The Basic Structure of Legal Precedent
In legal argumentation, lawyers constantly use this reasoning: In case Y, the situation was such-and-such, and the defendant was acquitted. In our current case X, the situation is like the situation in case Y. Therefore, the same argumentation that led to acquittal in case Y should apply, and the defendant in case X should also be acquitted.
Let me give you a concrete example. In case Y from 1990, John found a wallet on the street, took the cash, and left the wallet behind. The court ruled that John was not guilty of theft because he had no legal duty to return found property. Now in case X from 1991, Mary finds a purse on a bench, takes the money, and leaves the purse behind. Mary’s lawyer argues that case X is like case Y—both involve found property in a public place, both defendants took money, neither attempted to find the owner. Since the defendant in case Y was acquitted, the defendant in case X should also be acquitted. The judge agrees because the situations are legally identical.
This is the essence of legal precedent. You take an already decided case, show that your current case has the same relevant features, and argue that the same conclusion must follow. The power comes from the principle that like cases must be treated alike—if we decided one way before, we must decide the same way now when the facts are the same.
How Mathematical Proofs Work the Same Way
In mathematics, we can use exactly this structure to write proofs. We say: In proof Y, we proved statement S using method M. Our current problem X is like problem Y in all the relevant ways. Therefore, the same method M that worked in proof Y will work for problem X, and we can prove the same type of statement.
Here’s a simple example. In proof Y, we need to prove that the sum 2 + 4 + 6 + 8 equals 20. We notice all terms are even, so we factor out 2 and get 2(1 + 2 + 3 + 4) = 2(10) = 20. Now in problem X, we need to prove that 5 + 10 + 15 + 20 equals 50. We observe that problem X is like problem Y—all terms share a common factor, just like in Y. In Y, we factored out the common factor 2. In X, we factor out the common factor 5 to get 5(1 + 2 + 3 + 4) = 5(10) = 50. The same argumentation holds.
Notice how we explicitly referenced the previous proof and explained why the situations are alike. This is exactly what lawyers do in court, and it’s a perfectly valid way to structure mathematical arguments.
A More Complex Example: Proving Infinitude
Let’s see how this works with a famous mathematical theorem. In proof Y, Euclid proved there are infinitely many prime numbers. His method was to assume there are only finitely many primes—let’s call them 2, 3, 5, 7, up to some largest prime p. Then he constructed the number N equals the product of all these primes plus one. This number N cannot be divisible by any of the primes in our list because it leaves remainder 1 when divided by any of them. So either N itself is prime, or it has a prime factor not in our list. Either way, we’ve found a prime not in our supposedly complete list, which is a contradiction.
Now consider problem X: prove there are infinitely many primes of the form 4k + 3, that is, numbers that leave remainder 3 when divided by 4. We can argue that problem X is like problem Y. In Y, Euclid created a number that couldn’t be divisible by any prime in the assumed finite list and derived a contradiction. In X, we’ll create a number that requires a prime factor of the form 4k + 3 that’s not in our assumed finite list.
Here’s how the argument transfers. Assume there are only finitely many primes of form 4k + 3, and call them q₁, q₂, up to qₙ. Consider the number M equals 4 times the product of all these primes minus 1. This number M has the form 4k + 3. Now M must have prime factors, and at least one must have the form 4k + 3 (because a product of primes of form 4k + 1 always has form 4k + 1). But M is not divisible by any of our primes qᵢ because it leaves remainder 3 when divided by 4 times their product. So we’ve found a new prime of form 4k + 3, contradicting our assumption. The same argumentation that worked in Euclid’s proof Y works in our proof X.
Writing Complete Mathematical Arguments in Legal Style
When mathematicians write proofs, they often hide the fact that they’re adapting an existing proof. But we can make this explicit and actually strengthen our arguments by acknowledging the precedent. Consider how we prove statements about derivatives.
