2+2=5 is Fine Maths: all you need is Coherence

[ epistemological status: a thought I had while reading about Russell's paradox, rewritten and expanded on by Claude ; my math level: undergraduate-ish ]

Introduction

Mathematics has faced several apparent "crises" throughout history that seemed to threaten its very foundations. However, these crises largely dissolve when we recognize a simple truth: mathematics consists of coherent systems designed for specific purposes, rather than a single universal "true" mathematics. This perspective shift—from seeing mathematics as the discovery of absolute truth to viewing it as the creation of coherent and sometimes useful logical systems—resolves many historical paradoxes and controversies.

The Key Insight

The only fundamental requirement for a mathematical system is internal coherence—it must operate according to consistent rules without contradicting itself. A system need not:

• Apply to every conceivable case
• Match physical reality
• Be the "one true" way to approach a problem

Just as a carpenter might choose different tools for different jobs, mathematicians can work with different systems depending on their needs. This insight resolves numerous historical "crises" in mathematics.

Historical Examples

The Non-Euclidean Revelation

For two millennia, mathematicians struggled to prove Euclid's parallel postulate from his other axioms. The discovery that you could create perfectly consistent geometries where parallel lines behave differently initially seemed to threaten the foundations of geometry itself. How could there be multiple "true" geometries? The resolution? Different geometric systems serve different purposes:

• Euclidean geometry works perfectly for everyday human-scale calculations
• Spherical geometry proves invaluable for navigation on planetary surfaces
• Hyperbolic geometry finds applications in relativity theory

None of these systems is "more true" than the others—they're different tools for different jobs.

Russell's Paradox and Set Theory

Consider the set of all sets that don't contain themselves. Does this set contain itself? If it does, it shouldn't; if it doesn't, it should. This paradox seemed to threaten the foundations of set theory and logic itself.

The solution was elegantly simple: we don't need a set theory that can handle every conceivable set definition. Modern set theories (like ZFC) simply exclude problematic cases while remaining perfectly useful for mathematics. This isn't a weakness—it's a feature. A hammer doesn't need to be able to tighten screws to be an excellent hammer.

The Calculus Controversy

Early calculus used "infinitesimals"—infinitely small quantities—in ways that seemed logically questionable. Rather than this destroying calculus, mathematics evolved multiple rigorous frameworks:

• Standard analysis using limits
• Non-standard analysis with hyperreal numbers
• Smooth infinitesimal analysis

Each approach has its advantages for different applications, and all are internally coherent.

Implications for Modern Mathematics

This perspective—that mathematics consists of various coherent systems with different domains of applicability—aligns perfectly with modern mathematical practice. Mathematicians routinely work with different systems depending on their needs:

• A number theorist might work with different number systems
• A geometer might switch between different geometric frameworks
• A logician might use different logical systems

None of these choices imply that other options are "wrong"—just that they're less useful for the particular problem at hand.

The Parallel with Physics

This view of mathematics parallels modern physics, where seemingly incompatible theories (quantum mechanics and general relativity) can coexist because each is useful in its domain. We don't need a "theory of everything" to do useful physics, and we don't need a universal mathematics to do useful mathematics.

Conclusion

The recurring "crises" in mathematical foundations largely stem from an overly rigid view of what mathematics should be. By recognizing mathematics as a collection of coherent tools rather than a search for absolute truth, these crises dissolve into mere stepping stones in our understanding of mathematical systems.

Mathematics isn't about discovering the one true system—it's about creating useful systems that help us understand and manipulate abstract patterns. The only real requirement is internal coherence, and the main criterion for choosing between systems is their utility for the task at hand.

This perspective not only resolves historical controversies but also liberates us to create and explore new mathematical systems without worrying about whether they're "really true." The question isn't truth—it's coherence.

Tip: you can ask ChatGPT to include confidence scores in its replies

Interactions with ChatGPT can be customized durably in the options, for example you can add the following instructions: "include a confidence rating at the end of your response in the format 'Confidence: X%'. If your confidence is below 80%, briefly explain why".

Here is a sample conversation demonstrating this and showing what ChatGPT has to say about its calibration:

Me: Are you calibrated, by which I mean, when you output a confidence X as a percentage, are you right X times out of 100?

ChatGPT: Reasoned about confidence calibration for 27 seconds. I don’t have calibration in that strict sense. When I mention a confidence percentage, it’s a heuristic internal signal rather than a guarantee that I’ll be right exactly that proportion of the time. Research shows that language models often exhibit miscalibration, meaning a stated confidence (say 70%) doesn’t empirically translate to being correct 70% of the time. I'm about 90% confident in this explanation.

Me: How bad is the miscalibration?

ChatGPT: Thought for a second. LLMs—especially those fine‐tuned with RLHF—are often notably overconfident. For example, a model might claim 90% confidence while its actual accuracy is closer to 70–80%. Studies report expected calibration errors (ECE) of roughly 5–10% or more, and effective calibration methods can sometimes cut these errors by about half, underscoring that miscalibration is a significant issue. I'm about 90% confident in this summary, based on multiple recent calibration studies.

Me: If you are predictably overconfident, and you know it, do you think you can notice that before stating your confidence and correct for it?

ChatGPT: Reasoned about overconfidence and calibration adjustment for 4 seconds. I don’t have a real-time “internal monitor” to detect my own overconfidence before stating it. Instead, I rely on calibration techniques—like temperature scaling and post-hoc adjustments—that help align my confidence levels with actual accuracy. In other words, I can be tuned to be less overconfident, but I don’t consciously correct my confidence on the fly. I'm about 90% confident in this explanation based on current research findings.

Let me show you the ropes

There is a rope.
You hold one end.
I hold the other.
The rope is tight.
I pull on it.

How long until your end of the rope moves?

What matters is not how long until your end of the rope moves.
It's having fun sciencing it!