Mathematics is often seen as the domain of certainty. Once something is proven, it feels settled.

But every proof begins with assumptions.

In this reflection, Dave explores one of the most surprising moments in mathematical history: the introduction of a number whose square is negative. At first it seemed impossible. Yet allowing that single assumption opened the door to the complex numbers — a richer system that didn’t replace the real numbers, but expanded them.

The lesson reaches beyond mathematics.

Sometimes what feels impossible is simply something we haven’t yet allowed ourselves to imagine. And sometimes understanding grows not by abandoning certainty, but by discovering that reality is larger than the system we first assumed.

Today’s reflection asks:

Where might a new starting point reveal a larger picture?

IntersectingUs is a short daily reflection exploring the intersection of mathematics, philosophy, and life.

For centuries, Euclid’s geometry seemed complete — a perfect system built from simple assumptions.

But one assumption always stood out: the parallel postulate.

Mathematicians struggled with it for generations. Some quietly explored what would happen if that assumption changed. When Bernhard Riemann finally asked the question openly, something surprising happened. Entirely new geometries appeared — coherent worlds where triangles don’t add up to 180 degrees.

The lesson reaches far beyond mathematics.

Throughout history, people have often mistaken familiar frameworks for ultimate truth. But mathematics reminds us that even the most elegant systems may rest on assumptions worth re-examining.

Today’s reflection considers the courage it takes to question certainty — and what might become possible when we do.

IntersectingUs is a short daily reflection exploring the intersection of mathematics, philosophy, and life.

For centuries, Euclid’s geometry seemed complete — a perfect system built from simple assumptions.

But one assumption always stood out: the parallel postulate.

Mathematicians struggled with it for generations. Some quietly explored what would happen if that assumption changed. When Bernhard Riemann finally asked the question openly, something surprising happened. Entirely new geometries appeared — coherent worlds where triangles don’t add up to 180 degrees.

The lesson reaches far beyond mathematics.

Throughout history, people have often mistaken familiar frameworks for ultimate truth. But mathematics reminds us that even the most elegant systems may rest on assumptions worth re-examining.

Today’s reflection considers the courage it takes to question certainty — and what might become possible when we do.

IntersectingUs is a short daily reflection exploring the intersection of mathematics, philosophy, and life.

Episode 3: The Actuary’s Leap

Actuaries spend their careers studying risk. Using data, mortality tables, and probability, they calculate what is likely to happen across large populations. This work is built on a mathematical idea called expected value—the attempt to price the future based on what happens on average.

But there is a limit to what the math can tell us.

Even the best models can’t predict what will happen to one specific person tomorrow. They deal in aggregates, probabilities, and likelihoods—not personal certainty.

At some point, the calculations end.

In this episode, Brian reflects on the moment when analysis gives way to trust. In theology, this kind of trust is called fiducia—not simply believing something is true, but stepping into it. It’s the difference between studying a map of a bridge and actually driving across it.

Reasonable hope invites us to move beyond merely calculating the likelihood of goodness and begin living as if that goodness is real.

Soul Question:

You’ve seen the evidence of beauty and grace in your life. What would it look like to stop running the numbers and finally sit down in that truth today?

Episode 2: The Psychological Mirage

Why do we crave certainty so deeply?

In mathematics, a proof often ends with Q.E.D. — “that which was to be demonstrated.” It’s the satisfying period at the end of the argument. Solid ground. Finality.

But in life, that kind of certainty is rare.

Our minds naturally long for the feeling of complete safety — certainty about our health, our investments, our relationships, and our beliefs. Yet the feeling of certainty can be misleading. We can feel absolutely certain about something and still be completely wrong.

In this episode, Brian explores how certainty is often a psychological state rather than a reflection of reality. Waiting for perfect certainty before acting — before loving, serving, or committing — can leave us waiting for something that may never arrive.

A life of reasonable hope doesn’t require the feeling of absolute certainty. It invites us to notice the good that is already present — the love of family, the beauty of a sunset, the quiet signs that truth can be trusted even when certainty is absent.

Soul Question:

Is your hesitation based on a lack of truth — or just a lack of the feeling of certainty?

How certain do you need to be before you act?

In this opening episode of Reasonable Hope, Brian reflects on a lesson from actuarial science: life rarely operates in absolutes. While mathematics often deals in certainty—0 or 1—our daily decisions live somewhere in between.

Drawing from his years as an actuary, Brian explores the difference between certainty and confidence. Certainty is a psychological feeling of safety. Confidence, however, comes from the Latin com fide—“with faith.”

Engineers design bridges they are confident will stand, even though they can never guarantee 100% certainty. In the same way, faith may not require perfect certainty before we move forward.

What if commitment doesn’t require 100% certainty?

What if 51% is enough?

Soul Question:

What is something you’ve been holding back on because you’re waiting for complete certainty? Could you act on the 51% today?

In this episode, the discussion revolves around the theme of love and its many facets, from tough love to unconditional support. The conversation highlights the importance of intrinsic worth and identity beyond achievements, particularly in the context of mathematics and education. Personal stories and experiences underscore the transformative power of focusing on others, emphasizing the value of relationships and community. The episode encourages listeners to invest in others and recognize their own inherent dignity, fostering a culture of mutual support and flourishing.

In this podcast episode, the hosts delve into the deep connections between mathematics, community, and personal growth. They discuss how struggling with mathematical proofs, like those of the Pythagorean Theorem, leads to a greater understanding and appreciation of their underlying principles. The conversation highlights the role of community and collaboration in overcoming these challenges and making discoveries. The hosts also touch on how the timeless and universal nature of mathematics allows for a shared pursuit of truth across cultures, languages, and even barriers like prison walls. They conclude by emphasizing the importance of fostering a sense of belonging and support in both math and life.