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[ epistemological status: a thought I had while reading about Russell's paradox, rewritten and expanded on by Claude ; my math level: undergraduate-ish ]
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 only fundamental requirement for a mathematical system is internal coherence—it must operate according to consistent rules without contradicting itself. A system need not:
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.
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:
None of these systems is "more true" than the others—they're different tools for different jobs.
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.
Early calculus used "infinitesimals"—infinitely small quantities—in ways that seemed logically questionable. Rather than this destroying calculus, mathematics evolved multiple rigorous frameworks:
Each approach has its advantages for different applications, and all are internally coherent.
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:
None of these choices imply that other options are "wrong"—just that they're less useful for the particular problem at hand.
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.
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.
Hypothetically one could spend a few decades researching how to make people smarter (or some other long term thing), unlock that tech, and all that is really good.
But what if you plan your path towards that long-term goal such that it is the unlocking of various lesser but useful techs that gets you there?
Well now that's even better: you get the benefit of reaching the end goal + all the smaller things you accomplished along the way. It gives you some hedge: in case you don't reach the end goal you still accomplished a lot. And cherry on top: it's more sustainable as you get motivation (and money?) from unlocking the intermediary tech.
So it looks like it's worth going out of your way to reap benefits regularly as you journey towards a long term goal.
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.
A paperclip maximizer would finally turn itself into paperclips after having paperclipped the entire universe.
And probably each local instance would paperclip itself when the locally-reachable resources were clipped. "local" being defined as the area of spacetime which does not have a different instance in progress to clippify it.
decisionproblem.com/paperclips/index2.html demonstrates some features of this (though it has a different take on distribution), and is amazingly playable as a game.
Chatting with ChatGPT I learned that latest organoids have about 1 million neurons.
Wondering whether that's a lot or not, it tells me that bees and other insects have on the order of 10^5, fish, like zebra fish, have on the order of 10^6. So we are engineering fish brains, at least in terms of number of neurons. That's concerning, as far as I know zebra fish are conscious and can hurt.
What about humans? ChatGPT says humans have around 10^11 neurons, however 10-12 weeks embryo have about 10^6. It so happens that 10 to 12 weeks is the median limit to abort in European countries... and labs want to scale.
I barely know anything about the subject, but imagining a 12 weeks embryo's brain driving my roomba sure feels weird compared to the silicon version.