The Geometry of Riches: Why Mathematical Intuition is the Key to Personal Finance Success

In the realm of personal finance success, few skills prove as critical yet misunderstood as mathematical intuition. It represents the ability to grasp abstract concepts—compound interest, convexity, or the impact of time—not merely as equations, but as tangible, predictable forces shaping our reality. For many, this intuition remains elusive, a casualty of a mathematical education system focused heavily on formalization.

The central challenge in personal finance is not in understanding the simple concepts of the benefit of savings or how delayed gratification builds long-term wealth. The true obstacle lies in consistent execution. People often fail to achieve financial goals, which is a direct result of lacking mathematical intuition. This leads to a lack of both a motivating vision and the resulting confidence necessary for achieving their future wealth.

The challenge of building wealth is less about knowing the formula for compound interest and more about possessing the mathematical intuition making the growth path feel real, immediate, and achievable. Addressing this gap is a core foundation of our personal finance class.

This article describes why mathematical intuition is essential. Along the way, examples are provided from music and other famous intuitive math brains. The challenge of the American math education approach is described. We conclude with our approach to building personal finance mathematical intuition as required to truly see and achieve the path to getting rich.

About the author:  Jeff Hulett leads Personal Finance Reimagined (PFR), a decision-making and financial education platform dedicated to empowering individuals and businesses. PFR offers specialized services for entrepreneurs, including Founder's Copilot support and comprehensive business startup services, helping visionaries turn ideas into thriving ventures. Jeff also teaches personal finance at James Madison University and delivers impactful seminars on financial literacy. Check out his book—Making Choices, Making Money: Your Guide to Making Confident Financial Decisions.

Jeff is a career banker, data scientist, behavioral economist, and choice architect. He has held leadership roles at Wells Fargo, Citibank, KPMG, and IBM, bringing decades of experience to his mission of transforming financial decision-making.

Math as a Language of Abstraction

Many see math as separate from language. But our brain does not see it that way. To the brain, language acquisition is how it grows and improves concept precision. Math is just one method it uses to acquire language. But math as a language serves a very important purpose, different than our day-to-day spoken language. It is the language of abstraction.

The mathematician David Bessis defines math as providing a precise language for things we cannot point to in our day-to-day lives. This makes mathematics fundamentally a language of abstraction and imagination. The formalization of math becomes necessary because the objects of study remain beyond our direct, physical reach—whether non-physical concepts (like probability) or distant, physical entities (like the gravity impacting stars in a far-off galaxy). Since we cannot physically manipulate or interact with these objects, a strict formalization becomes necessary. The notation, the symbols, and the theorems serve as the rigorous words communicating our imagination.

Developing mathematical intuition helps expand our imagination. Math is inherently visual. It helps us see beyond three dimensions, grasp the relativity of time the way Einstein did, or imagine future demand the way a visionary entrepreneur does. It helps us see music the way Bach did, structuring complex, interlocking musical forms with a mathematical structure. It helps us imagine a starry sky the way Picasso did, transforming the visible world through geometric abstraction. It turns the unseen forces of the world into clear paths and predictable outcomes.

The Intuitive Power of Seeing Music: Expanding Dimension

Think of a song from a favorite band—perhaps Led Zeppelin's Kashmir. A song serves as the listener's vessel holding a set of sounds prefabricated to align with their brain's right hemisphere. A "good" song intuitively sparks a positive emotion, and "good" is unique to the listener. For me, Kashmir instantly lifts my mood whenever it comes on. Perhaps it brings you similar happiness. If not, I invite you to think of a song creating that same feeling.

While music initially registers as this holistic, sensory feeling (a "fast brain," right-hemisphere focus), over time, someone with mathematical intuition begins changing how they see their favorite music. Often, it still resides in the "it just feels good" part of the brain, but as our math intuition grows, it morphs into "seeing" the music and the more language-intensive part of our brain. Then, as the seeing part of our brain sharpens, it evolves back to the intuitive part of our brain as improved intuition. It is a powerful feedback loop. This loop is the essence of how we become more intelligent, sharper thinkers.

