Boltzmannbrain

2 hours ago, wtf said:

Step 1: The limit of the sequence 1/2, 3/4, 7/8/ 15/16, ... is 1

Step 2: The sum of an infinite series is defined as the limit of the sequence of partial sums.

Step 3: The infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... has the associated sequence of partial sums 1/2, 3/4, 7/8, 15/16, ...

Step 4. Therefore the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... is 1, by Step 2.

Which part of that logic is giving you trouble?

If you can focus on the logic of these steps you will understand the process. You can have private intuitions about "the end" or whatever your intuitions may be, but when doing math, you need to focus on the math itself.

I think I am just missing the point of the partial sums part.  I don't really know the point of it.  Why is it part of the process?  

5 minutes ago, wtf said:

Yes. The partial sums of 1/2 + 1/4 + 1/8 + ... are

1/2, 3/4, 7/8, ...

and the limit of that sequence of partial sums is 1. So the sum of the original series is 1 by definition. 

 

Definitions can't be correct or incorrect, only useful or not, insightful or not. 

The entire point of the formal definition is to bypass meaningless and unanswerable questions involving "the end." There are no answers to those kinds of speculations nor is there really any meaning in them.

I am still confused. 

I would understand if the definition was something like an arbitrary symbol like they did with an imaginary number i.  That would make sense to me with what your saying because it wouldn't be a value that already has a definite meaning. 

You are calling it meaningless, but 1 has meaning.  This is why I am still stuck with the problem in my OP.

4 hours ago, wtf said:

The notation says that the sum is defined as the limit of the partial sums.

Ok, I agree, but isn't it also saying that the infinite sum of 1/2^n equals 1?

Quote

 

It's defined to be 1, the limit of the partial sums. 

That's actually the clever part of the definition. We can't make sense of "what is the sum after infinitely many operations?" or "Isn't there a tiny little bit left over?" and so forth. The limit definition avoids those problems by providing a precise definition of the sum of an infinite series.

 

Then getting back to my original issue, if the definition is correct, doesn't this mean that there is an end?  This end would seem to be at 1.    

4 hours ago, Genady said:

Perhaps it might be clearer if instead of

they have said

Is it wrong?  Other sources that I am reading have the equal sign with only two bars too. 

9 hours ago, wtf said:

End of 4th paragraph.

Isn't the notation on the bottom saying that the limit is the total sum?  If it doesn't actually equal 1 like the notation suggests, then why is it even used in the first place?

1 hour ago, wtf said:

The sum is defined as the limit of partial sums. 

So if we have 1/2 + 1/4 + 1/8 + 1/16 + ..., the partial sums are:

1/2, 3/4, 7/8, 15/16, ...

The limit of the sequence of partial sums is 1. So by definition the sum of the original infinite sum is 1. 

It's explained here on Wiki.

https://en.wikipedia.org/wiki/Series_(mathematics)

"it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series."

 

I could not find the part that you quoted in the link that you gave me.  But I did read in the same Wiki link (under the heading "Convergent Series") that the infinite sum of 1 + 1/2^n = 2.  They don't seem to be saying it is just a definition either.  What is going on here?

4 hours ago, Genady said:

By definition, it means the following:

for every real number d>0 there exists such natural number N that

|(sum from 1 to m of 1/2^n) - 1| < d

for any m>N.

This is just the limit, right?  I want to know if the sum of all n fractions actually equals 1, but I see on your next response that you answered what I wanted to know.  Thanks.

The only thing is that I was told on a different forum that there is a solution, being 1.  They seemed quite knowledgeable too.  

Hmm, it's not letting me put the link to the other forum.

If you go to mathforums dot com and go to real analysis, scroll down about 21 threads to my thread (from mathmath) called "How close to 1 does this infinite sum get".  They seem to be agreeing that the sum does actually equal 1.

My new issue in my journey to try to understand infinity concerns the "ends" of infinity.

I was told on here that the infinite sum of 1/2^n = 1, and not just gets close but actually equals 1.  

