Both idealists and realists can agree with the Ontic-Structural Realism of Tegmark. For example, British idealism's doctrine of internal relations is in logical agreement with OSR, without jumping the shark to conclude that only unthinkable and unperceivable mathematical structure exists in a way that is divorced from the Lockean secondary qualities of perception — sime

Whatever. As I mentioned in another thread, a simple isomorphism between physical reality and mathematical structures provides a way of saying they are the "same" without being identical. But if this is truly what Tegmark had in mind he overdid his arguments - as do some posters on this forum. :cool:

Rather than insist that our intuitions are wrong and that the mathematics is right, we can instead insist that our intuitions are right by switching to an arguably more realistic axiomatization of geometry in which the paradox is dissolved or doesn't arise in the first place, such as computational geometry or intuitionism. — sime

Try the Taxi Cab metric.

Please die, thread. Poor thing, I should never have started you. This is wandering in the direction of threads having hundreds of posts devoted to the meanings of "2+2=4" :worry:

The instability, ambiguity and uncertainty that characterises mental imagery and perception complements the realities of mathematical undecidability and finitistic reasoning that intuitionistic geometry recognises and which classical geometry ignores, while Brouwer's theory of choice sequences parallels how one visualises or recognises "infinity" (i.e. as a finite random truncation of a vaguely sized process). — sime

Whatever this means, the paradox is not that big a deal. Aristotle may have been aware of the fact that arc length is not preserved in this kind of process.

3.333 The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself.

alcontali is taking the expression F(F(x)) as function composition, ie compute x1 = F(x) and then compute F(x1) , and x could be an integer and F(x) returns an integer. That is different from the question considered by Wittgentstein, which is could there be a function which takes the function itself (the mapping) as an argument? — Robert Durkacz

As a retired mathematician I read the OP and thought, This is utter rubbish. I have worked with infinite compositions of functions for years. But some degree of clarification followed. Thinking of a function as a collection of ordered pairs helps.

There is no god battering the eaarth with his wrath, it's simply a natural fact that if we impose an artificial technological order on nature, the natural order will get fucked up and natural chaos will replace it — Raymond

Imagine if all those vehicles stranded on I95 for 24 hours in snow and ice had been electric.

By your logic, when I walk a mile on a flat surface, I should say I walked more than a mile. — T Clark

Only if it pleases you to do so.

This is a mathematical can of worms, with definitions and counter-definitions. In the physical world certain phenomena may exhibit what we normally think of as randomness. What comes to mind is recording very slight changes of atmospheric pressure in some sort of controllable context.

Since there are algorithms predicting successive digits of the square root of two, from one perspective those digits - the sequence - would not be considered random. But for practical purposes the sequence could be considered random.

How is an infinite line between 0 and 1 constructed? — Raymond

A Short Note: Extending the Diagonal Paradox

It's not a paradox — T Clark

:sad:

My feet don't follow the discontinuous path of the stair. — T Clark

The path is continuous. Sounds like you just float up. :gasp:

It's amazing how vast math is; the same goes for other disciplines too I suppose — Agent Smith

Over 24K physics articles on Wikipedia. :gasp:

So, on a purely logical basis, a "mathematical universe" makes no sense to me — Alkis Piskas

It's a stretch, isn't it? I think Tegmark's ideas somehow hinge upon the notion of "isomorphism" - the physical universe is the "same" as a purely mathematical structure up to an isomorphism. But I haven't really read his works.

Also, I think you’ve mentioned some mental health concerns before; struggles with happiness, etc. This should be a great opportunity to see what makes you happy — Pinprick

Buy a Boeing 707 and hire a crew on standby. Just a thought.

The New Age holds so much promise and this is such an exciting time. There is a chance of a new enlightenment that will be manifest in consciousness so different from the past, the people of the future will not be able to relate to the primitive lives we have had . . . — Athena

This is dialogue right out of the wonderful 1936 film Things to Come

I thought mathematicians were very keen to find some practical use for their theories. — Agent Smith

Most are not concerned about this, being immersed in their explorations of the subject. Staggering isn't it? Over 20K math topics on Wiki.

You're only the second billionaire I've spoken with in my life, incidentally. (That I'm aware of.) — Xtrix

Congratulations on your good fortune. My only experience with a billionaire was to drive around Jackson Hole sixty years ago in a model T with him picking up pieces of scrap metal from junk piles. He was a dirtbagger back then.

This eerily reminds me of Alexandre's paper. A breakthrough in perspective, this time not in physics but art and literature.

Is it like this all the time and everywhere? All sizzle, no steak!? — Agent Smith

Try this on for size: Catastrophe theory

I had a math friend, a fellow climber in fact, who worked with Thom. This topic made a stir when it first came out. Not so much now. I recently discovered there are 22,705 math articles on Wikipedia. Catastrophe theory ranks mid-importance and has a daily average of 4 views. My math page is low importance and has 16 views a day. Trivia.

