God created the real numbers • Hacker News Board

Re: God created the real numbers

#201

Post by zarzavat » Sun, Aug 31, 2025, 3:58 AM UTC

baxtr wrote: post #193

zarzavat wrote: post #21

God created the rational numbers. 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.

A circle seems quite ordinary at first glance, yet its area is pretty irrational.

The area of a circle is a computable number so it can be put into one-to-one correspondence with the rationals. It's much more like a rational number than a real number, insofar as it doesn't require infinities to represent it.

The set of real numbers is almost all extraneous junk that the universe definitely doesn't care about but is very important to mathematicians.

Re: God created the real numbers

#202

Post by adrian_b » Sun, Aug 31, 2025, 10:11 AM UTC

drdeca wrote: post #196

adrian_b wrote: post #136

Earlier quoted context omitted.

The right magnitude for things to get weird must be very small, but nobody can say whether that scale is a million times greater than the Planck length or a million times smaller than the Planck length. Therefore using the Planck length for any purpose is meaningless. For now, nobody can say anything about the value of a Schwartzschild radius in this range, because until now nobody succeeded to create a theory of gra…

By “things get weird” I meant “our current theories/models predict things to get weird”. So, like, I’m saying that if Einstein’s model of gravity is applicable at very tiny scales, and if the [p,x] relation continues to hold at those scales, then stuff gets weird (either by “measurement of any position to within that amount of precision results in black-hole-ish stuff”, OR “the models we have don’t correctly predict…

I get what you mean, but one thing about which we are certain is that you cannot apply Einstein"s model of gravity at these scales, because his theory is only an approximation that determines the metric of space from an averaged density of the energy and momentum of matter, not from the energy-momentum 4-vectors of the particles that compose matter.

So Einstein's theory depends in an essential way on matter being continuous. This is fine at human and astronomic scales, but it is not applicable at molecular or elementary particle scales, where you cannot approximate well the particles by an averaged density of their energy and momentum.

Any attempt to compute a gravitational escape velocity at scales many orders of magnitude smaller than the radius of a nucleus are for now invalid and purposeless.

The contradiction between the continuity of matter supposed by Einstein's gravity model and the discreteness of matter used in quantum physics is great enough that during more than a century of attempts they have not been reconciled in an acceptable way.

The offset of the spin is likely to be caused by the fact that for particles of non-null spin their movement is not a simple spinning, but one affected by some kind of precession, and the "spin" is actually the ratio between the frequencies of the 2 rotation movements, which is why it is quantized.

The "action" is likely to be the phase of the intrinsic rotation that affects even the particles with null spin (and whose frequency is proportional with their energy), while those with non-null spin have also some kind of precession superposed on the other rotation.

Re: God created the real numbers

#203

Post by adrian_b » Sun, Aug 31, 2025, 10:36 AM UTC

Dylan16807 wrote: post #24

chasd00 wrote: post #14

Earlier quoted context omitted.

> The idea of arbitrary precision is intrinsically broken in physical reality. 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.

Can you name a physical thing that is a circle even to the baseline precision level of a 64 bit float?

The most perfect things from this POV that have been made by humans are spheres of monocrystalline silicon, which have been made for the purpose of counting how many atoms they contain, for an extremely accurate determination of the mass of silicon atoms.

The accuracy of their volume and radius did not reach the level of a 64-bit float, but it was several orders of magnitude better that of 32-bit FP numbers.

While you cannot build a thing made of molecules with an accuracy better than that of a FP64 number, you can have a standing wave in a resonator, which stays in a cryostat, where the accuracy of its wavelength is 4 orders of magnitude better than the accuracy of a FP64 number, and where the resonator is actively tuned, typically with piezoelectric actuators, so that its length stays at a precise multiple of the wavelength, i.e. with the same accuracy. Only the average length of the resonator has that accuracy, the thermal movements of the atoms cause variations of length superposed over the average length, which are big in comparison with the desired precision, which is why the resonator must be cooled for the best results.

However, it does not really matter whether we can build a perfect sphere or circle. What it matters that modelling everything while using a geometry that supposes the existence of perfect circles we have never seen errors that could be explained by the falseness of this supposition.

The alternative of supposing that there are no perfect circles is not simpler, but much more complicated, so why bother with it?

Re: God created the real numbers

#204

Post by moi2388 » Sun, Aug 31, 2025, 12:14 PM UTC

griffzhowl wrote: post #191

moi2388 wrote: post #127

I am a finitist and constructionist at heart. Sure, mathematical abstractions and infinite structures are fun to play around with.. But go ahead and actually provide me the list of all naturals. You can not. Ever.

> But go ahead and actually provide me the list of all naturals. You can not. Ever. But how did you come to this conclusion unless by assuming that there are infinitely many natural numbers?

Proof by contradiction. Heard mathematicians like that.

Re: God created the real numbers

#205

Post by griffzhowl » Sun, Aug 31, 2025, 1:01 PM UTC

moi2388 wrote: post #204

griffzhowl wrote: post #191

Earlier quoted context omitted.

