Entropy can be reset to initial or previous state

Oh yes, the hyperreals are a legitimate number set, but that doesn’t mean you can just do whatever you want with them and expect it to be valid.

There’s plenty of introductory material on the subject floating around on the internet, but actual examples of their proper use seems to be sparse.

Most of the actual usage of hyperreals in arithmetic seems to be done through sets, for example here where every term in the set is operated on sequentially with the corresponding term in the second set. So it would be legitimate to represent a specific divergent series within a set, and operate mathematically upon that. Dealing with specific infinite series (tending to infinity) is done all the time and is fine.
Other usage seems to be in conjunction with sets of numbers like the real numbers, in order to get a real number result. Concepts like ‘dx’ are used all the time in calculus, which is fine because they represent a tendency towards infinites.

James may have been perfectly aquainted with their usage at the time he wrote what you quoted, but none of it shows. He said he had his “own proof” so apparently intentionally deviated from the legitimate pathway, and what he came up with is extremely dubious.

You say you don’t see the complication with infA + infA = 2 * infA because you’re using your intuitions about finite quantities. Infinities easily result in nonsense and have to be dealt with very carefully.

I’m hearing things like “densities” in posts on this thread, but it doesn’t matter how relatively closer numbers are when they’re added to infinity, for example, the sum is still divergent and not-finite i.e. can’t be defined. It goes on forever so never reaches a point at which it can be compared with another way of going on forever, such that one can be called “bigger” than another. Size presupposes finitude: something “infinitely big” has no bounds to compare to anything else - it can’t even be ascertained to be a specific thing. The only thing it can be ascertained to be is a set of smaller things - maybe even a specific infinite series, which like I said is fine. Care is needed.

From Dr Math, the article that you linked to:

That author points out that people have had a lot of trouble understanding the issues of infinity to the point of banning them from professional maths only to have to put them back in later.

I’m sure that I’m not in a position to argue with Euler, Hewitt, Newton, Gödel, Robinson and probably hundreds of others who James seemed to have agreed with. You seem to have trouble with it all just as the author of that article, Hermoso, admitted to having. I wouldn’t be able to competently take up the argument between you and them. But I do have a couple of questions for you.

  1. Do you have trouble with the idea that one infinite set can be known to be larger than another?

  2. Do you find something specific, very specific, that you consider to be invalid reasoning or usage from James? Please exactly, precisely, quote an example of the error.

James only believed in a six dimensional reality, I never saw him attempt a proof for this.

Personally, I think there is a ‘divine’ sequence that orders the reals in 1:1 correspondence.

An algorithm that is infinitely chaotic won’t formulate an expansion, it’d be indecipherable… I could make a very long post to show this, that you can’t prove higher orders of infinity. Maybe I’ll have the energy later.

The author of that article I linked, and both you and I aren’t in a position to argue with Euler, Hewitt, Newton, Gödel, Robinson and probably hundreds of others too. How about James?

The difference between his usage and the last 70 years of the number set being accepted to the extent that it has been seems striking to me, as does the logic behind his usage. You, James and I may not be world famous mathematicians, but does that mean we are equally amateurish? It’s not without problems for each of us to argue the relative legitimacy of our arguments, and regardless even of the full potential that we each achieved at any given point during our respective lives, the standard of what James wrote on his blog some years ago is of the sort I used to play around with when I was a child. Having achieved top marks throughout my mathematical education, and maintaining significant interest and exposure to much higher levels over the decades since, I know there are at least some objective measurements to justify at least a higher level of amateur ability within myself - and I have no interest in overstating this subjectively, but I do have interest in criticising content that appears to me to be exhibiting lower levels of amateur ability, yet is gaining traction and influence over others who may be susceptible to mathematical sophistry due to their own standard being insufficient to see past it. Hell, I might be wrong, but I know there’s plenty of reason for me to not be. I will put what I have out there, and you can take it or leave it, though I recommend you take at the very least a healthy amount of skepticism with you that you don’t yet appear to be exhibiting.

