Super discussion about negative numbers on the BBC
 

In article , Jon Yaeger
wrote:

in article , Rich Wilson at
wrote on 3/12/06 9:50 PM:


"Jon Yaeger" wrote in message
...
in article , Rich Wilson at
wrote on 3/12/06 9:23 PM:


"Jon Yaeger" wrote in message
...
in article , Rich Wilson at
wrote on 3/12/06 9:14 PM:


"flipper" wrote in message
...
On Mon, 13 Mar 2006 00:12:55 GMT, "Rich Wilson"
wrote:


"flipper" wrote in message
...
Which, btw, is exactly the same situation as those who don't think i
exists.

It doesn't. But then, "numbers" don't exist in any meaningful sense.

If you go back and look at the context that's been removed you'll see
that I said it's as 'real' as the 'real' numbers are. The point is
it's 'reality' is conceptually no different regardless of what common
conceptualization, or lack thereof, you now wish to obscure the
matter with.

I wasn't disputing your argument! Yes, i is as real as any other
number.

You can
have, say, five apples, but you can't just have five. It's just an
idea.

Ideas 'exist'. They must since you apparently have one ;)

I have an idea of, say, a machine that turns lard into banknotes. That
doesn't mean it exists. Equally, I have an idea of the number "five".
The
idea exists, the number doesn't necessarily exist.


Yes, but the idea of numbers is a priori; the lard converter is not.

I'm not familiar with the latin phrase and the Wikipedia definition ain't
helping... care to explain your argument in English?

This link should do it:

http://www.importanceofphilosophy.co...l_APriori.html

OK... so "a priori" knowledge is knowledge that has no reference to
reality... i.e. knowledge about things that don't exist.

My understanding of a priori as it applies to math is that we have a concept
of numbers that seem to be beyond argument, but on closer inspection, is
actually based upon faith. We can't really "prove" the number 5, for
example, but we operate on the assumption that "5" is true. As an article
of faith, perhaps it is "more empirical" (if such a non-sequitur can be used
to describe something that is based upon "faith") than, say, the imaginary
concept of a lard converter. We don't really depend upon concept of a lard
converter for understanding other concepts as we do numbers.

It is a bit of an abstract and subtle distinction.

Perhaps a philosopher or professional parser such as Mr. Byrnes can correct
me or shed more light on the distinction, if indeed there is one?

My job here is picking nits, "philosopher" and "professional parser" is
Peter Wieck's job.


Regards,

John Byrns


Surf my web pages at, http://users.rcn.com/jbyrns/

Super discussion about negative numbers on the BBC
 

Is there an advantage over just essentially representing the 'area' as a
vector result perpendicular to the surface and giving it a size of width x
length? (in effect, using a cross-product vector approach.)

I have no idea of whether there is (was, more aptly) any advantage in
using one method over another in the calculations for box-making. But
the guy who gave the lecture stated that for him, not having to deal
with sign-changes shortened the time it took to make the calculation
and set up the machines. Using " i " allowed him to ignore sign
changes... I am sure today that all of this is done by computer, and
the set-up will be largely automated. Further, that the guys on the
floor very likely do not understand how those settings were derived.
Back then, at least some of them did.

Peter Wieck
Wyncote, PA

Super discussion about negative numbers on the BBC
 

"flipper" wrote in message
...
On Mon, 13 Mar 2006 02:14:18 GMT, "Rich Wilson"
wrote:


"flipper" wrote in message
. ..
On Mon, 13 Mar 2006 00:12:55 GMT, "Rich Wilson"
wrote:


"flipper" wrote in message
m...
Which, btw, is exactly the same situation as those who don't think i
exists.

It doesn't. But then, "numbers" don't exist in any meaningful sense.

If you go back and look at the context that's been removed you'll see
that I said it's as 'real' as the 'real' numbers are. The point is
it's 'reality' is conceptually no different regardless of what common
conceptualization, or lack thereof, you now wish to obscure the
matter with.

I wasn't disputing your argument! Yes, i is as real as any other number.

You can
have, say, five apples, but you can't just have five. It's just an idea.

Ideas 'exist'. They must since you apparently have one ;)

I have an idea of, say, a machine that turns lard into banknotes. That
doesn't mean it exists. Equally, I have an idea of the number "five". The
idea exists, the number doesn't necessarily exist.


