Mathematical Identity
H. de Bruijn
Article posted originally to 'sci.skeptic':
Several silly stories about Perpetuum Mobilae appeared in this newsgroup,
kicking in an open door. Virtually nobody seems to be skeptic about REAL,
established science. Well, I surely am skeptic.
I would like to start with the science of mathematics. First question:
which symbol is the most frequently used in mathematical formulas ?
I am pretty sure that it is the sign for "equality" or "identity": = .
Now, I don't think that it is possible to develope Predicate Calculus,
as a part of mathematical logic, without the whole notion of identity.
Nevertheless, let us assume this for a moment. Then we can find a kind of
mathematical definition in Russell and Whitehead "Principia Mathematica",
chapter "Identity". It sounds like this:
( a = b ) :<-> All P: P(a) <-> P(b) .
a equals b means that, for all properties P:
P is a property of a if and only if P is a property of b .
This sounds reasonable. Consider, however, the expression ' 1 = 1 ' .
Then, the '1' on the left of the equality sign is not equal to the '1' on
the right of the equality sign, because the property 'being on the left'
is obviously not equivalent with the property 'being on the right'.
Therefore we conclude to a paradox: 1 <> 1 .
Identity is NOT properly defined by the logic in "Principia Mathematica".
But now we are in trouble; I have NEVER seen another valid definition of
Equality, the most important of all mathematical concepts.
This is not the end of the story. So called "Equivalence Relations" are
supposed to share the following properties:
a == a ; a == b -> b == a
( a == b and b == c ) -> ( a == c)
Quoting a (Dutch) textbook, equivalence relations should be conceived as
a "generalization" of "ordinary" equality. But tell me how something can
be generalized, if it is not even defined ?
Now it is time to give my own opinion.
There is no such thing as an "absolute" identity. Every identity is only
"in some respect". Equivalence relations cannot be distinguised, at all,
from "ordinary" equality. Just replace == by = . What's in a word ?
Equivalence relations and "definition by abstraction" are good examples
of how to make mathematics unnecessarely complicated.
To be continued in 'sci.skeptic' ...
--
* Han de Bruijn; computer Graphics | "Mathematics is trivial; and *
* TU Computing Centre; P.O. Box 354 | in so far as it is nontrivial, *
* 2600 AJ Delft; The Netherlands | it is not important." (HdB). *
* Fax: +31 15 78 37 87 =================================================
John Franks
>
> Consider, however, the expression ' 1 = 1 ' .
>Then, the '1' on the left of the equality sign is not equal to the '1' on
>the right of the equality sign, because the property 'being on the left'
>is obviously not equivalent with the property 'being on the right'.
>Therefore we conclude to a paradox: 1 <> 1 .
>
Hans de Bruijn = Hans de Bruijn
Then, the Hans de Bruijn on the left of the equality sign is not equal
to the one on the right of the equality sign, because the property
'being on the left' is obviously not equivalent with the property
'being on the right'. Therefore we conclude the original poster must
have been (at least) two people.
John Franks Dept of Math. Northwestern University
Internet jo...@math.nwu.edu
Bitnet j_franks@nuacc
postp...@alien.enet.dec.com
>This sounds reasonable. Consider, however, the expression ' 1 = 1 ' .
>Then, the '1' on the left of the equality sign is not equal to the '1' on
>the right of the equality sign, because the property 'being on the left'
>is obviously not equivalent with the property 'being on the right'.
You err in talking about the "1" on the left side or the "1" on the right
side. The equation is not a statement about the "1"s. It is a statement
about 1. For any property, P(1) is true if and only if P(1) is true, so
1 = 1.
-- edp (Eric Postpischil)
"Always mount a scratch monkey."
j.nienhuys
>
>I am pretty sure that it is the sign for "equality" or "identity": = .
>
> ( a = b ) :<-> All P: P(a) <-> P(b) .
> a equals b means that, for all properties P:
> P is a property of a if and only if P is a property of b .
>
>This sounds reasonable.
Your B.Sc. thesis for the degree of Engineer of Technical Physics
started the same way.
After that you went on to prove something like:
Lemma (?). If eps_1 << eps then also n. eps_1 << eps for all natural eps.
In this lemma the numbers eps_1 and eps were reals, as you will remember.
You used it to prove that almost-equality is transitive.
Of course, you used a remarkable order relation <<, that had the property
((a<<c) & (b<<c)) -> (a+b<<c) for positive a,b,c.
An ordinary mathematician would think that the lemma (?) would show this
to be a contradictory concept if applied to R, and if a<<b would entail
a<b .
I see that you have retained both your old interests and your
approach to this type of problem.
>Consider, however, the expression ' 1 = 1 ' .
>Then, the '1' on the left of the equality sign is not equal to the '1' on
>the right of the equality sign, because the property 'being on the left'
>is obviously not equivalent with the property 'being on the right'.
>Therefore we conclude to a paradox: 1 <> 1 .
This comes from using a formal language (in this case the language
of Principia) to talk about itself, namely about the syntactical structure
of the formulas. It is known that doing so always risks contradictions.
