Re: Need help in interpreting symbol 225e (measured by)

From: Hans Aberg (
Date: Fri Mar 10 2006 - 14:09:02 CST

  • Next message: Keutgen, Walter: "RE: Need help in interpreting symbol 225e (measured by)"

    On 9 Mar 2006, at 05:51, Gusztáv Jánvári wrote:
    > I’m in localizing Unicode symbol names to Hungarian, and I have to
    > translate the symbol name “measured by”: I studied incredibely lot
    > of Math :), but unfortunatelly I’m not sure when this symbol is used.
    Perhaps the fellows adding it may be able to provide a mathematical
    context; I have no immediate recollection of seeing it. If it is an
    older symbol, as the name suggests a connection to measure theory,
    perhaps one might check the Bourbaki team's book on integration.
    > My not-too-educated guess is that it is used to express that the
    > fact of the equality is determined by measuring something (for
    > example, when you fill in two glasses with the same amount of water
    > and both looks like it is completely full, you could say, they are
    > equal by measure).
    In measure theory, one says two functions defined on a measurable set
    are almost equal (a.e.) if they differ on a set of measure zero. But
    then one is using "a.e." or something, anyway not "m".

    On 10 Mar 2006, at 18:06, Philippe Verdy wrote:

    > From: "philip chastney" <>
    >> HAZARD WARNING: this is pure conjecture . . but the
    >> description MEASURED BY suggests a specific metric
    > If not even the UTC members know the exact meaning or usage of the
    > symbol, but only acknowledge the existence of the symbol, mostly as
    > a glyph, then the normative English or French names defined by ISO
    > is also probably suggestive or conjectured regarding this usage.
    > So why not defining a name that does not attempt to translate
    > "measured by", but instead a name that is descriptive of its
    > graphic aspect? (additionally, if one would want to translate
    > "measured by", it could be a word that does not even contain any
    > "m" letter, so using this symbol in a mathematic document written
    > in another language than french or english wouldprobably be a bad
    > choice for authors.)
    > After all, mathematic symbols most often have various contextual
    > usages and are often defined specifically by different authors. So
    > why not translating instead by "equal with m"

    Though true, these Unicode names are just identifiers, that, as in
    other common language usage, just tries to capture some aspect pof
    the symbol, but not all. So even if the description would be
    definitely wrong in other contexts, there is no point in attempting
    to change that. And as I recall, these identifiers are fixed in
    Unicode anyway.

    But this "abuse of language", as the Bourbaki team would have said,
    is common in working math. To illustrate it, the common symbol "="
    changes meaning according to context, and there is usually no attempt
    to clarify this in detail in notation. So for example, I could
    declare two functions to be almost equal if they differ on a set of
    measure zero and use say the symbol "=" with an "a.e." over it, but
    in noting that this is an equivalence relation, I can divide out with
    that, and use just "=". Context and personal preference will dictate
    the choice.

    > In fact the symbol looks more as if it actually meant "equals by
    > measure to". I would say that mathematically, two entities can be
    > different but still equal by measure, and "m" symbolizes the
    > application of an implicit measurement (or projection on a measure
    > axis, such as a norm, or an angular argument) on both sides of the
    > equality. This suggests that items on both sides are comparable
    > using a projection in some metric space, and that inequalities
    > could complement this operator (so we would also have "lower by
    > measure than", and so on...)
    > This U+225E operator is then just another variant of another
    > similar symbols of the same family, like:
    > * U+2251 (geometrically equal to: equal by measure of geometric
    > proportions, i.e. isometric)
    > * U+2256 (ring in equal to: equal by measure through projection
    > symbolized by this ring)
    > * U+2257 (ring equal to: equal by measure through projection
    > symbolized by this ring, ignoring infinitesimal differences)
    > * U+2259 (estimates, corresponds to: equal by measure through
    > projection in a cartesian space, ignoring infinitesimal or
    > unmeasurable differences in this space such as probablistic variance)
    > * U+225A (equiangular to: equal by measure of an angle)
    > * U+225B (star equals: equal by measure through a projection or
    > injection in some space symbolized by this star)
    > * U+225C (delta equal to, equiangular, equal to by definition, see
    > also U+225A: equal by measure of an angle or of measurable
    > definition characteristics...)

    Since "measure" has a very common formal definition in pure math, one
    would avoid any other usage of this word.

    But you have listed a host of symbols that perhaps would have
    benefitted form a non-mathematical description, instead favoring a
    more rendering description. Perhaps someone sometime used them in a
    context indicated by their names above, but the usage is not
    sufficiently wide to be viewable as common standard.

    But then again, there is no point in attempting to change that. A
    working mathematician will just use the symbols according to
    traditions and contextual needs to clarify notations supported by
    notation, anyway.

       Hans Aberg

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