↪jgill

From a mathematical pov, does prime number theorem support or act against the Boltzmann brain proposal? — universeness

It was a counter to the number of primes being 50%.

In an infinite duration, aren't all possible outcomes equally likely to occur? — Down The Rabbit Hole

An infinite duration could include an infinite number of stages, the possibilities of something happening in a particular stage might not exist in other stages. In other words, its all babble. :smile:

In an infinite duration, and as all possible existents are of finite duration, then everything would have happened already. — Wayfarer

Why is that?

your argument sounds like an AI generated device — Paine

:up:

Countable infinities are equal, so the infinite set of worlds where we're Boltzmann brains is equal to the infinite set of worlds where we're not. It's a 50/50 chance, epistemically speaking. Given an infinitely large multiverse, of course. — RogueAI

Prime Number Theorem

The phenomenon you call depth is generally known as scale — T Clark

:up: :roll:

It's odd that a thread on the S-B expansion arises in this forum. S-B is an outlier in number theory - the Wiki page is classified as low interest. And since a primary interpretation of its generation are continued fractions of a special type, one immediately loses easy addition and subtraction - which decimal expansions of the reals have.

The value of continued fraction representations is that chopping them off at various levels give rational approximations to what one is expanding. This is easily seen when expanding a real number, like the Golden Ratio. But where it is of greater value is expanding a complex or real function as a continued fraction, providing rational functions (one polynomial over another) as approximations to the expanded functions.

$Tan(z)=\frac{z}{1-\frac{{{z}^{2}}}{3-\frac{{{z}^{2}}}{5-\ddots }}}$

Although I have little knowledge of this kind of number theory (S-B) I feel the line of inquiry expressed here would be of little interest to the mathematical community. But I could be wrong.

Edit: After doing an internet search for "area" in S-B defined in the Wiki article on Farey sequences, I probably am wrong about the interest shown in S-B by mathematicians. Embarrassingly so as I find that two former colleagues of mine have included it in their book. :yikes:

I thought I saw this problem posted before in the Lounge? — L'éléphant

Yes. It's been discussed before. It can be considered a problem in decision making.

Sleeping Beauty

Wonder when Invicta's period of suspension will be over? — jgill

Never, as it turns out. — Banno

It's hard to know if he was putting me on. has a great deal of patience dealing with these things. Thanks, Tones. :cool:

But the diameter is a straight line… — invicta

And the importance of this obvious fact is . . . . ? :roll:

Make a circle with a line one foot long and the circumference is one foot (if no contraction/expansion). This simply means the diameter of the circle is irrational, 1/pi .

Apparently, in the face of such sniping, Invicta bailed on his own thread. :smile: — Gnomon

Apparently he/she has just been released from suspension.

I thought this post on Quora from Victor Toth was relevant — universeness

Toth is excellent. Sometimes Quora clarifies issues. A recent article explained entanglement in a way both appealing to intuition and illuminating: Two coins entangled, moved light years apart. One coin is tossed, coming up, say, heads. The other coin tossed comes up the opposite. Etc.

Invoking mathematics into a philosophical argument deserves not mangling that mathematics. Posting incoherently about the mathematics is a set up for degraded discourse from the start. — TonesInDeepFreeze

:up: :clap:

In uneasy times many look towards an authoritarian figure, in marked contrast to our current president, who is widely seen as only a figurehead, perhaps senile, controlled by a progressive cabal. We drift towards Banana Republic-hood with every video clip of Trump - no matter if they portray him as good or bad. He exudes confidence. :sad:

Corncob pipe? — L'éléphant

I'm not complaining. If this guy can puff away, so could I !

George Wallace demonstration and the date it happened. — L'éléphant

1963. There was turmoil all around, with the Klan playing the crowds. At one point there was an explosion, which someone said was one of the confederate canons at the ROTC building going off. One of the civil rights demonstrators yelled, "I hope they hit the bastard this time!" (meaning Wallace).

Edit: That's how I remember the incident, although the Klan perhaps was there before that day and not on the day he stood in the doorway. And the crowds may have been smaller on that day. The canon going off does stick in my mind, however.

