[oraphalous]

> If you think there are others, please exhibit one.

If everyone like you attempts to rail road the imaginative process at the beginning of hypothesis formation, then we'll never get to the point of being able to exhibit one, should one be possible.

The demand for rigour at this point in a discourse - which was pretty clearly signalled by the commenter to be offered at a stage prior to substantive hypotheis formation - just shuts down the imaginative process. It's not constructive.

[rachofsunshine]

There's a difference between being closed-minded and saying "yes, we've obviously thought about this thing that you, someone with no apparent background in our field, thought of in ten seconds". And if you're an expert in any field that gets a lot of people who are interested, but not a lot of people who are experts, you hear these kinds of half-baked theories all the time, often with this exact "oh you orthodox experts just can't handle my field-disrupting free-thinking!" kind of framing.

I'm a mathematician by education, and I cannot tell you how many people insist on things like 0.999... < 1 without an understanding of (a) what the left side of that expression even means, (b) what a real number is, or (c) what basic properties the real numbers have. Going "no, you're wrong, and it would take me a couple of full lectures to explain why but trust me we're pretty sure about this" is a reasonable answer to that, provided that you have indeed established that to your own satisfaction, at least.

[gryfft]

https://en.m.wikipedia.org/wiki/0.999...

From Wikipedia, an intuitive explanation of an elementary proof:

> If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1. For any number that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than . So, it does not make sense to identify 0.999... with any number smaller than 1. Meanwhile, every number larger than 1 will be larger than any decimal of the form 0.999...9 for any finite number of nines. Therefore, 0.999... cannot be identified with any number larger than 1, either. Because 0.999... cannot be bigger than 1 or smaller than 1, it must equal 1 if it is to be any real number at all.

And then:

> The elementary argument of multiplying 0.333... = 1/3 by 3 can convince reluctant students that 0.999... = 1.

[bawolff]

Sure, but we are talking about a physical theory here not a mathematical one. There are always alternative physical theories, infinitely many. They may not be good theories, but they clearly exist. This has nothing to do with cosmology but more is a fundamental principle of logic.

To put another way, there is a big difference between saying some specific alternative theory is wrong/unlikely/bad, and claiming there exists no alternative theories at all regardless of quality.

[psychoslave]

It's just that 0.999... is an awful notation, in the sense that it invites people to complete these allusive ellipsis with whatever fit their intuition, possibly even different meaning depending on the context.

If we mean 9×10^-i for i from 1 to infinity, then let's just state so. Let's not blame people to interpret towards other direction when they are provided misguiding hints.

Regarding infinity, there is a constructive proof that it doesn't exists which work perfectly provided that there is an infinite amount of resources allocated to it's execution.

[rachofsunshine]

I don't blame people for finding it counterintuitive. Lots of things in math are counterintuitive. I spent like three months learning Galois theory and I'm still pretty sure someone just snuck in there and changed a 1 to a 0 to make that sorcery work.

My point is that it's not closed-minded of me if I fail to provide a complete answer to someone making such a claim, particularly if that person hasn't done any of the research or rigor to handle what is - by the standards of an expert - a pretty easy question to answer. Outsiders can occasionally result in great insights, but they do that through very hard work, not from ten seconds of thinking about a field they haven't learned anything but the pop-science version of.

Most of the theories being speculated about in this thread are veering into "not even wrong" territory, in that they're not even necessarily well-defined. When you're talking about cosmology you'd better bring your general relativity game, which means you better bring your differential geometry game, which means you better have a graduate-level mathematics education. I have a graduate-level mathematics education and on a good day I could half explain to you what a metric tensor is and what the hell it has to do with curvature ("it's, uh, it's kinda like a Jacobian I guess, except the dx and dy are local vectors that can't be translated globally around the space").

Without those tools, you don't even have meaningful notions of what "distance" even is on a cosmological scale, much less how it changes with time! It's like speculating about biology without knowing what a protein is, or speculating about computer science without knowing what a for loop is. It's just not going to get you anywhere.

