Do you believe this statement is true:
(n – n = 0) = (n = n)
It says: "To say that the difference
between n and n = 0, is to say that
n and n are the same (n = n)."
It seems quite clear to me that
(0 difference) = (same as), and
it also seems that that is just
what the statement above says.
Perhaps the right question is not
if the statement is true or false,
but is it mathematically proper?
**************************************************************
VISIT Ian Williams Goddard ——–> http://Ian.Goddard.net
______________________________________________________________
I would think that this is an abuse of the
"=" symbol.
I generally use something like this:
(n-n=0) <==> (n=n)
That little double headed arrow thingy does not
translate to ASCII very well but it should be read
as:
(n-n=0) if and only if (n=n)
The arrows going both ways — that’s meant to
symbolize implication both ways:
(n=n) implies that (n-n=0)
and the other way
(n-n=0) implies that (n=n)
But then again, I am not a mathemetician and
this stuff is only a hobby.
Jason W. Paul
Ian Goddard <I…@GODARD.NET> wrote in article
<35ecd1df.130899…@news.erols.com>…
- Hide quoted text — Show quoted text -
> Do you believe this statement is true:
> (n – n = 0) = (n = n)
> It says: "To say that the difference
> between n and n = 0, is to say that
> n and n are the same (n = n)."
> It seems quite clear to me that
> (0 difference) = (same as), and
> it also seems that that is just
> what the statement above says.
> Perhaps the right question is not
> if the statement is true or false,
> but is it mathematically proper?
On 2 Sep 1998, "Jason W. Paul" <gi…@fuse.net> wrote:
>> Do you believe this statement is true:
>> (n – n = 0) = (n = n)
>I would think that this is an abuse of the
>"=" symbol.
>I generally use something like this:
>(n-n=0) <==> (n=n)
IAN: Mutual implication was also my
first thought, but it seems to me
that (0 difference) = (same as),
in that the two are the same
thing, and thus are not two
things that imply each other.
**************************************************************
VISIT Ian Williams Goddard ——–> http://Ian.Goddard.net
______________________________________________________________
On Wed, 2 Sep 1998, Ian Goddard wrote:
> Do you believe this statement is true:
> (n – n = 0) = (n = n)
This notation is very nonstandard, but the text (clipped) is clearer.
The way one would say this is
For any number n, n-n=0 if and only if n=n.
For "number", one can substitued "real number", etc, or even "vector".
"If and only if" is often abreviated "iff" [sic--two f's].
Amusingly, n-n is not always 0 on computers! If n is a floating point
type, then it is possible for n to be obtain the value NaN, meaning
"not a number", and then lots of things are special cased. This
can happen if n is a floating point type assigned to 1.0/0.0. (Note
that for integer types the correspoding division will generally
result in your program stopping with a division by zero error, but
for floating point numbers on some machines the division gives
"NaN" or maybe "positive infinity". There are several relevant
standards and I’m not an expert, but I figured you might want to
be aware of the issue).
Best wishes,
Mike
In article <Pine.GSO.3.95.980902215610.15020A-100…@esther.rad.tju.edu>, "Dr.
Michael Albert" <alb…@esther.rad.tju.edu> writes:
>On Wed, 2 Sep 1998, Ian Goddard wrote:
>> Do you believe this statement is true:
>> (n – n = 0) = (n = n)
>This notation is very nonstandard, but the text (clipped) is clearer.
>The way one would say this is
> For any number n, n-n=0 if and only if n=n.
>For "number", one can substitued "real number", etc, or even "vector".
>"If and only if" is often abreviated "iff" [sic--two f's].
It depends on what you mean by "number", by "n – n", etc!
For example, addition of ordinal numbers is not commutative,
and so sometimes a distinction is made between the left
difference, – b + a, and the right difference, a – b. If this is
done (and we let w represent the infinite ordinal omega), then
w – w is not well defined (!) (although – w + w is 0) and yet it
is true that w = w.
Cheers,
David Cantrell
Im Artikel <35ecd1df.130899…@news.erols.com>, I…@GODARD.NET (Ian Goddard)
schreibt:
>Do you believe this statement is true:
> (n – n = 0) = (n = n)
==> Before you read on: excuse my BAD english! I’m trying to
express as good as possible. Since I do know the mathematical
terminology only in my native language (german), some expressions
seem to be unnecessarily long and complicated. Sorry! <==
In my opinion the question should be "Do you think the statement
(n-n=0)=(n=n) makes any sense?" and not if it is true!
