math

[Vraith]

Yea...I'm not sure "subjective" applies...but I know that many countable infinities are the same "size" of infinity.
Like, odd as it seems, the set of whole numbers, set of even numbers, set of odd numbers are the same size, even if it seems like the even and odd should be half the size of the whole.
That might be true for ALL countable ones? But I'm not sure I recall that correctly.

It never occurred to me to look or think about it...maybe I will now...but are many or all uncountable ones also the same size as each other?

But the math and material get into all kinds of trouble all the time.
Related to your quantum thing...
People often talk about Planck length being the smallest possible size in the universe. But that's not what physics/physicists really say---it is the smallest measurable/meaningful size, and that's a direct, physical result of energy/observation/uncertainty.
But if you ignore the material, it is relatively easy to say "Hmmm...times 10^-35? Well, I'm thinking now of 10^-36!, I BROKE PLANCK!"

Heh, Peter, if you're listening, energy is also one of the reasons the Throwing out of Rubbish in Infinite-Model Hotels is cool for math-thought, but ridiculous in any and all material terms.

[peter]

Before I go to the link V, can I just ask: as a youngster I was always told the universe was infinite. Is that infinite in a 3D way in the same way that the surface of a sphere is infinite in a 2D way (to an ant walking on the surface), and if so is that not a material infinity, albeit one that it is difficult for us to comprehend?

(V. - I'm always listening: if nothing else, this much at least I've learned. )

[Avatar]

I've always had issues with that "infinite universe" thing. If nothing else, surely it has a finite duration...

--A

[Vraith]

Before I go to the link V, can I just ask: as a youngster I was always told the universe was infinite. Is that infinite in a 3D way in the same way that the surface of a sphere is infinite in a 2D way (to an ant walking on the surface), and if so is that not a material infinity, albeit one that it is difficult for us to comprehend?

(V. - I'm always listening: if nothing else, this much at least I've learned. )

Yea, most people, I think, grow up surrounded by various statements that rope in the claim that the universe is infinite.
If I were forced to place a bet, I'd wager it probably isn't. But I wouldn't be entirely happy with my odds of winning, and I'm very open to persuasion.
Thing is, the front line of our knowledge now is, in several ways, at the point where we really need to know---for the first era of human history, and for practical purposes---which is true.
For most of human history, it didn't matter much---or at all---if the universe was merely incomprehensively vast or literally infinite. Now, it kinda does...and fairly soon it will be essential.

[Hashi Lebwohl]

The universe isn't literally infinite but it does extend out farther than we are able to see at this time, which makes it functionally infinite. Even if we improved the design of that ion propulsion engine to give a satellite more constant acceleration, by the time it reaches the nearest galaxy to our own the Sun will have gone nova and we will have transitioned into whatever we are going to be once we stop being human beings (in the genetic sense).

I have two weird suspicions. One is that the universe is older than we think it is--yes, we know that expansion can result in objects being 20 billion light years out in the space of only 13 billion years (hooray for relativity) but what if the universe really is that old? The second is that the Big Bang didn't happen like we thought, that there wasn't a singularity which exploded; rather, those darned membranes collided like two or more three-dimensional objects intersecting to create another three-dimensional space, which would result not only in an expanding universe but one which does not have a "center" from which all other things are retreating. Can I prove this? Of course not...but it does appear to fit observable facts at a quick glance and can thus be a legitimate alternate explanation.

[peter]

The idea I've had goes something along the lines of "If you head up into the air and just keep going in a straight line, sooner or later you burst up through the ground at the point you left from." This would be a sort of closed loop 3D infinity: you haven't deviated from your straight line, but you're back where you started from (due to the curvature of space maybe?). Is this how it's supposed to work or do I have it muddled up?

[Vraith]

The idea I've had goes something along the lines of "If you head up into the air and just keep going in a straight line, sooner or later you burst up through the ground at the point you left from." This would be a sort of closed loop 3D infinity: you haven't deviated from your straight line, but you're back where you started from (due to the curvature of space maybe?). Is this how it's supposed to work or do I have it muddled up?

That "happens" theoretically in a universe that eventually has a Big Crunch.
I THINK also possible in some ever-expanding, but slowing over time, versions.
But the one we have has some pretty solid evidence of accelerating expansion. On top of that, the curvature of space seems not to exist.
I mean, there are enormous numbers of "local" curvatures---every star and planet, black holes, galactic super-clusters, anywhere there is mass....but on the universal scale, apparently, it is exceedingly flat. Like that old analogy---the Earth, despite mountains and canyons is actually far smoother than precision ball-bearings. [I'm not sure that's true nowadays---we might have made smoother surfaces by now.]

H.--you could well be right. There are still some problems caused by the Big Bang that need solutions. Its biggest advantage right now is that every other kind of origin has far, far more problems. For now.

[peter]

The theorists seem to be outpacing the capacity of the experimentalists to verify or refute in this branch of cosmology at present - or do we just not have the available technology (either coz it doesn't exist or simply because of lack of quantity) to be able to kick some of these competing theories into the long grass?