[hindsightbias]

Like everywhere else in the intertubes, people think their experience domain is relative to every other specialty domain. We live in a conspiracy world now, where RTFM or "do the work" is a micro aggression.

[eszed]

Whoa! What? Not a mathematician in any way (in case that isn't obvious), but I'd have totally thought 0.999... asymptomatically _approaches_ 1, but never reaches it, and so is <1. Is there a short-form explanation (that I might have a chance of understanding, lol) of why that's incorrect? I'd love to have my mind blown.

[techno_tsar]

There's a rigorous proof on Wikipedia, but there's simpler ways to show it.

For example, we know that 1/3 = 0.333...

3 * 1/3 = 3 * 0.333...

1 = 0.999...

You can also do it with subtraction. For example, 1 - 0.999... = x. Assuming x is greater than 0, then it should evaluate to 0.000...1.

But we can't have the digit 1 after an infinite number of zeros. If there truly were a "1" after infinite zeros, it implies reaching the end of infinity, which is a logical contradiction. So x can't be greater than 0.

[atq2119]

An alternative to the sibling comments:

In this context, the notation 0.999... does not represent a process. It represents a fixed number.

Which number? Well, if you reason through it you'll find that it has to be the same number as that represented by the notation 1.

An insight that is crucial (and pretty obvious in hindsight, though many people seem not to be exposed to it) is to distinguish between a number and its representations. "1" is not a number, it is merely the representation of a number. And numbers have many different representations. As a member of this forum, you can probably appreciate that "12" and "0xC" are two different representations of the same number. In the same way, 0.999... and 1 are different representations of the same number.

[evanb]

Sequences can approach things. The sequence 0.9, 0.99, 0.999, 0.9999 and so on asymptotically approaches 1. The difference between 1 and the Nth term in the sequence is 1e-N, which goes to 0 with N.

0.999...[forever] is not a sequence, it is a number. Numbers have values, they don't approach things. The misleading part is that 'forever' is not something about evolution or the passage of time. It's not 'happening' or 'sequential' like the sequence. There is no 'and then another 9'. All the 9s are really there, at once. And it is closer to 1 than any term in the sequence. Since the sequence gets closer and closer to 1, converging to it asymptotically, 0.999...[forever] cannot differ from 1; if it did the sequence wouldn't converge.

[eszed]

Thank you, and everyone else who answered (I hope they see this reply). Your distinction between "sequence" and "number", along with the mathematics of 0.333... = 1/3, convinced me - and my mind is successfully blown.

Follow-up: is it the same for other repeating sequence-looking numbers? As in, would 0.9333... = 0.94?

[eigenket]

Its true of numbers whose with a decimal representation which ends in an infinite string of 9s, so 0.939999... = 0.94. This is because we write numbers in base 10. If you write numbers in base 2 its equal to numbers whose binary representation ends with an infinte number of 1s e.g. 0.11111... (base 2) = 1.

[evanb]

0.9333... is equal to 9/10 + 1/30. To get 9/10 + 4/100 I think what you're aiming at is 0.93999...

[justsomeshmuck]

1/3 + 1/3 + 1/3 = 3/3 = 1

.3… + .3… + .3… = .9… = 1

https://en.m.wikipedia.org/wiki/0.999...

[tsimionescu]

0.xxx... is just a notation for certain fractions (specifically, the fraction x/9). If we set x = 9, then the notation 0.9999... is just a notation for 9/9 = 1. So it's just a silly notation for 1.

[nkrisc]

This actually makes the most sense to me. It’s an artifact of our chosen numerical notation system.

[xypage]

I think the simple one is x=0.999..., 10x=9.9999..., 10x-x=9.999...-0.999..., 9x=9, x=1

[aguaviva]

The simplest argument I can think of is to ask yourself: "Are there any numbers between 0.999... and 1?"

If not then it's logical to conclude they wind up at the same "place", that is, the same number. Or equivalently: it can't have any other value besides 1.

If you care about such things, then you're a mathematician.

[ForOldHack]

It is a limit. A very powerful tool.

You can choose any arbitrary finite number in the sequence, and I can find a number greater. So the value of 1 exists, and its a continuous function, so therefore the limit exists.