  1. I have zero trouble with how people can think they understand one “infinite quantity” to be larger than another even though it’s undefinable, using their intuitions about finite quantities. The Hyperreals meet the transfer principle with respect to the Reals, but that does not make them equivalent - especially in how to treat their results.
    For example, I have zero trouble with representing two infinite sums added together as twice the initial sum, particularly if it is a convergent series. However, to gain meaning from doing the same to divergent series is not without problems that need to be approached with due respect and caution. You can “represent bigger infinities”, giving a semblence of size comparison between two or more, but this still makes no real-world sense as they both diverge forever and therefore never get to the point where they can be compared. Any specificity in constructing and comparing infinites is in their means of construction, not in the end itself - which you can only physically get to, by definition, if it’s a finite value. Do you have trouble accepting this logic?

  2. I’ve been giving specifics all this time, in particular in this post, which starts off with the most glaring contradiction so far:

But I’m re-reading previous posts of mine as well and they too are specific, exact, precise and with quotes for reference - as you requested… Do you have trouble accepting their logic?

Why am I thinking that your objections to James are far more about you than him.

So yes, you do have trouble with it. I asked nothing about sums, divergent or otherwise.

And yes, I do see what seems like a gaping hole in your logic. That hole seems to be centered around your inability to comprehend adding to infinity. And then you hold onto the idea that if you cannot understand something, it can only be because the other people are ignorant, amateurish, or childish. I suspect that your protective defense attitude prevents you from growing as quickly as you otherwise might.

Well okay. That isn’t an entirely invalid complaint. But I could easily call it “amateurish” of you. Is that all you have?

Having been a professional observer for years, I can tell you that one of the first things to learn is to not take anything you hear or read too seriously until you have investigated the perspective of the speaker or writer. We called it “linguistic grace”. In politics it is the way of conservatives and the opposite of the way of liberals looking for any excuse to express and propagate their hatred.

Probably years before James came to this site, he refused to try to discuss his understanding of the make of the universe with online posters. He expressed two reasons. First he stated that without a good grasp of the infinities, no one is going to understand it. Most people agreed with that idea. Secondly he expressed concern as to what influential people might do with it, especially if they misunderstood it. I see at this site he posted a thread on that issue.

Later on a Catholic site, he attempted a brief tutorial on the general concepts of the cardinality of infinity - how you get from an endless, infinite list to 2 times that list to an endless list of endless lists, “infA^2”. He used a story about God calling a meeting of all angels and requiring that his accountant count each and every angel.

My point in mentioning that is that James, knowing that he was talking to people who were certainly not mathematically inclined, spoke to them in simpler terms that perhaps they could more easily understand. It seems that you would have called him childish for using such language. I thought it was smart of him to not use elite sounding verbiage in an attempt to impress them with his brilliance, as you seem to require of people.

In the blog quote that you mentioned, James did use the word “infinity” in a maths formula. Of course, a year prior to that he had explained that in order to use maths properly, you first must rigorously define your “infinity”, which he had done as “infA”. So why didn’t he use his infA notation in that blog?

From James’ perspective, a year would have amounted to over 2000 posts on this site discussing both his ontology as well as many other issues. He would have seen his audience as regulars who were probably tired of his explanations. He would have known that this audience knew what he was referring to when he said either “infA” or “infinity”. It wouldn’t have been an issue. But his blog is a different audience.

You are accustom to interrupting trains of thought for sake of extraneous details. I have observed you doing it greatly in merely the short time that I have been here. Perhaps James knew that such interruptions make it difficult to follow a newly presented idea. So rather than confuse a new audience or further bore the old audience, he said what they all would understand most easily.

People who get pedantic with their language and have no linguistic grace usually don’t go far unless they are speaking to an elitist audience of highly educated high brows. I’m sure that James would not have seen this audience that way.

So actually I think he did the right thing by NOT using the “infA” notation in a place that would have just led to more confusion and need of explanation, even if it was technically insufficiently defined enough to impress the non-amateur elites.