The difference is 'numbers' are part of a conceptual (mathematical)
system that is defined, consistent , well regulated, transportable,
and testable. That's enough for the conceptual world but for the 'hard
core realist' numbers, and the mathematical system, also apply to the
'real world' in useful ways. You may claim that '5' doesn't 'exist'
but it does, in fact, correlate to 5 'real world' apples. It's simply
the 'real world' conceptualized so that '5' works equally well with
horses, bails of hay, neutrinos, or any other 'real world' thing, as
well as it does for apples. Further, we'd find it impossible to
understand or explain the quantify of apples without it.

OK, but it's still just an abstract thought. Agreed?

I think the problem here is that "the number 5 exists" doesn't actually mean
anything. It's not made of matter or energy so it can't exist in the same
way that the apple can. There's no way you can directly experience "5" with
any of your senses. If "5" exists I'd like you to put one in the mail and
send it to me, on its own, without any apples or any other item that you
might want to apply the idea of "5" to.

On the other hand, you may claim to have the 'idea' of a money lard
machine but it isn't well defined, much less testable, even in the
conceptual sense, not to mention it contradicts reality as we
currently understand it.

OK, OK, I'll come up with a more sensible idea next time. But proving an
idea makes sense doesn't prove its existence.

Or, put another way, we accept the reality of things we can, by some
means, verify even if we can't 'see' them, like electrons and protons.

Stand in front of a source of beta radiation for long enough and you'd be
able verify the existence of electrons.

We can devise tests that 'confirm' their existence and, similarly, we
can confirm the 'existence' of 5 by it's adherence to mathematical
principles and useful, even necessary, applicability to the 'real
world'.

Go ahead, prove it. I think all you're going to show is that it's a useful
idea.

One the other hand, if we flip the coin over we could argue that
there's no such thing as 'reality' since everything we pretend to
'know' about it is simply our mind's own conceptualization.

I see where you're going with that but there's still a distinction between
abstract concepts such as numbers and non-abstract concepts such as apples.

Super discussion about negative numbers on the BBC
 

"flipper" wrote in message
...
On Mon, 13 Mar 2006 23:09:13 GMT, "Rich Wilson"
wrote:


"flipper" wrote in message
. ..
On Mon, 13 Mar 2006 02:14:18 GMT, "Rich Wilson"
wrote:


"flipper" wrote in message
m...
On Mon, 13 Mar 2006 00:12:55 GMT, "Rich Wilson"
wrote:


"flipper" wrote in message
news:n114125cjsid15km22ea3sml22u6u4p6vj@4ax. com...
Which, btw, is exactly the same situation as those who don't think i
exists.

It doesn't. But then, "numbers" don't exist in any meaningful sense.

If you go back and look at the context that's been removed you'll see
that I said it's as 'real' as the 'real' numbers are. The point is
it's 'reality' is conceptually no different regardless of what common
conceptualization, or lack thereof, you now wish to obscure the
matter with.

I wasn't disputing your argument! Yes, i is as real as any other number.

You can
have, say, five apples, but you can't just have five. It's just an
idea.

Ideas 'exist'. They must since you apparently have one ;)

I have an idea of, say, a machine that turns lard into banknotes. That
doesn't mean it exists. Equally, I have an idea of the number "five".
The
idea exists, the number doesn't necessarily exist.


The difference is 'numbers' are part of a conceptual (mathematical)
system that is defined, consistent , well regulated, transportable,
and testable. That's enough for the conceptual world but for the 'hard
core realist' numbers, and the mathematical system, also apply to the
'real world' in useful ways. You may claim that '5' doesn't 'exist'
but it does, in fact, correlate to 5 'real world' apples. It's simply
the 'real world' conceptualized so that '5' works equally well with
horses, bails of hay, neutrinos, or any other 'real world' thing, as
well as it does for apples. Further, we'd find it impossible to
understand or explain the quantify of apples without it.

OK, but it's still just an abstract thought. Agreed?

I think the problem here is that "the number 5 exists" doesn't actually
mean
anything. It's not made of matter or energy so it can't exist in the same
way that the apple can. There's no way you can directly experience "5"
with
any of your senses. If "5" exists I'd like you to put one in the mail and
send it to me, on its own, without any apples or any other item that you
might want to apply the idea of "5" to.

You're using a self fulfilling argument by first defining 'exist'
specifically to exclude what you will then argue doesn't 'exist'
because you first defined it away.

I'm not, I'm trying to find a definition of "exist" that could actually give
any meaning to the statement "numbers exist".