In case Y, we find the derivative of x². We use the limit definition, which gives us the limit as h approaches 0 of [(x+h)² – x²]/h. Expanding the numerator gives us [x² + 2xh + h² – x²]/h = [2xh + h²]/h = 2x + h. As h approaches 0, we get 2x. So the derivative of x² is 2x.
In case X, we need to find the derivative of x³. We observe that case X is like case Y—both involve finding the derivative of a power function using the limit definition. In Y, we expanded (x+h)², simplified, and took the limit. In X, we expand (x+h)³ to get x³ + 3x²h + 3xh² + h³. Following the same argumentation as in Y, we compute [(x+h)³ – x³]/h = [3x²h + 3xh² + h³]/h = 3x² + 3xh + h². As h approaches 0, we get 3x². The defendant in case Y was “acquitted” with derivative 2x, and the defendant in case X is “acquitted” with derivative 3x², following the same legal process.
When Legal-Style Argumentation Is Most Powerful in Mathematics
This approach is particularly effective in number theory. In case Y, we might prove that n² + n is always even. The proof factors this as n(n + 1), observes that consecutive integers have different parities, so one is even, making the product even. In case X, we prove that n³ – n is always divisible by 6. We argue that X is like Y—both involve factoring into consecutive integers. We factor n³ – n as n(n-1)(n+1), which gives us three consecutive integers. Among three consecutive integers, one is divisible by 3, and at least one is even, so the product is divisible by 6. The same argumentation about consecutive integers that worked in Y works in X.
The legal style also clarifies induction proofs. In case Y, we prove that 1 + 2 + … + n = n(n+1)/2 by induction. We verify the base case for n=1 and show that if the formula holds for k, it holds for k+1. In case X, we prove that 1² + 2² + … + n² = n(n+1)(2n+1)/6. Case X is like case Y—both are sum formulas proved by induction. In Y, we used the inductive hypothesis to simplify the sum for k+1. In X, we do the same: assume the formula holds for k, add (k+1)² to both sides, and algebraically verify we get the formula for k+1. The same induction structure that proved Y proves X.
Why This Method Actually Improves Mathematical Writing
When you write “case X is like case Y” in a mathematical proof, you’re doing several valuable things. First, you’re acknowledging your sources, which is intellectually honest. Second, you’re helping readers understand your proof by connecting it to something they might already know. Third, you’re showing that you understand the deep structure of the mathematics, not just the surface manipulations.
More importantly, this approach makes you think carefully about why two problems are really the same. When you claim that case X is like case Y, you must identify exactly which features matter and which are superficial. This is precisely what lawyers do when arguing precedent—they must show that the differences between cases are immaterial while the similarities are essential. In mathematics, this means identifying the core structure that makes a proof method work.
Consider one final example. In case Y, we prove that the square root of 2 is irrational by assuming it equals p/q in lowest terms, showing this leads to both p and q being even, contradicting “lowest terms.” In case X, we prove that the square root of 3 is irrational. Case X is like case Y—both involve assuming a rational expression in lowest terms and deriving a contradiction about divisibility. In Y, we showed both numerator and denominator were divisible by 2. In X, we show both are divisible by 3. The same argumentation structure that proved irrationality in Y proves it in X.
The Deep Unity of Reasoning
What we’re seeing is that legal precedent and mathematical proof by analogy are not just similar—they’re the same logical process applied to different domains. Both rely on the principle that if two situations share the same essential structure, they must have the same outcome. In law, this principle ensures fairness and consistency. In mathematics, it ensures logical coherence and allows us to extend known results to new territories.
The next time you write a mathematical proof, consider structuring it explicitly as a legal argument. State your precedent (the known theorem or proof), argue that your current problem is like the precedent case, and show how the same argumentation leads to the same type of conclusion. Not only will this make your proofs clearer, but it will also reveal the beautiful unity between legal reasoning and mathematical thinking—two fields that, despite their different subject matters, share the same fundamental logic of human reasoning.
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