A statement can have the logical value "true" or "false" only if it actually
is a correct and meaningful statement in a certain language.
Thus the first question is: What is the language? Is it the language of
A) mathematical equations or B) a certain computer language or etc., etc.,?
A)
In the language of mathematical equations the operator "=" is used in two
diffent forms of sentences:
1) in "statements" which can be true or false, e.g. 2+5=7 (true)
or 3-2=4 (false).
2) in "statements with variables", e.g. 2+x=7 or x*x = -1. Here we are looking
for solutions x to solve the equation. "Statements with variables" are not true
or
false; they are solvable or unsovable within a certain set of allowed values.
In the language of mathematical equations "n-n=0" and "n=n" obviously are
"statements
with variables" and both are solvable, e.g. within the set of natural numbers.
A meaningful question concerning these equations is if they have the same
set of values which solve them, e.g. 2+x=7 and 5*x=25 have the same set
of solutions ( –> 5 ), 1+x=2 and x*x=25 don’t have ( 1 resp. 5).
In the mathematical language we have a particular operator "<=>" ("is
equivalent to")
to express this. We can construct new sentences in the mathematical language
using this operator. This sentences again are true or false, i.e. they are
statements.
>It seems quite clear to me that
>(0 difference) = (same as), and
>it also seems that that is just
>what the statement above says.
Obviously this operator should be used to express the quoted expression:
(n-n=0) <=> (n=n)
This is a true statement, where (n-n=0)=(n-n) in our mathematical language is
a meaningless sentence.
============
B)
In certain programming languages (e.g. BASIC) the discussed expression may be
syntacticaly correct and meaningful.
Here "=" is used as a binary operator which leads to a boolean result (true or
false)
when applied to operands (expressions) of the same type. The result is true if
the
evaluated expression on the left side results in the same value as the
evaluated
expression on the right side of the operator, otherwise false.
The "="-operator in a programming language like BASIC is on the same
syntactical
level as any other binary operator, e.g. the "+"-operator.
Note that the BASIC-construct "n-n" is on the same logical level as the
mathematical
"5-5". Even if we call "n" a "variable" in programming languages it is used
like any
other constant value in expressions (you may call a variable in programming
languages
a variable constant, since it always has a value (when asigned to it, otherwise
the value
"undefined" or "nil" or a default value) which it is evaluated to before the
expression
is evaluated).
The BASIC (n-n) is logicaly like the mathematical "statement" and not like the
mathematical "statement with variables" !
So the expression (n-n=0) = (n-n) can be evaluated like any other expression
with binary operators. First evaluate the left side: n-n=0 results in true. The
right
side as well results in true. Now we have to evaluate the expression true =
true,
which again results in true.
==============
All this shows that we always have to be clear on which language we are using
and
how the operators in the particular languages are defined.
==============
Best regards
Frank.
P.S.: I’m looking for a english/german-dictionary on mathematical terminology.
Thank’s in advance for all hints!
On Wed, 2 Sep 1998, "Dr. Michael Albert" wrote:
>On Wed, 2 Sep 1998, Ian Goddard wrote:
>> Do you believe this statement is true:
>> (n – n = 0) = (n = n)
>This notation is very nonstandard, but the text (clipped) is clearer.
>The way one would say this is
> For any number n, n-n=0 if and only if n=n.
IAN: If the equals, or "same as," symbol isn’t
indicated, does that mean that 0 difference is
not the same as same as? But obviously it is.
It seems to me that mutual implication, or <=>,
is indicated between two things that are not
exactly the same, for example, good <=> evil.
But <=> isn’t indicated if the two are the
same, such as in this case: 12 <=> 12,
which is properly stated 12 = 12.
As such, while it could be said that good <=>
evil, it would also be said that good =/= evil,
but we could not say 0 difference =/= same as.
**************************************************************
VISIT Ian Williams Goddard ——–> http://Ian.Goddard.net
______________________________________________________________
Im Artikel <35eda0ee.183915…@news.erols.com>, I…@GODARD.NET (Ian Goddard)
schreibt:
>[]
>but it seems to me
> that (0 difference) = (same as),
> in that the two are the same
> thing, and thus are not two
> things that imply each other.