The left hemisphere, or the "seeing" part of our brain, is where Wernicke's area and Broca's areas are located. These neurobiological areas are focused on language comprehension and acquisition. This is where "seeing" is developed with language for promoting imagination. Keep in mind, the brain is incredibly dynamic, but the left/right hemisphere-based logical model of the brain helps recognize the essential centers involved in imagination and "seeing."

For instance, you might see Kashmir not just as a singular sound, but as the rich base and drum support from John Bonham and John Paul Jones, distinct from the soaring guitar and vocals of Jimmy Page and Robert Plant. This shift reflects how a developing math brain allows the whole sound to evolve into a disaggregated, visual structure.

For me, music often registers as a vector defined by rhythm, tempo, and differential—a manifestation of mathematical beauty with a fractal quality. The perception breaks down the whole into dimensions:

• X is Time.

• Y is Music Direction.

• Pitch represents Music Speed or intervals.

• Line thickness and order represent the relative significance of the instrument type.

• Color differentiates the instrument type.

(Please note: these dimensions just "showed up" in my imagination when I was younger. I do not know if it is technically correct, but I did not really have a choice. The dynamic imagery is just there.)

Notice this multi-dimensional system uses familiar concepts like the X-Y plane. It requires no formal mathematical notation to expand perception well beyond the standard three-dimensional world. This ability instinctively applies dimensional thinking and structure to something non-spatial—mapping concepts like pitch and color onto a geometric model. It illustrates the true power of mathematical intuition. It represents the imaginative leap allowing us to expand our seeing beyond the constraints of our standard 3D world.

This vision aligns with the idea, expressed by George Dyson, the son of the Physicist Freeman Dyson: “Through music, we are able to share four-dimensional structures we are otherwise only able to observe in slices, one moment at a time.” Furthermore, Neuroscientist Dean Buonomano states: “Time is to speech and music recognition as space is to visual object recognition.”

Each instrument flows as a separate, colored stream in this vision. The current music appears vibrant, while the past fades. This ability mentally translates a temporal, auditory experience into a multi-dimensional, structured visual. It results directly from mathematical concepts promoted from the technical side of the left hemisphere to the intuitive side of the right hemisphere. However, as Hofstadter notes in Gödel, Escher, Bach, difficulty arises when simultaneously following individual components (mathematical) and hearing the whole effect (holistic). We "flip back and forth" between the two modes, enjoying the music either analytically or intuitively.

The Flawed "Geometry Sandwich"

Mathematics in the U.S. often receives instruction through the lens of strict formalization—the high school "geometry sandwich" of Algebra I, Geometry, Algebra II, and Calculus. This approach emphasizes a timeboxed, technical scaffolding of theorems and procedures instead of nurturing imagination. Students may feel discouraged because:

1. The technical scaffolding is presented before the learner is ready to internalize it. One "bad" teacher can create a discouraging, difficult-to-overcome gap in the scaffold.

2. The crucial imaginative part of math—the new "thing" the formal structure helps us imagine—remains absent.

For many, math truly works only when tied to a progression of imaginative outcomes. For example, the formal math like the Pythagorean theorem helps us imagine a fundamental relationship in the real world, allowing us to see how the mathematical structure of a right triangle underlies the creation of harmonious musical scales. The emphasis should fall on the abstraction, the new truth or reality a formal concept helps us unlock. Why teach math unless it links directly to our imagination?

To clarify, I am not saying the formalization in the traditional geometry sandwich is categorically bad; it is more a matter of math curriculum weighting and incentives. Both the training of teachers and the curriculum expectations create incentives that may lead to student discouragement. Adding a bigger focus on the imaginative part of math requires de-emphasizing something else. The "something else" often impacts teachers' jobs and requires the work of education administrators to change. The American education system is notoriously resistant to change.

