I noticed years ago too that AI doomers and rationalist type were very prone to (infinity * 0 = infinity) types of traps, which is a fairly autistic way of thinking. Humanity long time ago decided that infinity * 0 = 0 for very good practical reasons.
> Humanity long time ago decided that infinity * 0 = 0
I'm guessing you don't mean this in any formal mathematical sense, without context, infinity multiplied by zero isn't formally defined. There could be various formulations and contexts where you could define / calculate something like infinity * zero to evaluate to whatever you want. (e.g. define f(x) := C x and g(x) := 1/x, What does f(x) * g(x) evaluate to in the limit as x goes to infinity? C. And we can interpret f(x) as going to infinity while g(x) goes to zero, so we can use that to justify writing "infinity * 0 = C" for an arbitrary C... )
So, what do you mean by "infinity * 0 = infinity" informally? That humans regard the expected value of (arbitrarily large impact) * (arbitrarily small probability) as zero?
It's true in the informal sense. Normal people, when considering an "infinitely" bad thing happening (being killed, losing their home, etc) with a very low probability will round that probability to zero ("It won't happen to ME"), multiply the two and resultantly spend zero time worrying about it, planning for it, etc.
For instance, a serial killer could kill me (infinitely bad outcome) but the chance of that happening is so tiny I treat it as zero, and so when I leave my house every day I don't look into the bushes for a psycho murderer waiting there for me, I don't wear body armor, I am unarmed, I don't even think about the chance of being killed by a serial killer. For all practical intents and purposes I treat that possibility as zero.
Important to remember that different people gave different thresholds at which they round to zero. Some people run through dark parking garages and jump into their car because they don't round the risk of a killer under their car slashing their achilles tendons down to zero. Some people carry a gun everywhere they go, because they don't round the risk of encountering a mass shooter to zero. Some people invest their time and money pursuing spaceflight development because they don't round a dino-killing asteroid to zero. A lot of people don't round the chance of wrecking a motorcycle to zero, and therefore don't buy one even though they look like fun.
The lesswrong/rationalist people have a tendency to have very low thresholds at which they'll start to round to zero, at least when the potential harm would be met out to a large portion of humanity. Their unusually low threshold leads them to very unusual conclusions. They take seriously possibilities which most people consider to be essentially zero, giving rise to the perception that rationalists don't think that infinity * 0 = 0.
> It's true in the informal sense. Normal people, when considering an "infinitely" bad thing happening (being killed, losing their home, etc) with a very low probability will round that probability to zero ("It won't happen to ME"), multiply the two and resultantly spend zero time worrying about it, planning for it, etc.
Is this the kind of thing that is part of the Less Wrong cult? I see this "multiply" word being used which I understand is part of the religious technology of LW. It all seems very sophomoric. I don't know what talking about "Infinity * 0" means in an informal sense means. What I can tell you is that "Normal people" are not multiplying "infinitely bad" with a "very low probability rounded to 0". For one, this is conflating multiple senses of infinite. I'm not sure anyone thinks likes bad outcomes are "infinitely bad", maybe in a schoolyard silly-talk kind of way, they just think it is bad. I think that's basically what Less Wrong is, a lot of fancy words and Internet memes and loose-talk about AI all strewn together in a "goth for adults" or some other kind of nerd social club.
> "I don't know what talking about "Infinity * 0" means in an informal sense means"
I'm not a rationalist, I'm only using their language to make the mapping to their ideology simpler. A comet striking earth would be "infinitely bad". The chance of that happening is, as far as I'm concerned, zero (its not zero, but I round it down.) If you multiply the infinitely bad outcome by the zero percent chance of it happening, you result is that you shouldn't waste your time and emotional resources worrying about it.
Normal people don't phrase this kind of reasoning with math terminology as rationalists do, but that terminology isn't where the rationalists go wrong. Where the rationalists go wrong isn't the multiplication, it's the failure to ignore very unlikely outcomes as normal people would. They think themselves too rational to ignore the possibility of unlikely things, but ironically it is normal people who don't spend their time dwelling on extremely unlikely bullshit have a more rational approach to life.
The rationalists spend hours discussing scenarios like "What if a super AI manipulates people into engineering a super virus that wipes out humanity? Its technically possible; there's no law of physics which prevents this!", to which a normal person would respond by wondering if these people are on drugs, why would they spend so much time worrying about something which isn't going to happen?
Yeah pretty much. If I was to write it out further: "near infinity bad thing could happen but it has a near infinitesimal chance of it happening, what is the amount of finite resources you should spend to prevent it?". The numbers are probabilities and how much of an effect it is. It really is infinity * epsilon but that would confuse more people so I decided to say infinity * 0.
