Grothendieck's Use of Equality

Michael H. Brill

 

Send contributions to mhbrill2001@gmail.com.

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].

___________________________________________        

1. Caroline Delbert, “Mathematicians are suddenly rethinking the equals sign,” Popular Mechanics,12 June 2024.
   https://www.popularmechanics.com/science/a61042424/mathematicians-rethinking-equal-sign/

2. Kevin Buzzard, “Grothendieck's use of equality”
   https://arxiv.org/abs/2405.10387 posted to ArXiv May 2024.

3. M. H. Brill, “Joseph B. Keller at NJIT: A more precise color-matching theory,” ISCC News #394, 16 (November 2001).

4. https://en.wikipedia.org/wiki/Alexander_Grothendieck.) 

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

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 ^† 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.

How They Measured Color 100 Years Ago

Guest author John Seymour

I am sure all my fellow color aficionados have had this experience. I woke up at 11:30 this morning and found myself wondering how they measured color a century ago. I mean, before they had these fancy spectrometers, and before they had these fancy computers powerful enough to control automated corkscrews, and before they had blue teeth to connect the spectrometers to the computers.

I start the story with James Clerk Maxwell, a physicist whose name is right up there with Isaac Newton and that other guy whose name I can’t remember. Maxwell built a color mixing instrument in 1855 that was in a 6-½ foot long box that some people call his color coffin. He revised this in 1860 to be more compact. This was a big improvement to the user interface, since the person looking into the box could actually reach the apertures!   In his words:

I am so well satisfied with the working of this form of the instrument, that I intend to make use of it in obtaining equations from a greater variety of observers than I could meet with when I was obliged to use the more bulky instrument.

 

 

Maxwell’s second color box

The basic idea is to select (at positions X, Y, and Z in the diagram) the amounts of three different wavelengths of light. The mixture is seen on one side of the field of view. The other side of the field of view is a slice of the rainbow. What is not obvious from the diagram is how those three slices of the rainbow get mixed.

Any fan of Pink Floyd knows that if you pour light into one side of a prism, you get a rainbow draining out on the other. You could put three adjustable apertures on the rainbow side, but the different wavelengths of light are diverging from each other, so Maxwell would have needed some additional stuff to combine these rays into one.

Faster than you can say Helmholtz Reciprocity Principle, Maxwell had a clever solution. Figure 1 shows how we normally think of prisms working. But Helmholtz taught us that any arrow in an optical diagram can be reversed, and everything will all magically still work. So, on the right, the arrows have been reversed. Another feature – an aperture – was added with slits in the positions corresponding to where the red, the green, and the violet rays would normally be draining out.

 

Figure 1 

 

Figure 2

As clever as Maxwell’s color box was, it was not a color measurement device. He set it up so that people could match a combination of red, green, and blue light against the color of slices of the rainbow. Maxwell was ultimately after the spectral response of the cones. Maxwell did not use his color box as a color meter, but it inspired two developments in colorimetry.

Frederic Ives is best known for his contributions to color photography, but he also developed the first additive trichromatic color mixing colorimeter in 1907 [Ives 1907a]. He makes it clear, though, that it was not his original idea:

It has long been recognized that a universal color meter, capable of measuring all colors and expressing them in numerical terms, must be based upon the principle of Clerk Maxwell's "color box," in which half of a divided field is illuminated with ordinary white light, while the other half is illuminated by an adjustable mixture of the three simplest colors of the spectrum, isolated bands of pure red, green and blue-violet.

In the diagram below, the telescope-looking-thing has a diffraction grating at the lower left with three adjustable apertures for the three primaries. The upper portion of the light passes directly through it (or what?)the device. To measure the color of a reflective sample, the sample is placed on something white (like paper) with the edge of the sample aligned so that the reflection from the sample enters the top portion and the reflection from the paper enters through the three apertures.