Up to this point, we have discussed the biological mechanisms for math intuition. We also discussed connecting math teaching to the practical abstractions math is intended to describe. We have also discussed the way math is typically taught in America and the need to evolve the math curriculum. But what does this mean for Personal Finance? The core challenge is most of my students come to class with little onboard mathematical language. So we are challenged to build mathematical intuition with little, if any, of the formalization helpful for building mathematical intuition.

Next, we discuss how the challenge is overcome and how Personal Finance math intuition is delivered.

Making Financial Success Visible: Promoting Math to Intuition

The core problem in Personal Finance—and indeed, in any discipline dealing with long-term, abstract outcomes—is the challenge of saving. Living below your means requires suppressing immediate gratification for a benefit which remains mathematically abstract. Students struggle to grasp the time value of money and compound interest because they cannot easily point to them. Long-term, abstract outcomes are subject to availability bias, the essential cognitive bias causing us to overweigh what we can point to today and underweight things we cannot easily point to in the future. A fundamental goal of our class is to help students overcome cognitive biases subverting their personal finance success.

We teach the same Personal Finance curriculum at several Virginia public universities. We teach a wide variety of students, both business school students and many from outside the business school.  While there are a handful of “mathy” students, my experience suggests the majority have not developed much more than basic high school mathematical abilities. So this is tricky. Given these various math levels, I need to find the most effective approach for the most students.

My Personal Finance class approach is to teach mathematical intuition with little formalization.  This approach is grounded in visualization and practice to help students build a mathematical intuition.  The intention is to help the personal finance students visualize the mathematical concepts necessary for personal finance success.

The mission of developing mathematical intuition becomes an act of empowerment. My job in the classroom involves elevating math in an imagination-provoking kind of way. I make it accessible and visible. We systematically work through the visible. This makes the mathematics of wealth less of an abstraction, rendering it something they can intuitively grasp. The teaching of mathematical intuition has become easier because of apps and other technologies executing the notational math in the background. For example, we use Definitive Choice, a decision-making and choice architecture technology. It relies on matrix algebra, eigenvalues, and eigenvectors associated with the mathematics developed by Thomas Sattay called the Analytical Hierarchy Process ("AHP"). The AHP math is elegant, but more important is the intuition of what the math does to help the students make the best decision. We use visual imagery to help the students make the best economic trade-offs for themselves and determine their criteria models. We use graphics to display the cost benefits of multiple alternatives.

By practicing a consistent, repeatable decision process with various personal finance examples and using the decision technology, we are improving mathematical intuition. We help the students imagine an otherwise abstract concept and build a mathematical intuition of living below our means and achieving the benefits of compound interest.

Visualizing the Wealth Curve

I focus on showing them the long-term compound interest curves as paths to wealth. We do not just calculate a future value; we chart the convex, nonlinear path. We are born with an innate understanding of linear relationships. This makes internalizing the nonlinear relationship between savings and long-term wealth a difficult, non-intuitive mathematical abstraction. In almost every class and based on the many practical financial decisions they will soon face, the students interact with the same long-term view of their desired financial journey.

Starting on the first day of class, I share a path to getting rich—a path based on achievable incomes, doable savings rates, and common diversified investments. The point is simple: none of this is miraculous; it is VERY ACHIEVABLE. The biggest challenge remains the mathematical intuition for making getting rich less of an abstraction, rendering it more intuitive.

I illustrate the dramatic difference between simple interest and the convexity of compound growth. They see the curve remains relatively flat in the early years—the work of saving feels slow and unrewarding. As time progresses, the curve begins accelerating exponentially. By making small, consistent investments today, they effectively steepen the slope of their future wealth curve. I introduce the intuitive theorem based on Jensen's Inequality to lay the basis for understanding why nonlinear functions increase at an increasing rate, emphasizing the immense upside of being convex to time.