I do care about such things, and yes, a degree in math for me was just slogging through every math course the college had, with a long string of high grades, and honor roll achievements.

[floxy]

     1/6 = 0.166666...6
     1/6 = 0.166666...6
     1/6 = 0.166666...6
     1/6 = 0.166666...6
     1/6 = 0.166666...6
  +  1/6 = 0.166666...6
  ----------------------
         = 0.999999...6

...so that 6 times 1/6 is zero point 9 repeating, with a 6 on the end ;-)

[rachofsunshine]

There are a lot of proofs of this, but they all rely on a certain level of rigor about what a real number is - and that, it turns out, is a much more difficult question than it sounds like. You don't typically get a rigorous definition of the real numbers until well into a college-level math education.

-----

First, you're making a category error. "0.9999..." is a single value, not the sequence of values 0.9, 0.99, 0.999, 0.9999... Single values cannot "asymptotically approach" anything, any more than the value 2 or the value 7 can asymptotically approach anything. It's just a number like any other.

To show what value 0.9999... takes on, we need to do two things. First, we need to show that this notation makes sense as a description of a real number in the first place, and second, we need to show what that real number is (and it will happen to be 1).

-----

So, why is it a real number?

Well, remember what we mean by place value. 0.9 means "0 ones, 9 tenths[, and zero hundredths, thousandths, and so on]". 0.99 means "0 ones, 9 tenths, 9 hundredths, [and zero of everything else]". Another way to say this is that 0.9 is the value 0 * 1 + 9 * 0.1 [plus 0 times 0.01, 0.001, and so on], and that 0.99 is the value 0 * 1 + 9 * 0.1 + 9 * 0.01 + [0 of everything else].

What that means is that if 0.9999... means anything, it means 9 tenths, plus 9 hundredths, plus 9 thousandths, plus 9 ten-thousandths, plus 9 hundred-thousandths, and so on and so forth forever. In other words, 0.9999... is the value of an infinite sum: .9 + .09 + .009 + .0009 + ...

Infinite sums, in turn, are by definition the limit of a sequence. This is where that "asymptotic" thing comes back, but notice the distinction. 0.9999... is not the sequence, it is the LIMIT OF the sequence, which has a single value.

To show that it's a real number, then, we need to show that the limit of the sequence 0.9, 0.99, 0.999, 0.9999... does in fact exist. But this sequence is clearly increasing, and it is clearly not greater than 1, so we can (among other things) invoke the Monotone Convergence Theorem [1] to show that it must converge (i.e., the limit exists). Alternately, you can think back to your algebra 2 or calculus classes, and notice that this is the geometric series [2] given by sum 9 * 10^-n, and this series converges.

-----

Now, why is it equal to 1?

Well, there's a few ways to prove that, too. But the simplest, in my book, is this: given any two different real numbers x and y, I'm sure you would agree that there is a value z in between them (this is not a difficult thing to prove rigorously, provided you've done the hard work of defining the real numbers in the first place). The average of x and y will do. But we can flip that statement around: if there is NOT a value between two real numbers, those two real numbers MUST be equal.

In more symbolic terms, we claim that for all real numbers x and y such that x < y, there exists z such that x < z < y. So if there ISN'T such a z, then we must not have x < y in the first place. (This is the contrapositive [3], if you're not up on your formal logic.)

So consider the values of 0.9999... and 1. What value lies between them? Can you find one? As it turns out, no such value exists. If you pick any real number less than 1, your 0.99[some finite number of nines]9 sequence will eventually be bigger than it - and therefore, since the sequence is increasing, its limit must be bigger than that value too.

Since there are no numbers between 0.9999... and 1, they must be equal.

-----

[1] https://en.wikipedia.org/wiki/Monotone_convergence_theorem [2] https://en.wikipedia.org/wiki/Geometric_series [3] https://en.wikipedia.org/wiki/Contraposition

[eszed]

This is my (new) favorite answer. You've made something counter-intuitive seem simple and obvious. Fantastic!