Actually, I doubt whether you can formulate 'being on the left' in the language
of the Principia, anymore than you can refer to the color or the brand name
of the ink used to print the formula, or the font of the letters used.
Similar games can be played with the equality a=b, to prove that b is the
first letter of the alphabet or from Venus=Aphrodite that Venus has 9 letters.
>Identity is NOT properly defined by the logic in "Principia Mathematica".
>But now we are in trouble; I have NEVER seen another valid definition of
>Equality, the most important of all mathematical concepts.
If you do not understand the distinction between a logical system (a
language) and what the language talks about, you NEVER will.
>To be continued in sci.skeptic ...
Please. Sci.skeptic is not the place for undergraduate courses in
logic and set theory.
Jan Willem Nienhuys (wsa...@win.tue.nl)
Doug Merritt
he reached a point that had to do with math:
In article <9...@dutrun.UUCP> rct...@dutrun.UUCP (H. de Bruijn) writes:
> [ The symbol '=' ...]
>is the most frequently used in mathematical formulas [...]
>mathematical definition in Russell and Whitehead "Principia Mathematica",
>chapter "Identity". It sounds like this:
>
> ( a = b ) :<-> All P: P(a) <-> P(b) .
> a equals b means that, for all properties P:
> P is a property of a if and only if P is a property of b .
>
>This sounds reasonable. Consider, however, the expression ' 1 = 1 ' .
>Then, the '1' on the left of the equality sign is not equal to the '1' on
>the right of the equality sign, because the property 'being on the left'
>is obviously not equivalent with the property 'being on the right'.
>Therefore we conclude to a paradox: 1 <> 1 .
The spot where you went wrong is in considering 'being on the left/right'
to be valid predicates, for in terms of algebra, they are not. In fact,
this is a specific instance of a very general confusion about mathematical
systems, which is that the language used to *describe* a mathematical
system generally does not follow the same rules as the system being
described.
Formal axiomatic systems (the ultimate in mathematical rigor) use a meta
language to describe the language of the system being set up.
This dual language system has been explored very extensively, and there
are some very interesting results, such as the impossibility (I believe
it's been proved) of using a single meta language to define both itself
*and* the usual axiomatic systems of analysis.
Everything else that can arise tangentially from this sort of discussion
has to do with very nonrigorous philosophy, and this would certainly
not be the newsgroup in which to discuss it. Before I go, I will say
that there is an extensive literature in philosophy on the subject of
"identity".
>Identity is NOT properly defined by the logic in "Principia Mathematica".
Wrongo.
>To be continued in 'sci.skeptic' ...
Thanks for not carrying on with philosophy here.
Doug
--
Doug Merritt {pyramid,apple}!xdos!doug
Member, Crusaders for a Better Tomorrow Professional Wildeyed Visionary
Richard O'Keefe
> Consider, however, the expression ' 1 = 1 ' .
>Then, the '1' on the left of the equality sign is not equal to the '1' on
>the right of the equality sign, because the property 'being on the left'
>is obviously not equivalent with the property 'being on the right'.
>Therefore we conclude to a paradox: 1 <> 1 .
In article <11...@accuvax.nwu.edu>, john@hopf. (John Franks) rebuts:
> Consider, the expression
> Hans de Bruijn = Hans de Bruijn
Do we have to drag out the type/token/quotation stuff all over again?
In the equation
1 = 1
^ ^ ^
a b c
there are three parts, labelled a, b, and c. a labels a token belonging
to the type ``1'', b labels a token belonging to the type ``='', and
c labels a token belonging to the type ``1''. a and b clearly label
different tokens (as de Bruijn observes), but the equation is not
mentioning those tokens but USING them to refer to the number 1.
If de Bruijn's argument were valid, we would never be able to talk
about anything at all. (There used to be another occurrence of that name
in this sentence but my deleting it didn't delete de Bruijn himself.)
Peter Ladkin
as some subsequent replies have made it out to be. he asks how one may define
identity for mathematical objects. it is not a trivial question, although it
may not be of great concern to most practicing mathematicians. the comment
about the equation `1=1' may indeed involve a type/token distinction, in
which case one can rephrase the question as `how do we define the type of
numbers?'. if one tries to define them axiomatically, one would have to use
a henkin-type construction from the language in order to obtain a model
(i.e. the logical analogue of modding out by a suitable ideal in a ring
e.g. of polynomials). however, such methods involve non-constructive
mathematics of the sort unlikely to appeal to a constructivist (especially
a dutch one!) and may even be circular. note that under standard
mathematical logic the identity of indiscernibles is axiomatic, so
standard books on mathematical logic are unlikely to address doubts
about its validity. maybe the proper place to carry out this discussion
is sci.logic, to avoid boring the geometers with yet more logic.
peter ladkin
Col. G. L. Sicherman
>
> This sounds reasonable. Consider, however, the expression ' 1 = 1 ' .
> Then, the '1' on the left of the equality sign is not equal to the '1' on
> the right of the equality sign, because the property 'being on the left'
> is obviously not equivalent with the property 'being on the right'.
