We seek out patterns since they’re shortcuts to understanding. They provide an abbreviated means of making sense of things. When they work, they can save time and effort. But we sometimes find patterns in instances where none exist. Over time, we come to learn that searching for patterns can be a more complex task than anticipated. Take, for example, your basic multiplication table.
1 x 1 = 1 2 x 2 = 4 3 x 3 = 9
11 x 11 = 121 22 x 22 = 484 33 x 33 = 1,089
111 x 111 = 12,321 222 x 222 = 49,284 333 x 333 = 110,889
1111 x 1111 = 1,234,321 2222 x 2222 = 4,937,284 3333 x 3333 = 11,108,889
Someone searching for a pattern will notice that the number 1 has the property of producing a palindromic number sequence. Surely there should be another number that follows the same pattern. But no such sequence emerges with the numbers 2 and 3. In fact, the number 1 is the only number that holds this unique property.
4 x 4 = 16 5 x 5 = 25 6 x 6 = 36
44 x 44 = 1,936 55 x 55 = 3,025 66 x 66 = 4,356
444 x 444 = 197,136 555 x 555 = 308,025 666 x 666 = 443,556
4444 x 4444 = 19,749,136 5555 x 5555 = 30,858,025 6666 x 6666 = 44,435,556
7 x 7 = 49 8 x 8 = 64 9 x 9 = 81
77 x 77 = 5,929 88 x 88 = 7,744 99 x 99 = 9,801
777 x 777 = 603,729 888 x 888 = 788,544 999 x 999 = 998,001
7777 x 7777 = 60,481,729 8888 x 8888 = 78,996,544 9999 x 9999 = 99,980,001
Looking at the number 5 it can be seen that the number 25 is always present in the result obtained. Perhaps this provides a method, but when we look at the number 6 we find that the number 36 does not recur. Well, perhaps it works only with odd numbers, but when we look at the number 7 we see the number 29, but nowhere do we find the number 49. We note that the last number of each product is uniform and consistent, but the first numbers vary. Etc.
Our effort to draw easy parallels between numbers quickly reveals its limited reach. Patterns can be detected for each individual number, but when we try to derive laws from these our simple observational methods of looking for obvious patterns tend not to be easily generalized.
As in number theory so too in legal theory. One would think that a set of reasons which works well in one context can be easily applied to another. We deceive ourselves into believing that simply because a set of reasons makes sense in one area it will necessarily make sense in a different area. We simplify reality to keep it comprehensible, detecting patterns we believe must be universal, without recognizing how reality is more complex and multi-faceted than we’ve assumed.
“The world looks like a multiplication-table, or a mathematical equation, which, turn it how you will, balances itself.” ––Ralph Waldo Emerson
Patterns do not necessarily conform to our expectations. This is why each case must be looked at separately. This is why overstatement and generalization are ineffective and why universality can never be assumed. Proving a disputed fact or conclusion should be looked at mathematically––in much the same way axioms are set forth. Holding an idea up to the light––only then will its true pattern reveal itself.
