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Challenger Lift Off, January, 1986

Why do mathematicians think the way they do?

Why are good physicians for most part horrible investors despite being otherwise successful?

Mathematicians think in generalities called abstractions one famous mathematician was John von Newmann.

John von Neumann (1903 — 1957) was a Hungarian-American mathematician, physicist, computer scientist, and engineer known for his extraordinary intellect and contributions across various fields. A child prodigy, von Neumann mastered complex mathematical concepts at a young age, and his work spanned quantum mechanics, game theory, computing, and economics. He played a key role in the Manhattan Project, developing mathematical models for nuclear weapons, and contributed to the design of the first modern computer architecture.

von Newmann wiki photo

Von Neumann was also deeply involved with U.S. defense strategies during the Cold War, consulting for the RAND Corporation and shaping missile programs. His work in game theory and decision-making influenced military strategies and may have inspired the character Dr. Strangelove in Stanley Kubrick’s film.

Higher levels of abstraction in mathematics and other disciplines attempt to connect lower levels because and it is this ability to recognize and connect patterns or abstractions which provides a more general framework for understanding and integrating specific phenomena. Plato called these abstractions “Forms” and

Aristotle created schema which integrated the Platonic Form into his categories as a series of interconnected causes.

In Metaphysics (1013a), Aristotle explains, “We do not have knowledge of a thing until we have grasped its why, that is to say, its cause,” outlining the completeness of his causal framework.

The four causes address not only physical composition and processes but also purpose and form, making them foundational to Aristotle’s broader metaphysical and scientific work.

Why Only Four Causes?

Aristotle settled on four causes because he believed they covered the necessary dimensions for explaining any phenomenon — what something is made of (material cause), what form or pattern it follows (formal cause), what initiates or produces it (efficient cause), and what purpose or goal it serves (final cause).

Aristotle built on earlier Greek philosophers’ inquiries into the elements and forces of nature but went beyond them to create a more comprehensive system of explanation that could apply to both natural and human-made objects.

In mathematics, this is evident in how abstract concepts like set theory, category theory, and formal logic seek to unify diverse branches of thought under one overarching structure. The aim of higher levels is to capture the essence of relationships, which allows multiple lower-level frameworks to interact coherently.

Conversely, lower-level systems, often tied to empirical and concrete observations, do not naturally seek to connect to higher abstractions because their focus is on direct, immediate applications rather than overarching principles.

In political discourse, a similar dichotomy appears between higher-level, abstract ideologies and lower-level, pragmatic approaches. For example,

Democrats, particularly in their more ideologically driven factions, often engage with abstract frameworks aimed at creating idealized or utopian futures.

These frameworks may focus on broad principles like equality, justice, and social welfare. Such abstractions resemble how mathematics seeks to connect different domains under a universal structure, aiming for a future that aligns with these abstract ideals. This higher stack level of thought integrates various political and social issues under unifying goals and theories.

Republicans, on the other hand, often demonstrate a more pragmatic, lower-level empiricism, focusing on tangible, immediate solutions to specific problems. Their political discourse is more grounded in practical outcomes, policies driven by observable data and experience rather than abstract ideals.

This aligns with how lower stack levels in mathematics focus on solving specific problems within a constrained framework without necessarily attempting to connect with broader, more abstract structures. For instance, fiscal conservatism, a common Republican stance, is rooted in immediate concerns about budgetary limits and economic growth, reflecting an empirical mindset more focused on the present than future ideal states.

This contrast between the two political ideologies mirrors the mathematical comparison between higher and lower levels of abstraction. Higher levels attempt to generalize and find universal connections — whether in political discourse or mathematical formalism — while lower levels remain grounded in specific, immediate concerns.

Philosophically, mathematics has long sought to connect cognitive frameworks through abstract structures, paralleling how political ideologies try to connect various issues into coherent visions. However, lower-level approaches in both politics and mathematics prioritize concrete, present realities over unifying theories.

In both mathematics and political discourse, abstraction enables broader connections by creating a common framework that unites diverse components. Whether in Plato’s view of mathematical objects as abstract entities or the Democratic tendency toward idealized visions of society, higher levels of thought seek to provide universal structures that encompass various lower-level concerns. In contrast, empirically focused, lower-level approaches, like those often seen in Republican politics or certain pragmatic mathematical methods, prioritize immediate applications and problem-solving over the pursuit of universal frameworks.

A Rubric to Isolate Cognitive Problems Using Aristotle’s Four Causes

Understanding complex concepts, especially in subjects like mathematics, physics, or philosophy, often leads students to experience moments of confusion or aporia — a mental block where progress seems impossible.

To overcome these cognitive challenges, it is essential to identify the level at which the misunderstanding occurs. By using **Aristotle’s four causes** — material, formal, efficient, and final — we can create a structured rubric to help students analyze and isolate the source of their cognitive struggles.

This rubric will help students think differently, using language and fundamental cognitive structures to break down and reorganize their thought processes. If a student doesn’t understand a concept, perhaps they are thinking at the wrong cognitive stack level. This tool can help locate the correct level.

