Opinion – Marcelo Viana: George Boole and the mathematics of thought

The dear reader is a candidate for a great job. The company asks a series of questions to which the reader must answer in binary form: Yes or No. Based on these responses, the contractor decides on hiring, also in a binary way.

Unfortunately, the answer is no! And the reader wonders —hypothetically, of course!—if it wouldn’t have been better to “do something” with some answers… How many little lies would have been enough for the result to become positive?

This type of procedure is an example of what mathematicians and computer scientists call a “Boole function”, which is any rule for transforming a certain number of binary data (the reader’s responses) into a result that is also binary (the decision to hire -there). The simplest interesting example is “negation”, which uses only one die: if its value is Yes, the result is No; and if the die value is No, the result is Yes.

The name honors the English mathematician and philosopher George Boole (1815-1864), author of the book “The Laws of Thought”, published in 1854. In it, Boole introduced what we now call “Boolean algebra”, which is a mathematical formulation of the logic. In this theory, the binary values ​​can be Yes and No, as in our example, or True and False, or even 1 and 0, which are my class favorites: it makes no difference.

It’s not a particularly difficult area of ​​mathematics: I studied the basics of it in early high school. But to say that it is extremely important, useful and current would be an understatement: all that electronic computers do is calculate Boolean functions and, therefore, Boolean algebra is at the base of all computer science and technology.

The number of changes in the data (the reader’s hypothetical white lies) enough to invert the result is called the “sensitivity” of the Boole function. It’s a way of measuring how complex the function is. There are others, but computer scientists have shown that all the others give similar results: the only one that resisted fitting into this theory was, precisely, sensitivity.

In 1992, researchers Noam Nisan (Israel) and Mario Szegedy (United States) conjectured that sensitivity also agrees with other complexity measures, but for almost three decades no one has been able to prove this fact.

Until in 2019… (continued next week).