I can't help but notice that we are giving infinity a definite beginning point at 1/2 and a definite end point at 1.  What could n possible equal to get to this point?

If this last point really is a solution to the equation, then wouldn't it have to be 1/infinity, or in  other words, the "infinity-ith" point?  If so, how can it be said that the natural numbers can numerate all points of a set of size aleph-null? 

On 8/29/2023 at 10:10 PM, wtf said:

Yes it's very strange. Yes it is allowed. In fact it's the defining property of infinite sets. We can't do that with a finite set! One way to define an infinite set is to say that it's a set that can be placed into bijection with a proper subset of itself. Only infinite sets have that property. And yes it is strange!

This was noticed by Galileo in 1638. He observed that we can correspond the natural numbers with the perfect squares: 0 <-> 0, 1 <->1, 2 <-> 4, 3 <-> 9, etc. 

So the whole numbers must at the same time be more numerous and equally numerous with the squares. 

https://en.wikipedia.org/wiki/Galileo's_paradox

If he had only stuck to math he would not have gotten into trouble with the Pope. There's a lesson in there somewhere.

Very interesting, thanks for this.  It is a little clearer.  However, I wouldn't be honest if I said that infinity makes sense to me now.

4 minutes ago, wtf said:

Here is what's going on. 

One day Alice eats a cheeseburger. The next day her vegetarian friend Bob says to Alice, "Alice, you are a meat eater." Alice indignantly replies: But no, TODAY I have not eaten any meat. I only eat meat sometimes. And Bob explains that a meat eater is someone who SOMETIMES eats meat. A vegetarian is someone who NEVER eats meat. If someone is not a vegetarian, they are a meat eater. Since Alice sometimes eats meat, she is clearly not a vegetarian. She is by definition a meat eater, by virtue of the fact that she SOMETIMES eats meat.

Ok that's a bit of a shaggy dog story and if it's unclear I'll try to come up with a bette example. But here is the relevant definition for our mathematical purposes:

* Definition: Two sets are said to have the same cardinality if THERE EXISTS a function between them that is a bijection. 

By this definition, we see that N and Z have the same cardinality. Because THERE EXISTS some function between them that is a bijection: namely, the function that corresponds them as follows:

0 <-> 0

1 <-> -1

2 <-> 1

3 <-> -2

4 <-> 2

5 <-> -3

6 <-> 3

and so forth. 

It is certainly the case that there are SOME functions between N and Z that are NOT bijections. But that doesn't matter. To have the same cardinality, there only needs to be a single bijection between the two sets; just as to be a meat eater, you only have to have one cheeseburger.

Another example is a guy who is convicted of robbing a bank. For the rest of his life he'll be labeled a bank robber, even if he hasn't robbed a bank in years. Doing it once is enough to earn the label. Likewise, a single bijection between two sets is all it takes to declare them to have the same cardinality.

 

I understand that there only needs to be some bijection.  But doesn't this seem a bit strange to you that we can exhaust all elements of I and we also can't exhaust all elements of I (using N)?  It is a yes and no answer.  I never see that in math.  Is it allowed?

The two sets N (naturals) and I (integers) have a one-to-one correspondence and are said to have equal size/cardinality.  

But if we put them one-to-one in a specific way, such as the naturals to the naturals from I, we see that the naturals of I get used up leaving 0 and the negative integers.

This seems to show that a correspondence from N to I can also not be one-to-one.  

The curiosity I get from this is just too much.  It almost seems like this is an example of something that can be proved to be true and can be proved to be false. 

I would have to think that my problem is that I am not allowed to correspond the naturals to only the naturals of the integers, but why not?  

I think since there are so many dust storms, and thus probably a lot of charged particles, this video makes the most sense.  Start watching at 1:20 

And here is just an example of what lightning does in dirt/sand

1 hour ago, TheVat said:

BBrain is just a solipsistic philosophical conundrum and not a serious theory.  You can't argue against it any more than you can argue against Descartes's Demon or the flying spaghetti monster.  

Speaking as someone who used to post as BrainInaVat, I do appreciate Boltzmann Brain as a forum handle, though.  Got a LOL from me the first time I saw it.