Hi there Ian,
this is my second reply to your question. I sent the first before I read your
reply to Jason. If my replies overlap, please read #1 first!
As a mathematically thinking reader I feel a bit angry about your
argumentation,
so I can’t wait with my reply. ;-)
> the two are the same thing []
a) I can’t see "two things", I can’t even see any "things"! I just see 7
symbols on
the left side, (n-n=0), and 5 symbols, (n=n), on the right side!
b) And these two sequences of symbols are obviously not the "same"
Now the question is, how do we interpret these sequences of symbols?
One interpretation was an equation with variables vary over natural numbers
and to ask for the set of natural numbers which make it to a true statement
when n is substituted with these values.
After that we can compare the two sets of values and look if there is some
relation between the two equations. There is! Both equations have the same(!)
set of values solving them.
You see, we had two different "questions": The first indicated with the symbol
"=" in the equations (n-n=0) and (n=n); the second indicated with the symbol
"<=>" between the two equations.
In my opinion, we don’t have "identity" in mathematics!
We only have sequences of symbols we have to interpret in a particular way.
And we have rules for sequences of symbols to transform to or substitute with
other sequences of symbols.
So 2+2 and 4 are not identical. But we have rules to transform the sequence
of symbols "2", "+", "2" into the symbol "4".
I don’t mean this as an use- or senseless game of symbol shifting. There always
is the interpretation of symbols.
But never forget the difference between the symbol and its interpretation.
And never forget that we may have different symbols/operators (e.g. "=", "<=>")
for "same" words in natural language.
Best wishes
Frank.
Im Artikel <35ef62f3.7273…@news.erols.com>, I…@GODARD.NET (Ian Goddard)
schreibt:
Datum: Fri, 04 Sep 1998 04:07:39 GMT
>IAN: If the equals, or "same as," symbol isn’t
> indicated, does that mean that 0 difference is
> not the same as same as? But obviously it is.
After reading your last reply I suppose to understand where the problem is:
We are not using the same language!
You asked a "philosophical" question:
"if there is no difference between … does it mean that … is identical to
…" (or so)
But you tried to express it in a quasi "mathematical" language (at least you
used mathematical symbols).
My answer considered only the mathematical aspects of your presentation. I
didn’t see that you used
a mathematical symbolism in a more or less inadequate way, i.e. you used the
mathematical
symbolism in a ad hoc manner to represent your philosophical question
neglecting that there is
a well defined calculus and theory behind all the symbolism.
Don’t mix up languages!
You can’t use a mathematical symbolism with (in its context) well defined
meaning
to express ad hoc natural language / philosophical questions, e.g. good => evil
, good+evil=the almighty god,
(jesus christ – holy ghost) >= man, …
By the way: I had this inspiration after I visited your homepage, learning that
you are not a mathematician,
but a philosopher and poet.
Greetings
Frank.
On 04 Sep 1998 08:19:37 GMT, rom…@aol.com (Romeni) wrote:
>I…@GODARD.NET (Ian Goddard) wrote:
>>[]
>>but it seems to me
>> that (0 difference) = (same as),
>> in that the two are the same
>> thing, and thus are not two
>> things that imply each other.
>Hi there Ian,
>this is my second reply to your question. I sent the first before I read your
>reply to Jason. If my replies overlap, please read #1 first!
IAN: Yes I did, sorry, I was
downloading from sci.math only,
as it went to sci.math,sci.logic.
Thanks for your replies! I think
you address the issue quite well.
>As a mathematically thinking reader I feel a bit angry about your
>argumentation,
>so I can’t wait with my reply. ;-)
IAN: I don’t think asking a question and
looking for an answer is nonmathematical
thinking, even if it gets you angry.
- Hide quoted text — Show quoted text -
>> the two are the same thing []
>a) I can’t see "two things", I can’t even see any "things"! I just see 7
>symbols on
>the left side, (n-n=0), and 5 symbols, (n=n), on the right side!
>b) And these two sequences of symbols are obviously not the "same"
>Now the question is, how do we interpret these sequences of symbols?