 

Figure 3 – Ives first trichromatic colorimeter

This design proved to be a bit fidgety because the slits were very narrow. Ives later hit on the idea of using “dyed gelatine films” to provide red, green, and blue-violet light instead of doing this with a diffraction grating. He also found he needed a way to mix the primaries uniformly across the field of view and provided several means for accomplishing this. He filed for a patent on this [Ives 1907b] and presented a paper on it [Ives 1907c].

Several similar devices were developed in the next 20 years, including Alfred Bawtree [1919], Frederick Ives’ son Herbert [1921], Frank Allen [1924], John Guild [1925], and Carl Keuffel [1926]. Anyone who did engineering, chemistry or physics before 1980 will recognize that last name. Carl Keuffel worked for Keuffel and Esser, manufacturer of the most coveted slide rules in the world and manufacturer of color measuring devices.

Figure 4 – Adam Hilger (known today for textbooks) sold Guild’s colorimeter [Anderson]

So it is that Maxwell’s scientific apparatus for investigating color vision inspired a variety of devices to measure color.

But these trichromatic colorimeters did not get glowing reviews. A. Ames [1921] was dismissive about the role of instruments in the practical measurement of color. “As any instrument is too complicated for general use…” Luckiesh [1921] stated that measurements from trichromatic colorimeters “can hardly be considered more than approximately comparative and of limited usefulness.” Martin and Gamble [1926] were slightly more positive when they said the accuracy was “just significantly good to make the instrument of practical use.”

Irwin Priest [1918] directed us toward a solution.

Of whatever value the so-called “Colorimeter” may be in special cases, it must be admitted that the fundamental basis of color specification is spectrophotometry.

If you don’t believe Priest, here is what K. S. Gibson [1919] said:

… for it is generally admitted that the fundamental basis of color specification is spectrophotometry…

Sometime in the 1920s, someone invented a way to use a spectrometer to emulate a trichromatic colorimeter. I have yet to find the first reference. It may have been first described by Priest, or maybe Deane Judd, but it was certainly known by John Guild in 1926 when he started using a color mixing system based on Maxwell’s work to determine the Standard Observer, which is the bridge between spectral data and trichromatic colorimeters. But that’s another story.

References

Frank Allen, A New Tri-Color Mixing Spectrometer, J. Opt. Soc. Am. 8, 339-341, 1924

A. Ames, Systems of Color Standards, J. Opt. Soc. Am. 5, 160-170, 1921

J. S. Anderson, Descriptions of the Exhibits - Optical instruments, J. Sci. Instrum. 4 196, 1927

Alfred Bawtree, An instrument for the quantitative analysis of colour, British patent GB2083219, filed Aug 25, 1919

K. S. Gibson, Photo-Electric Spectrophotometry by the Null Method, J. Opt. Soc. Am, Vol. 2, Issue 1, pp. 23-26, 1919

John Guild, A trichromatic colorimeter suitable for standardisation work, 1925 Trans. Opt. Soc. 27 106

Frederick E. Ives (a), A new color meter. J. Franklin Inst. 164, 47–56, April 25, 1907

Frederick E. Ives (b), Color-meter, US Patent #894,694, filed Oct 22, 1907

Frederick E. Ives (c), A color screen color meter, J. Franklin Inst. 164, 421-423, Nov 7, 1907

Herbert E. Ives, A proposed standard method of colorimetry, JOSA Vol 5, No. 6, Nov, 1921

Carl W. Keuffel, A trichromatic additive colorimeter, J. Opt. Soc. Amer., 12, 479A, 1926

Luckiesh, Color and its Applications, 3rd printing, Van Nostrand 1921

Martin, Louis Claude, and William Gamble, Colour and Methods of Colour Reproduction, Blackie and Son Limited, 1926

James Clerk Maxwell, On the Theory of Compound Colours, and the Relations of the Colours of the Spectrum, Philosophical Transactions of the Royal Society of London, Vol. 150 (1860), pp. 57-84

Irwin G. Priest, The Work of The National Bureau of Standards on the Establishment of Color Standards and Methods of Color Specification, Transactions of The Illuminating Engineering Society, Vol. XIII, January – December, 1918