens. No one ever dreams of pi.
We discover pi as an indwelling relation the in same way that we
discover a skeleton when we tease apart an animal body.
>
> If the moon does not exist until someone looks at it, it's certainly
> clear that pi and square root of minus one do not exist if there are no
> humans to think about them! That is simple modern physics.
>
As I already mentioned, just ask your dog. Ask a farmer's horse.
They can count. They can understand "two-ness," and they existed
long before the humans evolved from the apes.
Perhaps we will one day make contact with other intelligent beings.
(This is extremely unlikely but not completely impossible.)
If/when this happens, what will we talk about? We'll talk about
pi, of course, and all of the other mathematical objects that
we each have discovered. Then we'll share a laugh over the
mathematical reality deniers.
Just like wild flowers or juniper berries, mathematics is there
for the picking.