I'm always sceptical of studies which look at a single diagnosis rather than pooling multiple related diagnoses, because the boundaries between them are so fuzzy
Your response essentially assumes formalism - mathematics is a game with rules (axioms, inference rules, etc), and all rules are in themselves equally valid, it is just a question of whether the game they produce is playable (i.e. produces interesting or useful theorems). Formalism has no objection to infinities: the axiom of infinity is just another axiom, in itself as valid as any other-but one which produces a near-endless array of interesting results.
Formalism is a very common approach in the philosophy of mathematics-but it isn’t the only one, and it is not the philosophy which motivates ultrafinitism.
Another viewpoint is that mathematical objects somehow really exist; mathematics is more than just a symbol manipulation game. One variation of this is (mathematical) Platonism, which believes they exist in some timeless realm beyond this physical universe; that view has no issue with infinities either, since adherents of this view generally believe that realm to be infinite and filled with infinities.
Yet another view is conceptualism-mathematical objects really exist, but in the human mind. And this is the viewpoint that motivates ultrafinitism - the human mind is finite, so infinite mathematical objects cannot really exist in it, or at least not in the fullness of the sense that finite objects can; and that turns out to be true, not just for infinities, but also for overly large finitudes.
This idea that some mathematical objects are in a philosophical sense “more real” than others is a big motivator of mathematical constructivism-trying to find axioms which respect that philosophical distinction, and work out what the consequences of those axioms are. Ultrafinitism is just a particularly extreme form of constructivism, which adopted a stricter “criterion of reality” for mathematical objects than most constructivists do
I think you're conflating opinions about when math is useful with opinions on the nature of math itself. Formalism does not assume that "all rules are equally valid". You can be a staunch formalist and yet still believe that X set of axioms are the only useful ones and everyone who assumes different axioms is wasting their time. You could be a formalist and still believe that the concept of infinity is leading math astray from useful math. Many of the differences you lay out seem to just be in people's opinion on which axioms are useful and which aren't. That's still formalism.
Setting that aside, it's very difficult for me to take non-formalist views of mathematics seriously. I strongly suspect that anyone who subscribes to those views has some deep-seated confusion in their heads.
> Platonism, which believes [mathematical objects] exist in some timeless realm beyond this physical universe
This is equivalent to formalism, except perhaps in how the mathematician feels about it. What could any possible difference be? In what way could it ever matter in the slightest whether something "really exists", if we define that to be so weak as to include "in some timeless realm beyond this universe"? Surely pink goblins "really exist" in this sense as well. With such a weak definition, the difference between your "really exists" and my "really exists" is purely emotional.
> Yet another view is conceptualism-mathematical objects really exist, but in the human mind.
You can be formalist and still argue about whether humans invented or discovered math. Beyond that, this is again just relying on the weakest possible definition of "really exists", with some added human-centric arrogance added in. Crows can count to 5; it's patently absurd to claim they are using something that is "not mathematics" or some completely alien form of mathematics that humans cannot access, because it's crow-brain math rather than human-brain math. This sounds like the Copenhangen Interpretation but for math: humans brains are magic! What are we doing? What are we talking about?
> This idea that some mathematical objects are in a philosophical sense “more real” than others is a big motivator of mathematical constructivism
Yet again, this is still formalism. Up until here, you've used the word "real" in such a weak tautological sense as to have no connection to our (or any possible) universe. But here, you've switched back to "real" meaning "having any bearing on our universe". So you're saying "constructivists consider different axioms useful than ZFC mathematicians do." More often they don't even really think about usefuless at all, it's just something that caught their interest and they decided to explore it.
> I think you're conflating opinions about when math is useful with opinions on the nature of math itself. Formalism does not assume that "all rules are equally valid"
I think you're misinterpreting what I was saying. Of course, a formalist will say that some rules are "more valid" in the sense that they produce more interesting or useful theorems. My point was, to a formalist, there is nothing more to be said about the validity of axioms than the value of the theorems they produce. Whereas, from certain other perspectives in the philosophy of mathematics, that is not the only grounds on which axioms can be judged.
> This is equivalent to formalism, except perhaps in how the mathematician feels about it. What could any possible difference be? In what way could it ever matter in the slightest whether something "really exists", if we define that to be so weak as to include "in some timeless realm beyond this universe"? Surely pink goblins "really exist" in this sense as well. With such a weak definition, the difference between your "really exists" and my "really exists" is purely emotional.