Statistically, the vast majority of Boltzmann brains would see nothing but chaotic mush and internal confusion.  

Yeah, I definitely do not take the BB to be realistic.  Though I did think it was quite interesting when I first heard of it.  

29 minutes ago, joigus said:

Why? The point has been settled. You don't get it, that's all. I can see that.

Ok

25 minutes ago, joigus said:

I hope my brain lasts for something more than a few seconds yet. It remains to be seen if my brain's attention to your arguments will last that long --last time we discussed something I didn't find it very promising.

Again (because you missed it the first time): A world made up of false, fluctuation-generated, memories would not display correlations like those manifest when I watch my family album, legal documents, history books, etc, and compare them to my sensorial memories. Have you actually listened to Susskind's explanation? His point about Boltzmann's wife? You could have 1st-order coincidences, so to speak. Much more unlikely would be to have 2nd-order. Let alone 3rd, 4th, and apparently unlimited in the order or depth --if you will--. This is not a world of Boltzmann brains, only too obviously. This is a world in which what I see has been seen by many other 'processes' out there. You cannot simulate that with a thermal fluctuation.

Yes I watched the lecture.  I am getting frustrated that you are understanding what I am saying.  If your brain only lasts for a few seconds, or even a millisecond, you only have memories of the structures.  They won't actually be out there because your memories may not be of the actual structures. And what you see maybe illusions, a dream, a picture, i.e. a facade

I don't know why Susskind went in the direction he did.   

This is a type of solipsism.  It is quite difficult to rationalize your way out of it.

But let me guess, you are going to keep arguing anyways?

9 hours ago, joigus said:

If by "not very convincing" you mean "not convincing you", I would agree instantly.

You have proven to be... How should I put it... very resilient to solidly understanding many ideas involving infinity, or perhaps very stubborn in your own views about them.

Here's a piece of conversation between a student and Susskind about Boltzmann brains, elaborating on what they would be and why they wouldn't explain the world as wee see it.

https://www.youtube.com/watch?v=3hh0lJZbUfo&t=1680s

Upon the student stubbornly insisting on them and their properties, he ends at about t=1800s,

"Don't worry about it: This is not the right theory of Nature."

The reason is a world of Boltzmann brains spontaneously popping out of a thermally-dead universe would not bring about the correlations we see in the real world. Comments concerning George Washington and the cherry tree.

I take it you did not read what I said carefully enough.  

What if your Boltzmann Brain only lasted for a few seconds?  How would you know that the universe/structure actually exists, and are not just false memories?  I am not actually asking this question to get a response.  I am just saying that your argument against a BB was not convincing.

2 hours ago, joigus said:

Obviously he means quark. 

 

And we're not Boltzmann brains. Structure formation in our world is not to do with fluctuations, obviously. The contents of my mind come from events in the past. It obviously cannot be the case that the contents of my mind, and my feeling of them having to do with events in the past both!!! arise from thermal fluctuations.

And not everything should be considered. Silly ideas don't have to be considered, when they're silly for obvious reasons.

I get the feeling that you think I am arguing for a Boltzmann Brain.  I'm absolutely not.  Moreover, you did not give a very convincing reason why a Boltzmann Brain is not the answer.  The BB would explain everything that you mentioned.  The structure out there, would actually be just your BB.  Everything including your memories, conclusions, arguments, etc. would only exist for as long as the brain exists.  It is like proving solipsism false; it is very hard to do so. 

When we are this far down the rabbit hole trying to explain away constants, laws etc. and how they came to be, a Boltzmann Brain may be a lot less crazy than the actual answer.  

2 hours ago, exchemist said:

What has this to do with the fine structure constant?

I felt we were in fringe-theory territory to explain away fine-tuning mysteries.  No matter how distasteful, everything should be considered.

You may also be interested in reading about the Boltzmann Brain.  Matt O'Dowd does a good summary of it.

On 5/23/2023 at 10:27 AM, phyti said:

 I.e. a missing sequence is only relative to an individual list.

