Same old same old (not)

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.

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Additional illustrations:

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 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

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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”.

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A supplement to that essay, of quotations recently encountered.

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)

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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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