(I've never done this before, but: a pox on those down-voting my original question. Learning is the very essence of "hacker"-dom. Thank you to all who have seen this and taken their time to teach me something.)

[ForOldHack]

This needs to be repeated time and time again for people who deny the basic tools of Calculus, and will suffer the misuse of them. ( specifically the sum of the sequence of the reciprocals of the natural numbers is equal to 1/12. I get that it is a useful tool for quantum chromodynamics, but it makes my skin crawl. )

[eszed]

It's one of the regrets of my life that I didn't take start calculus my first term in college. I'd done pre-calc in high school, but I'm a Humanities guy, and maths - even when I "get" it (and I'd done fine in high school) - drops out of my head pretty quickly. By the time I had an opportunity to continue, and signed up for a "refresher" pre-calc course, I realized I'd have had to go right the way back and re-learn enough algebra and geometry stuff that it was too heavy a lift. Maybe when I'm retired and have time on my hands I'll sign up for some courses at a community college. I still remember how satisfying it was to grok a problem or a concept that had seemed impossible.

So, ahem: is there any possibility that I'll be able to understand "the sum of the sequence of the reciprocals of the natural numbers is equal to 1/12"? I understand (I think!) each of the words in that sentence, but I can't make them make sense! I've got as far as

1 + 1/2 + 1/3, etc.

So how does something that starts off with 1 + [something] end up < 1?

[rachofsunshine]

I suspect the downvotes are just because this is a well-known result whose proofs are rather easy to google. But I like running into one of today's 10,000, I guess. https://xkcd.com/1053/

[eszed]

Fair enough. I'm enough not-a-mathematician that I didn't know it was well-known, and wouldn't even have known how google the proof! (Nor have much confidence I'd understand the real thing once I found it, lol.)

Anyway, I love that concept, and enjoy being on either side of the exchange. Thanks again.

[autoexec]

> So consider the values of 0.9999... and 1. What value lies between them? Can you find one?

As long as it's true that you can always add two real numbers, there's always infinity + 1, or ultimately 0.9999... + 0. ...9

[thrance]

No one prevents new ideas from being presented, but simply suggesting the universe does not expand without giving any arguments for this position nor trying to explain the observed red shifts, contributes exactly nothing to the discourse. This, is not constructive.

[ants_everywhere]

> at the beginning of hypothesis formation

But the expansion of the universe has been thoroughly studied for over a century. We're past the brainstorming phase.

I generally think people should brainstorm to generate ideas and then filter them down. And it's true that filtering too early can significantly decrease the quality of ideas.

And it's also true that in a place like Hacker News there will be smart people from all sorts of backgrounds getting to experience the joy of exploring a new topic that they're not fully up to speed on yet.

The risk though is that somebody who thoroughly understands the field is reading your comment. So for that reason I think it's a good practice to always be aware that technical fields we're not expert in are usually more subtle than we initially think.

[dtgriscom]

When a random techie comes up with a completely novel hypothesis that contradicts a broad range of theories accepted by the vast majority of practicing physicists, the proper response is not to stop and say "Hmmm. I wonder if he's right. Let's talk about it."

[alexey-salmin]

Thanks, but I don't think it's a fair description of what happened here. I'm a mathematician who noticed that the statement "The totality of all the observations we have cannot be explained in any other way" is obviously false.

Explaining is neither hard nor useful and it's not what science is normally concerned with. The goal is to predict new observations not to explain known ones.

[JasonSage]

Why not? Seems perfectly normal to just talk about it.

I learn things all the time by wondering if somebody else is right. Much better than just thinking everything is simply the way I think it is now.

Even if you know somebody is wrong, talking about it is absolutely harmless.

[kstenerud]

Because it's noise, and noise is distracting, and there's a LOT of noise out there.

Part of being an expert is knowing how to filter out the noise so that you can actually get some work done.

If one wanted to deal with noise all day, they'd join SETI. Or parliment.

If someone has a novel theory, let them come up with evidence to support it, and clear identification of how the theory can be invalidated.