1. Material Cause: Understanding Basic Cognitive Inputs

• Key Question: Do you have the necessary foundational knowledge (facts, definitions, formulas) to approach this problem?
• This step involves determining if the cognitive “raw materials” necessary to understand the concept are present. Material causes include memorized facts, historical context, mathematical definitions, or basic data required to build on more complex ideas.
• Example: If a student doesn’t understand a calculus problem, the first question to ask is whether they have mastered the fundamental concepts such as limits or derivatives. In philosophy, it may involve understanding the basic tenets of a thinker like Descartes before grappling with his metaphysical arguments.

**Action**: Review the core definitions, basic principles, and key data. If these are missing or not well-understood, further study at this basic level is required.

2. Formal Cause: Recognizing Patterns and Structures

• Key Question: Are you recognizing the patterns or relationships between the material inputs?

Formal cause refers to how the basic material knowledge is organized into a larger structure or pattern. This involves pattern recognition, categorization, and seeing how individual parts fit into a greater whole. Often, misunderstanding arises when students cannot connect the basic facts or rules to a broader structure.

• Example: In mathematics, failing to recognize how different equations are related through a shared principle (e.g., algebraic and geometric relationships in analytic geometry) could hinder understanding. In philosophy, it could mean not seeing the logical structure that connects different arguments in a broader theory.

**Action**: Ask yourself how the individual components relate to one another. Can you identify a pattern or structure? Try visualizing or diagramming the problem to see connections that are not immediately apparent.

3. Efficient Cause: Understanding the Cognitive Process**

• Key Question: Are you applying the correct process to analyze or solve the problem?

Efficient cause deals with the processes that transform raw knowledge into understanding or action. This includes problem-solving methods, mental modeling, and critical thinking strategies. Students may struggle because they are not using the right cognitive strategies to apply their knowledge effectively.

• Example: In solving a physics problem, a student may know all the formulas but fail to apply them correctly due to a poor problem-solving approach. In philosophy, they may have read the primary texts but fail to engage in critical analysis that challenges the underlying assumptions of the arguments.

**Action**: Re-evaluate your approach to solving the problem. Are you following a logical process? Break down the steps and consider whether a different method (visualizing, trial and error, reworking assumptions) might yield better results.

4. Final Cause: Clarifying the Purpose or Goal

• Key Question: Are you clear about the end goal or purpose of your task?

The final cause refers to the purpose or goal behind the learning or problem-solving activity. Confusion may stem from a lack of understanding of the ultimate goal, such as why a particular concept is important or how it fits into a broader framework. If students cannot see the “why,” they may struggle to engage with the material.

• Example: In mathematics, a student may understand the steps to solve an equation but not grasp why it matters in the larger context of the subject. In philosophy, a student may follow the logic of an argument but fail to see its implications for ethical theory or metaphysics.

**Action**: Reflect on the purpose of the problem or concept. Why is it important? How does it fit into the larger subject? Understanding the “final cause” can provide motivation and clarity, making it easier to overcome obstacles.

Cognitive Stack Dynamics: Locating the Problem

Once you have analyzed the four causes, consider where your cognitive processing is stuck within the **cognitive stack**:

Lower Levels (Material Cause)**: If you’re struggling with basic facts or definitions, you’re likely at the material level. Focus on reviewing foundational knowledge before progressing.

• Mid-Level (Formal Cause)**: If you understand the components but fail to see how they relate, you’re dealing with formal cause issues. Use pattern recognition or visualization techniques to form connections between the elements.
• Higher Levels (Efficient Cause)**: If you have the necessary knowledge but struggle with how to apply it, the problem may lie in the cognitive processes you’re using. Reassess your approach and methods.
• Top Level (Final Cause)**: If you can’t understand the purpose of the concept or task, you may be missing the final cause.

flect on the broader purpose or goal of the task. Try to understand how it fits into the larger academic or theoretical framework. Clarifying this can provide the motivation and insight needed to continue.

Top Level (Final Cause): If you can’t understand the purpose of the concept or task, you may be missing the final cause.

Reflect on the broader purpose or goal of the task. Try to understand how it fits into the larger academic or theoretical framework. Clarifying this can provide the motivation needed to move forward and can help bridge the gap between what is currently understood and what remains unclear. When students can see the **final cause** — the overarching reason for engaging with a particular concept or problem — it not only makes the immediate task more meaningful but also helps to align their efforts with larger academic goals. This clarity can transform confusion into focus, as understanding the purpose allows students to apply their cognitive efforts more effectively, overcoming obstacles that previously seemed insurmountable.

P.S. Good physicians often struggle as investors due to the nature of their training, which emphasizes trust in authority and a rigid hierarchy of knowledge. From day one, their education is highly structured, progressing from gross anatomy to cell biology, pathology, and eventually board-certified specialties. This foundational, concrete knowledge operates at the bottom of the cognitive stack, where diseases, treatments, and clinical conditions remain relatively constant. Physicians are trained to recognize patterns, but these patterns are narrowly defined by their specialization and patient base, limiting their exposure to broader, more dynamic systems.

