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[malloc]

It seems rather ironic that dogs cannot eat puppy chow because it contains chocolate.

[Raphael]

It seems rather ironic that dogs cannot eat puppy chow because it contains chocolate.

Agreed.

[malloc]

It's already September and the sun has set and yet it's still 90°F / 32°C outside. Global warming has definitely arrived.

[zompist]

Bret Devereaux is on a roll. This primer on naval power is amazing; it'll explain things you didn't know needed explaining.

Why the US is choosing to blow up global trade, which makes itself and the whole world richer, is hard to say, but of course by now we expect stupidity from the Trumpists. His other piece on why fascists are bad at war is relevant.

But what may be news to many, especially on the left, is why global trade is important and beneficial to everyone else. E.g. anyone who thinks "America bad, fall of America good" should be discomfited to realize that all but two countries in Africa import food. Less global trade may be an inconvenience in the US (more expensive fuel, less out-of-season fruit), but in Africa it means people starving to death.

The other main lesson: no one is going to step up to replace American naval power. It is physically impossible. China cannot do it; Europe cannot do it; no one else is even close. Naval power is spectacularly expensive. European nations have only token navies, and China will take decades to reach a similar power if it even wants to.

[linguistcat]

It's already September and the sun has set and yet it's still 90°F / 32°C outside. Global warming has definitely arrived.

Oh mood. During the summer, I was seeing temps that I was used to in SoCal back in the 90s. Now, in Northern Utah and Idaho.

[Raphael]

Going back to this from a couple weeks ago...

Someone named Elizabeth Nolan Brown, who's an editor at the libertarian rag Reason, posted on social media:

My 4-year/old just learned about the concept of infinity and is now throwing a tantrum over it. “It’s weird! I don’t like it. I just want numbers to stop!”

https://bsky.app/profile/enbrown.bsky.s ... euj52eb22s

...would it ever, under any circumstances, lead to any kind of practical application problems to simply declare, let's say, a googolplex to be the officially highest number, and then go on pretending that there aren't any higher numbers than that?

[Travis B.]

Going back to this from a couple weeks ago...

Someone named Elizabeth Nolan Brown, who's an editor at the libertarian rag Reason, posted on social media:

My 4-year/old just learned about the concept of infinity and is now throwing a tantrum over it. “It’s weird! I don’t like it. I just want numbers to stop!”

https://bsky.app/profile/enbrown.bsky.s ... euj52eb22s

...would it ever, under any circumstances, lead to any kind of practical application problems to simply declare, let's say, a googolplex to be the officially highest number, and then go on pretending that there aren't any higher numbers than that?

The problem with ultrafinitism is that it leads to not being able to use useful tools such as limits which are based on the concept of infinity. Yes, there are in many cases ways around the resulting problems, but they are essentially self-inflicted in nature.

[Richard W]

...would it ever, under any circumstances, lead to any kind of practical application problems to simply declare, let's say, a googolplex to be the officially highest number, and then go on pretending that there aren't any higher numbers than that?

Well, as Travis B implies, it would lead to us not knowing what we were talking about for calculus. However, we could just accept its results as axioms, and then I can't think of any problems that would result.

[zompist]

You'd also lose the concept of irrational numbers, as well as the concept of a continuum. You couldn't say that 1/3 = 0.3̅, or that 0.9̅ = 1. Most of mathematics would have to be redone.

Zeno's Paradox would have to be reopened.

I suspect, though I'm not sure, that relativity would be broken too. E.g. accelerating a particle (or anything else) only approaches the speed of light, it never reaches it. The googolplex idea means a particle has a maximum speed less than that of light: what happens if you try to accelerate it more? You have one principle saying one thing (acceleration makes things go faster) and one saying another (there is a maximum integer).

(I hope it's clear that if there is no infinity, there are no infinitesimals. The interval between 0 and 1 could not be divided infinitely, but only into a googleplex of divisions. Somehow, those divisions can't be divided in half. There is no longer a way to relate fractions to decimal notation, without heresy,)

Note that there are ancient proofs that (say) √2 is not a rational number. That proof doesn't disappear just because of an ideological bias against infinities.

A googolplex is a pretty big number ^{[citation needed]}. So I can't think of a physical process that would be mismeasured, or whatever. But breaking math and physics is a practical problem.

[Raphael]

Thank you, zompist, Richard, and Travis!

This reminds of one of the few things I remember from the one semester I unsuccessfully tried to study math. That it, the basics of the concept of computational numbers. The three of you certainly already know that, but for anyone else reading this: turns out every computer has only a limited number of numbers with which it can do calculations. A really high limited number of numbers, but still limited. That was quite an illusion-shattering thing for me to learn back then.

[zompist]

This reminds of one of the few things I remember from the one semester I unsuccessfully tried to study math. That it, the basics of the concept of computational numbers. The three of you certainly already know that, but for anyone else reading this: turns out every computer has only a limited number of numbers with which it can do calculations. A really high limited number of numbers, but still limited. That was quite an illusion-shattering thing for me to learn back then.

Not quite. Built-in numbers are inherently limited, especially floats (used to represent non-integers). But you can use arbitrary precision arithmetic. Think of a representation that uses one ordinary byte (16 bits) for every digit of a rational number. The representation is not hard: use an array if you know the maximum size you'll be working with, otherwise an extensible structure. Writing all the arithmetic operations is harder but not too hard.