I think we can all agree on this.  I don't understand what the problem is.  

On 6/16/2023 at 2:15 PM, wtf said:

You keep referring to "infinity" but that's a vaguely defined word. It's better to talk about infinite sets, which do have a clear definition. A set is infinite if it can be placed into one-to-one correspondence with a proper subset of itself. [Pedantry note, that's the definition of Dedekind infinite, but it will do for present purposes].

By that definition, the natural numbers 0, 1, 2, 3, 4, ... are an infinite set, because they can be placed into one-to-one correspondence with their proper subset the even numbers. 

Now, any set can be ordered in many different ways. Consider a class full of school kids. You can ask them to line up in order of height. You can ask them to line up in order of age. You can ask them to line up alphabetically by last name.

In each case you have the same set, but it's ordered differently. So we see that there are two distinct concepts: The elements of a set, which don't change no matter how you reorder them; and order properties, which can change depending on how you line up the kids, or the elements.

So we can reorder the natural numbers as 1, 2, 3, ..., 0. It's still the same set, but we just ordered it differently. In the usual order 0, 1, 2, 3, ... the ordered set as a first element but no last element. In the reordered set 1, 2, 3, ...,0 there is both a first and last element.

The order properties of a set can vary depending on how we line up the elements.

 

That's not the definition of an infinite set. It's very common in online discussions for people to say that an infinite set is one that has no end. But this is simply false. Dictionary definitions are not helpful in mathematical discussions.

As we've seen, many infinite sets have ends. The funny ordering of the natural numbers 1, 2, 3, ..., 0 has an end. The closed unit interval of the real numbers [0,1] has an end, namely 1. 

Lots of infinite sets have ends. Circles are infinite sets that have no ends at all, yet have a finite length.

So the trick here is for you to unlearn the wrong definition of infinite sets that you've been using. Infinite sets can sometimes have beginnings and ends, other times not.

 

Line up the kids by height, line up the kids by weight (and get sued by the parents). Two different orderings on the same set. 

Set membership is one thing. Set orderings are a different thing. You can put many different orderings on a given set. 

Now I am talking about mathematical infinity. I am not talking about physics or the real world (whatever that is, ask a quantum physicist if there even is one). I'm only talking about math. 

But math is a good place to start, because it's the one area of human learning where we have a clear, logical theory of infinity.

Thanks for this +1

I definitely see what you are saying.  I was not specific enough in the OP. 

But I think you answered my questions for the other "types" of infinity.   

20 hours ago, wtf said:

You just define it that way. You make up a relation called the "funny order" on the natural numbers that says, that if n and m are both nonzero, then their new funny order is the usual one.

Except that zero is larger than any other number. 

This new funny order satisfies the axioms of an ordered set: Reflexivity, antisymmetry, and transitivity.

https://en.wikipedia.org/wiki/Partially_ordered_set

It's really no different than taking a bunch of school kids and having them line up by height; and then taking the shortest one and telling then to go to the tallest end.

Another more familiar mathematical model is the closed unit interval [0,1] consisting of all the real numbers between 0 and 1, inclusive. That is an uncountably infinite set that has a smallest and largest value.

Or just think about the points on a circle. That's an uncountably infinite set with no beginning and no end that has a finite length, namely the circumference. 

+1  Thanks for this, but it is very unsettling for me.

What I am interpreting this to mean is that infinity can have a final element (or end) but it also cannot have a final element.

It is also hard to grasp that something that is defined as having no end, can end.

I suppose that infinity ends in one respect and does not end in another.  I am struggling to find the difference between the two "ends".

9 hours ago, wtf said:

Sure. Just reorder the natural numbers from their usual order, 0, 1, 2, 3, 4, ... by taking 0 and putting it at the end to get the ordered set:

1, 2, 3, 4, ..., 0

That's an infinite, ordered set of of numbers that's the exact same set as the natural numbers in their usual order, but has both a first and last element. 

Interesting +1

But I have to ask, how can 0 come after numbers > 0 in an ordered set?