You sound like a logical positivist. And that's the issue – if your philosophical assumptions are positivist, then non-positivist philosophies of mathematics (and of anything else) simply aren't going to be intelligible to you. They can only make sense if you are at least willing to doubt for a moment your positivist assumptions.
> Crows can count to 5; it's patently absurd to claim they are using something that is "not mathematics" or some completely alien form of mathematics that humans cannot access, because it's crow-brain math rather than human-brain math. This sounds like the Copenhangen Interpretation but for math: humans brains are magic! What are we doing? What are we talking about?
Conceptualism claims that mathematics exists in the mind–but it doesn't claim necessarily only human minds. If animals have minds too, then mathematics can exist in animal minds as well, even if in a much more rudimentary form. I doubt any conceptualist would say, that if intelligent extraterrestrial life were discovered to exist, that their minds wouldn't contain mathematics simply because they are a different species from homo sapiens.
> So you're saying "constructivists consider different axioms useful than ZFC mathematicians do." More often they don't even really think about usefuless at all, it's just something that caught their interest and they decided to explore it.
There are different types of constructivists: (a) those who have a philosophical commitment to constructivism; (b) those who are interested in constructivism for practical reasons (related to computer science); (c) those who are just interested in it as an interesting mathematical system to explore. You can be (b) or (c) without needing any philosophical commitments at all, and they are completely compatible with a formalist philosophy of mathematics. And, quite possibly, the majority working in constructive mathematics today are (b) or (c) not (a). But, historically, the founders of constructive mathematics (e.g. Brouwer) were very much (a) not (b) or (c).
> There simply is no "non-formalist" mathematics.
I think you are conflating mathematics with the philosophy of mathematics – they are two distinct disciplines. Disagreements about the philosophy of mathematics make no direct difference to mathematics itself; at the margins, they can influence judgements about which problems are interesting – although, even there, a person can find ultrafinitist mathematics interesting without needing any philosophical commitment to an ultrafinitist philosophy of mathematics.
A USB floppy drive behaves almost identically to a USB hard drive-yet another SCSI block device. The cost of keeping support for them is minimal
This is very different from legacy PC floppy drive controllers which spoke a completely different protocol, which was very complex and full of footguns
Legacy floppy controllers also had various legacy features almost nobody used, like soft deletion of sectors (IBM added this in the 70s for use with primitive database systems), or attaching tape drives using the floppy interface (nowadays if you buy a brand new tape drive, the interface options are SAS or Fibre Channel)
> There's no mystery here, it's basic physics and chemistry that this will change things, and it's accepted that we don't know exactly _how_ things will change. The alternative: "adding gigatons of carbon to the atmosphere will _not_ change anything" is simply non-sensical. It goes against the basic rules of physics and causality. I'm happy to be proved wrong here, I just legitimately can't see how an alternative position makes any sense.
With any position, you have to distinguish between its thoughtful advocates and its thoughtless ones-every position has both
Any thoughtful “climate change sceptic” is going to say (a) of course the climate is changing-it always has and always will; (b) of course it is implausible than human activity has literally zero impact on that change. But that still doesn’t tell us: (i) the relative scale of anthropogenic versus natural causal factors; (ii) the validity of any specific predictions of future change; (iii) the likely socioeconomic impacts of any future changes that may occur. It is totally possible that a person may affirm (a) and (b) while questioning the “consensus” on (i) and (ii) and (iii)
Personally, I don’t have a strong opinion on the substantive issue - but I wonder about the extent to which mainstream discourse on the topic represents good epistemic hygiene. It is even possible that the sceptics are on the whole more wrong than right, but simultaneously the mainstream response to them is more irrational than rational.
> Its really just a matter of degrees. There are 1 million, 1 million, 1 trillion parameter LLMs... and you keep scaling those parameters and you eventually get to humans.
It isn’t because humans and current LLMs have radically different architectures
LLMs: training and inference are two separate processes; weights are modifiable during training, static/fixed/read-only at runtime
Humans: training and inference are integrated and run together; weights are dynamic, continuously updated in response to new experiences
You can scale current LLM architectures as far as you want, it will never compete with humans because it architecturally lacks their dynamism
Actually scaling to humans is going to require fundamentally new architectures-which some people are working on, but it isn’t clear if any of them have succeeded yet
> LLMs: training and inference are two separate processes
True, but we have RAG to offset that.
