Hᵤ(2, 2) = 4 for all u>0

A bit of recreational mathematics. I’d be (pleasantly) shocked if this is novel.

2+2 = 2×2 = 2² = 4 🤔

Hᵤ is the set of hyperoperations, e.g.
H₁(a, b) = a + b
H₂(a, b) = a × b

H_u(a, b) = H_u-1(a, H_u(a, b-1)) when u≥3 & b≠0
∴ H_u(2, 2) = H_u-1(2, H_u(2, 1)) when u≥3

also H_u(2, 1) = H_u-1(2, H_u(2, 0)) when u≥3

also H_u(2, 0) = 1 when u≥3

∴ H_u(2, 1) = H_u-1(2, 1) when u≥3
∴ H₃(2, 1) = H₂(2, 1) = 2
∴ H₃(2, 2) = H₂(2, 2) ⇒ 2² = 2 × 2 = 4

also H_u(2, 1) = 2 when u≥3
∴ H_u(2, 2) = H_u-1(2, 2) when u≥3
∴ ∀ u ≥ 3 H_u(2, 2) = 4

as H₁(2, 2) = 4 and H₂(2, 2) = 4, ∀ u ≥ 1 H_u(2, 2) = 4

i.e., for any hyperoperation ∘ other than the zeroth (successor) operator, 2 ∘ 2 = 4

Trek head-cannon

In the real world, the “vaporise” setting in SciFi ray-guns comes from a desire to make extras disappear quickly when their characters are killed off.

As countless of pedants have noticed, a real-life weapon which vaporised a target would have all sorts of unpleasant side-effects, from the merely icky of inhaling your enemies to the potentially fatal of suddenly adding 80 cubic-meter-STP of (lethally hot) gas to your room in less than a second. There are also, shall we say, artistically convenient behaviours such as one scene in Star Trek VI where a pan is vaporised while the mashed potato inside is untouched.

None of these things ought to spoil your experience of Star Trek or similar — they’re character-driven space soaps, not hard SciFi. That doesn’t mean it’s not fun to invent some plausible-sounding rationale for how it fits into the in-universe technobabble.

So, phasers, with a digression via subspace.

In the Star Trek universe, “subspace” is treated as an extra dimension — not up-down, not left-right, not front-back, not future-past. If you’re “deep” in subspace, you’re “further” from the reality we know about. How this works precisely is never described, so let’s pretend that there are a bunch of different stable layers that matter can occupy, one for normal space and one more for each different integer warp speed; and lets assume that in the absence of anything pushing you between layers, you just stay on your current layer.

What if the “rapid nadion effect” is a nudge in the subspace-realspace direction?

All the atoms in your body are connected by relatively strong interatomic forces, while the interatomic forces between different objects are much weaker (not zero, but much much weaker). Let’s say you’re hit by a beam which nudges you in the direction of subspace: if its a weak push, the atoms hit by the beam are briefly a little bit outside normal space, but they rapidly return. This effect propagates through your body in exactly the same way that a sound wave would — each out-of-place atom drags nearby atoms with it, but they’re quickly restored to their original place. This could stun you or kill you, depending on how much it interferes with the chemistry that keeps you alive, in much the same way that a punch or a grenade both send waves through your body yet have very different impacts on your life expectancy.

If this nudge is strong enough to push your body to the next subspace layer, the part of your body first hit by the beam will seem to disappear entirely, without the inconvenience of exploding! All the atoms bound to the nudged-into-subspace patch of flesh will be dragged with it onto the next subspace layer, which does not have any air. If this happens at the speed of sound in water (5336 km/h), a 2m humanoid hit in the middle would disappear completely in about 0.7 ms. The effect you see on screen is a far more prosaic 8-15 km/h — again, don’t worry to much about that: Trek has very little in the way of scale or time consistency, but even if it did you should pretend it’s a dramatic slow-mo.