>One interpretation was an equation with variables vary over natural numbers
>and to ask for the set of natural numbers which make it to a true statement
>when n is substituted with these values.
>After that we can compare the two sets of values and look if there is some
>relation between the two equations. There is! Both equations have the same(!)
>set of values solving them.
>You see, we had two different "questions": The first indicated with the symbol
>"=" in the equations (n-n=0) and (n=n); the second indicated with the symbol
>"<=>" between the two equations.
>In my opinion, we don’t have "identity" in mathematics!
>We only have sequences of symbols we have to interpret in a particular way.
>And we have rules for sequences of symbols to transform to or substitute with
>other sequences of symbols.
>So 2+2 and 4 are not identical. But we have rules to transform the sequence
>of symbols "2", "+", "2" into the symbol "4".
IAN: I like your analysis. 2+2 and 4 are two
different ways of saying the same thing. We
could say "Washington D.C." or we could
say "70-miles south of Baltimore," and we
point to the same thing in two different
ways (assuming D.C. is 70 miles S of B).
>I don’t mean this as an use- or senseless game of symbol shifting. There always
>is the interpretation of symbols.
>But never forget the difference between the symbol and its interpretation.
>And never forget that we may have different symbols/operators (e.g. "=", "<=>")
>for "same" words in natural language.
IAN: I understand what you said in your
other post (one of them) about mixing a
philosophical statement and a mathematic
statement, but "0 difference" and "same
as" are, shall we say, members of the
set of mathematically relevant concepts.
The Webster’s New World Dictionary of
Mathematics uses the example of the =
symbol "Mark Twain = Samual Clemes,"
and in total implies that = is equal
to "same as." But that may be a crude
example. In the case that we are equat-
ing numerical identities, it may follow
that the = should only apply to the
numeric values, as opposed to meta-con-
cepts like difference and similarity,
which apply to and/or can be expressed
with those same numbers.
Obviously, if (n-n=0) = (n=n) and n is
not equal to 0, we have a contradiction;
the exploration of that possibility was
the reason that I raised the question.
**************************************************************
VISIT Ian Williams Goddard ——–> http://Ian.Goddard.net
______________________________________________________________
Why did you switch this discussion from sci.logic (where I found it the first
time)
to sci.math. My answers can can found in sci.logic. I’m not going to switch the
NG.
Greetings Frank.
In article <1998090504261200.AAA20…@ladder03.news.aol.com>,
Romeni <rom…@aol.com> wrote:
>Why did you switch this discussion from sci.logic (where I found it
>the first time) to sci.math. My answers can can found in sci.logic.
>I’m not going to switch the NG.
It was not switched, it was crossposted, and you may have removed
sci.math later. An article can have more than one newsgroup on the
Newsgroups: line.
I think I read somewhere that the AOL newsreader does not allow more
than one newsgroup if you change the line, but at least some versions
will preserve the list of newsgroups if it is already present in the
article that you follow up, possibly up to a limit (5 or so?).
Greetings,
Ørjan.
–
A pro-spam bill has passed the US Senate, and is now in the House of
Representatives. See <http://www.cauce.org> and
<http://www.ybecker.net/>.
Ian Goddard wrote..
- Hide quoted text — Show quoted text -
> Do you believe this statement is true:
> (n – n = 0) = (n = n)
> It says: "To say that the difference
> between n and n = 0, is to say that
> n and n are the same (n = n)."
> It seems quite clear to me that
> (0 difference) = (same as), and
> it also seems that that is just
> what the statement above says.
> Perhaps the right question is not
> if the statement is true or false,
> but is it mathematically proper?
Dear Ian William
It is not mathematically proper.
If it where, then (n-n=0)-(n=n) = 0
more about my opinion in this case : visit newsgroup sci.math, where the
same question(?) is posed.
It doesn’t really matter what format your numbers are in ( floating
point, integer, etc. ) on a computer. The statement is true. Whatever 0
is, n-n equals it. Therefore, the statement:
n-n=0 evaluates to 1 ( boolean TRUE )
n=n evaluates to 1 ( boolean TRUE )
Therefore
(n-n=0) = (n=n) is a 1.
I’m also assuming your = are really ==.
JSV
Why isn’t (n-n=0)-(n=n) = 0?