The only thing left to do is to accept the wrath of the almighty Creator, succumb to His Divine Word, bow to His Infinite Creation, and show eternal gratitude for His Wisdom, given to us by His Word, to be heard by submissive prayer only. Let's pray He will get a grip and restore Natural order. If not, we are doomed for sure. — Raymond

Maybe you should skip Bible Study Group next week. Take a breather. :roll:

Conceptual reconstructionism in science? Mathematics?

Chaos theory is BS! :grin: — Agent Smith

Quite possibly. I prefer interesting quasi-symmetric patterns emerging from dynamical systems.

If we don't want birds to fall from the sky, seas to devour, superstorms to rage, sweet water to taste bitter, unworldly screaming to be heard from within, the last trees to burn, the dark to enter daylight, and the light to ruin the night, the pace must be lowered this very moment. It will be too late tomorrow. — Raymond

Truly Biblical! I'm speechless. :scream:

I am wondering how the discussion would go if we thought the Creator manifested our reality by giving chaos order and that human activity can either maintain that order or destroy it? — Athena

If you speak of chaotic systems, then chaotic theory searches for underlying patterns and deterministic formulae that depend upon initial conditions in a very sensitive manner, so that iteration leads to what is generally thought of as chaos. Are you saying the Creator made all chaos deterministic at the beginning of time?

To destroy that order would mean human activity could somehow alter these initial conditions, traveling back in time.

In general meteorology is far too complicated to involve such deterministic systems. Ecosystems lie on what is referred as the "edge of chaos".

I suspect you meant to trigger dialogue in a much less sophisticated scenario. Initial conditions being in play at the present time, such as global temperatures rising at a rate determined by conditions prevailing minus humankind, then humans messing that up with fossil fuels.

Perhaps it would clarify the arguments if placed in the following stark scenario: A scientist creates a robot that he programs to kill the first human it encounters. This happens and the robot is secured and returned to the laboratory, but the scientist is nowhere to be found. Other scientists confer, some wanting to dismantle (destroy) the robot, others wanting to attempt to reprogram the creature. The latter group prevails. So the computer brain of the robot is scrutinized closely to see if it can be altered and the critter returned to society in a manner in which it helps humans. Several scientists say, yes, it's possible, but another few say, no, we can't be sure - if there is the slightest chance it might revert to its previous condition we must not proceed.

It is finally agreed the robot is to be destroyed. :meh:

Lounge perhaps? :chin:

I think a distinction between my unconscious decisions and actions and conscious ones is artificial and pointless. — T Clark

Perhaps a conscious decision is one where we logically weigh all factors and do some sort of analytics to determine a course of action, whereas most decisions just seem to come to us after pondering. I see a difference.

I have my doubts about free will. Many decisions seem to pop into consciousness from hidden areas of the brain. However, we do have agency - the ability to act on these decisions. A potential criminal may have a strong desire to commit a crime due to circumstances affecting his mental state, but he has the agency to act or not act on that impulse. If the former, off to prison.

How is identifying as a Democrat equal to CNN being left? — Manuel

Ask the researchers at MIT, I suppose. Makes no difference to me.

Why is there not therefore a paradox? — hypericin

Probably because points have no dimension. The real numbers are complicated and fascinating.

Look, suppose I pick at random a point in the interior of the cube: (.251956... , .629435..., .194735...) Then that point corresponds to a unique point on the x-axis edge: .261529194947...

You just alternate digits. Here, you do it by going the other way: find the point inside the cube corresponding to the point on the x-axis: .381402258639...

The process is continuous in the epsilon-delta sense.

https://web.media.mit.edu:

Just like the American average, our survey and the published surveys we studied show the
majority of CNN’s audience identifies itself as Democratic. The largely leftist audience has
probably had a significant influence on CNN’s reputation as being a left- leaning news
organization.

Yep. It's an algorithm for you to guess demonstrated by the example. Standard stuff. :cool:

Then there's that hypothetical spaceship we are in, approaching a Black Hole at a constant speed. We sail in and if not annihilated continue our explorations. But a friend is watching our progress from a great distance and sees us going slower and slower until it appears we have stopped before penetrating the anomaly. To him, we will sit there for eternity, gradually inching towards the periphery - like a mathematical limiting process, never quite making it.