There’s no narrative requirement for subspace layers to be limited to three spacial dimensions, so we can also posit that subspace is (e.g.) 4 spacial dimensions. In 4D, a creature like us built in 3D space would fall to pieces in much the same way as you might expect a creature built out of a single layer of atoms sandwiched between two plates would almost instantly disintegrate if you took the plates away. One idea which would allow “being pushed into subspace” to be much more dangerous than “being in a starship when the warp drive is switched on” would be another fairly ambiguous piece of Star Trek tech: the inertial dampeners. The inertial dampeners are supposed to be space-filling forcefields which push every atom in your body at the same rate the ship accelerates so that you don’t feel any G-forces — vitally important when you go from zero to 0.25c (74,770 km/s, full impulse) in one second. Those very same force fields could (with enough technobabble) keep the crew from disintegrating even if they were in a 4D (or 5D or 6D or…) subspace domain.

Post-scarcity

There are many different ways to discuss “post-scarcity”.

The traditional idea is that all material goods are available at no cost, kinda like the replicators in Star Treks TNG and DS9. However, even in the Trek universe, replicators used power, and this allowed replicator rationing to be a plot point in Star Trek Voyager.

Even without a magic Santa Claus machine, you could say post-scarcity happens per-resource and per-location, rather than as a single one-time-covers-everything event. I would argue that Switzerland is post-scarcity for water because it’s available for free in public fountains throughout the country.

By the measure “does it have second-hand value?”, the G7 is post-scarcity for biros and paper, because nobody keeps track of which biro belongs to who or cares if someone steals a pen or a sheet of photocopier paper.

You could even say the G7 is post-scarcity for cups, because you can’t give them away (I’ve tried) — you only pay money for cups because you want that one in particular or you can’t be bothered collecting the free ones other people are throwing out; likewise, the G7 is post-scarcity for hairbands because there are enough clean ones lying on the street you never need to buy them (that observation courtesy of the ridiculous degree of penny-pinching thriftiness which I inherited from my father).

There is at least one more category: things which we have so much of that we harm ourselves by having it. Artificial light — light pollution is a a thing; Food — obesity and conditions associated with it cause 14% of premature deaths in Europe; Communications — spam, personalised propaganda, attention economics.

I wonder what the world would look like if we all had too much of the very things we still strive for precisely because they are not attainable. What could “too much room” in our houses even mean? How could we “travel too much” or “learn too many things”?

Hypocrisy

Hypocrisy confuses me.

How can someone someone be angry to be called a “TERF” (Trans Exclusionary Radical Feminist), how can they denounce it an anti-feminist insult, when in the next breath deny the existence of Trans-* people? That’s what the acronym means.

How can someone claim that it’s justified for a police officer to shoot dead a child holding a toy gun in an open-carry state where it is not only legal to be walking around with a real gun, but the freedom to be so armed is considered a right worth fighting for specifically to protect against a government that will oppress you?

How can someone denounce one politician for having a multi-million-pound patch of land while they boast about another politician also having multiple millions in property? Or similar with different people for getting close to breaking quarantine rules.

Why do these people call me an “ex-pat” for leaving their country, but denounce anyone arriving as being an “economic migrant”?

Why is it that some will cheer when one person who donated a lot of money to charity and got a knighthood as a result has his statue and name removed from public use when his crimes came to light, yet boo and hiss when the statue of someone who killed tens of thousands is pulled down?

Then there’s the way one presidential candidate was criticised because her husband committed adultery, but the other candidate personally committed adultery and that was apparently fine?

It’s not just right-wing talking points.

I’ve watched people condemn Tesla as if it were unadulterated evil while being adamant that the only reason electric cars aren’t more popular is that billionaires exist.

I’ve seen people scoff at the idea that the UK might not be unified behind Brexit while simultaneously being exasperated that not everyone in the UK is unified behind Brexit. (Brexit isn’t a left-right split, even though quite a lot of loud-mouthed right-wing people think any dissent has to be from the left).

And I’ve seen supporters of the UK Labour party’s previous leader, Jeremy Corbyn, insist that Corbyn was doing better than Blair despite the fact that Blair won elections and Corbyn did not.

I know my left-wing list is shorter than my right-wing list. I can’t tell if that’s me, or the media I consume, or if it’s a difference between the typical supporters of left-wing and right-wing politics. (I certainly doubt the leadership are different, but the supporters must be).

Mistakes

It is important to keep track of one’s mistakes — you can’t learn to be better if you don’t.

Here is one of mine from 2016:

“Trump and Clinton are both equally awful”.

