Michael H. Brill

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I have recently been bothered by the multiple meanings of “=” or its common-language proxies “equates to,” “is equal to,” “comprises,” or “is.”

A clear example from computer programming is the distinction between “A equals B” and “If (A equals B).” The first instance is an instruction to copy the contents of B into the location A so as to over-ride A’s original contents; the second instance is to use the current contents of A and B to make a decision. The dichotomy had to be solved by a distinction in notation. Obviously the first instance received the prized “=” sign, and the second made do with “.EQ.” (in FORTRAN), and “==” in C.

Another example is the usage “z comprises x,” which in common language means “z consists of x,” but in the legalese dialect (particularly in patents) means “z contains x.” Legal documents betray the second meaning when, after “z comprises x,” they say “z further comprises y.” Unlike the computer-program example, this dichotomy is never explicitly clarified. (Let inventors be warned!)

In color science, there is a one-two punch of examples.

Punch 1 is elementary. The distinction between metamers and isomers^{†}
is a reason to use a squiggly equals sign for “match in color” and an ordinary
equals sign for “match in spectrum.” Simple linear algebra connects one with
the other, and the result simplifies the general problem of color from n
dimensions to 3. Both in 3 dimensions and in n dimensions, the match is
transitive: If A matches B and B matches C, then A matches C. That is one of
Grassmann’s laws, which are axioms of colorimetry.

Punch 2 is more problematic: A metameric match is not at a point
in a 3-dimensional space but has a significant uncertainty. The
uncertainty-convolved match (the best you can get from our present experiments)
can be expressed by “A is less than a just-noticeable difference (JND) from B.”
This violates transitivity. Joseph B. Keller [3] described an (impractical) experimental
plan to banish the need for the uncertainty-convolved match, or alternatively to
coin the term “*really* match” for the unattainable Grassmann match.

Despite this conundrum, color scientists have been able to find practical ways to proceed. Two paradigms collide (color as a linear vector space and color as a space whose points are ill-determined within a JND). Somehow, the mathematical intuition of color scientists finds an acceptable way to distinguish the term “color match” as used by these paradigms.

Such discussion touches uncomfortably on philosophical questions in ontology, entification, and epistemology.

A nearby neighbor of philosophy is pure mathematics. I recently learned that serious mathematicians are also concerned with the multiple meanings of the equal sign. It started when I encountered Caroline Delbert’s article in Popular Mechanics [1]. In turn, Delbert cited a research-level article (actually more of an editorial) by Kevin Buzzard from the Imperial College of London. That article is “Grothendieck's Use of Equality,” a title so stylishly enigmatic that I adopted it for this column. Here is the Abstract:

“We discuss how the concept of equality is used by mathematicians (including Grothendieck) and what effect this has when trying to formalise mathematics. We challenge various reasonable-sounding slogans about equality.”

(I am reminded of a slogan from Orwell’s Animal Farm: some animals are more equal than others.)

Buzzard focuses on the problem of applying AI to prove theorems automatically, making intuitive decisions just as human mathematicians do. AI‑based theorem-proving software is now unable to invent the right categories within which to define “equality” of members of the category. An example of category selection is the proposition 2+2 = 4. Can I say this is always true? If “2+2” is numerically evaluated and compared to the numerical evaluation “4,” the answer is yes. But the alpha-numeric string you have to type is more arduous for “2+2” than for “4.” That would make the answer no. The designations “numerical evaluation” and “alpha-numeric string” are categories that may or may not be felicitous in proving theorems.

In pure mathematics, the ambiguity of “equal” is much more subtle, sometimes arising from collisions of paradigm (perhaps resembling the one I mentioned for color matching). There’s a whole field called category theory that ponders this sort of problem.

Mathematicians now seem to want a more rigorous definition of equality. Buzzard says that as soon as we have this sorted out, the way will be cleared to an AI system that can prove theorems as well as mathematicians do. Other mathematicians agree with him.

I am not an expert in pure mathematics, but it seems to me that the way ahead may not be as easy as Buzzard implies. Reaching the Grail of AI is not a foregone conclusion to this story of equality.

Note: Alexander Grothendieck (1928-2014) was a German-born French mathematician who co-created the field of algebraic geometry. Many consider him to be the greatest mathematician of the twentieth century, but he is not as well known as others because of his lower publication output and because of his controversial political views [4]. Buzzard cites one work by Grothendieck [5].

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Caroline Delbert, “Mathematicians are suddenly rethinking the equals sign,” Popular Mechanics,12 June 2024.

https://www.popularmechanics.com/science/a61042424/mathematicians-rethinking-equal-sign/Kevin Buzzard, “Grothendieck's use of equality”

__https://arxiv.org/abs/2405.10387__posted to ArXiv May 2024.M. H. Brill, “Joseph B. Keller at NJIT: A more precise color-matching theory,” ISCC News #394, 16 (November 2001).

A. Grothendieck, “Elements de geometrie algebrique. I. Le langage des schemas,” Inst. Haute Etudes Sci. Math Sci. Publ. Math. (1960) no. 4, 228.

^{† }The word *isomer* is an underutilized word that was borrowed by Ostwald
from the chemistry lingo. In color science, it refers to a color match where
the two colors match at all wavelengths, so will match for any illuminant.