Mathematics, Mapping, and the Limits of Intuition

The Conceptual Boundaries of Mathematics, Statistical Reasoning, and Its Representation in the Real World

Note: As usual, I opine on things outside of my domain of expertise. I guess I’m like a journalist in that regard. Anyway, it is something like this, but being only a bargain basement polymath, I guess I might be full of it. Blame it on ChatGPT, not my coaching. Not sure why Dall-E added an hourglass to the illustration. Extra points if you note that the circuit could not possibly work (removes tongue from cheek).

Introduction

Mathematics is a formal system of symbols and deterministic rules, built upon foundational axioms. It is often seen as the universal language of nature, capable of describing everything from fundamental physical laws to abstract probabilistic models. However, the assumption that mathematics inherently corresponds to reality is flawed. While certain mathematical models—like Ohm’s Law—map well to observable phenomena, many others exist purely within formalism, with no clear connection to the real world.

This issue, known as the "mapping problem," highlights the difficulty of determining when and where mathematical structures meaningfully correspond to reality. The problem extends beyond deterministic equations and is especially prominent in statistical reasoning, where probabilistic models introduce further ambiguity.

Statistical methods, including but not limited to Bayesian inference, require significant judgment in their application. This judgment appears at multiple levels:

1. Choice of inputs – How data is selected and structured.

2. Method of analysis – What statistical framework is applied.

3. Interpretation of results – How conclusions are drawn from statistical outputs.

Contrary to common misconceptions, Bayesian statistics is not uniquely subjective—all branches of statistics require human judgment at multiple stages. Whether frequentist, Bayesian, or another statistical framework, subjectivity enters at the same points: in setting assumptions, selecting data, and determining how to interpret findings. This essay will examine the broader conceptual challenges in statistics, including Bayesian reasoning, while also addressing the fundamental limitations of mathematical representation, graphical intuition, and accessibility.

Mathematics as a Symbolic and Deterministic System

Formalism and Symbolic Manipulation

Mathematics is a system of symbols manipulated by deterministic rules. Some equations describe real-world behavior, while others exist purely as formal constructs with no practical application.

Well-Mapped Mathematics – Ohm’s Law and Newton’s Laws map cleanly to physical reality.

Purely Formal Mathematics – Many abstract mathematical structures lack real-world correspondence.

A common misconception is that because mathematics is internally consistent, it must necessarily apply to reality. However, mathematical truth does not imply physical truth—valid equations can exist without any connection to observable phenomena.

Graphical Representation and Its Limits

Mathematical ideas are often visualized to aid intuition, but such representations are limited:

Two-dimensional graphs effectively represent simple functions.

Three-dimensional models introduce additional complexity but remain interpretable.

Higher-dimensional representations become impractical, often misleading rather than clarifying.

This is why illustrations of curved space and hyperspace are often difficult to interpret—they attempt to map higher-dimensional relationships onto lower-dimensional visualizations, which do not preserve the full conceptual structure. As a result, these illustrations frequently confuse rather than illuminate.

Cases Where Mathematics Maps Well to Reality

Some mathematical structures do correspond clearly to the physical world:

Ohm’s Law – Simple and directly measurable.

Classical physics equations – Newtonian mechanics, conservation laws.

However, beyond these straightforward cases, the connection between mathematics and reality becomes tenuous, particularly in statistics and probability.

The Mapping Problem in Mathematics and Statistics

Mathematics and the Limits of Direct Mapping

While deterministic equations can sometimes be directly applied to reality, probabilistic models function differently. They do not describe absolute physical laws but rather patterns of uncertainty.

Statistical methods do not establish strict cause-and-effect relationships. Instead, they offer inferential tools that must be interpreted carefully. This introduces ambiguity and judgment at multiple levels, affecting all statistical approaches.

The Problem with Probability and Interpretation

Probability theory, while computationally simple, is conceptually challenging. The difficulty arises not in performing calculations but in assigning meaning to statistical terms.