If you want to play the game of pedantic, “no linguistic grace”, you might find yourself steeped in issues and far less respected than you would have been. Just a very brief example (try not to get carried away) is your use of the phrase “infinite sum”.

The term “infinite sum” is an oxymoron. A sum is a finale, end point. Infinite things don’t have ends. How stupidly amateurish of you to not know even that elementary detail.

Of course with a slight bit of linguistic grace, I can accept that you were referring to the sum of infinite series’. Although I don’t see why you brought up summations of infinite series’ since that has nothing to do with the question at hand. An “amateurish” distraction perhaps?

But again, using a touch of grace, it is easy to accept that you were reminded of something very slightly related and chose to get the thought off of your mind without concern of its distraction. An “amateurish” compulsion?

To sum all of this up, what I observe is that you have a problem grasping the idea of adding to an infinite quantity. I surmise that you instinctively feel that you have to add things end to end and thus cannot add anything to something that doesn’t have an end. And if you were right about that, not only would Euler, Hewitt, Newton, Gödel, Robinson and probably hundreds of others be wrong, but the entire universe would have to be considered the same size as a 1 inch line segment. Both would have the same number of point locations within.

And that is the problem that I see in your logic.

Hey, any chance you babies are going to stop whining to each other and address the actual different orders of infinity which I showed you?

You both have spent great slabs of text and god knows how much time saying precisely nothing about the actual issue of different sizes of infinities.

Go back to my posts, which are empty of rambling and to the mathematical point.

Or just answer this:

Is an infinite line equal to an infinite 3d space?

They’re both adequate the the term “infinity”. So, since “A”=“A”, how come they aren’t identical concepts?

Can you boys please stop posting your resumes and address this question?

I don’t know what you mean by “adequate”.

According to the experts, they are both endless, but not equally endless.

Adequate is a basic logical term.

If A is adequate to B, it means we can say that A is B.

It doesn’t mean that A=B, because B might not be adequate to A.

Like say, a “left shoe” is adequate to “shoe”, but “shoe” is not adequate to “left shoe”.

“According to the experts, they are both endless, but not equally endless.”

Yes, according to logicians as well.

James and I understood each other well as logicians not having to show each other our CV’s, but sufficing with our logic.
Even if James never bragged about his credentials and IQ, you can be sure that he, just as I, always scored at the top percentile of any significant intelligence tests. Because he is intelligent, he doesn’t need to refer to his diploma or “experts” but can just argue a case directly.


Depth of infinity.

The infinity of rational numbers is deeper than the infinity of integers.

Can you “stop posting your resume and just answer the question”:

“Entropy can be reset to initial or previous state?”

:slight_smile:

:slight_smile:

Entropy has an echo.

And to pingpong it back to you - that is the part of my post which you choose to respond to.

As Ierrelus says I guess the teacher has to leave for the student to learn.

:laughing:

So long Boyz.

I honestly have no idea, since logic has nothing to do with the writers. This isn’t the only instance of argumentum ad hominem that I’ve seen from you. Here is another:

Again, I understand adding to infinity perfectly - I’ve been trying to explain to you how to do wit all this time. I also understand (I just don’t accept) how people are doing it incorrectly, and when they do this it makes them look amateurish whether they are or not: this is not an accusation, it’s just a comparison. As a “professional observer for years”, I don’t get how it’s passed you by that what I’m explaining is the legimitate way that all these names you’re dropping either came up with or used themselves - and it’s not how James has been treating infinities. I don’t really have an interest in “growing” in my capacity to accept the illogical, so I’m fine with your suspicion, but there’s no accounting for taste.

I mean, you should have been? This is my whole point?

This is another argumentum ad hominem: again, the writers (or speakers) have nothing to do with the soundness and validity of the logic being presented. It’s interesting that you find this fallacy in conservative politics - I do too.