> But go ahead and actually provide me the list of all naturals. You can not. Ever. But how did you come to this conclusion unless by assuming that there are infinitely many natural numbers?

Proof by contradiction. Heard mathematicians like that.

Ironically, finitists and constructivists don't like proof by contradiction...

I agree though, that you have come up with a contradiction. Specifically, because you seem to believe these two statements:

There are finitely many natural numbers.

Given any finite list of natural numbers, we can always produce another natural number not on that list.

Re: God created the real numbers

#206

Post by Someone » Sun, Aug 31, 2025, 2:08 PM UTC

BeetleB wrote: post #93

omnicognate wrote: post #37

Earlier quoted context omitted.

As in why bother using real numbers in physics? Mostly because you need them to make the maths rigorous. You can't do rigorous calculus (i.e. real analysis) on rationals alone.

We don't need reals to make the math rigorous. Only to make the math a lot more tractable. I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result - you of course get the same result if you had simply used regular integration/differentiation, and it's a lot easier to use calculus than combinatorics. But the point is the "rationa…

> I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result

But taking the limit of a sequence of rationals isn’t guaranteed to remain in the rationals (classic example: https://en.wikipedia.org/wiki/Basel_problem. Each partial sum is rational, but the limit of the partial sums is not)

So, how does that statement rebut “You can't do rigorous calculus (i.e. real analysis) on rationals alone.”?

Re: God created the real numbers

#207

Post by Dylan16807 » Sun, Aug 31, 2025, 2:54 PM UTC

adrian_b wrote: post #203

Dylan16807 wrote: post #24

Earlier quoted context omitted.

Can you name a physical thing that is a circle even to the baseline precision level of a 64 bit float?

The most perfect things from this POV that have been made by humans are spheres of monocrystalline silicon, which have been made for the purpose of counting how many atoms they contain, for an extremely accurate determination of the mass of silicon atoms. The accuracy of their volume and radius did not reach the level of a 64-bit float, but it was several orders of magnitude better that of 32-bit FP numbers. While yo…

> However, it does not really matter whether we can build a perfect sphere or circle.

When talking about whether arbitrarily precise numbers are real in the universe, it extremely matters.

Sadly, atoms exist. In some ways that makes things more complicated, but it's the truth. Anything made of discrete chunks in a grid can't have arbitrarily precise dimensions.

Re: God created the real numbers

#208

Post by tboyd47 » Sun, Aug 31, 2025, 3:33 PM UTC

BalinKing wrote: post #58

tboyd47 wrote: post #30

> If the something under examination causes a sense of existential nausea, disorientation, and a deep feeling that is can't possibly work like that, it is divine. 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 itsel…

I want to push back on this, because the Christian conception of God definitely includes the idea that God created all good and comforting things, and is indeed their ultimate source. Like, just because God is transcendent[0] does not mean He cannot create things that are perfectly approachable, understandable, and enjoyable. [0] Jesus being human changes the calculus quite a lot, of course, as elaborated in e.g. Heb…

Then what’s your view on the OP, as a Christian? Can you “see” God in one set of numbers but not the other? What’s your take?

Re: God created the real numbers

#209

Post by tboyd47 » Sun, Aug 31, 2025, 3:35 PM UTC

thechao wrote: post #55

tboyd47 wrote: post #30

> If the something under examination causes a sense of existential nausea, disorientation, and a deep feeling that is can't possibly work like that, it is divine. 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 itsel…

> This is a Muslim 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? ... > 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.

Is it bigoted to discuss nuanced differences between belief systems?

Re: God created the real numbers

#210

Post by BeetleB » Sun, Aug 31, 2025, 5:42 PM UTC

Someone wrote: post #206

BeetleB wrote: post #93

Earlier quoted context omitted.

We don't need reals to make the math rigorous. Only to make the math a lot more tractable. I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result - you of course get the same result if you had simply used regular integration/differentiation, and it's a lot easier to use calculus than combinatorics. But the point is the "rationa…

> I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result But taking the limit of a sequence of rationals isn’t guaranteed to remain in the rationals (classic example: https://en.wikipedia.org/wiki/Basel_problem . Each partial sum is rational, but the limit of the partial sums is not) So, how does that statement rebut “You can't…

> But taking the limit of a sequence of rationals isn’t guaranteed to remain in the rationals

I'm not saying it does. What I'm saying is that you can make a correspondence with the reals by using only rationals.

You can define convergence without invoking the reals (Cauchy convergence). If you take any such sequence, you give that sequence a name. That name is the equivalent of a real number. You can then define addition, multiplication - any operation on the reals - with respect to those sequences (again, invoking only rational numbers).

So far, we have two distinct entities: The rationals, and the converging sequences.

Then, if you want, you can show that if you take the rationals and those entities we're calling "converging sequences" together, you can make operations involving the two (e.g. adding a rational to that converging sequence) and eventually build up what we know to be the number line.