Ask HN: Why hasn't x86 caught up with Apple M series?
438 points by stephenheron 3d ago 616 comments
Ask HN: Best codebases to study to learn software design?
106 points by pixelworm 5d ago 92 comments
God Created the Real Numbers
33 Bogdanp 63 8/29/2025, 3:31:15 PM ethanheilman.com ↗
Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but representing the state even of a very simple system involves continuous quantities.
But the Cantor vision of the real numbers is just wrong and completely unphysical. The idea of arbitrary precision is intrinsically broken in physical reality. Instead I am off the opinion that computation is the relevant process in the physical universe, so approximations to continuous quantities are where the "Eternal Nature" line lies, and the abstraction of the continuum is just that -- an abstraction of the idea of having perfect knowledge of the state of anything in the universe.
But that's tatamount to the belief that the minutest particle of the universe requires the equivalent of an infinite number of bits of state.
The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?
My point was that it is possible that all values in our universe are rational, and it wouldn't be possible for us to tell the difference between this and a universe that has irrational numbers. This fact feels pretty cursed, so I wanted to point it out.
I think the conceit is supposed to be that analysis—and therefore the reals—is the “language of nature” more so than that we can actually find the reals using scientific instruments.
The idea that a quantity like 1/3 is meaningfully different than 333/1000 or 3333333/10000000 is not really that interesting on its own; only in the course of a physical process (a computation) would these quantities be interestingly different, and then only in the sense of the degree of approximation that is required for the computation.
The real numbers in the intuitionalist sense are the ground truth here in my opinion; the Cantorian real numbers are busted, and the rationals are too abstract.
The continuum is the reality that we have to hold to. Not the continuum in the Cantor sense, but in the intuitionalist or constructivist sense, which is continuously varying numbers that can be approximated as necessary.
Citation needed.
Especially since there are well-established math proofs of irrational numbers.
A skeptic in what way? He said a lot.
Math is math, if you start with ZFC axioms you get uncountable infinites.
Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.
So yes, generally not starting with ZFC.
I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tarski) from the axioms but have no way to map these wildly non-constructive ideas back into the real world. Or where you construct a version of the reals where the reals that we can produce via any computation is a set of measure 0 in the reals.
Well you can be skeptical of anything and everything, and I would argue should be.
Addressing your issue directly, the Axiom of Choice is actively debated: https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_...
I understand the construction and the argument, but personally I find the argument of diagonalization should be criticized for using finities to prove statements about infinities.
You must first accept that an infinity can have any enumeration before proving its enumerations lack the specified enumeration you have constructed.
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
> Math is math, if you start with ZFC axioms
This always bothers me. "Math is math" speaks little to the "truth" of a statement. Math is less objective as much as it rigorously defines its subjectivities.
https://news.ycombinator.com/item?id=44739315
The axiom of choice is not required to prove Cantor’s theorem, that any set has strictly smaller cardinality than its powerset.
Actually, I can recount the proof here: Suppose there is an injection f: Powerset(A) ↪ A from the powerset of a set A to the set A. Now consider the set S = {x ∈ A | ∃ s ⊆ A, f(s) = x and x ∉ s}, i.e. the subset of A that is both mapped to by f and not included in the set that maps to it. We know that f(S) ∉ S: suppose f(S) ∈ S, then we would have existence of an s ⊆ A such that f(s) = f(S) and f(S) ∉ s; by injectivity, of course s = S and therefore f(S) ∉ S, which contradicts our premise. However, we can now easily prove that there exists an s ⊆ A satisfying f(s) = f(S) and f(S) ∉ s (of course, by setting s = S), thereby showing that f(S) ∈ S, a contradiction.
One could argue that nature always deals in discrete quantities and we have models that accurately predict these quantities. Then we use math that humans clearly created (limits) to produce similar models, except they imagine continuous inputs.
There have been attempts to create discrete models of time and space, but nothing useful has resulted from those attempts.
Most quantities encountered in nature include some dependency on work/energy, time or space, so nature deals mostly in continuous quantities, or more precisely the models that we can use to predict what happens in nature are still based mostly on continuous quantities, despite the fact that about a century and a half have passed since the discreteness of matter and electricity has been confirmed.
you said a lot and i probably don't understand but doesn't pi contradict this? pi definitely exists in physical reality, wherever there is a circle, and seems to be have a never ending supply of decimal points.
Is there a circle in physical reality? Or only approximate circles, or things we model as circles?
In any case, a believer in computation as reality would say that any digit of π has the potential to exist, as the result of a definite computation, but that the entirety does not actually exist apart from the process used to compute it.
What does it mean to "exist in physical reality"?
If you mean there are objects that have physical characteristics that involve pi to infinite precision I think the truth is we have not a darn clue. Take a circle, that would have to be a perfect circle. Even our most accurate and precise physical theories only measure and predict things to 10s of decimal places. We do not possess the technology to verify that it's a real true circle to infinite precision, and many reason to think that such a measurement would be impossible.