> it architecturally lacks their dynamism
We'll get there eventually. Keep in mind that the brain is now about 300k years into fine-tuning itself as this species classified as homo sapiens. LLMs haven't even been around for 5 years yet.
In practice that doesn’t always work… I’ve seen cases where (a) the answer is in the RAG but the model can’t find it because it didn’t use the right search terms-embeddings and vector search reduces the incidence of that but cannot eliminate it; (b) the model decided not to use the search tool because it thought the answer was so obvious that tool use was unnecessary; (c) model doubts, rejects, or forgets the tool call results because they contradict the weights; (d) contradictions between data in weights and data in RAG produce contradictory or ineloquent output; (e) the data in the RAG is overly diffuse and the tool fails to surface enough of it to produce the kind of synthesis of it all which you’d get if the same info was in the weights
This is especially the case when the facts have changed radically since the model was trained, e.g. “who is the Supreme Leader of Iran?”
> We'll get there eventually. Keep in mind that the brain is now about 300k years into fine-tuning itself as this species classified as homo sapiens. LLMs haven't even been around for 5 years yet.
We probably will eventually-but I doubt we’ll get there purely by scaling existing approaches-more likely, novel ideas nobody has even thought of yet will prove essential, and a human-level AI model will have radical architectural differences from the current generation
The term “fascist” has been watered down to the point it doesn’t really mean anything the way many people use it now
I think the real standard for “fascist” has to be - how similar is what someone is doing to what Mussolini did? If there’s a genuine similarity there, the term “fascist” may be appropriate; otherwise, it isn’t
The Australian federal government is planning to build a high-speed rail line from Sydney to Newcastle (medium-sized city two hours drive north). Their solution to property rights, is >50% of the line will be underground. It will cost >US$50 billion, but if the Australian federal government wants to spend that, it can afford it. The US federal government could too, but it isn’t a priority for them
> local regulations make it prohibitively expensive
Local regulations can be pre-empted by state or federal legislation. The real problem is lack of political will to do it.
They could let you nominate an S3 bucket (or Azure/GCP/etc equivalent). Instead of dropping data from the cache, they encrypt it and save it to the bucket; on a cache miss they check the bucket and try to reload from it. You pay for the bucket; you control the expiry time for it; if it costs too much you just turn it off.
A lot of this is issues Microsoft could fix if they were sufficiently motivated
e.g. Windows lacks a fork() API so cygwin has to emulate it with all these hacks
Well, technically the NT API does have the equivalent of fork, but the Win32 layer (CSRSS.EXE) gets fatally confused by it. Which again is something Microsoft could potentially fix, but I don’t believe it has ever been a priority for them
Similarly, Windows lacks exec(), as in replace the current process with new executable. Windows only supports creating a brand new process, which means a brand new PID. So Cygwin hacks it by keeping its own PID numbers; exec() changes your Windows PID but not your Cygwin PID. Again, something Microsoft arguably could fix if they were motivated
> A lot of this is issues Microsoft could fix if they were sufficiently motivated...
They did fix it, in a sense, with WSL1 picoprocesses. Faster and more compatible than Cygwin. Real fork and exec on the Windows NT kernel. Sadly, WSL2 is even faster and more compatible while being much less interesting. WSL1 was pretty neat, at least, and is still available.
In any event, this diversion doesn't change my analysis of Cygwin. Cygwin still sucks regardless of whose fault it is. I intentionally left this stuff out of my post because I thought it was obvious that Cygwin is working around Windows limitations to hack in POSIX semantics; it's the whole point of the project. None of us can change Windows or Cygwin and they're both ossified from age and lack of attention. We have to live with the options we've actually got.
If you need a Windows build of a Linux tool in 2026 and can't use WSL, try just building it natively (UCRT64, CLANG64, MSVC, your choice) without a compatibility layer. Lots of tools from the Linux ecosystem actually have Windows source compatibility today. Things were different in the 90s when Cygwin was created.
I really dislike Notion. Its public API is full of bizarre arbitrary limitations, like a rich text database field can only contain max 100 “child blocks”, where each change in formatting consumes one child block-but its web UI doesn’t have this issue. Yes, I realise the undocumented private API that the web UI uses doesn’t have this issue either-but I shouldn’t have to, and I haven’t.
I don’t love Confluence, but at least it doesn’t do this to me.
Here's a similar study from some years back which doesn't have that flaw: https://pmc.ncbi.nlm.nih.gov/articles/PMC6880188/
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