In contrast, successful investing requires **global thinking**, adaptability, and the ability to navigate between different knowledge domains and abstraction levels. Physicians are rarely required to shift between knowledge stacks, as their professional focus is largely confined to their board certification criteria. They operate within a stable framework where most challenges fall within well-defined, predictable parameters, and where authority — whether medical literature or clinical guidelines — plays a central role in decision-making.

Investing, on the other hand, is inherently fluid and uncertain, requiring a different cognitive approach. Markets shift constantly, driven by economic, political, and global trends. In this domain, success is often tied to independent analysis, risk assessment, and recognizing emerging patterns across diverse sectors — skills that are less emphasized in the medical profession. Physicians, accustomed to relying on structured, evidence-based knowledge, often find it difficult to operate in the ambiguous and speculative environment of investing, where no established protocols or authorities provide definitive answers. This cognitive mismatch between the structured, stable world of medicine and the dynamic, high-risk nature of investing explains why even the most skilled physicians may struggle as investors.

Coda

Higher levels of abstraction in mathematics and other disciplines attempt to connect lower levels because it is this ability to recognize and integrate patterns that provides a more general framework for understanding specific phenomena. Plato referred to these abstractions as “Forms,” which he believed were the perfect, unchanging ideals behind the material world. As Plato states in *Phaedo* (100b),

“There is such a thing as absolute beauty and absolute good, and absolute greatness, and all the rest, and they are the cause of all that is beautiful and good and great in this world.”

These Forms, for Plato, were the highest level of reality, and all physical manifestations were merely imperfect copies of these ideal Forms.

Aristotle, however, sought to integrate Plato’s Forms into a more grounded framework through his theory of **four causes**.

In *Metaphysics* (1013a), he outlines these causes: the **material cause** (what something is made of), the **formal cause** (its “shape” or essence, the schematic diagram or blueprint ), the **efficient cause** (the agent or process that brings it into being), and the **final cause** (its purpose or goal).

These four causes provided a way of understanding the world not just through abstract ideals, but through the interaction of different elements in a more interconnected, empirical system. Aristotle writes in *Metaphysics* (1041b): “The explanation of all things is through these causes, whether we speak of what is made by nature or what is made by art.”

In integrating these perspectives, both Plato and Aristotle provided frameworks that allow higher levels of abstraction to connect with and explain lower-level phenomena.

Plato’s Forms offer a universal ideal, while Aristotle’s causes provide a systematic way of understanding how these ideals are realized in the material world. These foundational ideas have shaped the way science and mathematics are structured today, focusing on both the recognition of underlying patterns and their practical application in the physical world.

Aristotle’s four causes — material, formal, efficient, and final — were developed as a way to explain the various aspects of existence and change in the natural world. Aristotle formulated these causes to provide a comprehensive framework that could explain both physical objects and abstract concepts. The origins of the four causes are linked to earlier philosophical inquiries about nature, especially those of the Pre-Socratic philosophers, who sought to understand the fundamental elements and principles of the universe.

The four causes were not directly based on the seasons or elements, but they were influenced by earlier thinkers who proposed that the basic building blocks of reality were composed of fundamental elements (earth, water, air, fire) or natural cycles. Aristotle’s causes were an intellectual evolution of these ideas, transforming them into a more abstract and structured explanation of reality.

Pre-Socratic Philosophers Influencing Aristotle’s Four Causes:

1. Thales of Miletus (c. 624 — 546 BCE) — Thales is often considered the first philosopher, and he proposed that water is the fundamental substance of all things, an early attempt at identifying a material cause (Aristotle’s Metaphysics 983b).

2. Anaximander (c. 610 — 546 BCE) — A student of Thales, Anaximander proposed the concept of the apeiron or the “boundless” as the source of all things. His idea of an indeterminate principle influencing existence hints at an early notion of the formal cause (Physics 203b).

3. Anaximenes (c. 586 — 526 BCE) — He suggested that air is the primary substance from which everything emerges. Anaximenes’ theory pointed to an efficient cause in the way that air changes into other substances through processes of rarefaction and condensation (Metaphysics 984a).

4. Heraclitus (c. 535 — 475 BCE) — Famous for his doctrine of flux, Heraclitus believed that everything is in constant change, driven by conflict or strife, which could be connected to the efficient cause. He also believed in a unifying principle called the Logos, which can be seen as a precursor to Aristotle’s formal cause (Metaphysics 984a).

5. Empedocles (c. 494 — 434 BCE) — Empedocles proposed that all matter is composed of four elements — earth, water, air, and fire — mixed by the forces of love and strife. These elements acted as material causes, while love and strife could be seen as early ideas of efficient causes (Metaphysics 984a).

6. Anaxagoras (c. 500 — 428 BCE) — Anaxagoras introduced the concept of Nous (mind) as a cosmic force that organizes matter, an early precursor to Aristotle’s final cause, as it introduced the idea of purposeful order in nature (Metaphysics 984b).

7. Democritus (c. 460 — 370 BCE) — A pioneer of atomic theory, Democritus proposed that everything is made up of indivisible atoms, which interact to form the material world. This is another precursor to Aristotle’s material cause (Metaphysics 985b).