That's absolutely correct.
That's why you have to have a business address, and get all your business admin ducks in a row, even if it's your first real monetized app. Your future self will always thank you!
355 karma · joined October 11, 2024
That's why you have to have a business address, and get all your business admin ducks in a row, even if it's your first real monetized app. Your future self will always thank you!
On the other hand, using the axioms of ZFC, one can say any real number exists without having a function to compute it, or a proof to construct it.
For an ultrafinitist, or any finitist for that matter, we say that you only need the minimum of ingredients to produce math -- you do not need to assume anything over and above that, as it's not even helpful in the verification process.
So assuming only finitely many symbols and finitely many numbers, I can produce what we call sqrt(2). We only ever verify it numerically and finitely anyways. We can never reach decimals at infinite ordinals.
So it makes no sense to say, "Hey I assume transfinitely many entities, and my assumption says these numbers exist even though the proofs and decimal expansions are only ever finite."
I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation.
The whole point is that as a theory for the foundations of mathematics, you do not need to assume numbers with infinitely long decimal expansions in order to do math.
The point is to not confuse the notational convenience with the underlying concept that makes such numbers comprehensible in the first place.
After all, the Cayley-Dickson construction is not an infinite affair.
An ultrafinitist is still allowed to call that 'i'.
To defend Wildberger a bit (because I am an ultrafinitist) I'd like to state first that Wildberger has poor personal PR ability.
Now, as programmers here, you are all natural ultrafinitists as you work with finite quantities (computer systems) and use numerical methods to accurately approximate real numbers.
An ultrafinitist says that that's really all there is to it. The extra axiomatic fluff about infinities existing are logically unnecessary to do all the heavy lifting of the math that we are familiar with. Wildberger's point (and the point of all ultrafinitist claims) is that it's an intellectual and pedagogical disservice to teach and speak of, e.g. Real Numbers, as if they're actually involving infinite quantities that you can never fully specify. We are always going to have to confront the numerical methods part, so it's better to make teaching about numbers methodologically aligned with how we actually measure and use them.
I have personally been working on building various finite equivalents to familiar math. I recommend anyone to read Radically Elementary Probability Theory by Nelson to get a better sense of how to do finite math, at least at the theoretical level. Once again, on a practical level to do with directly computing quantities, we've only ever done finite math.
The UK is not a democratic or even liberty-focused state anymore. It's always been ruled by a crowd of people who went to privately-funded schools that cost a fortune. Half the government's politicians and staffers can trace their relations back to the same historical personage.
They aren't afraid for their kids with these laws. They're afraid that this ossified, stunted system of power that's been built over 800 years will break, and they will be out of a job with pitchfork-wielding crowds chasing them out of London.
What happens is generational shifts over longer periods of time mean that draconian law or feature has more and more chances to be used by someone with bad intentions. It's the law of large numbers or murphy's law in full effect, it's not just 1 or 2 people.
It doesn't even get that hot with LLMs running with max fans, where the SoC is about 80º C.
Aside from those use cases, the M4 Max runs 43º C or less even in summer conditions.
These matters are always problems of organization, and of prioritizing what the job is, what are the inputs/outputs, how do you efficiently parameterize them into messages and data packets, where do they go and how will you send it, etc.
It's not that much of a stretch to imagine ultra dense wafers that can have compute, storage, and memory all in one SoC.
First, unify compute and memory. Then, later, unify those two with persistent storage so that we have something like RAM = VRAM = Storage.
I don't think this is around the corner, but certainly possible in about 12 years.
Something happened in 15.1 onwards for me where Spotlight has become way faster and way better. But yes, Alfred used to dominate in search and speed as well.
I know, but hear me out: it's a decent hook for teaching people about Geometry, Recursion, and Dynamic Programming.
Nvidia GPU: spin up OS, run your sims or load your LLM, gather results.
AMD GPU: spin up OS, grok driver fixes, try and run your sims, grok more driver fixes, can't even gather results until you can verify software correctness of your fixes. Yeah, sometimes you need someone with specialized knowledge of numerical methods to help tune your fixes.
... What kind of maddening workflows are these? It's literally negative work: you are busy, you barely get anywhere, and you end up having to do more.
In light of that, the Nvidia tax doesn't look so bad.
Once again, my point is that people are trying to take shortcuts with abstractions that are not grounded in reality. That is a matter of self-discipline, of priorities, of putting the cart before the horse. Consider string theories: we have worked out so many ways in which strings can behave, etc. with so many possibilities and permutations. However, we never proved the ground reality for strings, we just ran with a bunch of assumptions and then parameterized them, went meta a bunch of times, and called that a research program.
All of that mathematical sophistication and model-building could have went to, e.g. perfecting QCD, or even in other directions.
Math is all you've got to work with, we wouldn't have modern day physics without math.
The issue is that people think they can find some kind of magic shortcut by playing around with abstractions without reference to or grounding in physical observables. That's not a math problem, that's a psychology problem.
You can also vary infinitesimals and utilize them not just in nonstandard analysis, but in fractional calculus, such as for inferring stock market motions.
They have helpful applications in physics, especially field theory.
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I can imagine, a long time from now, many elegant mathematical constructs simplified by the use of, e.g. infinitesimals, Clifford algebras, category theory, etc. There's a lot of complicated ideas that are nicely simplified, and are even more intuitive, easy to teach the fundamentals of, rather than the standard approach.
I think it's important to understand that the canonical calculus approach came from rather mechanical questions in analysis and proofs, and the math is layered with that, as well as the notational conveniences of forms of calculus commonly used for electromagnetism, classical mechanics, etc. There's a lot of legacy syntax there, and we just live with it, but it's not optimal. Infinitesimals are a way to go back to applications and to better syntax.
I know I'll be gunning for the 42" 8K's whenever they actually reach a decent market price. Sigh, still too many years away.