Some of the most misunderstood statistical concepts include:

• Conditional probability – The probability of an event given another event has occurred.

• P-values – Misinterpreted as measures of truth rather than as thresholds for rejecting hypotheses.

• Confidence intervals – Frequently mistaken for probability ranges rather than estimates of uncertainty.

These conceptual misunderstandings affect all branches of statistics, not just Bayesian methods.

The Role of Judgment in All Statistical Methods

One of the most persistent misconceptions in statistics is that Bayesian reasoning is uniquely subjective. In reality, all statistical methods require judgment:

Frequentist statistics requires setting significance thresholds, selecting models, and determining confidence levels.

Bayesian statistics requires setting priors, interpreting posteriors, and defining likelihood functions.

The distinction between these approaches is methodological, not in their level of subjectivity. Both require:

1. Assumptions about data (whether explicit in Bayesian priors or implicit in frequentist models).

2. Decisions on interpretation (whether assessing posterior probabilities or P-values).

3. Choices on application (whether using a likelihood ratio or hypothesis test).

Thus, the notion that Bayesian statistics is inherently more subjective is incorrect—all statistics involves human judgment at multiple levels.

The Role of the Denominator in Bayesian Inference

Bayesian reasoning, like all statistical methods, presents conceptual challenges. One area of particular confusion is the denominator in Bayes’ theorem.

The denominator ensures that probability calculations remain proportionally correct relative to all possible outcomes. This is critical because Bayesian probability is not computed in isolation—it is always relative to prior probabilities and observed evidence.

However, while the denominator has a valid conceptual foundation, estimating it in real-world applications is often impractical:

In simple cases, like signal detection theory, where possible outcomes sum to 100%, it is intuitive.

In complex cases, estimating it reliably is often impossible, requiring questionable approximations.

Thus, while the mathematical justification for the denominator is sound, its application often depends on assumptions that cannot be reliably verified in practice. This is not a problem unique to Bayesian statistics—it reflects a broader issue in all statistical reasoning.

The Limits of Explaining Mathematics and Statistics in Plain Language

The Accessibility Problem

Some mathematical ideas can be effectively translated into plain English:

Ohm’s Law can be explained using analogies of water pressure and flow.

Basic probability can be illustrated with dice rolls and coin flips.

However, many concepts resist simplification:

Order of operations cannot be clearly explained without notation.

Multi-term equations become unreadable in text format.

Statistical models require assumptions that cannot be easily expressed in common language.

Why Many People Struggle with Statistics and Mathematics

Mathematics and statistics remain inaccessible to many people due to:

1. Lack of formal training – Without proper instruction, even simple statistical concepts are misunderstood.

2. Intellectual variability – Some people grasp abstraction naturally, while others struggle.

3. Poor explanations – Many statistical and mathematical concepts are explained poorly, even in formal education.

Statistics, in particular, suffers from misleading terminology—terms like "significance," "confidence," and "likelihood" have technical meanings that differ from their common-language interpretations.

Conclusion: Mathematics, Statistics, and Human Intuition

Mathematics and Statistics as Tools, Not Absolute Truths

Both mathematics and statistics serve as descriptive tools, not absolute representations of reality. Some structures map well to the physical world, while others function purely as inferential systems.

All Statistics Requires Judgment

Contrary to popular belief, Bayesian reasoning is not uniquely subjective—all statistical methods require judgment in inputs, analysis, and interpretation. The distinction between different statistical frameworks is methodological, not one of objectivity versus subjectivity.

The Inherent Limits of Explanation

Some mathematical and statistical concepts can be expressed in plain English, but many cannot. Probability, statistical inference, and higher-dimensional reasoning often defy intuitive understanding, requiring formal notation and structured frameworks.

Ultimately, mathematics and statistics are powerful but imperfect tools—exceptionally useful in some cases, but fundamentally limited by the judgment required to apply them meaningfully.