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Re: Must equality "=" be defined as the identity relation or as "the same"?
Posted:
Apr 25, 2012 2:59 PM
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On Wed, Apr 25, 2012 at 10:23 AM, Paul A. Tanner III <upprho@gmail.com> wrote:
> Proof. The proof of the equivalence (R&S&T) <-> (R&S&T&A) will be by
> direct proof, where for each implication in this equivalence we assume the
> antecedent and derive the consequent and then infer the implication, where
> this last step we will call Direct Inference. Where "=" is the binary
> relation:
>
> (1) R&S&T&A Assumption
> (2) R&S&T Conjunction Elimination on (1)
> (3) R&S&T&A -> R&S&T Direct Inference on (1) and (2)
>
Note: I'm mostly just amused by "foundationalists" who think workaday STEM
is holding its breath for these logicisms to be applied, as if they
add certainty.
On the topic of equality, equivalence and so on, what STEM is seeing is
an increasing need to explain how "=" may be used as an "assignment operator"
with "equal in value" replaced by "==". Of course that's not communicated in
such a way as to break backward compatibility. Notation comes in "pools"
and we flit from pool to pool. Any idea that "mathematics" = "one unified
notation for all time that all agree upon" is dismissed as romantic BS at best,
dangerously misleading PR on average (another reason to shift to STEM
as more stable than any of the parts taken alone).
In the Python notation (one of many), we have assignment (=), equal (==)
and identical (is). Students grapple with this notation (among others) in the
course of a busy day. Emphasis is on open documentation, easy lookup
abilities. Memorizing everything is not a goal. Net Logo is another popular
package. Also SAGE. Not forgetting about Mathematica.... not much time
left for calculators though, unless you're still Cult of TI.**
The cool thing about operator overrriding, in languages that permit it, is you
can have __eq__ mean something more like "equivalence class" if you like.
This gives students experience with polymorphism, so important in STEM,
in mathematics in particular, although we don't have to rely on mathematicians
to tell us that.
Kirby
** many USA secondary schools are Cult of TI. For the most part, this
does not matter much, as upgrading USA education is not an achievable
or expected goal. A main function of USAers these days is to serve as
object lessons to others: how NOT to turn out. A kind of museum for
bad / stupid life choices and their consequences. A kind of prison camp.
If you can't fix it, exhibit it. Tourists clap when the plane takes off, like
they used to do in the USSR. Of course there are pockets where the
light of intelligence still flickers. Santa Clara is nice... Mark Twain was
sure cool (friend of Homer Davenport of Silverton, Oregon).
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