5 'exists' as a member of a mathematical system

....and *only* as a member of a mathematical system...

that meets the
mathematical criteria for 'existence'

Which are what exactly? Your definition of mathematical "existence" seems to
be what I'd call validity.

and it is in that same context
that the matter of whether the square root of -1 'exists' was being
discussed..

Your parochial definition of 'exist' seems intuitive on the surface
but, to turn a phrase, "an idea can change the world."

Not without people to think it.

However, that's
hard to stomach for something that doesn't exist

I don't have a problem with that.

and I submit that
your narrow definition leads to more mysticism, with non existence
changing the world, than accepting a broader view of it.

What I'm trying to say is that "existence" has no meaning when applied to
numbers.

On the other hand, you may claim to have the 'idea' of a money lard
machine but it isn't well defined, much less testable, even in the
conceptual sense, not to mention it contradicts reality as we
currently understand it.

OK, OK, I'll come up with a more sensible idea next time. But proving an
idea makes sense doesn't prove its existence.

Or, put another way, we accept the reality of things we can, by some
means, verify even if we can't 'see' them, like electrons and protons.

Stand in front of a source of beta radiation for long enough and you'd be
able verify the existence of electrons.

Well, at least the observed results will, hopefully, be consistent
with present day theory.

I'm mean they'd have a direct effect on your body that you could experience
with your senses.

One the other hand, '5' is consistent with present day mathematical
theory too.

We can devise tests that 'confirm' their existence and, similarly, we
can confirm the 'existence' of 5 by it's adherence to mathematical
principles and useful, even necessary, applicability to the 'real
world'.

Go ahead, prove it. I think all you're going to show is that it's a useful
idea.

Which I contend exists, in the broader view.

Meaning what exactly? How do you differentiate between an idea that exists
and one that doesn't?

One the other hand, if we flip the coin over we could argue that
there's no such thing as 'reality' since everything we pretend to
'know' about it is simply our mind's own conceptualization.

I see where you're going with that but there's still a distinction between
abstract concepts such as numbers and non-abstract concepts such as
apples.

If you're willing to accept subsets of concepts why not subsets of
existence?

I'm not sure what you mean by that.

As far as I can tell, some people have a notion of some kind of "ether" of
mathematical laws and rules that had to exist before any matter could exist
in the universe. That's what I'm trying to argue against.

Super discussion about negative numbers on the BBC
 

"flipper" wrote in message
...
On Tue, 14 Mar 2006 20:57:30 GMT, "Rich Wilson"

If you'd like to study on the philosophy of mathematics this might be
a place to start.

http://en.wikipedia.org/wiki/Philosophy_of_mathematics

Wow, I hadn't realised there was so much religion in mathematics. Platonic
heavens full of numbers?! I thought mathematicians were meant to be
*logical* people!


Your parochial definition of 'exist' seems intuitive on the surface
but, to turn a phrase, "an idea can change the world."

Not without people to think it.

And according to Schroedinger the cat is neither physically alive nor
dead till observation fixes the quantum state..

http://www.lassp.cornell.edu/ardloui...e/Schrcat.html

But to your 'complaint' with an idea changing the world, the mechanism
of implementation doesn't alter the fact that the world does change
and we have a 'classic', or for you - mystical, cause-effect
situation: the 'real world' being altered by what you claim doesn't
exist.

Would you mind not telling me what I think...? ;-)

I think you need to take a step back - maybe a few hundred of them - forget
all the philiosopy you've learnt and look at what's actually happening. I
guess this is one of those things that's really obvious until you start
thinking about it, and then you've got to do an awful lot more thinking to
make it obvious again.

This is what's obvious to me:. numbers are abstract ideas. They are concepts
we use to help us think about the world, and they're nothing more than that.
The fact that they're useful doesn't imply that we "discovered" them from
somewhere. The human brain is quite capable of coming up with useful
abstract ideas and generalisations on its own; in fact, that's exactly why
it's so useful.

Your cause-and-effect argument makes no sense to me. You could trace the
chain of effects from a real-world phenomenon, through the abstract idea
stage via a chain of electrical and chemical processes in the brain, and
back to the real world as movement caused by the person's nerve impules.
Where's the problem?


Or, put another way, we accept the reality of things we can, by some
means, verify even if we can't 'see' them, like electrons and protons.

Stand in front of a source of beta radiation for long enough and you'd
be
able verify the existence of electrons.

Well, at least the observed results will, hopefully, be consistent
with present day theory.

I'm mean they'd have a direct effect on your body that you could
experience
with your senses.