They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in "reality".
On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically; two quantities are interestingly "unequal" only at the precision where an underlying process can distinguish them. Turing tells us that any underlying process must represent a computation, and that the power of computation is a law of the underlying reality of the universe (this is my view of the Universal Church-Turing Thesis, not necessarily the generally accepted variant).
The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. It leads to rabbit holes that are just uninteresting; trying to distinguish inifinities (continuum hypothesis) and leading us to counterintuitive and useless results. Fun to play with, like writing programs that can invoke a HaltingFunction oracle, but does not tell us anything that we can map back to reality. For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.
it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment
I am not sure what you are arguing here. We’ve been teaching this to all undergraduate mathematicians for the last century; are you trying to make the point that this part of the curriculum is unnecessary, or that mathematics has not contributed to the wellbeing of society in the last hundred years? Both of these seem like rather difficult positions to defend.
I didn't mean to suggest that the reals are the floor of reality, rather that they are more floorlike than the integers.
> The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no.
Tools are created by transforming nature into something useful to humans. Is Cantor's conception of infinity more natural? I can't really say, but the uselessness and confusion seems more like nature than technology.
In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can tell me all its arithmetical properties, you can write down its construction as a set in ZFC set theory… but can you really know the number – not a representation of the number, not its properties, but the number itself? Of course not: mathematical objects are their properties and nothing more. It doesn’t even make sense to consider the idea of a “raw” object.
Can it? We can only work with things we can name and the real numbers we can name are an infinitesimal fraction of the real numbers. (The nameable reals and sets of reals have the same cardinality as integers while the rest are a higher cardinality.)
Nah, you're likely thinking of the rationals, which are basically just two integers in a halloween costume. Ooh a third, big deal. The overwhelming majority of the reals are completely batshit and you're not working with them "just fine" except in some very hand wavy sense.
(And the discrepancy might not be in the physical continuum being simpler than the mathematical reals, as some here postulate, but rather in the continuum being far stranger than the reals, in ways we may never observe nor comprehend.)
I'm so used to thinking this way that I don't understand what all the fuss is about, mathematical objects being "real". Ideas are real but they're not real in the way that rocks are.
Whenever there's a mysterious pattern in nature, people have felt the need to assert that some immaterial "thing" makes it so. But this just creates another mystery: what is the relationship between the material and the immaterial realm? What governs that? (Calling one or more of the immaterial entities "God" doesn't really make it any less mysterious.)
If we add entities to our model of reality to answer questions and all it does is create more and more esoteric questions, we should take some advice from Occam's Shovel: when you're in a hole, stop digging.
The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.
This is a problem of modeling optimization. The models based on uncountable "real" numbers are logically consistent and simple to use, so they are adequate for predicting what happens in natural or artificial systems.
All attempts to avoid the uncountable infinities produce models that are both more complicated and also incomplete, as they do not cover all the applications of traditional infinitesimal calculus, topology and geometry.
Unless someone will succeed to present a theory that avoids uncountable infinities while being as simple as the classic theory and being applicable to all the former uses, I see such attempts as interesting, but totally impractical.
The real numbers exist and are approximable, either by rationals or by decimal expansion. The idea of approximability and computability are the critical things, not the specific representation.
Integers come into existence long before god - as the only presumption required is a difference between one thing and another (or nothing). The integers also create infinite gaps. The primes.
So no - I do not think reals are closer to the divine. They require we import infinity twice to be defined, and I'm undecided on whether our reality has unbounded 'precision' like that - or if 'just' an infinite number of discrete units.
Caveat: former Catholic; 50+ years of fervent atheism.
ps. Various numerology phenomena have a similar vibe, and no wonder so many people who go off the deep end tend to get trapped by them. Maybe I will be one of them as I become old and senile :-D
This is a Jewish and Christian conception of God. How can this be true when so many things that give us comfort in the natural world: fresh fruit, shade trees, sunshine and warm sand between our toes, etc., were not created by man?
Even in mathematics itself: how improbable, how ludicrous, how miraculous is it that the 3rd, 4th, and 5th natural numbers -- numbers you could discover by looking at your own hands -- have the amazing property of demonstrating the Pythagorean theorem?
The Islamic ideal of God (Allah) is so much more balanced. God created both the integers AND the reals. He created everything, some things for our comfort and rest, some things to drive us close to madness, and a lot of stuff in between. Peel back enough layers of causality and all of creation has the stamp of the divine.
[0] Jesus being human changes the calculus quite a lot, of course, as elaborated in e.g. Hebrews 4:14–16. God, who was fully transcendent, became human, hence why Jesus is also called Immanuel/Emmanuel (lit. “God with us”) in the Bible.
...
> The Jewish [Christian] ideal of God (YHVH) is so much more balanced.
There's enough bigotry out there. Let's not make assumptions about people's beliefs.