> Therefore we conclude to a paradox: 1 <> 1 .
> Identity is NOT properly defined by the logic in "Principia Mathematica".
> But now we are in trouble; I have NEVER seen another valid definition of
> Equality, the most important of all mathematical concepts.
You're right about mathematical equality. Have you read any
'pataphysics? Loosely speaking, 'pataphysics is the study of the
errors inherent in representing knowledge in print.
Somebody has already posted an illustration:
H. de Bruijn = H. de Bruijn
The poster concluded that you are not you--a valid inference from a
false premise. _Mathematical_ equality does not hold for people, or
for anything but symbols. This is obvious if you have studied
mathematical equality in its own right. Like privacy, mathematical
equality is worst understood by those who use it most.
This argument will not appeal to platonists, who seldom appreciate
'pataphysical observations. But if you're interested, check out
Raymond Queneau's essay on 2 + 2 = 4 and wind velocity.
-:-
'Patageometry, n. The study of those mathematical
properties which are invariant under brain transplants.
--
G. L. Sicherman
g...@odyssey.att.COM
H. de Bruijn
Jan Willem Nienhuys, from Eindhoven University of Technology, writes:
> Your B.Sc. thesis for the degree of Engineer of Technical Physics
> started the same way.
The work was premature, however. Much of it should be abandoned.
> I see that you have retained both your old interests and your
> approach to this type of problem.
> This comes from using a formal language (in this case the language of
> Principia) to talk about itself, namely about the syntactical structure
> of the formulas. It is known that doing so always risks contradictions.
CORRECT, but nevertheless harmless, argument.
> If you do not understand the distinction between a logical system (a
> language) and what the language talks about, you NEVER will.
> Please. Sci.skeptic is not the place for undergraduate courses in
> logic and set theory.
I'm sorry, Doug !
Eric Postpischil writes:
> The equation is not a statement about the "1"s. It is a statement
> about 1. For any property, P(1) is true if and only if P(1) is true,
> so 1 = 1.
and I will prove you that 1 = 2 .
John Franks writes:
> Hans de Bruijn = Hans de Bruijn
Dale R. Worley writes:
> For a practical way to do this, I would recommend Bourbaki's "Theory
> of Sets" (available in French and English). They get around such
> oddities as R&W's definition of "=" by having "=" be a primitive
> relation in the system, and then adding axioms like:
>
> For any property P, for any x and y, if x = y then P(x) iff
> P(y).
>
> (Or was it a theorem?) This gives you all of the power of R&W's
> definition, without having to deal with its difficulties.
Don't believe the last statement, but I will read the reference.
Doug Merritt writes:
> are some very interesting results, such as the impossibility (I believe
> it's been proved) of using a single meta language to define both itself
> *and* the usual axiomatic systems of analysis.
> that there is an extensive literature in philosophy on the subject of
> "identity".
Richard O'Keefe writes:
> Do we have to drag out the type/token/quotation stuff all over again?
Why not ? Since everybody does it.
Peter Ladkin writes:
> folks, just a note to say that de bruijn's comment is by no means as
> trivial as some subsequent replies have made it out to be.
Finaly someone who seems to understand ...
Quoting myself:
> There is no such thing as an "absolute" identity. Every identity is only
> "in some respect". Equivalence relations cannot be distinguised, at all,
> from "ordinary" equality.
Actually THIS is the serious argument, the bullet I am loading my gun with.
Nobody seem to have noticed ...
G.L. Sicherman writes:
> equality is worst understood by those who use it most.
Think there is no better way to close off.
Vicki Powers
SET THEORY!
In article <9...@dutrun.UUCP>, rct...@dutrun.UUCP (H. de Bruijn) writes:
>
> "Mathematics is trivial; and *
> in so far as it is nontrivial, *
> it is not important." (HdB). *
Maybe, but at least mathematics is a way to get paid for doing something I
love. Otherwise I would probably be a (argg!!!!) computer scientist.
Signed,
A die-hard algebraist
P.S. Come back, Monty Hall, all is forgiven.
--
Vicki Powers | vi...@mathcs.emory.edu PREFERRED
Emory University | {sun!sunatl,gatech}!emory!vicki UUCP
Dept of Math and CS | vicki@emory NON-DOMAIN BITNET
Atlanta, GA 30322 |
mcc...@s.cs.uiuc.edu
I seem to recall that the notion of "equivalence" was to single out multi-named
objects, with the classic example being equivalence-classes, where any member
could serve as a name (i.e., the 'class' being the object); if so, then the
identity-property is immediate: the same name should always identify the same
object, which is all (I believe) "1 = 1" really means. In other words (with-
out getting too philosophical here), "=" is a property of the naming process,
and where it holds, testifies to the disambiguity of that process.
Scott McCaughrin
University of Illinois
Urbana, Illinois.
Jeremy Teitelbaum
-- Wittgenstein
--
Jeremy Teitelbaum jer...@math.lsa.umich.edu
Math Dept.
U. of Michigan
Ann Arbor, MI 48109
(313)-763-0294