3 hours ago, Intoscience said:

I'm not sure but it seems like you are confounding mathematical concepts with physical objects.

For instance you have been given examples of mathematical concepts based on infinite sets. In physical reality however the idea of "infinite number of objects" might not be possible. 

Mathematics can lead to infinities when modelling physical reality. One example being the singularity predicted at the centre of a black hole this brings into question the possibility of physical infinities.

However, space itself maybe infinite, it exists (we are part of it) but it may not end. In which case there would be room to fit an infinite number of objects, in which case we may assume but not confirm that they "all" exist without end. Counting them "all" would take an infinite amount of time though, so in this instance there would be no way to confirm there is an infinite number of them. 

Mind blowing stuff and a subject that is discussed profusely across all science/math forums. 

 

Yeah, thinking about how these strange mathematical concepts may cross over into the real world is almost scary.  

My math professor told me something that was absolutely mind-blowing to me.  I asked him what he researches as a faculty member with a doctorate in mathematics.  He said that they look at physical phenomena (exotic phenomena I presume) to understand more about math.  

So in some limited sense, or maybe not even limited, it seems to me that there is almost no difference between math and physics.  Maybe eventually we will find that they are both the same thing.   

3 hours ago, Genady said:

I don't know if it matters here, but generally mathematical objects can be considered in a variety of perspectives. For example, a complex number can be considered as built of real and imaginary parts or of phase and magnitude.

+1  I find that very interesting and important to remember.

4 hours ago, Genady said:

Sorry, I've tried but failed to understand the analogy. Thus, don't know what to say.

Forget that analogy. 

I was thinking that maybe it matters in what perspective the infinite objects are being considered.  But we probably don't have to get into that.

4 hours ago, Genady said:

I am reasonably clear about what you mean saying "exist", i.e., the mathematical existence. However, I still don't know what "end" means.

For example, number 3 exists. Does it "end"?

Good point (+1 ), but there is still something unsettling about an infinite "row" or "list", in particular.  The whole row somehow exists, but also doesn't exist from a certain perspective.   Does perception play into mathematics sort of how observation plays into physics?

3 hours ago, studiot said:

OK so I respectfully suggest your vocabulary of concepts is too limited.

 

For instance 'end' is a one dimensional instance of a boundary.

Go to two dimensions and you have for instance the edge of  rectagular piece of paper.

If that paper is now an infinite roll you have two edges but no ends

Carry this line of thinkin g inot higher dimensions.

 

So I have introduced some important new terms for you, boundary, edge, dimension.

But the complexity of the matter does not end there.

 

It is necessary not to confuse boundary with bounded. 

They are quire different concepts with confusingly similar names.

For instance the function f(x) = sin x is bounded, yet x is unbounded. Neither have a boundary.

 

Then again we introduce infinity. Some Astrophysicists like to  argue that the Universe is 'finite yet unbounded', with no end or beginning.

How can that be ?

Well consider na circle.

Does it have a beginning or an end ?

If you travel round it is that journey finite or infini9te ?

But have far do you travel if you make an infinite count of circumnavigations?

 

If you wish your considerations to enter the later 20th/early 21 centuries then you need to consider porous and fractal boundaries (or ends).

 

I don't claim my list of additional terms is exhaustive, just a good start.

 

+1 for  good topic by the way.

+1  Yeah, I am trying to keep all of this in mind.  When I said that there may be a gap in our understanding (or at least in my understanding) of infinity, I was sort of referring to your points in your post.  There might just be a property of infinity that allows it to somehow exist in and out of the ether.  I will look at those properties that you mentioned.  Thanks.  

16 hours ago, Genady said:

In the mathematical sense we apply now, yes, they do.

I was afraid of that answer.  I don't understand how something can exist but not end.  

This is why I am so interested in this topic.  It doesn't make clear sense.  Something seems wrong, or at least there is a gap of knowledge to be filled.

8 minutes ago, Genady said:

I think that infinite set is an object that exists in mathematics in the same sense as 3.

Do all the objects exist that are in an infinite set?