I was very explicit when I said "humanity decided". It doesn't matter if one or the other is the actual formal system math system result either way, it was chosen out of practicality that in this kind of philosophical issue, the more pragmatic thing was to axiomatically choose that "infinity * 0 = 0" when faced with things like this. The rationalists in a more meta/broader sense have decided that it's infinity * epsilon = infinity even if they say it is not on the surface. Their actions show they believe the other direction.
In math infinity * epsilon is indeterminate until you decide what the details of infinity & epsilon is, which I find quite fitting.
> That humans regard the expected value of (arbitrarily large impact) * (arbitrarily small probability) as zero?
There are many arguments that go something like this: We don't know the probability of <extinction-level event>, but because it is considered a maximally bad outcome, any means to prevent it are justified. You will see these types of arguments made to justify radical measures against climate change or AI research, but also in favor space colonization.
These types of arguments are "not even wrong", they can't be mathematically rigorous, because all terms in that equation are undefined, even if you move away from infinities. The nod to mathematics is purely for aesthetics.
not exactly a rationalist thing, but a lot of bay-area people will tell you that exponential growth exists, and it's everywhere
i can't think of any case where exponential growth actually happens, though. exponential decay and logistic curves are common enough, but not exponential growth
The rats I hang out with know the difference between exponential and logistic just fine.
Hmm.
Not sure if it matters, but I'd note logistic curves can be hard to distinguish from an exponential for long enough that the difference isn't always very consequential — a nuke exploding doesn't keep doubling in power every few microseconds forever, but for enough doubling periods that cities still get flattened.
They actively look for ways for infinity to happen. Look at Eli's irate response to Roko's basilisk. To him even being able to imagine that there is a trap means that it will necessarily be realised.
I've seen "rationalist" AI doomers who say things like "given enough time technology will be invented to teleport you into the future where you'll be horifically tortured forever".
It's just extrapolation, taken to the extreme, and believed in totally religiously.
> Humanity long time ago decided that infinity * 0 = 0 for very good practical reasons.
Among them being that ∞ × 0 = ∞ makes no mathematical sense. Multiplying literally any other number by zero results in zero. I see no reason to believe that infinity (positive or negative) would be some exception; infinity instances of nothing is still nothing.
The problem is that infinity is neither a real nor a complex number, nor an element of any algebraic field, and the proposition that "x * 0 = 0" only holds if x is an element of some algebraic field. It is a theorem that depends on the field axioms.
The real numbers can be extended to include two special elements ∞ and -∞, but this extension does not constitute a field, and the range of expressions in which these symbols make sense is very strictly and narrowly defined (see Rudin's PMA, Definition 1.23):
(a) If x is real then
x + ∞ = +∞, x - ∞ = -∞, x / +∞ = x / -∞ = 0.
(b) If x > 0 then x * (+∞) = +∞, x * (-∞) = -∞.
(c) If x < 0 then x * (+∞) = -∞, x * (-∞) = +∞.
The extended real number system is most commonly used when dealing with limits of sequences, where you may also see such symbols appear:
3.15 Definition Let {sₙ} be a sequence of real numbers with the following property: For every real M there is an integer N such that n ≥ N implies sₙ ≥ M. We then write
sₙ ⟶ +∞.
In no other contexts do the symbols ∞ and -∞ make any sense. They only make sense according to the definitions given.
It's usually the case that when you see people discussing infinity that they are actually talking about sequences of numbers that are unbounded above (or below). The expression "sₙ ⟶ +∞" is meant to denote such a sequence, and the definitions that extend the real number line (as in Definition 1.23 above) are used to do some higher-level algebra on limits of sums and products of sequences (e.g. the limit of sₙ + tₙ as n becomes "very large" for two sequences {sₙ}, {tₙ}) to shortcut around the lower-level formalisms of epsilons and neighborhoods of limit points in some metric space, which is how the limits of sequences are rigorously defined.
In no case do the symbols ∞ and -∞ refer to actual numbers. They are used in expressions that refer to properties of certain sequences once you look far enough down the sequence, past its first, second, hundredth, umpteenth, "Nth" terms, and so on.
Thus when you see people informally and loosely use expressions such as "infinity times zero" they're not actually multiplying two numbers together, but rather talking about the behavior of the product of two sequences as you evaluate terms further down both sequences; one of which is unbounded, while the other can be brought arbitrarily close to (but not necessarily equal to) zero. You will notice that no conclusions can be drawn regarding the behavior of such a product in general, whether referencing the definitions comprising the extended real number system or the lower-level definitions in terms of epsilons and neighborhoods of limit points.
So much confusion today comes down to people confidently using words, symbols, and signs they don't understand the definitions nor meanings of. Sometimes I wonder if this is the real esoteric meaning of the ancient Tower of Babel mythos.
