# Mathematical Thinking and the Law

This Occupy Math is partly based on a More Perfect podcast entitled One Nation, Under Money. The More Perfect series tries to understand and explain US Constitutional law and this podcast is a doozy that looks at a very mathematical approach to the commerce clause:

The United States Congress shall have power “To regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes.”

– Article 1, Section 8, Clause 3, United States Constitution

The application of mathematical thinking to this clause increased the power of the clause far beyond what someone reading it for the first time might think it had. Consider this while you’re reading the rest of the post: is the interpretation of the clause that arises anything like what the people that wrote the clause thought of while they were writing it?

The first step: negative actions.

The commerce clause sounds no more than sensible at first. The strangeness started in 1942 and arose from the Agricultural Adjustment Act. This law was part of the New Deal legislation that was trying to help the agricultural sector get back on a firm footing. One of the things that the law did was to give farmers price support payments for wheat while limiting them to growing only twelve acres of wheat on their farm (restricting supply helps boost price). Mr. Roscoe Filbrun Sr. was fined for growing 23 acres of wheat. The law that limited farmers to 12 acres was based on the ability of Congress to regulate interstate commerce. Mr Filbrun sued, contending that he was feeding the wheat to his own livestock, and so no interstate commerce was involved.

The Ohio Supreme Court ruled in Mr. Filburn’s favor, but then, in Wickard v. Filburn, the Supreme Court ruled against him. This is where the mathematical thinking appears, in the court’s reasoning: by growing wheat to feed to his livestock, Mr. Filbrun was declining to buy wheat on the open market, which would have crossed state lines. The court said that Congress could regulate failure to participate in interstate commerce under the commerce clause.

You can be fined because of not buying something?

Someone trained in math will probably be more comfortable with this reasoning than anyone else — except possibly a lawyer. Doing nothing at all can have a huge impact! If you are not trained in a particular sort of logic — one used routinely in law and math and almost nowhere else — this may seem pretty bogus.

Let’s look at the effect mathematically. Congress can regulate commerce “among the several states”. There is no disagreement that that means any act of commerce that crosses state lines. This means we can think of the people and companies engaging in commerce as places in a network and any buying and selling between them as creating links in that network. This is a huge, dense network with millions of edges, and congress has authority over anything moving along those edges. So far this is big, complex, but pretty intuitive.

What the court decision did was add commerce that did not happen, but could have, to the collection of links. Said that way, the effect is almost terrifying. If you are connected to everyone who sold you something, that is a medium-size group. If you are connected to everyone who could have sold you something, that is a gigantic group. It makes the economic network far more connected and so enormously amplifies the regulatory power of Congress.

What else did this mathematical interpretation of commerce regulate?

A big decision that was anchored to the commerce clause was one that ended segregation in many places. The Civil Rights Act of 1964 forbade many types of discrimination, but was sort of hard to enforce on matters internal to a single state. In the case Katzenbach v. McClung, the commerce clause was used to create a whole new way of enforcing the Civil Rights Act. Ollie’s Barbecue in Birmingham, Alabama claimed that all their suppliers and customers were local. The federal government showed that their suppliers were getting their meat from out of state, the ingredients for the sauce were not raised in Alabama, and so on. Ollie’s was participating in interstate commerce and so was subject to the Civil Rights Act.

Mathematically this meant that Congress could not only regulate both present and potential links in the trade network that happen to cross state lines, but that the ability to regulate was contagious to adjacent links in the network!

The Affordable Care Act (Obamacare) required that people purchase insurance or pay a fine. This provision was challenged in National Federation of Independent Business v. Sebelius. The court ruled that, because uninsured people place a substantial burden on the economy, the commerce clause grants Congress the authority to enact the insurance requirement.

Mathematically, this decision makes the authority of the commerce clause contagious across the entire network. These three decisions leverage the commerce clause to grant regulatory authority over almost anything that involves money. There is a lot more to this — Occupy Math has highlighted the mathematical parts and recast them in the theory of networks — listen to the podcast linked up at the top for a lot more of the incredible story of the commerce clause.

Network theory and interpreting the law.

This is not Occupy Math’s first visit to networks. Earlier posts on this topic include Cookies or Calculus and the post on Influence Maps is another example of applying math to the real world. Looking at the way the commerce clause is interpreted from a network perspective gives you an additional point of view to think about what the court decisions did and let you see, in a simple way, how each decision increased the power of the commerce clause. Was that power there all the time, or did the power arise from the clever application of logic? Occupy Math is not sure.

Another matter that is not clear is how much the lawyers trying to win the three cases discussed in the post thought through the impact of their victories. Thinking of the matter in terms of where Congressional regulatory power applied in a network might have made it easier for them to understand what it was they were doing in the long term. Occupy Math also finds it concerning that fundamental civil rights require a hack involving commerce to be enforced.

What does this all mean?

Math is the right of all free people — in part because a lack of math can blind you to the consequences of your actions. Do you think the framers of the constitution would agree with the three court decisions we have discussed? Remember that this question is independent of the degree to which the decision were good ones, as measured by their immediate impact. Thinking about the different aspects of things independently is a core skill in math.

The commerce clause grants “the ability to regulate” under the assumption that everyone knows what the word “regulate” means. A scholarly publication in math typically has dozens of definitions. It has them because math people are terrified of not being as clear as humanly possible about their meaning (okay, there is also a lot of jargon, but once you master it, there is clarity). If you have a reaction to this post on network views of interpretation of law, please comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics