If you think there are others, please exhibit one.

> Probably what you meant is that the expanding universe theory is the simplest of them all and creates less problems then others.

There are no other theories that I'm aware of that account for all the data we have, even approximately.

I could easily do that (god tinkering with our measurements? we live in a simulation?) but explanations by themselves are worthless. Any set of facts can be "explained" but it doesn't really help.

The science is concerned with theories that don't just explain known facts and measurements but also predict new ones. These are also known as falsifiable theories because a theory is falsifiable IFF it predicts at least one new observation that can be actually made and tested.

Now even these can obviously be constructed ad infinitum. For your question of alternatives to expansion, take a theory from the same field (e.g. tired light), then take facts it doesn't fit and make specific ad-hoc carve-outs for these facts in the theory. Sure it makes it ugly and complex but it remains a scientific falsifiable theory. Furthermore this exact things happens very often in the scientific community because theories don't fall immediately when the first contradiction is found. They only fall when a better theory appears to supersede them and until then they just develop a "protective belt of ad-hoc assumptions" as Lakatos called it. No need to mention that the same thing will happen to the "better theory" in due time.

I write all of this not because I have a good cosmological theory sitting in my closet but rather because your statement "The totality of all the observations we have cannot be explained in any other way than an expanding universe" is outright false. As I show above it not only can be explained in an infinite number of ways, but you can also construct and infinite number of scientific theories that fit the totality of observations. Because this "totality" is alas finite.

This reminds me of my school math teacher who once told me that "the sequence 1, 2, 6, 24, 120, 720 cannot be explained other way than by n!". A pity I didn't know about Lagrange Polynomials at that age and had to spend the whole evening to construct a fitting polynomial by hand.

For example, to simulate a galaxy one should use a model based on General Relativity. But mathematics behind GR is too complex to allow simulations on the scale of the galaxy even with all that modern computation power. So instead of GR the calculations use Newtonian mechanics with minimal corrections for the light speed limit. Plus there are a lot of other simplifications like replacing star systems with hard balls that reflects when hit each other with no notion of matter transfer or ejection in the process.

Then we see that the simulation does not fit data. A typical explanation that is used is that of dark matter hypothesis. But this is unjustified.

We have no proof that numerical simplifications are mathematically valid and do not lead to a big error on a galactic scale. Moreover, there were recent papers that tried to account at least for some effect of General Relativity. Apparently it was enough to explain at least some effects previously attributed to the dark matter.

So it can be that the dark matter is just an artifact of inaccurate simulations.

However, generically, these extensions tend to have an under-determination problem frustrating attempts to arrive at a unique spacetime for a given distribution of matter or a unique distribution of matter which can exist in a given spacetime (or both problems). That makes them less attractive than GR, or even possibly outright unsuitable bases for initial-values formulations (and thus are unlikely to overthrow numerical relativity soon).

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.

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.

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.

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.

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.

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.

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.

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.

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

.3… + .3… + .3… = .9… = 1

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

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.

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.

     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

-----

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

(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.)

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?

The sum 1 + 1/2 + 1/3 + 1/4 ... - what we call the harmonic series - runs off to infinity, although it does so very slowly (the nth partial sum is approximately equal to ln(n) plus a constant whose value is about 0.58). Showing that this series diverges is standard calc-textbook stuff (it's the textbook example of the integral test for convergence, although there are plenty of other ways to show it).

However, in math well beyond basic calculus, there are methods for assigning meaningful values to series that don't converge in the conventional sense. Those methods assign the same value to convergent series as the regular old calculus arguments would, but they can also assign values to some divergent series in a way that is consistent and useful in some contexts.

For example, the series 1 + 1/2 + 1/3 + 1/4 + ... is a specific example of a more general series 1/1^z + 1/2^z + 1/3^z + 1/4^z + ..., for some arbitrary number z. This series only converges in the standard calculus sense when the real part of z is > 1, but it turns out that that's enough to define a function called the Riemann zeta function, whose input is the value z in the series and whose output is the sum of the series.