Infinity doesn't need to be in some "algebraic field" for it to be patently true that an infinite amount of nothing is still nothing, and that adding zero to itself over and over again for an infinitely long time will never give you a result other than zero. It's only impossible to define if you overthink it, and/or maintain a needlessly narrow definition of what a "number" is.
Or, if you really insist on speaking in mathematician-ese, an infinite series of zero is zero, and a zero-bounded summation is zero regardless of the summand:
x · y ≡ Σ(y, i = 1) x = y times { x + x + … + x } ≡ Σ(x,i=1) y = x times { y + y + … + y }
julia> i = 0
0
julia> while true
println(i)
global i += 0
end
0
0
0
0
0
0
0
0
0
...and on and on until the heat death of the universe or you hit Ctrl-C.
Either way, seems pretty straightforward to define if you have a clear definition of what multiplication is in the first place (and what either zero or infinite iterations of that definition will produce).
Ok, let's assume you are correct and that ∞ · 0 = 0. Consider then the two sequences sₙ = n, tₙ = 1/n.
By Definition 3.15 as provided in my last post, sₙ ⟶ +∞, and you will have to take it for granted that tₙ ⟶ 0 [0]. Intuitively we can see that the terms of {sₙ} are 1, 2, 3, ... tending to +∞; for {tₙ} we have 1, 1/2, 1/3, ... tending to zero, for progressively larger values of n.
Now I ask what happens if we multiply the "infinite'th" terms of both sequences together. The first few terms of this product would be 1 · 1, 2 · 1/2, 3 · 1/3, and so on; I ask what the value x is in the limit sₙ · tₙ ⟶ x as we evaluate further and further "nth" terms of both sequences.
You may have observed from the first three terms evaluated that sₙ · tₙ = n(1/n) = 1. Thus, as we continue to increase the value of n, it's always the case that sₙ · tₙ ⟶ 1 and the product tends to 1, because the product is constant and irrespective of n; we've "cancelled it out."
The limit of the product is the product of the limits [1]; that is, sₙ · tₙ ⟶ +∞ · 0, as we first established that sₙ ⟶ +∞, tₙ ⟶ 0.
If we thus take your supposition that +∞ · 0 = 0 for granted, we obtain sₙ · tₙ ⟶ 0, which contradicts our previous result that sₙ · tₙ ⟶ 1.
Thus we can either dispense with the cited established theorems of analysis used to deduce that sₙ · tₙ ⟶ 1, or conclude that the supposition +∞ · 0 = 0 must be false.
It might be the case that Σ(∞, i = 1) 0 = 0, but you can't extend this to conclude +∞ · 0 = 0 in general. Lots of intuitions from informal mathematics and even calculus start to break down once you examine the lower-level "machine code" of proof and analysis, especially once you start talking about concepts like infinity.
> Now I ask what happens if we multiply the "infinite'th" terms of both sequences together.
In that case, you would've reached their respective limits, and you're back to adding one of those limits into itself an other-limit number of times. If sₙ · tₙ ⟶ 1, then that only holds true if tₙ hasn't actually reached 0.
> Thus we can either dispense with the cited established theorems of analysis used to deduce that sₙ · tₙ ⟶ 1
You don't need to do that. You just need to accept that zero is just as much of a mathematical special case as infinity - unsurprisingly, since it's the inverse of infinity and vice versa.
> It might be the case that Σ(∞, i = 1) 0 = 0, but you can't extend this to conclude +∞ · 0 = 0 in general.
Sure you can, unless you've got some other definition of multiplication that's impossible to express as self-summation.
Even if you go with the alternative definition of multiplication as a scaling operation (wherein you're computing m × n by taking the slope from (x=0,y=0) to (x=1,y=m) and then looking up y where x=n), if m is zero then the line being drawn never stops being vertical, and if n is zero then you never leave (0,0) in the first place. Doesn't matter if the other factor is infinitely far in either the x or y axis; you're still ending up with zero no matter how hard you try and fight it.
> Lots of intuitions from informal mathematics and even calculus start to break down once you examine the lower-level "machine code" of proof and analysis, especially once you start talking about concepts like infinity.
Sure, but in this case, it's the intuition that multiplying something by its inverse (a.k.a. dividing something by itself) is always 1 that breaks down, not the above-verifiable and inescapable fact that multiplying something by zero is always zero. 0 ÷ 0 = n looks like it should correct for any value of n (incl. n = 1), since multiplying both sides by zero to eliminate that divide-by-zero will always produce a correct equation, but since m ÷ n ≡ m × (1/n), if m is zero then anything on the RHS must be zero, because of that inescapable nature of nothingness - thus, 0 ÷ 0 = 0 × (1/0) = 0, with all other possible alternatives having been rendered impossible.
> you're back to adding one of those limits into itself an other-limit number of times.
> some other definition of multiplication that's impossible to express as self-summation.
Ok. What happens if I multiply a number by pi? What does it mean to add something to itself, pi times?