Yes, I knew what you meant. But concluding that 'effect' is due to
beta radiation, as opposed to the gods being angry with you or some
other belief, is a matter of present day scientific theory.

You could say that about pretty much anything.

The only reason you accept the 'existence' of the electrons is because
the 'results' are consistent with a theory that's useful in what is,
apparently, the only kind of 'existence' you accept. But, similarly,
numbers exist because that (mathematical) theory is also useful, even
in the same 'reality' you insist is the only one. It isn't the same
kind of 'existence' you insist on but the usefulness is observable.

Why is it so hard to accept that a useful idea may still be just an idea?


One the other hand, '5' is consistent with present day mathematical
theory too.

We can devise tests that 'confirm' their existence and, similarly, we
can confirm the 'existence' of 5 by it's adherence to mathematical
principles and useful, even necessary, applicability to the 'real
world'.

Go ahead, prove it. I think all you're going to show is that it's a
useful
idea.

Which I contend exists, in the broader view.

Meaning what exactly? How do you differentiate between an idea that exists
and one that doesn't?

The same way one differentiates between 'beliefs' of what is 'real'
and, back to our electrons, it wasn't but a few hundred years ago
people would have told you they were nothing but your imagination.

Questions about the existence of sub-atomic particles have typically been
answered when someone's managed to isolate one of them. OK, you can question
their measuring equipment but that can be proved beyond reasonable doubt to
be working as expected.
I'd like to see you do the same for a number.


You would, of course, say they're 'real' whether man knew of them or
not

No I wouldn't.

but Schroedinger and Heisenberg will explain that, well, not
really.

Yes, taken to extremes, reality becomes "everything I am observing right
now". I think you'll find that my argument still holds, as numbers are not
observable.

At least not in your Newtonian view of it as they're a wave
function and don't 'exist' at any particular 'place and time'. To make
matters worse, beyond not ever being either 'here or there', they
also, apparently, regularly oscillate in and out of 'existence'
because they go places not possible in 'this reality' (loose use of
the term). One version is called "tunneling" and is the basis of a
'real world' device: the tunnel diode.

It's odd how people get so wedded to a model (like electrons being little
solid balls) that when the model is proved wrong they have to start
labelling things as "impossible". At the end of the day an electron is just
an electron and it can do whatever electrons do.


As far as I can tell, some people have a notion of some kind of "ether" of
mathematical laws and rules that had to exist before any matter could
exist
in the universe. That's what I'm trying to argue against.

Well, I don't know who those 'some people' are or what they said,

Try reading your own Wikipedia reference!

precisely, and I certainly didn't say it nor am I sure what the point
is.

Do you think the laws of physics exist?

No. Again, they're abstract generalisations and as such "existence" has no
meaning for them. Do you think they exist?

You seem to think electrons
exist but 'electron' is just a word. If I have a capacitor with 5
electron volts on it do the electrons exist? Does the electromagnetic
field exist? Does the magnitude 5 exist? They're all just words and
'concepts', you know. But don't they all 'relate' to the 'real world'?

Some are concepts to which the verb "to exist" can apply, some aren't.
Have you ever done any object-oriented computer programming? It forces you
into a way of thinking that makes problems like this very easy. "Electrons"
would be a subset of "Real Concepts", which would in turn be a subset of
"Concepts". "Numbers" would, again, be a subset of "Concepts". The property
of "Existence" would be a true-or-false value within "Real Concepts", and so
"Electrons" would inherit the property of "Existence" whereas "Numbers"
wouldn't. That'd probably be easier to understand on a diagram. Why can't
you draw stuff on Usenet?!


You may invoke the Shakespearian "a rose is still a rose by any other
name," fair enough (and my point) but the mathematical relationship
between field strength and the number of electrons is also the same
regardless of the symbols used. And they 'add' the same whether one
uses those words or not. Mathematics 'exists' because it relates to,
and describes, the 'real world' just as the word 'electron' represents
a thing you consider 'real'. One is what you think of as the
'physical' where the other speaks to properties and behavior of those
'physical' things, and I contend that their properties and behavior
are just as 'real' and that, indeed, you don't have 'reality' without
them.

And maybe that will help explain why I don't understand the 'which
came first' point you say you're arguing against.

It's not "which came first"!! It's "did one of them come at all"!

It's like asking
which came first, the electron or it's negative charge? huh?

