Once the student of math has labored to assimilate such concepts
as: holomorphic; homological; diffeomorphic; transfinite; functorial and what have you; or the student of life Being, Beauty, Deity, Causality, and all that lot; he may not imagine that the simple
Anglo-Saxon word same would give him much to munch on; it’s like Cordelia, after her sisters’
purple flourishes, saying she loved her father like “salt”. Lear felt it was a let-down.
But there are wrinkles, and folds, and even
profunditities.
Problematic right at the outset is the matter of continuity of identity over time; that essay must remain for another
day. Rather, we contemplate cases
where items are co-present and may be directly compared.
For the essay, click here: Categories and Sameness.
For an epigrammatic appendix, here: Same Difference.
~
Additional illustrations:
A practical mathematician (a lecturer in chemistry, in fact) notices the semantic niceties, without lingering over them:
A practical mathematician (a lecturer in chemistry, in fact) notices the semantic niceties, without lingering over them:
Eigenvectors are properly only
right- or left- eigenvectors.
A vector is only an eigenvector both on the right and the left of an operator if the operator is symmetric or Hermitian, and then only with the appropriate convention of the ‘same’ vector in each case.
A vector is only an eigenvector both on the right and the left of an operator if the operator is symmetric or Hermitian, and then only with the appropriate convention of the ‘same’ vector in each case.
-- Brian Higman, Applied
Group-Theoretic and Matrix Methods (1955), p. 24
Notice that the adverb “properly” here is not specially
related to mathematical uses of proper (as in, proper subset), but is metalinguistic: If we are speaking properly, we should not blithely
refers to “eigenvectors” without specifying the handedness.
And, a few pages later:
Elements which are conjugate are often described as equivalent, and sets of equivalent
elements are called classes [NDLR:
specifically, “equivalence classes”] To see just how justified the word equivalent is, we may return to the
matrix approach, in which conjugate elements can be regarded as the same element
in different coordinate systems.
-- Brian Higman, Applied
Group-Theoretic and Matrix Methods (1955), p. 36
~
In arithmetic, if a
has the same value as b, we say that
a “equals” b: a = b. Informally, we sometimes replace equals simply with is: “seven times eight is fifty-six”, “f(x,y) is zero at the origin”.
But now consider this expression:
(*)
f(x,y) is zero on the unit disc.
This is ambiguous between an existential and a universal
reading:
∃<x,y> in D such that
f(x,y) = 0.
∀<x,y> in D, f(x,y)
= 0
(The first reading might arise in a context where we’re
trying to determine if f ever reaches zero, and an existence proof shows that
this happens somewhere on the unit disk.)
So far so straightforward, reminiscent of the twin readings
of “A cat is on the mat” and “A cat likes milk”.
But now, the linguistics of the thing. (*) being ambiguous, the first reading
is typically phrased thus in
mathematical writing:
f
has a zero on the unit disc.
where the quasi-adjectival predication “is zero” is replaced
by the nominal construction “has a zero”.
The grammar of the latter is different; as, “f has only countably many
zeroes on the real line”.
The second reading becomes:
f
is identically zero on the unit
disc.
Thus, now the universal reading gets packed into an adverb, “identically”, leaving
mathematics proper and straying
into the semantics of the Wortfeld “same”.
~
Setting up equivalence classes of functions under the L2 norm:
We define the’distance’ from f to g
to be ||f - g||. Doing this, it is natural not
to distinguish between two functions whose ‘distance’ is 0, that is, which
coincide almost everywhere.
-- F. Riesz & B. Sz.-Nagy, Leçons
d’analyse fonctionelle [references to the English translation, Functional
Analysis, 1955], p. 58
(Italics in original.)
A quite different take on same vs. equivalent: here the functions will be wide apart
by the above criterion, but structurally identical:
We have not, so far, set up a
criterion by which solutions are to be distinguished. Clearly, a solution which is a constant multiple of
another is not to be regarded as different from it, since both are the
same after normalization, except for a physically insignificant factor of
absolute value 1.
-- -- Robert Lindsay & Henry
Margenau, Foundations of Physics (1936), p.
Not mathematical equivalences, but conflations based on
integer-mysticism:
The four Evangelists coalesce with four Irish annalists,
whose chronicle of ancient times is known as The Book of the Four Masters. These four again coalesce with four old
men, familiars to the tavern of HCE
[who are in turn] identical with the four “World Guardians” of the
Tibetan Buddhistic mandalas, who protect the four corners of the world -- these
being finally identical with the four caryatids, giants, dwarfs, or elephants,
which hold up the four corners of the heavens.
-- Joseph Campbell & Henry
Robinson, A Skeleton Key to Finnegan’s Wake (1944)
~
This “identified with” is tricky; it is not equivalent to (not the same as) “identical to”.
Thus, I was initially startled at Wikpedia’s description of
the hedgehog space: “the disjoint union of K real unit
intervals identified at the origin … a set of spines joined at a point”. How could this be a disjoint union if the spines are joined
at a point?
Then Wiki explains:
Although their disjoint union makes
the origins of the intervals distinct, the metric identifies them by assigning
them 0 distance.
.
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