> If sₙ · tₙ ⟶ 1, then that only holds true if tₙ hasn't actually reached 0.
I mean... it is in fact the case that tₙ never actually reaches zero; otherwise, if 1/n = 0 for some n, then by multiplying both sides by n we obtain 1 = 0.
What's meant by tₙ ⟶ 0 is that any neighborhood centered about 0 of any radius (call the radius "epsilon") always contains at least one point from the sequence {tₙ}.
To hammer the point that sₙ · tₙ ⟶ 1 home, and since you are fond of using a computer to perform arithmetic (note: not prove mathematical statements), here's what computers have to say about the limit of n · (1/n): https://www.wolframalpha.com/input?i=limit+as+n-%3Einfinity+...
> You just need to accept that zero is just as much of a mathematical special case as infinity - unsurprisingly, since it's the inverse of infinity and vice versa.
> the inverse of infinity
You again throw around words like "inverse" whose meaning you don't understand. Do you mean a multiplicative inverse, where a number and its multiplicative inverse yield the multiplicative identity, in which case +∞ · 0 = 1? Or an additive inverse that yields the additive identity, in which case +∞ + 0 = 0? Or some other pseudomathematical definition of "inverse" pulled out of a hat, like your definitions of +∞ · 0?
> if m is zero then the line being drawn never stops being vertical
Drawing pictures is different from putting together a formal, airtight proof in first-order logic that can be (in principle) machine-verified. Maybe I'll make an exception for compass-and-straightedge proofs, but that's not what you're presenting here.
Rudin was published in 1953, there are probably very good reasons for why this text has withstood refutation for over 70 years. Maybe you can rise to the task; publish a paper with your novel number system in which +∞ · 0 = 0 and 0 ÷ 0 = 0 and wait for your Fields Medal in the mail. Maybe you can collaborate with Terrence Howard and get a spot on Joe Rogan.
> Ok. What happens if I multiply a number by pi? What does it mean to add something to itself, pi times?
You add it to itself 3 times, then shift the decimal point and repeat with 1, then shift the decimal point and repeat with 4, and so on with each digit of π. 1 × π = 1 + 1 + 1 + 0.1 + 0.01 + 0.01 + 0.01 + 0.01 + 0.001 + 0.0001 + 0.0001 + 0.0001 + 0.0001 + 0.0001 and so on forever.
> To hammer the point that sₙ · tₙ ⟶ 1 home
That point doesn't need hammered. sₙ · tₙ ⟶ 1 can absolutely be true when you haven't yet reached zero. That doesn't mean it's true in the event that you do indeed manage to reach zero. It indeed can't be true in the event that you do indeed reach zero, because n × 0 = 0 for all values of n.
> Do you mean a multiplicative inverse, where a number and its multiplicative inverse yield the multiplicative identity, in which case +∞ · 0 = 1?
You obviously already know that's what I meant, since that's exactly what I described further down - including how ∞ × 0 ≠ 1 because the multiplicative identity breaks down when one of the factors is zero, specifically because having zero of something will always produce zero no matter what that something is.
> Or some other pseudomathematical definition of "inverse" pulled out of a hat, like your definitions of +∞ · 0?
If you're seriously calling multiplication-as-summation pseudomathematics, then you're in no position to assess whether or not I "don't understand" the meanings of words.
I've been nothing but civil toward you, and you've been nothing but condescending toward me. That normally wouldn't be a problem (condescension is par for the course on the Internet), but if you're going to be condescending, the least you can do is not be blatantly wrong in the process.
> Drawing pictures is different from putting together a formal, airtight proof in first-order logic that can be (in principle) machine-verified. Maybe I'll make an exception for compass-and-straightedge proofs, but that's not what you're presenting here.
That's exactly what I'm presenting here (since apparently you believe adding numbers together is a spook). You don't even need a concept of numbers to see plain as day that any multiplication wherein one of the factors is zero will always be zero.
> Rudin was published in 1953, there are probably very good reasons for why this text has withstood refutation for over 70 years. Maybe you can rise to the task; publish a paper with your novel number system in which +∞ · 0 = 0 and 0 ÷ 0 = 0 and wait for your Fields Medal in the mail. Maybe you can collaborate with Terrence Howard and get a spot on Joe Rogan.
You know what? Maybe I will. And I'm willing to bet you'll find some other pedantic reason to be a condescending prick when that happens.
Last word's yours if you want it. I have better things to do than argue with people engaging in bad faith.
There's no need for me to continue engaging you with formal mathematical arguments when you reply with the mathematical equivalent of climate change denialism or vaccine conspiracy theory and uneducated statements that are "not even wrong" [0], so instead I will just refer you to expert opinions on the topic; though at this point I doubt that your level of mathematical literacy is sufficient to understand any of this subject matter.