I demonstrate the mathematics of theoretical finance, such as Present Value and Future Value functions, underlying the visuals. We encounter the volatility of daily closing stock index prices to help them understand variability and dynamic risk. We "zoom out" to graphically show short-term losses nested in long-term gains. We introduce the idea of sophisticated financial risk measures called the "Greeks" (Delta, Gamma, Theta, Vega, and Rho) without naming them. We build personalized risk understanding by using common risk questionnaires to measure their own risk (volatility) tolerance and how it could change over time. The idea is to help personal finance students get comfortable with volatility and manage their downside via diversification. I do not test them on the formulas. The point is not rote memorization, but the internalization of the shape and behavior of these financial forces. The idea is to create visualizations to encourage and empower the students' personal finance success. This is how mathematical intuition presents itself in personal finance.

The Intuitive Shift

By repeatedly showing this visual evidence and linking it to their personal decisions, something profound occurs. The concept of compound interest and improving their relationship with risk transforms from an abstract formula into an intuitive roadmap. They begin internalizing the reality every dollar they save today works harder, faster, and more powerfully for their future than any dollar they save ten years from now. They can "see" the future value of delaying gratification.

This class does not require a deep mathematical background, other than the basic, high school-level math prerequisites required by the University. This is our challenge and opportunity. That is, to find ways to expand student math intuition, in the context of Personal Finance, without having a deep math education.

My ultimate goal by the end of the course involves promoting living below their means and saving as a habit, not an abstraction. This process makes getting rich via good personal finance decision-making more real and less like a fantasy. When they leave the class, they stand ready to make confident financial decisions—or, frankly, any decision requiring long-term foresight.

Being confident and setting a visible, achievable path aids in promoting abstract mathematics to their intuition. They have not just learned math; they have developed the imaginative power to fully "see" their own financial success. It offers the ultimate confirmation: the language of abstraction has become the language of their self-assured future.

After leaving my class, they now possess a higher level of math intuition than when we started. My great hope is that our personal finance starting point builds confidence for continuing to develop their math abilities for enhancing their life's success..

Resources for the Curious

This article draws upon principles from neuroscience, economics, and mathematics, applied through the lens of personal finance. For readers interested in exploring the original sources of these concepts, we recommend the following works and resources:

• Hulett, Jeff. "Wealth, Risk, and Time: The Hidden Power of Math in Smart Investing." Personal Finance Reimagined, Feb 2, 2025. (Connects mathematical intuition to achieving long-term financial success and managing risk.)

• Buonomano, Dean. Your Brain Is a Time Machine: The Neuroscience and Physics of Time. W. W. Norton & Company, 2017. (Explores the brain's construction of time, relevant to perceiving the long-term nature of compound growth.)

• Bessis, David. Mathematica: A World of Discovery. CreateSpace Independent Publishing Platform, 2023. (Defines mathematics as a language for communicating abstract ideas and imagination.)

• Dyson, George. Darwin Among the Machines: The Evolution of Global Intelligence. Basic Books, 1997. (Discusses the concept of sharing complex, four-dimensional structures through mediums like music.)

• Hofstadter, Douglas R. Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books, 1999. (Illustrates the difficulty in shifting between the analytical components and the holistic perception of complex structures like music.)

• Led Zeppelin. "Kashmir." Physical Graffiti, Swan Song, 1975. (A core musical example used to illustrate the shift from holistic listening to analytical "seeing" of rhythmic and melodic structure.)

• Picasso, Pablo. General Work on Cubism/Geometric Abstraction. (Source for the theme of geometric abstraction and transforming the visible world into new, ordered forms.)

• Einstein, Albert. General Work on Special and General Relativity. (Source for the concept of the relativity of time and its mathematical description.)

• Jensen, Johan. General Work on Jensen's Inequality. (Source for the mathematical principle of convexity, specifically concerning the relationship between the expected value of a convex function and the function of the expected value, crucial for understanding compounding.)

• Nitecki, Zbigniew. Calculus Deconstructed: A Second Course in First-Year Calculus. MAA Press, 2007. (Provides the formal basis for understanding non-linear functions and the concept that a function's second derivative determines whether the rate of increase itself is increasing.)