> Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man
The things this definition misses: First, 'intelligence' is a poorly defined and overly broad term. Second, machine intelligence is profoundly different than biological intelligence. Third, “surpassing humans” is not a single threshold event because machine and human intelligence are not only shaped differently, they're highly non-linear. LLMs are a particular class of possible machine intelligences which can be much more intelligent than humans on some dimensions and much less intelligent on others. Some of the gaps can be solved by scaling and brilliant engineering but others are fundamental to the nature of LLMs.
> an ultraintelligent machine could design even better machines
There is a huge leap between "surpass all the intellectual activities of any man" and "invent extraordinary breakthroughs and then reliably repeat that feat in a sequential, directed fashion in the exact way required to enable sustained iteration of substantial self-improvement across infinite generations in a runaway positive feedback loop". That's an ability no human or collective has ever come close to demonstrating even once, much less repeatedly. (hint: the hardest parts are "reliably repeat", "extraordinary breakthroughs" and "directed fashion"). A key, yet monumental, subtlety is that the self- improvements must not only be sustained and substantial but also exponentially amplify the self-improvement function itself by discovering novel breakthroughs which build coherently on one other - over and over and over.
The key unknown of the 'Foom Hypothesis' is categorical. What kind of 'difficult feat' this is? There are difficult feats humans haven't demonstrated like nuclear fusion, but in that example we at least have evidence from stellar fusion that it's possible. Then there are difficult feats like room-temp superconductors, which are not known to be possible but aren't ruled out. The 'Foom Hypothesis' is a third category of 'hard' which is conceptually coherent but could be physically blocked by asymptotic barriers, like faster-than-light travel under relativity.
Assuming Foom is like fusion - just a challenging engineering and scaling problem - is a category error. In reality, Foom requires superlinear, recursively amplifying cognitive returns—and we have no empirical evidence that such returns can exist for artificial or biological intelligences. The only prior we have for open‑ended intelligence improvement is biological evolution which shows extremely slow and unreliable sublinear returns at best. And even if unbounded self‑improvement is physically possible, it may be practically unachievable due to asymptotic barriers in the same way approaching light speed requires exponentially more energy.