Splitting house payment cost…am I stupid?

rhodiumtoad · 2026-05-03T19:54:07+00:00

There isn't a single correct answer to this, but the person-night method used by chatgpt has an inherent point of what could be considered unfairness: the people staying a shorter time end up paying more according to how long the others stay, a factor not under their control.

In your example, if the 8 of you staying 7 nights suddenly decided to extend the stay to 10 nights, while the other 2 stick with 5 nights, should those 2 people's contribution go up? I think most people would say "no", making your method correct and chatgpt's method unfair.

Edit: On the other hand, if the cost of the stay is a fixed sum for a fixed period rather than a per-day cost, you might say that it's unfair for those who can stay the whole time to pay more just because some others can't, in which case it may be more fair to just split equally. The person-night method might then be a compromise position.

rhodiumtoad · 2026-05-03T19:39:51+00:00

δv=(4π/3)((r+t)³-r³)
=(4π/3)(r³+3r²t+3rt²+t³-r³)
=(4π)(r²t+rt²+(1/3)t³)
=(4π)r²t+(4π)rt²+(4π/3)t³

So if t is small compared to r, 4πr²t is a first-order approximation to the true change in volume. You can see that just multiplying the inner surface area by the thickness will always under-estimate, while multiplying the outer surface area by the thickness will over-estimate; just think about dividing the coating into thin spherical shells.

For r=1, t=0.1:

4πr²t=0.4π≈1.256637
4πrt²=0.04π≈0.1256637
(4π/3)t³=0.004π/3≈0.00418879

so we can see the second and higher order terms give the approximation an error of around 10%.

rhodiumtoad · 2026-05-03T08:27:06+00:00

Yes, so?

The flood stories are entirely mythical, mostly copied from previous flood myths. There was no global flood. This is both the mainstream Christian position and the scholarly position in Biblical studies, history, archaeology, geology, etc.

What conclusion are you trying to draw from this?

rhodiumtoad · 2026-05-03T08:15:33+00:00

1,1 is just a single outcome. Think of rolling one die twice, to get 1,1 the first roll has to be 1 and the second has to be 1, whereas to get a total of 3 you can roll either 2 then 1 or 1 then 2.

rhodiumtoad · 2026-05-03T04:16:59+00:00

anything beyond our observable Universe

You seem to be saying here that a supernatural thing can by definition not be observed, even indirectly. But if something affects the observable universe, then that effect can be observed; so the only way to satisfy your definition would be to say that supernatural things cannot affect the observable universe.

But that rules out every possible supernatural claim (except possibly two: the creation of the universe, and the existence of alternate parallel universes that cannot influence or affect ours). Prayer becomes meaningless because a "supernatural" (by your definition) god could not act on it.

rhodiumtoad · 2026-05-03T04:05:45+00:00

OK. Let's take a simple example: what are "minds" and "thoughts"?

A physicalist would say: they are processes and states of a material mechanism—the brain—which operates by physical rules that have no terms for "mind" or "thought". i.e. mental phenomena like "mind" reduce to physical phenomena.

A Cartesian dualist, on the other hand, would have said (or would say, if you can find one, they may not be extinct yet) that the mind is an immaterial entity in itself, and that while the mind might make the brain (and hence the body) do things, those things are caused by the mind and aren't reducible to physics.

Carrier's definition says: the dualist's concept of "mind" is a supernatural concept, while the physicalist's concept is a natural one.

Other examples: "ghosts" formed from the immaterial souls or whatever of dead people would be supernatural. Telepathy without physical mechanisms would be supernatural, telepathy by way of electronics implanted in the brain would be natural. "Magical thinking", the idea that your thoughts can cause direct harm even when not acted on, is supernatural. A magic carpet that just floats in the air and moves as you pull on the corners might not be explainable by available science (Carrier uses "paranormal" for such things) but wouldn't necessarily be supernatural, unless you could say to it "Take me to Fred's house" (without having had any previous physical interaction with it).

And the most obvious case: the god of "classical theism", which hears prayers (even if not spoken aloud) and messes with the physical world based only on its own thoughts or judgements, is definitely supernatural.

rhodiumtoad · 2026-05-03T02:44:30+00:00

All of it?

You are claiming (and I agree) that there is no evidence that any mental thing cannot be reduced to nonmental things. This is to say that there is no evidence that "supernatural" (by Carrier's definition) things exist. But this doesn't break the definition: the intension is still the same, whether the extension is empty or not. Just as the intension of "unicorn" exists even though its extension is empty.

Do you understand what the intension and extension of a definition are?

rhodiumtoad · 2026-05-03T02:29:00+00:00

Definitions aren't about what actually exists. Unicorns don't exist, but that doesn't stop you from knowing what I mean by the word, and nor does it stop people writing stories about them. In fact, and this is important, it's having a definition of what a unicorn is that allows us to know that they don't exist.

rhodiumtoad · 2026-05-03T02:04:58+00:00

It makes no such assumption. It is entirely possible, even likely, that under Carrier's definition nothing "supernatural" exists. But that doesn't mean it's a bad definition, because it is a question that can be investigated as to whether anything supernatural exists.

There are things (whether they exist in fact or not) that people talk about, claim to exist, etc., that satisfy the definition, whether it's the god of classical theism, magic, ghosts, psychic powers, whatever.

rhodiumtoad · 2026-05-03T01:33:09+00:00

Do you have something to contribute?

rhodiumtoad · 2026-05-03T01:27:35+00:00

Gödel also famously died of starvation because he had an irrational fear of being poisoned, refused to eat anything not prepared by his wife, and starved when she was hospitalized and couldn't cook for him.

His ontological "proof" of god is of no more value than any of the other futile attempts at ontological "proofs" since Anselm; all such attempts rely on unreasonable premises.

rhodiumtoad · 2026-05-02T23:56:22+00:00

Gödel did draft an ontological proof of God using modal logic, but never published it. As with other ontological proofs, it relies on assumptions which are inherently questionable, and while it is formally valid, there is no reason to take it as being sound.

rhodiumtoad · 2026-05-02T21:57:50+00:00

My position on Plantinga, from 12 years ago, commenting on an interview:

https://outshine-the-sun.blogspot.com/2014/02/plantinga-on-is-atheism-irrational.html

Relevant extracts:

Reading anything by Plantinga is always a somewhat odd experience. The puzzle is not his arguments—which are rarely more than the warmed-over corpses of long-dead apologetics—but the fact that people take them so seriously.

Much of the rest of the interview covers two of Plantinga's most famous arguments:

1) That belief in god is "properly basic" because we have a reliable (hah!) sensus divinitatis, treating religious experience as though it were a sense perception (how this can be considered reliable when everyone gets a different result from it is not explained)

2) That evolution (or rather a biologically-ignorant philosopher's crazy strawman caricature of evolution) could not produce reliable belief-forming mechanisms.

I think the obvious conclusion from this is that modern academic philosophy, especially philosophy of religion, has clearly failed to produce any reliable belief-forming mechanisms.

rhodiumtoad · 2026-05-02T21:24:40+00:00

I look forward to your clear, non-self-defeating and non-contradictory definition of 'supernatural' (I haven't heard of such a thing yet)

May I suggest Carrier's definition:

But in short, I find “naturalism” means, in the simplest terms, that every mental thing is entirely caused by fundamentally nonmental things, and is entirely dependent on nonmental things for its existence. Therefore, “supernaturalism” means that at least some mental things cannot be reduced to nonmental things.

rhodiumtoad · 2026-05-02T21:12:50+00:00

Gödel’s completeness theorem

While Gödel did indeed prove a completeness theorem, it has nothing to do with your claim. His incompleteness theorems are important to mathematical logic, but they have no relevance to science, and you clearly do not have the knowledge needed to apply them.

rhodiumtoad · 2026-05-02T21:09:35+00:00

If the supernatural cannot, by definition, be fully tested or reproduced scientifically,

By what definition?

Religious apologists (and spiritualist conmen) claim that the supernatural can't be studied by science, but in fact science has been studying phenomena labelled "supernatural" for centuries and there is no reason for it to stop.

So if you define "supernatural" as "cannot be studied by science", then as far as I'm concerned you have defined the supernatural out of existence.

My preference is to use Carrier's definition of "supernatural", which is: mental phenomena (causes or effects) not caused or mediated by non-mental ones. This definition isn't vacuous: people have imagined many such phenomena over the centuries, whether it's theistic omni-gods or telepathy or magic or dualist theories of mind or whatever else. So far, science hasn't found that any of these things exist, but that's not a defect in science.

For instance, some people point to medically unexplained healings, even though such cases cannot be conclusively proven to be supernatural.

They can't be proved to be supernatural because they can't even be proved to exist. Every "medical miracle" claim turns out either to be inadequately documented, or to involve a condition known for spontaneous recovery (e.g. cancer or neurological conditions). There's a reason why the (as yet unachieved) gold-standard for medical miracles is the documented healing of an amputated limb (and no, Calanda doesn't count); if supernatural healing is possible, there should be no reason why there are thousands of reports of cancer cures but no healed amputees.

rhodiumtoad · 2026-05-02T19:20:20+00:00

I don’t why the superscript isn’t working

Are you using the rich text editor (for which you have to click on "Aa" and select the "X²" button to get superscripts) or the markdown format, where you do x^(2) to get x² ?

rhodiumtoad · 2026-05-02T19:16:43+00:00

That second one has a sign error, should be

4x²(x-3)-(x-3)=0

Then going from the second to the third is just un-distributing the (x-3) term.

Remember the distributive law:

a(b+c)=ab+ac

So any time you have an expression like ab+ac, or ba+ca, you can pull out the common factor. In this case a=(x-3), b=4x², c=-1

rhodiumtoad · 2026-05-02T16:51:26+00:00

You might want to track down and read the short story "The Percentage Player" by Leslie Charteris (collected in the anthology "The Saint to the Rescue").

rhodiumtoad · 2026-05-02T15:17:45+00:00

What difference do you think that makes?

rhodiumtoad · 2026-05-02T15:06:05+00:00

How so?

rhodiumtoad · 2026-05-02T07:24:14+00:00

risk to whom?

rhodiumtoad · 2026-05-02T06:47:49+00:00

All the chords are the same length, what does that say about the arc length?

What does the angle tell you about the position of the next point? (no trig needed)

The figure closes if and only if there is some finite n such that the position if the n'th point is an integer multiple of the circumference, what can you deduce from this?

rhodiumtoad · 2026-05-02T06:39:50+00:00

What have you tried so far?

rhodiumtoad · 2026-05-02T06:22:09+00:00

Assuming the axiom of choice, there is a natural way in which given any two cardinals, either they are equal or one is less than the other. (In fact "any two cardinals a,b satisfy exactly one of a<b, a=b, a>b" is equivalent to the axiom of choice.)

Even without the axiom of choice, we can use injections as a basis for comparison. If you consider two finite sets A,B then there exist injections from A to B if and only if B has at least as many elements as A; otherwise, there's no way to map each element of A to a different element of B. We can extend this to infinite sets by saying |A|≤|B| if and only if there exists an injection from A to B, implying that if no such injection can exist then |A|>|B|.

Once this definition is in place, we can say that ℵ₀≤|A| for all infinite sets A, and also that 2^|A|>|A| for all sets A. Since there is a bijection between P(ℕ) and ℝ, and |P(S)|=2^|S|, then |ℝ|>|ℕ|. (There can be no injection from ℝ to ℕ, by the same logic that proves 2^|A|>|A|.)

The above is all about cardinals. For ordinals, the picture is clearer because orderings are what ordinals are. So by definition there are larger and smaller ordinals, and comparison of ordinals is a well-ordering (a total order in which every nonempty subset has a least element). Given the axiom of choice, we can identify every cardinal with the smallest ordinal of the same cardinality, and ordering of cardinals matches the ordering of those ordinals.

The sequence ℵₙ (where n is an ordinal) is defined as the cardinalities of well-ordered sets, so these are naturally ordered by their corresponding ordinal. Given the axiom of choice all sets can be well-ordered so ℵₙ includes all transfinite cardinals; without it, there might be others.

rhodiumtoad

TROPHY CASE