The zeta function, as it turns out, can be extended to values where the original series didn't converge. And doing so gives you a method for assigning values to "sums" that aren't really sums at all.

-----

It turns out that even THAT won't get you a completely nice value for 1 + 1/2 + 1/3 + 1/4 + ..., because that sum corresponds to the zeta function's value at z = 1. But the zeta function isn't well-behaved at z = 1. If you do even more massaging to beat a number out of it, you don't get 1/12, you get the about-0.58 constant mentioned in the previous section.

Which brings me to the way the poster you were replying to is probably confused. I think the sum they meant to refer to was the even-more-obviously-divergent sum 1 + 2 + 3 + 4 + ... This sum happens to correspond loosely to the z = -1 value of the zeta function, since 1/2^-1 is just 2, 1/3^-1 is just 3, and so on.

Again, the sum 1 + 2 + 3 + 4 + ... does not converge, but if we choose to identify it with a zeta function value, we'd identify the sum 1 + 2 + 3 + 4 + ... with zeta(-1). And the value of zeta(-1) happens to be -1/12 (yes, minus 1/12).

----

So the answer here is just that no, 1 + a bunch of positive numbers is never < 1. It's actually a decent basic calculus exercise to prove that any series a_0 + a_1 + ... a_n, where all the terms are non-negative, either fails to converge or converges to a value >= a_0. But because we're taking some extra steps here to leave the domain of convergent series in the first place, it turns out we can get results that (for conventionally convergent series) would be impossible.

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

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

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.

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.

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.

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.

> 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.

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

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.

Comparing their work to the dross that AI produces is insulting

I'm actually excited for computers getting big enough to comprehend human language and science.

I hope one day Dross becomes Dos! AI needs more XP. VR AI and VR AI Science edu is what I'm most hyped for.

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

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.

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!

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.

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.

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

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.

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?

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 ?

One easy process for generating infinite explanations that fit a finite number of data points is taking the simplest theory you have, and adding extra rules that don't affect any of the existing data points.

e.g., if the standard explanation for observations like red shift, CMB, the abundance of light elements, etc. is H² = (ȧ/a)² = (8πG/3)ρ - kc²/a² + Λc²/3, One alternate explanation that fits all the data is H² = (ȧ/a)² = (8πG/3)ρ - kc²/a² + Λc²/3 + T, where T is the number of teacups in the asteroid belt. No observation has yet been made which would falsify this theory, and it fits the data just as well as the standard explanation. We reject it on the grounds of parsimony, not falsification.

One can construct such alternative theories quite easily: Everything is exactly as the "expanding universe theory" predicts except in a phone booth sized volume in a specific part of space 10 lightyears away from Earth where the universe is contracting.

Does not explain anything, and it is not testable in any practical sense. So it is not a good theory in any way, but it is a different one and it matches all the current observations.

Benevolent giant omnitech space squid manipulate EM radiation incident on the solar system to fool our primitive meat brains and sand computers into thinking we're alone in the universe so we don't go venture our and embarrass our local cluster.

See? There exist infinite theories explaining any set of data points. Parsimony, choosing the simplest theory that explains the facts, is what lets us make progress understanding and predicting the world around us. It's hard to predict space squid.

Sure, it is a hypothesis, but thanks to the Scientific Method the majority of people with knowledge in the field knows its most likely BS.

Going the same "it can be for ANY reason, guys!" in any field, will not get you much far, regardless if you feel justified in your ignorance or not.

Of course not. The point is logical correctness: as a matter of logic, infinite theories explain any set of facts. Are almost all of these theories useless? Of course. We should restrict our attention to plausible theories.

But how do we decide which theories are plausible? We look for the ones that require the fewest assumptions.

"That is like saying computers run because they want to, and only make us perfect the CPU, and secret processes inside it, to fool us into thinking we have a say in whether a computer will run."

I am going to quietly turn off my computer.

I am just thinking about Steven Wolfram's causal networks.

That's not a theory, nothing has been explained, merely further convoluted.

It’s not a scientific theory though. Which is a super important distinction in this discussion