Wonder when Invicta's period of suspension will be over?

Sorry. Just being frivolous! :smile:

Mind you that he never said anything else but what he felt or what background he's coming from. As soon as the man walked away, the woman called him a bigot and homophobic. — L'éléphant

I had a similar experience on another forum. I made a comment about how wonderfully women had progressed in a certain sport, given an opportunity to do so by Title IX. I am 86, so fair game for age discrimination. Two women replied, calling me misogynistic and demeaning, and referring to me as "puffing on a corncob pipe through withered lips" and avoiding the civil and women's rights movements in the 1960s. To which I replied I was on campus and had demonstrated against George Wallace as he stood in the doorway to the admissions office at the U of Alabama, denying entrance to a black man, and that, actually, I had joined the women's lib movement during that decade.

Age discrimination is thriving.

The 'something from nothing' proposal makes no sense. — universeness

Sure it does. The doughnut hole is something you eat and it is in 1:1 correspondence with the emptiness left by creating the hole in a chunk of cake.

No existent can satisfy the notion of nothing — universeness

Yesterday, upon the stair,
I met a man who wasn't there
He wasn't there again today
I wish, I wish he'd go away...

:cool:

Is there a representation of nothing, that as a maths prof, you accept as meeting/satisfying its concept? — universeness

I'm tempted to say, Nothing I can think of. But I won't. How about an element of the empty set? :cool:

Math definitions will not resolve the terminology disputes in this thread because ↪invicta is not making a mathematical proposition. "Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics — Gnomon

Wrong. I and others have studied infinite regress in detail, as infinite compositions or iterations.

In Mathematics, “infinity” is the concept describing something which is larger than the natural number. — Gnomon
Huh? — jgill

Sorry, I forgot to provide a link. That quote came from a math website. I didn't make it up. — Gnomon

What is "the natural number" ?

But for the cosmos to be, the end has to be before the beginning, or why and how could it begin. — Beena

This seems to be an assumption on your part. Please elaborate.

Welcome to the forum. :smile:

Reality is a donut-hole, or nothing out of something. — Thus Spoke 180 Proof

Or something out of nothing? This is a legitimate philosophical issue. :chin:

Fixed. :cool:

I've left out details to give an outline.

Here's an explanation of this process converging to the Golden Ratio. Leaving out the details, we begin with the

cf = $1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots }}}$, corresponding to the pattern of movement down the SB chart. This cf can be generated by iterating the function

$f(x)=\frac{1}{1+x},\text{ }{{f}^{(2)}}(x)=f(f(x))=\frac{1}{1+f(x)}=\frac{1}{1+\frac{1}{1+x}},\text{ etc}\text{.}$

Setting $f(x)=x\text{ }\Rightarrow \text{ }x=\alpha ,\beta$, where the first of these fixed points

is an attractor: $f(x)-\alpha =K(x-\alpha ),\text{ }\left| K \right|\lt1$, $K=K(x)$, for most values of x.


And ${{f}^{(n)}}(x)-\alpha ={{K}^{n}}(x-\alpha )\to 0$ as $n\to \infty$

Now, the value of the cf is $1+\alpha$ = Golden Ratio



This curiosity has a relation to arguments about endless cause/effect chains and first causes. Viewing the cf as going back in time, there is no end to the process, but stopping at any value of n , the x we have chosen at random, say, is a first cause.

In Mathematics, “infinity” is the concept describing something which is larger than the natural number. — Gnomon

Huh?

A little knowledge (from Wikipedia) can be a dangerous thing. :roll:

Thus, infinity, taking Pi as a currently known example is non-repeating and unpredictable — invicta

Not quite unpredictable, as the first 100 trillion digits in its expansion are now known. Why on Earth someone would make that their life's work is beyond me.

At the very least, causation causes more philosophical problems than it solves — Banno

I suppose so. But when context is specified it's not so problematic. I put a kettle on the stove and turn on the burner. The kettle boils over. Because I turned on the burner - just don't go down the rabbithole. I can set up a causal chain in math using iterations or compositions of functions, basic in dynamical systems.

