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Topic: Must equality "=" be defined as the identity relation or as "the
same"?

Replies: 12   Last Post: Apr 28, 2012 3:05 AM

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kirby urner

Posts: 2,762
Registered: 11/29/05
Re: Must equality "=" be defined as the identity relation or as "the same"?
Posted: Apr 26, 2012 2:02 PM
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On Thu, Apr 26, 2012 at 9:54 AM, Paul Tanner <upprho@gmail.com> wrote:
> No.
>
> Again: Theorems U, V, W, and X show that no matter how we interpret
> "=", there are some statements about "=" that we cannot make without
> contradiction.
>


Theorems that in turn depend on definitions, obviously.

My point was that "=" is proving confusing in that some STEM impaired
"math teachers" (remember those?) mock the tendency in computer
languages to say:

a = a + 1

They jeer at such statements and give them as reasons why students
should throw up their hands at those stupid "computer languages".

( If this happens to you, report this bigotry to the principal or vice
principal immediately, tell your parents, alert the phone tree. Pitch
forks should be well polished. )

What I was getting at in my post is you don't want naive acceptance of
symbols independently of their contexts, as if "meme pools" don't
exist and as if notations don't come and go.

Of course in our AMNs (animated math notations -- vs. the DOA ones)
the "=" symbol is often used as an *assignment operator*.

What we say about:

a = a + 1

is that the + is triggering the object on the left, looking for its
internalized understanding of addition. If nothing is found, the
right object is inspected for right-side addition (__radd__ to be
specific).

The objects in question might not be integers, we don't have a clue
yet, but the point is there may be an add method in the picture, the
result of which is, a new math object, the sum (a + 1).

This new object is then "reassigned the same name" i.e. we are
"rebinding the name 'a'" to the new object we have created.

So you see, according to our definitions, "=" is not some imperfect
implementation of the equals operator.

I don't know from the above theorems if a == b (translating to our
notation) is supposed to evaluate to an object if in fact a != b i.e.
if a is not equal to b.

Are we talking about boolean primitives? That's how it works over here.

I'd say there's not enough above to specify a game of addition, that's
for sure.

It may *not* be defined, and yet the idea of "equals" may well be, as
for triangles and "congruence".

You don't need addition to have binary operations.

> One such statement about "=" is any claim that we can view "=" as
> R&S&T&~I while also claiming that R&S&T&A -> I. This conjunction of
> claims is always false - it contradicts the above theorems.
>
> As I just said in a prior post, that there are some statements about
> "=" that we cannot make without contradiction no matter how we
> interpret "=" ultimately rests on the nature of the definition of the
> antisymmetric property with respect to binary relations - it includes
> "=" as part of the statement of its definition: The statement of this
> property is the implication (x#y & y#x) -> x=y.


If "=" is a naming device, used to assign names to objects, then all
the above theorems are blown out of the water, thanks to disagreement
on definitions.

Of course no one worries about this, as we're talking about different
formalisms, different meme pools.

On the other hand, on the ground, we *do* have to worry when "=" is
blithely assumed to be some kind of boolean operator resulting in True
of False.

It's not clear that 2 == 3 evaluates to anything interesting in Paul's
proto-system of some mythical calculus.

In the AMN I use, 2 == 3 evaluates to False.

In any case, we want Johnny (who is learning math through programming
some of the time) to be immune to the meme viruses propagated by those
cloddish math teachers.

>
> This means that when the symbol for a binary relation we are
> addressing is "=" - when we replace "#" with "=", the statement of
> this property becomes (x=y & y=x) -> x=y, a tautology of the form (p &
> q) -> p.
>


Rather than spend a lot of time on propositional calculus, which was
over-reliant on sets for foundations (Frege's mistake?), we should be
helping students understand that many math languages have repurposed
"=" as an assignment operator.

Then you have these ignorant know-nothing "math teachers" with a 1900s
background who speak mockingly and derisively of our languages and
refuse to share some of the basics of geometry we have come to know
and appreciate.**

Of course this is why we rule out having "math teachers" as respected
members of our faculty.

We have STEM teachers, certainly, but a "math teacher" would be
considered a throw back, more what the Imperial Schools do (the USA/UK
anglophones speak with a royal "we" a lot and are a source of so many
of our under-qualified applicants).

This doesn't mean we can't hire someone who went by "math teacher" in
a former job and have them teach STEM. I've done that myself many
times.

It's just that we're proud to be countering over-specialization where
it matters and having STEM be a unified curriculum is one of our
design features. Not every student is lucky enough to get STEM this
way.

Kirby

** this is a separate topic but the fact that the Common Core
standards are so weak in spatial geometry is a gold mine for us --
remediation jobs galore, with almost every math teacher in every state
in severe need of a real STEM education (especially after such a
severe dumbing down at the hands of our ancestors in the 1900s and
their pitiful curricula -- they should be compensated for what was
done to them).


Message was edited by: kirby urner



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