As such, as interested as you are in the history of some poster on this forum, I cannot say the same for myself - again no accounting for taste. But luckily, his past that you’re paying a suspiciously high amount of attention to has nothing to do with the logical content of what he’s said. Not sure where all this devout loyalty and empathy is coming from to just some guy and some things he said - again, where’s that healthy amount of skepticism that we should apply to all thinkers based on the logic behind what they’ve said? No ill will to the guy, intelligent or not, I don’t care - I only care about the logic of what he’s said. Let it be clear that anything I’ve suspected about him has nothing to do with what I’m making of the logic of his arguments.

Maybe he was dumbing down his language and presentation for the benefit of others, maybe not - again, amateurishness is just a comparison I’ve made and a suspicion I have, which I care not to confirm either way - the content is all I care about.

Whilst I sympathise with trains of thought being interrupted on an emotional level, if the foundations of your tracks are flawed - it’s better on a rational level that this is pointed out before you allow the train to reach its destination, or in some cases even begin its journey at all.

Such flaws may seem extraneous to you, but as a precise and unforgiving thinker, to me, every detail of the foundations must be examined and addressed to ensure what is built on top is sound and valid: another thing I’d recommend.
Try building computer programs with a missing piece of punctuation, some faulty logic, or a missed logical condition. If it compiles at all, it will be buggy.
Maybe this is only the realm of the highest level of thinking, and maybe this isn’t how James saw his audience, but either way I perhaps have a different approach to rigor, about which I attempt to be as clear as possible. Gaining respect is not my concern, I have no emotional investment, I just contribute what I feel ought to be contributed for the sake of illuminating flaws and improving thinking.

Feel free to read the first line of this section of the wiki article as I comment on the irony of your accusation.

Perhaps you misread me, but what you’re saying here was exactly my point in the first place? That infinities don’t have ends and they are a means of construction to represent the tendency towards the infinity that you never get to. An infinite sum is a construction to represent this as an infinite series.

I won’t accuse you of being amateurish for making a simple mistake in your reading, but I will express concern over how exactly wrong you were in reading what I wrote:

Infinite series have everything to do with James’s use of infinities, as this is how you mathematically operate on hyperreals, as he attempted to do. This is how you move away from an amateurish way of treating them.

All these names are just fine with the treatment of infinities as I am explaining - they treated them the same.

Honestly, I don’t think this discussion is going anywhere. I understand you’re invested in defending this guy for whatever reason, I’m just trying to help. Clearly you don’t want it, and I don’t want to bother you with it if you don’t want it.

I assume you mean “Sufficient”? I guess they’re similar in meaning and maybe it’s a translation thing, but just so you know, I believe the English convention is to use the term “Sufficient”. This is probably why obsrvr524 didn’t know what you meant.

By definition no, since a line represents 1d space.

I assume this is your basis for the concept in your next post:

But the distinction between a line and volume is irrelevant to the infinitude of either, because the information contained along an infinite line can also be represented in 3 dimensions.

Your argument would appear to be that the form of the representation of infinity affects the “quantity” of the infinity, for which you’re using the term “depth” - but comparing depths implies comparing quantities.

They’re both representations of that which has no bound and therefore can’t be defined. As I’ve been explaining to obsrvr and as you probably already know, finitude is the derivational root of the terms “finite” and “define”, meaning bounded. You can define finites, and you can represent the tendency towards the infinite through constructions that specify finites (infinite series), but strictly logically that’s the closest you can get to “defining” infinites such that you can compare them. But one definition of a divergent series that tends to infinity has no bound just the same as a definition of a different divergent series - there’s no final bound either way to compare the final result. All you can compare is the construction that uses finites. As such you can manipulate and perform arithmetic on the constructions because they use finites, but the final result is undefined whatever construction you use to represent a divergent series.

The difference in “depth” is in the representation, not in the result.
Even though this escapes obsrvr, perhaps you understand this?

As I explained to obsrvr, it doesn’t have an echo, nor does it need an echo.

