The Fundamental Circularity Theorem: Why Some Mathematical Behaviours Are Inherently Unprovable

post by Alister Munday (alister-munday) · 2025-01-22T18:20:25.697Z · LW · GW · 2 comments

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Dear LessWrong community,

Before diving into this paper, try something: Think of your favourite approach for potentially proving the Collatz conjecture - whether through dynamical systems, ergodic theory, or any other sophisticated mathematical framework. Now mentally trace that approach to its foundations. Notice how every path ultimately requires you to reason about basic arithmetic properties - the very things you're trying to prove things about. It's like trying to use a microscope to examine the microscope's own lens.

This paper proposes that this circularity isn't a failure of our proof techniques, but rather points to something fundamental about mathematical truth itself. It suggests there's a category of mathematical behaviours that are unprovable in a way distinctly different from Gödelian incompleteness or independence results - not because they're too complex or foundational, but because they emerge so directly from fundamental properties that any proof attempt becomes inherently circular.

When you first read about "fundamental properties" and their "emergent behaviours," you might be tempted to map this to familiar concepts like axioms or unprovability results. But the paper is pointing at something different: mathematical behaviours that aren't just foundational statements we must assume, but rather arise from the interaction of fundamental properties in a way that makes proof attempts inevitably circular.

The paper begins with concrete examples to build intuition (can you prove why even numbers divide by 2 without circular reasoning?), then moves toward formalization. While further formal development is certainly possible, the core insight - that certain mathematical behaviours resist proof because of their fundamental nature rather than their complexity - offers a new lens for understanding mathematical truth.

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Preface
This paper introduces a new and fundamental insight about mathematical unprovability. While Gödel showed us limitations based on self-reference, and Cohen showed limitations based on axiomatic choice, this work identifies a more basic form of unprovability: mathematical behaviours that emerge directly from fundamental properties cannot be proven without circular reasoning. The Collatz conjecture serves as a prime example - its behaviour emerges solely from how even and odd numbers interact under basic operations, making it inherently unprovable.
 

Abstract
This paper introduces the Fundamental Circularity Theorem (FCT), establishing that certain mathematical behaviours are inherently unprovable because they emerge directly from fundamental properties that cannot themselves be proven without circular reasoning. Using the Collatz conjecture as our primary example, we demonstrate how mathematical behaviours that arise purely from the interaction of fundamental properties resist formal proof not due to complexity or logical paradox, but because any proof would require proving unprovable fundamentals. This insight offers a new understanding of mathematical unprovability distinct from Gödel's incompleteness theorems or independence results.
 

1. Introduction
Mathematics rests upon certain properties that simply "are" - properties so fundamental that they cannot be proven without circular reasoning. Consider a simple question: why does any even number divide evenly by 2? Any attempt to prove this must use concepts that already assume this very behaviour.

From these fundamental properties emerge mathematical behaviours - not through some deeper pattern or law, but as direct consequences of how these properties interact under specified conditions. The Collatz conjecture provides a perfect example: its behaviour emerges solely from how even and odd numbers behave under basic arithmetic operations.

This paper demonstrates that such emergent behaviours are inherently unprovable. This is not due to complexity, self-reference, or axiomatic choice. Rather, proving these behaviours would require proving the fundamental properties from which they emerge - an inherently circular endeavour.

This insight differs fundamentally from previous results about mathematical unprovability:
   • Unlike Gödel's incompleteness theorems, it involves no self-reference 
   • Unlike independence results, these behaviours are consistently true 
   • Unlike complexity barriers, it affects even simple mathematical statements
 

2. Fundamental Properties
Some mathematical properties are truly fundamental - they form the irreducible basis of mathematical reasoning. These properties cannot be proven because any attempt at proof would require using the very properties being proven.

Consider evenness and oddness. When we say a number is even, we're expressing a fundamental property about how it behaves under division by 2. This behaviour isn't derived from deeper principles - it's part of what defines even numbers. Similarly, when we say a number is odd, we're expressing a fundamental property about what happens when we divide it by 2.

Other examples of fundamental properties include:
- The succession property of natural numbers (each number has a next)
- The behaviour of numbers under basic arithmetic operations
- The relationship between multiplication and division
- The distributive property of multiplication over addition

These properties are:
1. Unprovable without circularity - any proof must use the property being proven
2. Irreducible - they cannot be derived from simpler principles
3. Consistent - they behave the same way in all mathematical systems
4. Necessary - basic mathematics cannot function without them
 

3. The Fundamental Circularity Theorem
Theorem: If a mathematical behaviour emerges solely from fundamental properties, it cannot be proven within any mathematical system incorporating those properties.

Proof: Let B be a mathematical behaviour that emerges solely from a set of fundamental properties F.
Consider any mathematical system S capable of expressing B.
By definition, S must incorporate the fundamental properties F from which B emerges.
Suppose there exists a valid proof P of why B occurs.
Since B emerges purely from F, P must demonstrate why the interaction of properties in F produces B.

But this requires explaining why these fundamental properties behave as they do.
This explanation must either:
- Use circular reasoning (invalidating the proof)
- Fail to fully explain the behaviour (making the proof incomplete)

Therefore, no valid proof can exist.
 

4. The Collatz Case
The Collatz conjecture demonstrates this principle perfectly. The behaviour we observe emerges solely from how even and odd numbers interact with two operations:

1. If n is even, divide by 2
2. If n is odd, multiply by 3 and add 1

Why has this simple system resisted proof for so long? Because its behaviour emerges directly from fundamental properties about even and odd numbers. Any proof would require explaining why even numbers behave as they do under division by 2, and why odd numbers behave as they do under multiplication and addition - properties that are fundamental and thus unprovable.

Consider: we could define different systems (5n+1, 2n-1, etc.) that would produce different patterns. The Collatz sequence isn't special - it's just one possible expression of how even and odd numbers interact under specific operations. The fascinating patterns we observe are consequences, not causes.
 

5. Implications
This insight has profound implications for mathematics:

1. Some mathematical behaviours are unprovable not because we lack cleverness or technique, but because they emerge directly from unprovable fundamentals.
2. The search for proofs of such behaviours is fundamentally misguided - like trying to prove why even numbers divide by 2.
3. Other mathematical conjectures may be unprovable for similar reasons, including potentially:
  - The Twin Prime Conjecture (emerging from fundamental properties of primes)
  - Goldbach's Conjecture (emerging from fundamental properties of primes and addition)
  - Various patterns in number theory

This doesn't mean these behaviours are uncertain - they are consistently true. But they are true because they emerge directly from fundamental properties, not because of some deeper pattern waiting to be discovered.

Future work should focus on:
1. Identifying other mathematical behaviours that emerge from fundamental properties
2. Developing criteria for recognizing fundamentally unprovable statements
3. Understanding the boundary between provable statements and those emerging from fundamentals

The Fundamental Circularity Theorem suggests a new way of understanding mathematical truth - one that acknowledges some mathematical behaviours as direct expressions of fundamental properties, inherently resistant to proof by their very nature.


"You can't use a ruler to measure a ruler!"

 

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