You're still limited, but only by the size of your hard drive.

[Raphael]

This reminds of one of the few things I remember from the one semester I unsuccessfully tried to study math. That it, the basics of the concept of computational numbers. The three of you certainly already know that, but for anyone else reading this: turns out every computer has only a limited number of numbers with which it can do calculations. A really high limited number of numbers, but still limited. That was quite an illusion-shattering thing for me to learn back then.

Not quite. Built-in numbers are inherently limited, especially floats (used to represent non-integers). But you can use arbitrary precision arithmetic. Think of a representation that uses one ordinary byte (16 bits) for every digit of a rational number. The representation is not hard: use an array if you know the maximum size you'll be working with, otherwise an extensible structure. Writing all the arithmetic operations is harder but not too hard.

You're still limited, but only by the size of your hard drive.

Ah, thank you!

[Richard W]

You couldn't say that 1/3 = 0.3̅, or that 0.9̅ = 1.

Recurring decimals are simply a notation for the addition of a multiple of the reciprocal of 10^m × 10ⁿ⁻¹, so
0.3̅ is 3∕9, 0.9̅ is 9∕9 and 0.16̄ is 0.1 + 6∕90.

[Richard W]

I suspect, though I'm not sure, that relativity would be broken too. E.g. accelerating a particle (or anything else) only approaches the speed of light, it never reaches it. The googolplex idea means a particle has a maximum speed less than that of light: what happens if you try to accelerate it more? You have one principle saying one thing (acceleration makes things go faster) and one saying another (there is a maximum integer).

I wouldn't be surprised if as yet unknown quantum effects invalidated relativity here.

[Raphael]

About two years after giving up, I've now decided that I want to try another stab at getting somewhere with my SF novel.

Now, part of the problem back then had been that while I had a fairly clear general idea of what the novel should be about, who the most important characters should be, and what should happen at the most basic level, I had found it extremely difficult, ok, what I mean is, impossible, to think of more specific things that could happen during the plot, especially during the early parts of the plot.

So now, in my desperation, I've turned to the "Random entry" function on wiktionary.org, in the hope of finding inspiration. What it has given me so far is, after translating and cleaning things up a bit:

shame

district prosecutor's office

talented --- --- talentless

to air or ventilate

date of birth

humble pie

coffee break

to paint or describe

to run away or escape

to chant

strength or type of oak

to break

guanaco

to alliterate

belonging to or living in mud

wolf cub

diplomatic

to saddle


No idea what to do with that, I'm afraid.

(No, there will be no intentional "AI" stuff anywhere in the novel, if there ever is a novel.)

[malloc]

(No, there will be no intentional "AI" stuff anywhere in the novel, if there ever is a novel.)

Really? It seems hard to write science fiction these days without at least touching on that in some way.

[Raphael]

(No, there will be no intentional "AI" stuff anywhere in the novel, if there ever is a novel.)

Really? It seems hard to write science fiction these days without at least touching on that in some way.

You might have misunderstood me. My point is that I don't plan to intentionally use any "AI" in the making of the novel. What role fictional future "AI" or even AI might play in the plot is a different question.

In any case, I've tried to talk about something that's a bit of a practical problem for me right now, and would prefer it if you would pursue your hobby elsewhere.

[malloc]

Apologies for laboring this point. I realize it must sound incredibly tiresome by now. Given the stakes though, I feel obliged to say something and resist however possible.

[alice]

You're still limited, but only by the size of your hard drive.

WARNING: possible double-entendre detected

[zompist]

You couldn't say that 1/3 = 0.3̅, or that 0.9̅ = 1.

Recurring decimals are simply a notation for the addition of a multiple of the reciprocal of 10^m × 10ⁿ⁻¹, so
0.3̅ is 3∕9, 0.9̅ is 9∕9 and 0.16̄ is 0.1 + 6∕90.

You're not getting Raphael's idea, which is to posit a highest number, set at one googleplex. If there is no infinity, there is no infinite recursion.

1/3 in the current system:

/-------- a googolplex of 3's -------\
.333333333333...33333333333333333333333333....

In the no-infinities system,1/3 is

/-------- a googolplex of 3's -------\
.333333333333...333333333333

These are not the same number.

It's true that Raphael's system would allow a finite number of rational numbers— you can still write and calculate with1/3 etc.

This still leads to problems. The smallest fraction is 1/googolplex of the [0,1] interval. But consider the number n which is 1 less than googolplex. What is 1/n? It's a division of the [0,1] interval but not the same division, and it's impossible to measure 1/n in terms of 1/googolplexes.

We could definite both in terms of 1 / (googolplex * n). But this is heresy in the no-infinity system, since you cannot have numbers above googolplex, and there is no fraction smaller than 1/googolplex.

Basically every mathematical operation fails at some point and has to be limited. E.g. take this algorithm:

Start with n = 1.
Put n into pile A, and n*n into pile B.
Keep going till you get to googolplex.

You cannot fill pile B: you can only get to √googolplex, because higher numbers are heretical. Rather than multiplying two numbers, you can only multiply if the result is non-heretical. (This is rather like simple computer representations, in fact: you get overflow problems.)

One more fun deduction: I said you could only have a finite number of rationals. What is that number? Both numerator and denominator can vary from 1 to googolplex, so the number is googolplex * googolplex which is forbidden.