Entropy is non-linear, as demonstrated by the equation for change in entropy: ΔS = ∫₀∞ δQ/T.

The higher the δQ (difference in energy), the higher the ΔS (change in entropy) - a proportional relationship - and as spacetime uncurves and the constant energy of the system spreads out, there’s overall less and less differences between points of higher and lower energy, meaning there’s less and less change in entropy. In short, the rate of entropy increase slows down: therefore it’s not linear.

But the equation also shows that the higher the temperatures involved, the lower the change in entropy (they are inversely proportional, hence the T being on the denominator). So areas of high temperature gain entropy slower - which might give the impression of entropy being constant, or even resetting itself (echoing back on itself) when the temperatures are really high - such as in stars and black holes. But overall the entropy still increases, just slower (again, not linear).

You need to get out while you still can, Jake. If you stick around you’re gonna get pwned cuz sil ain’t playin’, homes.

It seems to me that if the universe has always existed and the universe is infinite in size, the average entropy level for the universe as a whole could never change. For every location where the entropy is increasing there must be a location where the entropy is decreasing. I don’t see any way around that.

And it probably never stays the exact same in any one place.

Interesting that the term “adequate” exposes so much literacy problems. A good indication of why it is so difficult to argue with “experts”.

Indeed. A line is 1 dimension.

No. It could be construed that way, with some effort, but thats not very elegant.

In fact the definition occurs at the formulation, at the outset.
Thats what the key to James’ calculations is. Work with the formulation, not with the results.

That goes for “infinity” in general. There is no “result”.

That was a joke dude.

Oh shut up you tool. Seriously. Don’t tell me to be impressed by someone who can recite the ABC.
I had a solid background in theoretical physics when I was 8. Fuck off.

“I had a solid background in theoretical physics when I was 8.”

Falling off the monkey bars doesn’t count, Jakester.

Alright alright. Jesus. I wuz just playin’.

Yeah temper temper

Actually Silhouette, what you said is wrong, or my saying it could be construed that way was wrong -

what I mean with a deeper infinity is one which expands quicker. The rate of adding up is greater with a deeper infinity. Why does this matter?
It matters because this is how James arrives at the idea that space is “larger” than time and cant be reduced to it, thus why there is no cyclical universe, no eternal recurrence of the same.

Let’s look at this “precise definition”:
infA = (1+1+1+…+1)

Now to perform some arithmetic:
infA ^ 2 = (1+1+1+…+1) * (1+1+1+…+1)

Time to sequentially multiply the terms as you do for multiplication of values in parentheses, let’s see…
11 = 1, ok. 11 = 1 as well, let’s keep going and what do we get?
(1+1+1+…+1) * (1+1+1+…+1) = (1+1+1+…+1)
infA ^ 2 = infA, huh…

Now, let’s compare this with infA * 2:
(1+1+1+…+1) + (1+1+1+…+1)
But instead of putting one after the other like you would normally do for addition of finites (which the very formulation of “(1+1+1+…+1)” tries to do in the first place!!), which would get “infA” just the same as if you multiply them, let’s remember there is no end to add the next term to, and sequentially add the terms the same as you do for multiplication:
1+1 = 2, ok. 1+1 = 2 as well, let’s keep going and what do we get?
(1+1+1+…+1) + (1+1+1+…+1) = (2+2+2+…+2)
infA + infA = 2 * infA

Ooo, have we got somewhere? Let’s check against the rule:
“x^2 > 2x” iff “x > 2”

Ok, so “infA > 2” check.
Therefore infA^2 should be “larger” (deeper?) than infA2
Recall:
“infA * infA” was “infA” and “infA + infA” was “2 * infA”, wait what?
Our results show infA^2 < infA
2…, which would only be true if infA was less than 2…
(1+1+1+…+1) < 2? Nope…

Looks like even being able to recite your ABCs and using the secret of James’ calculations and working with the formulations is a good start to being able to see how it makes absolutely no sense whatsoever.