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’ purples 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.
-- 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 pproach, 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

*L*2 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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