Super discussion about negative numbers on the BBC
 

Flipper:

You are trying wayyyyyyyyy to hard.

pi exists because we _can_ draw a circle. That there is no other part
of a circle that relates directly to the diameter or the area, yet we
can still continue to draw them and manipulate and calculate them (even
if a little bit fuzzy around the edges) is enough.

i exists because we _can_ calculate negative areas using square roots.
i takes care of those pesky signs. Keep in mind that something as
obvious as 0 did not exist as a written symbol until relatively
recently, historically.

And so forth. Obvious practical applications of real and imaginary
numbers happen every minute of every day. It hardly matters whether
they 'exist' or not. Their driving concepts do and have existed
forever.

I would refer you to Clarke's Laws here. Many real things exist because
someone somewhere had an idea that flew in the face of received wisdom.
So, as a codicil to "Question Authority", we should establish that
"Received Wisdom Isn't".

http://www.lsi.usp.br/~rbianchi/clarke/ACC.Laws.html

Peter Wieck
Wyncote, PA

Super discussion about negative numbers on the BBC
 

mega-snip

I give up. There's no way you're ever going to see that you're wrong! ;-)

I'd still be interested to know a few things about your view though:

- If numbers exist, when did they start to exist? At the Big Bang? (aka
Monday if you happen to be a creationist :-) Or did, say, 5 start to exist
the first time there were 5 things in the universe?
- When will they cease to exist?
- Who or what created them?

....and finally, without contradicting any of your responses to my last
post...
- What are they made of?

:-)

Super discussion about negative numbers on the BBC
 

wrote in message
oups.com...
Flipper:

You are trying wayyyyyyyyy to hard.

pi exists because we _can_ draw a circle.

Aaah but without circles there'd be no pi therefore the idea* of a circle
must have existed before the idea of the ratio of its circumference to its
diameter... I think you've got yourself into a chicken-and-egg situation
there. And Mr. Darwin can't help you with this one!

* "idea" isn't necessarily the right word there but I'm sure you see what I
mean...

Super discussion about negative numbers on the BBC
 

* "idea" isn't necessarily the right word there but I'm sure you see what I mean...

Amongst the almost uncounted thousands of English words, Idea is just
fine. And it is 'necessarily' the right word.

Accordingly, you trap yourself into necessarily-human concepts (no, I
am not going 'intelligent design'). Tree capillaries are nearly perfect
circles. Something dropped into still water causes near-perfectly
circular ripples which propagate based on pi-based relationships.
Molecules make up these circumferences and fit as perfectly as our
pen-or-ink efforts, or more-so. So, if nothing else, Nature understands
pi perfectly... and uses it. All the time. What humans did is merely
_describe_ it.

So, you need to understand that most of the concepts discussed in this
thread are entirely independent of us. What we do is describe what
*must* exist even if we cannot describe it perfectly given our fairly
rigid and absolutely limited understanding. Just keep in mind that if
our understanding were 'unlimited', no progress could be possible. We
push those limits (Clarke's second law) in order to increase those
limits.

Peter Wieck
Wyncote, PA

p.s.: Chicken and egg.... the egg is potential, the chicken is fixed.
Give me the egg every time.

Super discussion about negative numbers on the BBC
 

wrote in message
ups.com...
* "idea" isn't necessarily the right word there but I'm sure you see what
I mean...

Amongst the almost uncounted thousands of English words, Idea is just
fine. And it is 'necessarily' the right word.

Accordingly, you trap yourself into necessarily-human concepts (no, I
am not going 'intelligent design').

That's EXACTLY what I was trying to get you to avoid... never mind.

Tree capillaries are nearly perfect
circles. Something dropped into still water causes near-perfectly
circular ripples which propagate based on pi-based relationships.
Molecules make up these circumferences and fit as perfectly as our
pen-or-ink efforts, or more-so. So, if nothing else, Nature understands
pi perfectly... and uses it. All the time. What humans did is merely
_describe_ it.

What you're saying there is that our current way of understanding the
universe, involving numbers, circles, ratios like pi and so on - is the ONLY
way to understand it. You would, presumably, deny that some other race or
species could start from scratch and come up with a totally different but
equally valid way of thinking about the world.

That seems very unlikely to me.


By the way, I thought of a good analogy...

Your insistence that numbers exist is, to me, like insisting that numbers
are green. Or female. Or Welsh. The human brain lets us take a concept like
colour from one type of object and stick it on to something else, whether or
not it has any meaning in that situation. Before arguing about what colour
the number 7 is, we should consider whether the question is valid.