If it were merely a creation, there would be no reason for two independent mathematicians to land on the same creation given some directed effort. But of course we do see that. There is an objectivity to mathematics that must be accounted for.
"Where" mathematics exists is in the abstract combinatorical space of an infinite repeating application of logical rules. This space doesn't exist in a substantive sense, but it is accessible/navigable by studying the consequences of logical rules. It is the space of possible structure.
If this space of possible structure is real, but seemingly immaterial, how does our matter brain access it?
I think we create mathematics as thought structure in our mind. We can agree on things when we create the same structures. But this structure did not exist prior to creation.
I don't know what real means; I might call it real depending on one's definition. I definitely wouldn't call it immaterial (though it's not material either). We access it by construction: apply relevant rules and discover their consequences. Two people probing this structure are equally constrained by the requirements of consistency. There is no Benacerraf-style access problem.
It could be that RH is independent of current mathematical axiom systems. We might even prove that it is some day. But that means we are free to give it different truth values depending on the circumstances!
This is also true for established theorems! We can can imagine mathematical universes (toposes) where every (total) function on the reals is continuous! Even though it is an established theorems that there are discontinuous functions! We just need to replace a few axioms (chuck out law of the excluded middle, and throw in some continuity axioms).
What frequently happens when we recombine axioms like that is that they end up leading to inconsistencies or contradictions.
Do you know if this topos with every total function on real numbers is continuous has been constructed and proven to be a viable set of axioms? If so, I am curious about the source.
My go to example still remains the one of hyperbolic geometry and axiom of parallel lines, so the more approachable examples I can get, the better.
Sure. These toposes are well known, and proven to be consistent (relative to set theory). For instance Hyland’s effective topos, or Johnstone’s topological topos. The ideas are that these toposes either require everything to be computable, or continuous in some greater sense.
You're just using a different definition of "scientific". If Math isn't a science, then it cannot follow logical rigor. Math and philosophy are both sciences, even if they differ qualitatively from the natural and human empirical sciences.
People seem to use these tools very differently from each other. I value intelligence over speed any day. My programs are written in Haskell, so there are rarely any tasks which require thousands and thousands of lines to solve. Just intelligence. If there are rote tasks, I want the LLM to help me find intelligent ways of automating it: the right abstraction, the right meta-programming technique.
I constantly push Opus and GPT, and they are getting better. But still have to do the hardest parts myself. I would not mind waiting 10-15 minutes for the right 20 lines of code!
Why do you use Haskell? Why not something that produces a more predictable memory use at runtime? (I’m asking earnestly as a former Haskeller turned Rustacean who sees the value in “Boring Haskell”, but favours strictness for anything internet-facing and many things that aren’t compilers.)
I use Haskell because purity and strong typing gives so much control over what each part of the program does. This has huge benefits when it comes to security, and just general lack of bugs. Also, it makes the code easy to write, once the types are in place.
I use Haskell because I find laziness to be a super power. I can solve so many problems in the most straightforward way, and then laziness saves my butt w.r.t. performance.
I use Haskell because it is a better C than C is. The foreign function interface is brilliant, and I can take C primitives and apply all the abstraction mechanisms from Haskell to them. My latest project has been OpenGL based, so lots of caring about byte alignments and shovelling data to the GPU. But all this can be automated with clever use of type classes and Generics (Haskells super cool meta system of data types.)
I use Haskell because I love applying abstractions to make code which describes the problem, and then the compiler finds the solution.
I don’t do programming for embedded, so I am rarely memory constrained. I also understand Haskell memory usage quite well, and can get myself out of trouble.
As long as enough people keep the pirate bays open, it will be there as an alternative when the services start their inevitable enshittification.
I for one do not enjoy the “Which service has the classic film I wanted to watch this week?” Nor having to switch services every time I want to see a new TV series.
We need (and have!) similar “free” alternatives to the watermarked generative services. Just like I hate the yellow dots on my printed images, I am not happy to have my creative assets (I do nothing nefarious) stained with SynthID.
I see this with beginner programming students at university. They get AI to help them with assignments, with the intention of learning, but ultimately they do not get the understanding they would have if they had done the assignment themselves. Then they are at a deficit for learning more advanced topics.
My fear is that they never get to the level they need to be at to create good software even with the help of AI. So, although an expert with AI can create great software, that is not where we end up. In stead we will have vibe coded messes by people who barely have any grasp of what is going on.
I heard that air quality on planes was better back then (maybe someone who was alive then can confirm). Because of smoking they had to ventilate the whole aircraft much better. While these days I feel like they are just starving us for oxygen so as to not have to heat up fresh air.
Old person here. I think it's really hard to convey the extent to which smoke literally permeated everything. It's not just the immediate air quality aspects of it, but there was just a residue on all the surfaces, every cushion and fabric held onto the stuff.
I can recall the week that no-smoking indoors at restaurants/bars passed and it was literally shocking to walk into a place and not have it be hazy. It really felt weird.
Anyway, air quality + quality of life was much worse. Sometimes the future does get better.
Another old person here. At an office in Zurich I saw a layer of smoke filling the upper reaches of the atrium. I wondered how many working (i.e. smoking) hours it would take before it reached the balcony on which I was standing.
except for the dance bars. Dear lord the sweat smell during the transition was bizarre. It as always masked thanks to the smell of smoke. I think a lot places had to start thinking about adding nice parfumes, because almost at the end of that first year of zero tolerance inside bars, it was 'solved'.
Lol. I was 14 when I took a long distance international flight on a 747 in 1979. The family was sitting in the “non-smoking section”. I can tell you for a fact that the air quality in that plane was terrible. Possibly because a number of passengers in the non-smoking section still deigned to smoke. Whaddaya do eh?
It's an addiction, they're compelled to smoke, and so at the edges of the area they'll light up.
That's how the Kings Cross Fire started. Escalator full of potential fuel, smoker drops a used match, it falls inside the machine, fire. It wasn't legal technically to be smoking on that escalator, but it would have been legal in a few paces so "everybody" did it. The investigators found signs that such fires had likely started or almost started many times before, the disaster was just that this time it burned for long enough to create a pool of extremely hot gas flowing up the inclined ceiling for the escalator, and we got to discover the Trench Effect in the least fun way possible.
I flew to Japan from US in the "non smoking section" and which the smoking section started in the row immediately behind me... A woman smoked in the seat behind me most of the trip.
The airplanes were awful, usually with silly little signs stuck in some seats to designate the switchover which the smoke didn't seem to respect. I was in a train brought back to service from smoking times a few years ago and the stench still emanating from the fabric seats brought back those memories right away.
Turns out using less engine bleed air is good for fuel economy, so now it's 50% recirculated HEPA filtered (which does nothing for the co2 contents) air.
Haskell is a good language for LLMs! Claude knows it really well, and the type system catches so many mistakes. Just make sure to tell it to model the domain in the type from the start.
Also, Haskell can be really performant and low level, while still keeping the benefits of typing. With the C foreign function interface you can really do anything in Haskell!
I know of no realm where mathematical objects live except human minds.
No, it seems clear to me that mathematics is a creation of our minds.
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