That's not closed-mindedness; that's pragmatism. Could they actually be right? Yes, in the same way that a baby could beat Muhammad Ali - but you won't see anyone lining up to buy tickets.

You only have so much time in your life, so no need to waste it on peoples' 10-second "theories", like "How about achieving faster-than-light communication by stuffing so many photons down the fibreoptic cable that they "push" each other faster?". Some ideas are just plain dumb and obviously not worth a trained person's time.

[JasonSage]

> If one wanted to deal with noise all day, they'd join SETI. Or parliment.

> If someone has a novel theory, let them come up with evidence to support it, and clear identification of how the theory can be invalidated.

Ok, sure, but you can’t really be coming to _Hacker News_ expecting the latter, or none of the former, so it’s a moot point and poor expectations.

[kstenerud]

You absolutely can and should when they come at you accusing you of being "closed minded", "unimaginative", and "not open to new theories" (see the discussion thread).

It's one thing to politely ignore crackpot theories or state the established facts in response, quite another when the crackpots start attacking you.

[layer8]

The issue is that these physics threads always end up the same, with commenters having only popular-level background offering suggestions they came up in five minutes, mirroring obvious thoughts that actual physicists have of course already thought of decades ago, and in much more detail. It is really hard and quite unlikely to come up with novel ideas that haven't already been discussed and played out ad nauseum in that field.

[JasonSage]

I’m not suggesting the techie is correct, I just don’t think the right answer is complete dismissal instead of communication. Ok, so you know it’s obviously wrong, but there’s no obligation to then go and stifle their curiosity or imagination. Just don’t say anything or let them talk to somebody who has the time and care to indulge.

The original commenter I’m replying to, taken at their word, is ready to dismiss anything somebody says, regardless of merit with no further discussion, just because they think there’s “no way” that person could be right. Which is hilariously close-minded way to conduct oneself.

[layer8]

This is fine the first few times, but after a few dozens it gets exhausting. This is also not about stifling their curiosity or imagination. It's about understanding that in a field as advanced as physics and cosmology, how incredibly unlikely it is for some layman to come up with a worthwhile idea that hasn't already be tackled. To even be able to explain why an idea is impractical or beside the point, a solid knowledge and understanding of the field is often aready necessary. Articles like from Quanta Magazine dress the topics up in language that make them seem substantially simpler, and closer to human intuition, than they actually are.

[koolala]

Once we have open source AI training data and AI historians they could help make real science directly available instead of pop magazines.

[XorNot]

If you're not doing math, you're not doing physics.

Lookup what General Relativity actually is, what it looks like. The mass-energy tensor and the extremely complicated underlying partial differential equations it is actually encoding.[1]

Every parseable language explanation is irrelevant: the mathematics works. If you have an alternative idea...then the mathematics needs to work. What that means is irrelevant, provided it makes useful predictions and does not contradict established observations.

[1] https://en.m.wikipedia.org/wiki/Stress%E2%80%93energy_tensor

[kelnos]

> Ok, so you know it’s obviously wrong, but there’s no obligation to then go and stifle their curiosity or imagination.

Dismissing uninformed ideas does everyone a service. If you're a complete outsider to a field and have no training it it, decide to come up with random ideas about difficult unsolved problems, and then feel stifled when an expert dismisses your ideas... well, that's a level of arrogance and hubris that I think is more than a little infuriating.

> The original commenter I’m replying to, taken at their word, is ready to dismiss anything somebody says, regardless of merit with no further discussion, just because they think there’s “no way” that person could be right. Which is hilariously close-minded way to conduct oneself.

That is a hilariously uncharitable interpretation of what they said.

[scrubs]

Sorry, all this twaddle is like a culture war ... it's noise. Take the cosmologists and put them out of mind.

Now, exhibit a testable hypothesis. Better, try to explain the 67 to 75ish mpc/s ranges in other methods.

Then we can talk. Nuff said!

[ForOldHack]

I had the idea that each of these phenomena were being influenced by our local gravity well, in a different way. Then I remembered my physics. Then I read a few of their papers. This is not just good science, it's great science.