In article <35F3B410.6E203…@rtsys.com>,
Joseph Virzi <jvi…@rtsys.com> wrote:
:n-n=0 evaluates to 1 ( boolean TRUE )
n-n=0) = (n=n) is a 1.
:
:n=n evaluates to 1 ( boolean TRUE )
:
:Therefore
:
:
:I’m also assuming your = are really ==.
In my best Ambrose Bierce style (from “A Mathematica Crib Sheet: But It’s
OK, You’ll Need It”:
x == y Tests whether x is equal to y. You may think it a perversion to
take the innocent = sign, which is used in mathematics to denote
a statement of fact (“x is equal to y”), and turn it into a
verb (“x = y” becomes “set x equal to y”), after which you
must invent a monster == to regain your statement of fact (“x
is equal to y”), but there you have it–computer science majors
will recognize the C programming language.
I have never understood why K&R et al. didn’t use := for the assignment
statement. Because it required TWO symbols?!
Hunt-and-peck typists?
–Ron Bruck
What could possibly make anyone believe a number is equal to itself?
- Hide quoted text — Show quoted text -
Ian Goddard wrote:
> Do you believe this statement is true:
> (n – n = 0) = (n = n)
> It says: "To say that the difference
> between n and n = 0, is to say that
> n and n are the same (n = n)."
> It seems quite clear to me that
> (0 difference) = (same as), and
> it also seems that that is just
> what the statement above says.
> Perhaps the right question is not
> if the statement is true or false,
> but is it mathematically proper?
> **************************************************************
> VISIT Ian Williams Goddard ——–> http://Ian.Goddard.net
> ______________________________________________________________
In article <6t12aj$ci…@math.usc.edu>,
Ronald Bruck <br…@math.usc.edu> wrote:
@In article <35F3B410.6E203…@rtsys.com>,
@Joseph Virzi <jvi…@rtsys.com> wrote:
@
@:n-n=0 evaluates to 1 ( boolean TRUE )
@:
@:n=n evaluates to 1 ( boolean TRUE )
@:
@:Therefore
@:
@:(n-n=0) = (n=n) is a 1.
@:
@:I’m also assuming your = are really ==.
@
@In my best Ambrose Bierce style (from “A Mathematica Crib Sheet: But It’s
@OK, You’ll Need It”:
@
@x == y Tests whether x is equal to y. You may think it a perversion to
@ take the innocent = sign, which is used in mathematics to denote
@ a statement of fact (“x is equal to y”), and turn it into a
@ verb (“x = y” becomes “set x equal to y”), after which you
@ must invent a monster == to regain your statement of fact (“x
@ is equal to y”), but there you have it–computer science majors
@ will recognize the C programming language.
@
@I have never understood why K&R et al. didn’t use := for the assignment
@statement. Because it required TWO symbols?!
Because assignment is used roughly twice as often as equality test
is… so by right it ought to be half as long!
Ilias
Why isn’t (p<:97rhit&#whywhywhywhywhywhywhywhywhywhywhywhywhywhywhywhy
Joseph Virzi <jvi…@rtsys.com> wrote in article
<35F3B46F.B950D…@rtsys.com>…
- Hide quoted text — Show quoted text -
> Why isn’t (n-n=0)-(n=n) = 0?
Ian Goddard wrote:
> Do you believe this statement is true:
> (n – n = 0) = (n = n)
> It says: "To say that the difference
> between n and n = 0, is to say that
> n and n are the same (n = n)."
> It seems quite clear to me that
> (0 difference) = (same as), and
> it also seems that that is just
> what the statement above says.
You’re mixing semantics and syntax.
> Perhaps the right question is not
> if the statement is true or false,
> but is it mathematically proper?
In case you consider ‘=’ being a binary relation N -> {1,0}
then you could very well say (n – n = 0) = (n = n).
If I were you I’d say (n-n = 0) <=> (n=n) and avoid confusing
use (or abuse, if you wish) of the ‘=’ symbol. (As it seems to
me you’re mixing the symbols of your metalanguage with
those of your objectlanguage.)
But then again, I’m not a mathematician/logician so in case I’m
utterly off line here, please correct me instead of
flaming me
–
Aatu Koskensilta (squ…@seaga.org)
"Religion and politics – stigmata of a diseased society."