Math Terminology
quant
This is more of a curiosity than anything, but in regards to
terminology of the form Adjective+Noun where the Adjective+Noun is a
Noun that is an Adjective, for example:
abelian group
commutivative ring
normal subgroup
principal ideal
I was curious of these were thought of as single terms, as two terms,
both, or as open compounds, or as arguable. I have made a distinction
between open compounds (a compound word with a space separating the
two parts) and terms (a word or phrase used to name something).
English has a tendency to evolve compound words (flower pot -> flower-
pot -> flowerpot) out of common phrases and my thought was that there
would be disagreement about how these words were thought because maybe
they were in 'evolution'. I ask about these examples in particular
because I see the definition of a "normal subgroup" more often than
the definition of a group being "normal". Perhaps it sounds nicer to
define this way (it's certainly equivalent), but perhaps it is also
because "normal subgroup" is considered one term.
Thanks,
Leó
Victor Porton
As an other example, there were three distinct terms:
- Continuity
- Uniform continuity
- Proximity-continuity
Only in 21st century I discovered that these are three are kinds of
what I call "generalized continuity".
Before intuition was suggesting that these three continuity are
related. But only as a result of my recent discovery was found what
exactly they have in common.
See
http://www.mathematics21.org/algebraic-general-topology.html
particularly
http://www.mathematics21.org/binaries/continuousness.pdf
The same may happen with other "compound" math terms.
quant
Thank you for your e-mail, but I was referring to something entirely
different. I was asking whether (in a linguistic sense) common Adj
+Noun phrases such as 'commutative ring' was considered as one term or
two separate terms.
Sincerely,
Leó
cbr...@cbrownsystems.com
I think that linguistically speaking, there is less here than might
meet the eye.
Yes, one very frequently sees reference to "normal subgroup" or
"commutative ring"; but this is not analogous to your example
"flowerpot".
Typically, "pots" /never/ have a hole in the bottom, yet "flowerpots"
almost /always/ do (to allow water to drain from them, preventing root
rot). Normally, we would call a "clay vessel with one or more holes in
it" a sieve or colander or something of the like; because "pots" (in
the usual sense) do not have holes in them.
Conversely, without exception /every/ normal subgroup is, perforce, an
object completely satisfying the exact definition of a subgroup; every
commutative ring is perforce a ring, etc. There is considerable study
of the implications that some structure is a subgroup, ring, etc.,
that precedes the /additional/ "adjectival" constraint of being
normal, commutative, etc.
And despite your comments, I frequently see usages such as "because
the ideal I is principal, ...", and "because the subgroup H is
normal, ..." and so on used in proofs.
What you may be observing is that certain types of entities are more
frequently of use or interest: e.g., normal subgroups are more
commonly discussed than "non-normal" ones, because theorems
specifically about normal subgroups are quite useful (as they are
kernels of group homomorphisms); commutative rings are more often
studied than "non-commutative" ones because number systems such as the
naturals, the rationals, the reals, and complex numbers are all
commonly encountered examples of such entities; and so on.
Perhaps more in line with your thinking might be morphisms: in
English, we use single words for homomorphism, epimorphism,
monomorphism, isomorphism, automorphism, rather than compound terms
like "homologous morphism", etc.; even though a clear definition of a
"morphism" can be obtained, with certain morphisms having the
adjectival "homo", "epi", etc. prefix applied to them.
Although one occasionally sees ugly phrases like "because the morphism
T is epi, ..."
Cheers - Chas
jbriggs444
>
> - Show quoted text -
In my opinion.
The formal meaning of compound terms should be taken according to
their formal definition. While the linguistic structure may look like
"{adjective noun}", the meaning comes from the formal definition and
the combination should be taken as a primitive rather than as a
compound.
Certainly, "abelian group" can be successfully understood either way.
However, consider "natural number" or "real number".
There's no defined formal meaning for "natural" or "real" and "number"
that would allow one to produce a formal definition for "natural
number" or "real number" on a purely linguistic basis.
[Nor is there any reason to read something pejorative into the term
"irrational number"]
Mathematicians have historically borrowed ordinary words and have done
so in a way so that the words usually parse like the spoken language
and can be interwoven into ordinary natural language text.
But textual analysis only takes you so far. The words and word
combinations take on jargon meanings that are invisible to someone who
is trying to understand mathematics as if it were merely a
transliteration of English.
I'm not enough of a linguist to know if this is already obvious to
you.
Phrases like "as x increases without bound" may be understood as
suggestive mnemonics that can be easily understood and converted to a
more precise mathematical formalism if one needs to examine an
argument more closely. Often the phraseology captures some informal
intuitive sense of the underlying mathematical content.
In the case of "as x increases without bound" there's an underlying
formalism involving quantitifiers, epsilons and deltas or maybe open
and closed sets) which can put a formal grounding under the intuitive
notion.
quant
> the ideal I is principal, ...", and "because the subgroup H is
> normal, ..." and so on used in proofs.
> Cheers - Chas
Thanks, Chas. This makes sense as you wouldn't be able to use the
adjective 'principal' just after defining 'principal ideal' if we were
thinking of 'principal ideal' as a primitive object.
Sincerely,
Leó