Reasonable Hope (Math)

Every physical measurement contains uncertainty, but mathematics can describe exact relationships. A ruler gives us an approximation of a square’s diagonal; the Pythagorean theorem tells us exactly why its length is square root of 2.

An error of two dollars may be harmless on a grocery bill but enormous in another setting. Accuracy must be judged by the scale of the problem, the quality of the available information, and the decision we need to make.

From estimating a grocery bill to projecting future insurance claims, approximations help us form reasonable expectations. A useful estimate does not replace an exact answer—it helps us recognize whether that answer makes sense.

Claude Shannon showed that communication doesn’t require a perfect, noise-free channel. When we understand the channel, respect its limits, and protect the message, reliable communication remains possible. Noise is real, but what matters can still get through.

Every communication channel has a limited capacity. When an important conversation fails, the problem may not be the people involved—it may be the rushed, distracted, or incomplete channel through which they tried to communicate.

Error-correcting codes add carefully designed redundancy so messages can survive noise and mistakes. Repetition isn’t always wasteful. Sometimes an important message needs to be communicated again, more clearly, or in a different way.

Compression makes information smaller by taking advantage of predictable patterns or removing unnecessary details. We also compress experiences whenever we summarize them. But when might a short, accurate description be mistaken for the whole story?

Information entropy measures the uncertainty within a system. Sometimes uncertainty means we need to learn more, but sometimes it is an unavoidable part of the situation. Mathematics can help us describe honestly what we know—and what we don’t.

Why does an unexpected result carry more information than one we already anticipated? Claude Shannon found a mathematical way to measure surprise. The unexpected may not always be true or important, but sometimes it reveals that our understanding is incomplete.

Information theory begins with a simple idea: information reduces uncertainty. A surprising message may contain more information than a predictable one—but mathematics cannot tell us whether that message is meaningful or worth hearing.

Greater capability tells us how far we might go, but it cannot tell us which direction we should choose. As human and artificial intelligence become more powerful, the most important question may not be what we can do, but what we are moving toward.

Artificial intelligence can provide a polished, confident answer and still be wrong. Because human judgment also has limits, we need to combine curiosity with discernment, openness with skepticism, and confidence with the humility to check our conclusions.

Artificial intelligence tried—and failed—to solve the Riemann Hypothesis. Yet its attempt contributed to progress on a related mathematical question, which a human mathematician then clarified and extended. Sometimes recognizing a limit helps another perspective move the boundary.

Writing extended our memory, telescopes extended our sight, and calculators extended our ability to compute. Artificial intelligence may extend our ability to explore—but it doesn’t remove our responsibility to choose the questions that matter and decide what is true.

My limitations created space for other people’s strengths, and Coaching Actuaries became stronger because of it. Mathematics grows in much the same way: through people questioning, correcting, and building on one another’s work.

The Working Genius assessment helped me recognize the work that energizes me and the work that drains me. Knowing our limits doesn’t mean making excuses. It means understanding what we bring to the team—and making room for what others bring.

After starting my business, I gradually began to believe that because I could do many things, I should do everything. A mathematical limit points toward where something is heading—and our personal limits may show us when it’s time to ask for help.

Logic need not replace beauty, mystery, faith, or wonder. It can help us examine assumptions, follow evidence, and understand more clearly—while leaving us open to the possibility that there is always more to see.

Fractions can be arranged in a never-ending list, but the real numbers cannot. Cantor’s discovery offers a picture of reality exceeding our descriptions—and invites humility about the limits of logic and language.

The counting numbers and the even numbers appear to have different sizes, yet Cantor showed how to match them perfectly. Sometimes logic answers our question. Other times, it teaches us that we need a better definition.