If I elaborate further it will sound like bullshit so I must stop here. — invicta

I applaud your perspicacity. There are mathematical ways of avoiding circles, but not now, I think. :chin:

Wiki puts SB in the category of number theory. Perhaps it's analytic number theory since limits appear. Apart from the simple continued fraction expansion it's beyond me. :cool:

SB is a fairly deep mathematical subject, as seen here.

As we get older time seems to pass more rapidly. At the age of ten one hour seemed like an eternity. Is it possible the "rate" at which time passes actually is varied? But we only have time itself to compare it with, so we cannot tell, apart from rapid motion in special relativity. As I sit typing this, is time fluctuating, and if so is the fluctuation a local phenomenon?

In a typical spacetime graph (2D space + Time) how is it that passage of time is allowed at a purely spacial stationary "event"?

Is Elon Musk really a time traveler from the year 5000AD? :chin:

In the metrics of spacetime, should time be entirely independent of spacial change? — jgill

if space is removed from time then the notion of space loses meaning I’d say. — invicta

What I was referring to is seen in the Minkowski metric of spacetime, in which the time term is in fact a distance term $c^2\left(t_1 - t_2\right)^2 - \left(x_1 - x_2\right)^2 - \left(y_1 - y_2\right)^2 - \left(z_1 - z_2\right)^2.$

Agree with the idea that it’s not time that is being experienced but change, although the two concepts of time and change are inseparable. — invicta

In the metrics of spacetime, should time be entirely independent of spacial change?

If the universe is the mathematical universe of Tegmark, then perhaps it is a giant dynamical system that goes its merry way starting from a set of initial conditions. As to the nature of God's involvement I would not dare speculate.

Are there facts about reality that will forever be beyond the comprehension of humans, like my dog being unable to understand even the elementary aspects of calculus?

I tested the Bing AI in the following way: I have a low-priority mathematics page on Wikipedia, so I asked Bing what is known of this particular subject? Now, there are a smattering of papers on the internet on this subject; what Bing supplied was the first introductory paragraphs of my webpage, word for word. That's all.

Sorry, but that's an invitation to a crazy train — TonesInDeepFreeze

I know. But his quest for the smallest particle of space is no more absurd than Tegmark's mathematical universe IMHO. :cool:

So Bernie announced he's not running, and will be supporting Joe — Mikie

Maybe they could run as a geriatric team. But which one is Gandolf?

My impression is that finite SB strings describe 'destinations' (numbers) and infinite SB strings describe 'journeys' (unending processes with no destination). My issue is that I don't see how decimals are any different. Why can't we say that (non-repeating) infinite decimals are journeys that are described by unending processes (e.g. limits) and not 'destinations' (numbers)? — keystone

I don't see what the problem is here. You can say what you want, when you want. But asking mathematicians to go along with your ideas of synatax and grammar is another matter. You tell a math person such and such is a "journey" or a "destination" and they might nod their heads and say, Well, you might say that informally. The more precise language of set theory is what they would normally speak in this intellectual environment.

I was a math professor for many years and I am not offput or disturbed by your speculations. I have encountered such notions before, many times, and have never felt discomfited by these discussions. But if I were teaching a class in foundations (thankfully, I never did) I might discuss these ideas in more detail.

Who are you trying to convince here? Philosophers who consider definitions optional?

This sort of argument is at most peripheral to most serious mathematical discussions. If you are curious about deeper, somewhat more sophisticated concerns about the foundations of the subject, look up some of @Metaphysician Undercover's posts about the ultimate nature of mathematical objects, points vs contours, infinitesimals, etc.

:up:

I worked in the analytic theory of continued fractions years ago, and one of my forebearers was Omar Khayyam, the Persian poet and mathematician ca 1100 ad . He may have devised the continued fraction expansion of the equation

${{x}^{2}}-x-1=0$ => $x=1+\frac{1}{x}=1+\frac{1}{1+\frac{1}{x}}=\cdots =1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots }}}$

Which is a lot more palatable than Stern-Brocot. Well, in my opinion. :cool: