I think of my core audience as young economists, where “young” refers not to chronological age but rather to a mind still open to persuasion and new ideas from a random blogger (me). One advantage of writing for this audience is that I can assume that a large fraction of my readership has a familiarity with multivariable calculus. As it happens, the debate between Neil Gorsuch’s majority opinion and Samuel Alito’s dissent in Bostock v. Clayton County Georgia hinges on the discrete version of an important issue that comes up in multivariable calculus.

Before explaining the issue, let me be clear that as a policy matter, I am gladdened by the decision to give employment protections to gay and transgender individuals. Hereafter, I focus on the legal issue of whether the Civil Rights Act already enacted these protections, as the Supreme Court has now finally figured out, or whether establishing those protections should require additional legislation.

The First-Derivative Standard

The 1964 Civil Rights Act outlaws discrimination against an individual in the workplace (“with respect to his compensation, terms, conditions, or privileges of employment”) based on sex. Neil Gorsuch argues that this outlaws a nonzero first derivative (actually, first-difference) of treatment of an individual in the workplace with respect to sex. To back this up, let me first quote two passages (separated by a substantial vertical space) that give background:

Neil Gorsuch (majority opinion)

Sometimes small gestures can have unexpected consequences. Major initiatives practically guarantee them. In our time, few pieces of federal legislation rank in significance with the Civil Rights Act of 1964. There, in Title VII, Congress outlawed discrimination in the workplace on the basis of race, color, religion, sex, or national origin. Today, we must decide whether an employer can fire someone simply for being homosexual or transgender. The answer is clear. An employer who fires an individual for being homosexual or transgender fires that person for traits or actions it would not have questioned in members of a different sex. Sex plays a necessary and undisguisable role in the decision, exactly what Title VII forbids.

We must determine the ordinary public meaning of Title VII’s command that it is “unlawful . . . for an employer to fail or refuse to hire or to discharge any individual, or otherwise to discriminate against any individual with respect to his compensation, terms, conditions, or privileges of employment, because of such individual’s race, color, religion, sex, or national origin.”

But it is a third passage that makes the first-derivative standard clear. I have added bolding to the key two sentences:

The question isn’t just what “sex” meant, but what Title VII says about it. Most notably, the statute prohibits employers from taking certain actions “because of ” sex. And, as this Court has previously explained, “the ordinary meaning of ‘because of ’ is ‘by reason of ’ or ‘on account of.’ ” University of Tex. Southwest- ern Medical Center v. Nassar, 570 U. S. 338, 350 (2013) (cit- ing Gross v. FBL Financial Services, Inc., 557 U. S. 167, 176 (2009); quotation altered). In the language of law, this means that Title VII’s “because of ” test incorporates the “‘simple’” and “traditional” standard of but-for causation. Nassar, 570 U. S., at 346, 360. That form of causation is established whenever a particular outcome would not have happened “but for” the purported cause. See Gross, 557 U. S., at 176. In other words, a but-for test directs us to change one thing at a time and see if the outcome changes. If it does, we have found a but-for cause.

Changing one thing at a time to see what happens is looking at a first derivative or a first difference. Another passage appears to talk about the sign of the derivative:

What did “discriminate” mean in 1964? As it turns out, it meant then roughly what it means today: “To make a difference in treatment or favor (of one as compared with others).” Webster’s New International Dictionary 745 (2d ed. 1954). To “discriminate against” a person, then, would seem to mean treating that individual worse than others who are similarly situated.

However, if an individual were treated better because of her or his sex in the “but-for” sense, then it necessarily implies that others are treated worse because of having the other sex. So any nonzero first derivative of treatment with respect to sex is illegal.

In the Multivariable Case, First Derivatives and First Differences are Undefined without Specifying a Coordinate System

The trouble with a first-derivative or first-difference standard is that in a multivariate setting, first derivatives and first differences are not well-defined unless one specifies a coordinate system. In Bostock v. Clayton County, Georgia, Neil Gorsuch and Samuel Alito use and recommend different coordinate systems. Here it is important to realize that a first derivative or first difference depends on what the other coordinates are.

For Neil Gorsuch, the other coordinates are (a) sex of those an individual is attracted to and (b) sex for which an individuals behavior is stereotypical. Here is Neil Gorsuch’s direct description of the coordinate system he is using:

When an employer fires an employee because she is homosexual or transgender, two causal factors may be in play— both the individual’s sex and something else (the sex to which the individual is attracted or with which the individual identifies). But Title VII doesn’t care. If an employer would not have discharged an employee but for that individual’s sex, the statute’s causation standard is met, and liability may attach.

And here (in a passage that is somewhat difficult to read without all the context of the opinion) is Neil Gorsuch’s rejection of other coordinate systems. I have added bolding to the word “labels” because that is the way he refers to what I would call a “coordinate system”:

First, it’s irrelevant what an employer might call its discriminatory practice, how others might label it, or what else might motivate it. In Manhart, the employer called its rule requiring women to pay more into the pension fund a “life expectancy” adjustment necessary to achieve sex equality. In Phillips, the employer could have accurately spoken of its policy as one based on “motherhood.” In much the same way, today’s employers might describe their actions as motivated by their employees’ homosexuality or transgender status. But just as labels and additional intentions or motivations didn’t make a difference in Manhart or Phillips, they cannot make a difference here. When an employer fires an employee for being homosexual or transgender, it necessarily and intentionally discriminates against that individual in part because of sex. And that is all Title VII has ever demanded to establish liability.

In arguing for his coordinate system, Neil Gorsuch points out an especially unattractive coordinate system that is not directly at issue: a coordinate system that, instead of taking occupation as a coordinate takes “gender-appropriate occupation” as a coordinate:

Consider an employer eager to revive the workplace gender roles of the 1950s. He enforces a policy that he will hire only men as mechanics and only women as secretaries. When a quali- fied woman applies for a mechanic position and is denied, the “simple test” immediately spots the discrimination: A qualified man would have been given the job, so sex was a but-for cause of the employer’s refusal to hire. But like the employers before us today, this employer would say not so fast. By comparing the woman who applied to be a mechanic to a man who applied to be a mechanic, we’ve quietly changed two things: the applicant’s sex and her trait of fail- ing to conform to 1950s gender roles. The “simple test” thus overlooks that it is really the applicant’s bucking of 1950s gender roles, not her sex, doing the work. So we need to hold that second trait constant: Instead of comparing the disappointed female applicant to a man who applied for the same position, the employer would say, we should compare her to a man who applied to be a secretary. And because that jobseeker would be refused too, this must not be sex discrimination.

No one thinks that …

Samuel Alito Argues for a Different Coordinate System

By contrast, Samuel Alito argues that “homosexual/heterosexual” or “cisgender/transgender” are natural coordinates. For one thing, he argues these would have been components of a salient coordinate system in 1964 when the Civil Rights Act was enacted. And coordinate system is an important part of “original public meaning.”

Samuel Alito (dissent):

The Court tries to convince readers that it is merely enforcing the terms of the statute, but that is preposterous. Even as understood today, the concept of discrimination be- cause of “sex” is different from discrimination because of “sexual orientation” or “gender identity.” And in any event, our duty is to interpret statutory terms to “mean what they conveyed to reasonable people at the time they were written.” A. Scalia & B. Garner, Reading Law: The Interpretation of Legal Texts 16 (2012) (emphasis added). If every single living American had been surveyed in 1964, it would have been hard to find any who thought that discrimination because of sex meant discrimination because of sexual orientation––not to mention gender identity, a concept that was essentially unknown at the time.

Samuel Alito further argues that he doesn’t need to prove his coordinate system is the only possible one. If it is a reasonable coordinate system, then the Civil Rights Act is ambiguous about whether it makes discrimination on the basis of being gay or transgender illegal. And if a statute is ambiguous, then considerations such as legislative intent can be brought to bear. Samuel Alito:

The Court attempts to prove that point, and it argues, not merely that the terms of Title VII can be interpreted that way but that they cannot reasonably be interpreted any other way. According to the Court, the text is unambiguous. See ante, at 24, 27, 30.

In later passages, Samuel Alito directly illustrates how the first derivative (really first difference) is zero in his coordinate system. I reordered the passages in order to improve the flow:

Contrary to the Court’s contention, discrimination because of sexual orientation or gender identity does not in and of itself entail discrimination because of sex. We can see this because it is quite possible for an employer to dis- criminate on those grounds without taking the sex of an in- dividual applicant or employee into account. An employer can have a policy that says: “We do not hire gays, lesbians, or transgender individuals.” And an employer can implement this policy without paying any attention to or even knowing the biological sex of gay, lesbian, and transgender applicants. In fact, at the time of the enactment of Title VII, the United States military had a blanket policy of refusing to enlist gays or lesbians, and under this policy for years thereafter, applicants for enlistment were required to complete a form that asked whether they were “homosexual.” Appendix D, infra, at 88, 101.

In an effort to prove its point, the Court carefully includes in its example just two employees, a homosexual man and a heterosexual woman, but suppose we add two more indi- viduals, a woman who is attracted to women and a man who is attracted to women. (A large employer will likely have applicants and employees who fall into all four categories, and a small employer can potentially have all four as well.) We now have the four exemplars listed below, with the dis- charged employees crossed out:

Man attracted to men [crossed out]

Woman attracted to men

Woman attracted to women [crossed out]

Man attracted to women

The discharged employees have one thing in common. It is not biological sex, attraction to men, or attraction to women. It is attraction to members of their own sex—in a word, sexual orientation. And that, we can infer, is the employer’s real motive.

In sum, the Court’s textual arguments fail on their own terms. The Court tries to prove that “it is impossible to discriminate against a person for being homosexual or transgender without discriminating against that individual based on sex,” ante, at 9, but as has been shown, it is en- tirely possible for an employer to do just that.

The Court’s remaining argument is based on a hypothetical that the Court finds instructive. In this hypothetical, an employer has two employees who are “attracted to men,” and “to the employer’s mind” the two employees are “mate- rially identical” except that one is a man and the other is a woman. Ante, at 9 (emphasis added). The Court reasons that if the employer fires the man but not the woman, the employer is necessarily motivated by the man’s biological sex. Ante, at 9–10. After all, if two employees are identical in every respect but sex, and the employer fires only one, what other reason could there be?

As Neil Gorsuch does in one of the passages above, Samuel Alito calls a difference of coordinate systems a difference of “labels.” I bold the word “labels” in all the passages I quote in this post.

The problem with this argument is that the Court loads the dice. That is so because in the mind of an employer who does not want to employ individuals who are attracted to members of the same sex, these two employees are not materially identical in every respect but sex. On the contrary, they differ in another way that the employer thinks is quite material. And until Title VII is amended to add sexual orientation as a prohibited ground, this is a view that an employer is permitted to implement. As noted, other than prohibiting discrimination on any of five specified grounds, “race, color, religion, sex, [and] national origin.” 42 U. S. C. §2000e–2(a)(1), Title VII allows employers to decide whether two employees are “materially identical.” Even idiosyncratic criteria are permitted; if an employer thinks that Scorpios make bad employees, the employer can refuse to hire Scorpios. Such a policy would be unfair and foolish, but under Title VII, it is permitted. And until Title VII is amended, so is a policy against employing gays, lesbians, or transgender individuals.

Once this is recognized, what we have in the Court’s hypothetical case are two employees who differ in two ways–– sex and sexual orientation––and if the employer fires one and keeps the other, all that can be inferred is that the employer was motivated either entirely by sexual orientation, entirely by sex, or in part by both. We cannot infer with any certainty, as the hypothetical is apparently meant to suggest, that the employer was motivated even in part by sex. The Court harps on the fact that under Title VII a prohibited ground need not be the sole motivation for an adverse employment action, see ante, at 10–11, 14–15, 21, but its example does not show that sex necessarily played any part in the employer’s thinking.

The Court tries to avoid this inescapable conclusion by arguing that sex is really the only difference between the two employees. This is so, the Court maintains, because both employees “are attracted to men.” Ante, at 9–10. Of course, the employer would couch its objection to the man differently. It would say that its objection was his sexual orientation. So this may appear to leave us with a battle of labels. If the employer’s objection to the male employee is characterized as attraction to men, it seems that he is just like the woman in all respects except sex and that the employer’s disparate treatment must be based on that one difference. On the other hand, if the employer’s objection is sexual orientation or homosexuality, the two employees differ in two respects, and it cannot be inferred that the disparate treatment was due even in part to sex.

How to Pick a Coordinate System

Neil Gorsuch and Samuel Alito point to several considerations relevant to picking a coordinate system:

• What was the original public coordinate system considered as part of original public meaning? (Samuel Alito)

• In cases of ambiguity about the coordinate system, what coordinate system did the legislators have in mind? (Samuel Alito)

• Which coordinate system involves the most semantically primitive coordinates? For example, sex of the person one is attracted to must be determined in order to determine whether one is heterosexual or homosexual. (Neil Gorsuch)

• Which coordinate system is implicit in relevant precedents? (Neil Gorsuch)

To these I can add one grace note related to the criterion of being semantically primitive. Suppose someone could hide only their own sex and not anything else—not the sex of others and not behavior. Then an employer could not determine whether an individual was homosexual or transgender. This concealment test might help in defining what is semantically most primitive. (However, note that in this case, the employer could determine whether the stereotypical behavior matched the sex of those an individual was attracted to.)

Conclusion

Bostock v. Clayton County, Georgia not only yielded a policy result to be applauded, but raises fascinating legal issues that can be illuminated by the principle of multivariable calculus that, in general, the first derivative with respect to a variable cannot be defined without specifying the rest of the coordinate system. And a similar principle holds for first differences in a multivariate setting. The court system will be able to clarify these issues much better if it uses the language of calculus and finite differences.