(I'm not a physicist, but I recall reading of this. Could be wrong)

However, these companies are beginning to lose many of their viewers. It appears that audiences are tiring of combative and demeaning dialogue. — jgill

I think it's just polarization. Fox viewers aren't getting tired of Fox and CNN viewers aren't getting tired of CNN. But Fox gets few CNN viewers and CNN gets few Fox viewers. It's a tradeoff between loyalty and reach. — SophistiCat

From Forbes (1 Dec 2021):

Overall, the cable news networks saw ratings drop significantly from the same month one year ago, as the coronavirus pandemic coincided with the presidential election, driving heavy viewership. CNN experienced the most dramatic decline, dropping 77 percent from one year ago in prime, followed by MSNBC (down 59 percent) and Fox News (down 35 percent). The drop-offs were even bigger in the key demo, with CNN experiencing a year-over-year decline of fully 84 percent in prime time. MSNBC’s decline was 74 percent, and Fox News had the smallest year-over-year drop, down 49 percent.

The news media gives its viewers what they seem to want. CNN tilts to the left and FOX tilts to the right in their presentations (evening news, e.g., not their numerous and sometimes rabid commentators' shows). However, these companies are beginning to lose many of their viewers. It appears that audiences are tiring of combative and demeaning dialogue. ABC, NBC, and CBS don't seem to be caught up in this audience decline.

The fact that the line, plane or volume have the same cardinality is because of the attempt to reduce them to points — AgentTangarine

Ignoring the references to cardinality and accepted theory , it does seem like the unit cube has more of "something" than one of its edges.

The bijection between R and RxR is not continuous — AgentTangarine

Really?

This is the same crank whose banning you were lamenting earlier because (he says) he is a physicist and we should be grateful for him being here to educate us... Be wary of unhinged bullshitters confidently throwing around specialist terminology. — SophistiCat

I'm well aware of the situation. :cool: Having communicated with him I believe he has a graduate degree in physics. However, his concerns are with the "physical" or interpretive aspects of the science rather than mathematical descriptions. Even advanced math in QT seems not to have reached axiomatic set theory to a noticeable extent. Kenosha Kid is another genuine physicist. To the best of my knowledge neither of these gentlemen work in the profession at present.

Sounds good mr. Gill. Almost convincing. But you construct a new number from the both. Giving them both different decimal places. The diagonal proof of Cantor says you leave numbers out. Infinitely many. — AgentTangarine

Give me an example of a number you think is left out. I bet it's not. Avoid the .999... =1.0 thing. Have you figured out what the algorithm is?

Too, the definitions in math give me the impression that true understanding is being sacrificed for logical formalism. — Agent Smith

The definition you are seeing is the formal aspect. It's a kind of final touch to an idea that began as an interesting notion.

Take the square {(x,y):0<x<1,0<y<1} and map it one-to-one to the line {r:0<r<1) by using the procedure implied by the simple example (.329576914..., .925318623...) <-> .39229537168961243...

You can figure it out if you stay off the Xmas grog long enough. Although you are a smart physicist and may be pulling our legs. You and Agent Smith can work this out. It cropped up in the course I used to teach in Intro to Real Analysis.

Hence, there are exactly the same "number" of points in the (section of) the plane and on the unit interval. Same cardinality.

This is the correspondence between infinite and infxinfxinf. And if r goes to 0.04566..., or 0.005667... or 0.000556654... (you get my point, I guess...) even infinite times are included. So the map becomes one between inf^2 and inf^4. Aleph1 and aleph2 — AgentTangarine

Are you off your meds again?

You guys need to get a hotel room. — jgill

Disgusting. — TonesInDeepFreeze

Pardon me, Ma'am. Here in the bunkhouse on the prairie I tend to forget there are ladies present. :worry:

I jumped up a dimension to make a point. R^n has the same cardinality as R.

You guys need to get a hotel room.

Repeat for page 2:

That would imply that the covered piece has the same cardinality as the whole square. — AgentTangarine

This is getting painful to watch. A simple example shows that the "number" of points in the interior of a cube {p=(x,y,z):0<x<1,0<y<1,0<z<1} , is exactly the "number" of points on the line {r:0<r<1}:

1:1 correspondence demonstrated by r=.3917249105... <-> p=(.3795..., .921..., .140...)

Extending these ideas shows the cardinality of R^3 is the same as that of R. — jgill

Computers play music.

That would imply that the covered piece has the same cardinality as the whole square. — AgentTangarine

This is getting painful to watch. A simple example shows that the "number" of points in the interior of a cube {p=(x,y,z):0<x<1,0<y<1,0<z<1} , is exactly the "number" of points on the line {r:0<r<1}:

1:1 correspondence demonstrated by r=.3917249105... <-> p=(.3795..., .921..., .140...)

Extending these ideas shows the cardinality of R^3 is the same as that of R.

Ha! You think there is a connection? :) From my own experience, I've known a few physicist and astronomer climbers, but can't recall any mathematicians off the top of my head — SophistiCat

Yes, I do think there is a connection. I knew and climbed with ten or so fellow math guys from different locales over the years. For me it was problem solving and exploration. Short rock climbs are frequently referred to as "problems". A great combination of intellect and athletics.