Ye gods, how I wish that was so. I saw each as just two more in the same mould as all other American politicians: a rich narcissist, out of touch with the lived reality of the average person.

Here’s another, also from 2016 — my reply to a blogpost asking various questions about Brexit, with my 2020 annotations as emphasised text:

—

My expectations:

The UK votes to leave, Cameron resigns. One half of the population hates the other half. Redacted person name very happy. Correct.

France very relieved, they didn’t much like us anyway. Comme ci, comme ça.

France tries to set up a financial centre to attract all the business currently in the City of London; mostly this fails and they go to Zurich, Luxembourg, Geneva and Frankfurt in that order. Eh, close enough.

House prices collapse as Russian oligarchs move to New York, Frankfurt (or Zurich, if they are able to buy property there). Nope!

German citizens regard us as kids who threw our toys out of the pram. Politicians treat us accordingly. Correct.

Politicians, by necessity, have a number of sociopathic traits. Therefore they take a brutal approach to us in order to discourage other nations from leaving. (Our politicians are no better, and would be as obtuse as possible to Scotland if Scotland left the UK!). Greece is terrified, stays where it is. I’d say ‘wrong’, but it is notable that the most vocal Leavers say this is exactly what has happened.

We remain in NATO, Interrail. We probably don’t stay in EUHIC. NATO yes, left Interrail, will leave EUHIC at the end of the transition period, so 2/3 correct.

Businesses campaign to keep our standards in sync with EU standards to keep their costs down. Hard to tell. I keep hearing that Businesses are afraid to rock the boat, but hate everything that’s going on. How true that is, I do not know.

After 2 years, we leave EU. We stop paying into the CAP, receive no rebate. UK food producers upset their goods no longer given “protected region” status (or whatever the name is) in the EU. Two years? Incorrect. CAP/rebate? Correct. Geographical Indications? Looks like the UK is keeping them, despite some news stories saying otherwise. Either that or the UK government is giving incorrect advice, a possibility which I only even mention because of how it is acting throughout the Corona virus pandemic. 1/3.

Welsh economy collapses further as it no longer receives money from EU as a “severely deprived region”. Hard to tell, what with the Corona virus being a much bigger problem for all the economies everywhere.

British Islamic fundamentalist terrorists find it more difficult to reach EU countries, commit acts in UK instead. Again, hard to tell because of the Corona virus.

Britain replaces Human Rights Act with something that doesn’t mention the right to life, the right to privacy, the right to free association, or the right to trade unions. I still think this will happen, but after the transition period is over.

Commonwealth states further away than EU, not as rich. This makes for less effective trade. I still think this is the case, but no way to tell until after the transition period is over.

Fish stocks replenish as Spanish no longer allowed to fish in UK waters. This is enforced with at least one gunboat, leading to Spanish newspapers calling for the Spanish government to kick out all the Brits living in Spain. The Spanish government debates this, to the surprise of nobody except the English (on the one hand they no longer need to allow us, on the other the expats might be the only ones preventing further house price collapse). I am surprised to find I wrote this, so count this as “false” even if it later comes true — nobody deserves points for predicting a coin toss will be “either heads or tails”.

All those “twinned with” signs disappear, from a combination of vandalism and a lack of will to replace them. I think this is plausible, but less than 50% chance that more than a handful will be so affected in the next 12 months.

Meanwhile, rapid automation messes around with every economy at once. This is blamed on Brexit, despite having nothing to do with it. (Alternative: we remain, the economic mess from automation is blamed on the EU by the UK and the UK by France). Corona virus doesn’t count. Check again around 2025.

—

One of several reasons I no longer talk to the person I was replying to back then, is that their response to this was:

“Given your general stance seems to be that everything is doomed whatever anybody does, it’s hard to take seriously your claim that one course of action will doom us more than any other. :-p”

To believe falsely

“If there were a verb meaning ‘to believe falsely’, it would not have any
significant first person, present indicative.”

—Ludwig Wittgenstein

Would “confusion” not be such a first person, present indicative? Or am I confused about the meaning of the Wittgenstein quote?

(Update a few days later: Turns out I forgot the phrase “cognitive dissonance”, which is much better).

Ohne Kaffee

As anyone who has spent any significant time with me will know, I have a genuinely problematic relationship with coffee. If I don’t watch myself I can drink nothing but coffee all day, sometimes even double strength — 4 cups is one thing, 2 litres of double strength is too much.

Because of this, every so often I try to cut it out entirely. This time, I’m trying to keep a diary.

Day 1, Saturday

A bit tired, nothing special.

Day 2, Sunday

Really tired, bad headache, took ibuprofen.

Day 3, Monday

A bit tired, occasionally tempted to grab a coffee from the machine at work.

Day 4, Tuesday

Working from home along with everyone else because of Coronavirus. A bit tired, very unfocused. Not sure if the loss of focus is from the switch to home office or from the withdrawal. Definitely feeling like a coffee would help.

Leg ache developed in the evening, feeling like I’d run 5 km.

Day 5, Wednesday

Arm and leg muscles ache.

Day 6, Thursday

No issues.

Sufficient technology

Let’s hypothesise sufficient brain scans. As far as I know, we don’t have better than either very low resolution full-brain imaging (millions of synapses per voxel), or very limited high resolution imaging (thousands of synapses total), at least not for living brains. Let’s just pretend for the sake of argument that we have synapse-resolution full-brain scans of living subjects.

What are the implications?

• Is a backup of your mind protected by the right to avoid self-incrimination? What about the minds of your pets?
• Does a backup need to be punished (e.g. prison) if the person it is made from is punished? What if the offence occurred after the backup was made?
• If the mind state is running rather than offline cold-storage, how many votes do all the copies get? What if they’re allowed to diverge? Which of them is allowed to access the bank accounts or other assets of the original? Is the original entitled to money earned by the copies?
• If you memorise something and then get backed up, is that copyright infringement?
• If a mind can run on silicon for less than the cost of food to keep a human healthy, can anyone other than the foremost mind in their respective field ever be employed?
• If someone is backed up then the original is killed by someone who knows the person was backed up, is that murder, or is it the equivalent of a serious assault that causes a small duration of amnesia?

Hypothesise first, test later

Brought to you by me noticing that when I watch Kristen Bell playing an awful person in The Good Place, I feel as stressed as when I have to consciously translate to or from German in real-life situations and not just language apps.

Idea:

• System 2 thinking (effortful) is stressful to the mind in the same way that mild exercise is stressful to the body.
• Having to think in system 2 continuously is sometimes possible, but how long for is not a universal constant.
• Social interactions are smoothed by being able to imagine what other people are thinking.
• If two minds think in different ways, one or both has to use system 2 thinking to forecast the other. Autistic and neurotypical minds are one of many possible examples in the human realm. Cats and dogs are a common non-human example (“Play! Bark!” “Arg, big scary predator! Hiss!”)
• Stress makes personal touches and eye contact unpleasant.

Implications:

Autistic people will be:

• Much less stressed when they are only around other autistic people.
• Look each other in the eye.
• Be comfortable hugging each other.
• Less likely than allistic people to watch soap operas.
• Jokes based on mind state will not transfer, but puns will. (The opposite of the German-English joke barrier, as puns don’t translate but The Ministry Of Silly Walks does).
• Reality shows will make no sense, and be somewhere between comedic and confusing in the same way as memes based on “type XYZ and let autocomplete finish the sentence!”

Questions:

• How to interests, e.g. sports, music, painting, fit into this?
• Does “gender” work like this? Memes such as “Men are from Mars, women are from Venus” or men finding women confusing and unpredictable come to mind. Also, men’s clubs are a thing, as are women’s. Also the existence of transgender people would pattern-match here. That said, it might just be a cultural barrier, because any group can have a culture and culture is not merely a synonym for political geography.

The worst form of government except for all the others

(Does this sound fair? I’m not formally qualified in politics).

Democracy

The only way to get a government which wants to do the sort of things the public are OK with.

Technocracy

The only way to get a government of people who know what they’re talking about.

Industrialism

The only way to provide a government with the capability to do things.

(Not “capitalism” in general, industry in particular).

Diplomacy

The only way to provide a government with awareness that other nations can have their own desires and goals which differ from it’s own.

Journalism/Police

The only way to fight corruption.