I withdraw my idea, but continue to wave lengths of wire 11.8 inches long.

I do wish I could read a website called cosmogony news, every day.

[koolala]

Has anyone ever tested the theory of matter that is repelled by gravity instead of attracted? They would zoom away and seperate like helium escaping the Earth.

[tsimionescu]

What theory is that? How would it be tested? Gravity is not even a force, it is a consequence of how space deforms around mass&energy. So, even if something like "negative mass" existed, it wouldn't repel, it would just cancel out.

[koolala]

https://en.m.wikipedia.org/wiki/Exotic_matter#Negative_mass

The cancle out your talking about is what happens with magnets and dipoles. What if negative mass was a monopole?

Testing the theory would be be the same kind of tests created for testing the Big Bang.

[ForOldHack]

This would be way cool. A way to rid science of randos and their theories. For the fourth time today, I have to admit being facetious.

[koolala]

Today is a new day.

[tomrod]

Every day.

[koolala]

Same with room temperature highly conductive matter :( Every day I wish we had Hoverboards, Flying Brooms, and Magic Carpets. Instead... every day is another car ride to the grocery store.

[JasonSage]

Also

> The issue is that these physics threads always end up the same, with commenters having only popular-level background offering suggestions they came up in five minutes

… is this really an issue?

I speculate you’ve taken something you don’t want to see personally and dressed it up as “the problem” when you could instead just find a way to be okay with it.

But maybe there’s something genuinely problematic with that behavior which I don’t know about.

[kelnos]

Yes, it's an issue. I come here as a layperson who is interested in the universe; it's exhausting to read plausible-sounding but completely meritless theories brought up by people here with little more training than I have, and try to decide if these ideas are actually useful, or are just uninformed things some rando thought up in five minutes after reading a pop-sci cosmology book.

I'd much rather hear plausible theories made by people who actually know what they're talking about.

[ssklash]

If you've spent any amount on HN reading physics posts, you will see it is absolutely an issue.

[shigawire]

I remember the halcyon room temp superconductor days.

[Izkata]

> mirroring obvious thoughts that actual physicists have of course already thought of decades ago, and in much more detail.

They will eventually die and a younger generation will enter the field who don't know why those ideas were dismissed. For all you know you're shutting down a 14-year-old (either directly or someone observing) who is actually interested and may become one of those physicists.

It's roughly equivalent to https://xkcd.com/1053/

[kelnos]

> They will eventually die and a younger generation will enter the field who don't know why those ideas were dismissed

No, the younger generations are constantly entering the field, slowly learning and building up the tools required to think about these sorts of things. By the time they've done that, they don't need to have their silly ideas shut down, because they do that themselves, using the knowledge they've built.

And when they do have novel ideas, they have the mathematical and scientific tools to actually argue why their novel ideas deserve expensive, scarce telescope time, unlike the armchair pop-sci wannabe cosmologists (myself included) on HN.

[tqi]

On the one hand, I agree that those people are usually wrong and generally pretty annoying.

On the other hand, who cares? This is a random internet forum, not the Proceedings of the National Academy of Sciences, so maybe there is no such thing as a proper response?

[koolala]

What are you busy with? Are there any particular other subjects you would want to talk about instead?

[snowwrestler]

It seems pretty silly to think that we are collectively “at the beginning of hypothesis formation” about the structure of the universe today, in 2024.

[xwolfi]

BS, nobody has to listen to your imaginative process. Imagine away, build something that conforms with the data, then show it !

For now, we dont know any other way to explain than to say it expands, except maybe imaginative fantasies from amateurs on Hackernews, but does it count ?

[koolala]

What if it expanded from anti-mass? Couldn't we form testable hypotheses from that?

I looked it up on the Wikipedia page for Exotic Matter and there isn't much exploration or tested theories on it.

[BlarfMcFlarf]

You certainly could try. But don’t expect experts in the field to bother to do any of the legwork for you, and that includes learning about all the evidence that your model has to account for to be remotely as good as our existing ones.

[koolala]

It's exciting to me just even being able to ask experts if anyone has tried before! progress = legwork + new direction!