[Vraith]

The theorists seem to be outpacing the capacity of the experimentalists to verify or refute in this branch of cosmology at present - or do we just not have the available technology (either coz it doesn't exist or simply because of lack of quantity) to be able to kick some of these competing theories into the long grass?

Yes. I mean, there are at least 4 or 5 different big ones that end up with a multiverse. But for most of them, we don't know [and some say it is impossible even in principle to ever know] any test to falsify them.
I think String and Brane [some varieties] have some testable/falsifiable predictions---but no one has the slightest idea right now how to build a device---and even if did, the energy requirements for the tests seem to be absurd.

OTOH---there are a whole bunch empirical results out of the machines/methods we DO have that the theories, as they stand, cannot accommodate. For instance, there's some fun stuff going on with the size of the proton. The results of which [assuming the measurement is accurate, which is becoming more likely could be "oops, we've been slightly wrong on the value of a constant." OR "There is a new fundamental force, new physics."

https://www.quantamagazine.org/20160811 ... us-puzzle/

[peter]

That's interesting stuff V. There is a tendency to think of the 'nuts and bolts' stuff like measurements of atomic radii as being done and dusted - but as this article shows tweaking away at the small stuff (by this I mean the less publicised work - not the less important) could still yet radically alter the bigger picture of how the world works

[Vraith]

This is kind fun for some kinds of people.
I'm in favor of Tau. [[side fact===the Greek guy who discovered Pi did NOT use the Greek letter Pi to symbolize it. He had been dead for almost TWO THOUSAND YEARS before the letter became the notation.]
This link is the paper that started it all [it's kinda mathy, if that bugs folk in language, not just math]

www.math.utah.edu/~palais/pi.pdf

This one has an article link...but really, it's the on the page videos. The two mentioned are NOT very mathy. One labeled short explanation---but STILL almost 15 minutes---for knowledge. Presenter is a weird mix of mostly boring/awkward with some good lines. The last few minutes are pun-guments, which I like. Some may not. [[I didn't try the long explanation video.]]
There is also another vid I like "what tau sounds like." If you're musically inclined, it is pretty fucking cool.
Not as in "great song/composition" but---it is, as a base/root, much better/more implicative than other math/music transmutation/fundamentalizing I've heard. It LEADS places, SUGGGESTS things, sonically. [[by which I mean...it sounds like music EVEN IF you didn't know it was based on specific mathematical concept. Which has not been true of any other direct conversions I've heard.

tauday.com/

Here are just some quotes from some articles on Tau/Pi that cracked me up:

thinking in terms of pi is like reaching your destination and saying you're twice halfway there.



mathematics is the language with which we express and see certain underpinning truths to the universe. To clutter that language with superfluous twos would be as bad as littering a Shakespearean monologue with "likes" and "ummms" and "whatevers" As the Bard nearly wrote, "Knowledge is two of the half-wings wherewith we fly to heaven."


We Americans have almost a proud tradition of using poorly chosen units because of inertia: Fahrenheit instead of Celsius, miles instead of kilometers. Even the great Benjamin Franklin inadvertently established the convention of calling positive charge negative and vice-versa as a result of his experiments with electricity.


"But pie is yummy" remains one of the more compelling arguments for clinging to the traditional ways of 3.14. But tauists have a response for this as well: on Tau Day you get to eat twice as much pie!

[Hashi Lebwohl]

A new analysis of Plimpton 322, a Babylonian clay tablet containing a mathematical table, seems to indicate that it is a table of trigonometric calculations, meaning that the Babylonians had relatively advanced trigonometry about a millennium before the Greeks had it. Apparently they did not base their calculations relative to the angle but the ratio of the sides. An interesting read--the article is only 26 pages long.

[peter]

Hashi, while I have you here - can you quickly explain to me why Pythagoras theorem is provable, while pi is not? There must be some fundamental difference between them mustn't there for this to be so, yet I can't put my finger on what it might be?

(Very interesting post above, by the way (though I confess to not having read the 26 pages); I've always had a soft spot for the Babylonians, what with their trying to build a tower to heaven and confusing the Human toung and all!

[Hashi Lebwohl]

Truthfully, I do not understand what you mean by the phrase "pi is not provable". Pi doesn't require proof--it is merely the letter chosen to represent the ratio of the circumference of a circle divided by its diameter, nothing more. The fact that that ratio is a transcendental number (a transcendental number is a number which is not the root of any polynomial of any degree which has integer coefficients) is not trivial to prove; in fact, this was not proven until 1882 by a mathematician by the name of Lindemann. Even though the set of transcendental numbers is uncountably infinite, the list of known transcendental numbers is surprisingly short.

The Pythagorean Theorem is provable, usually by demonstrating it physically and is an excellent bonus problem for high-school geometry student to undertake.

[peter]

Yes, ok pi is the ratio of a circle's diameter to its circumference - and that ratio is always the same and equals a figure that can't be written down, but is invariably the same, and equates to something just over three. Why should I believe this?

If I take a right angled triangle and measure it's longest side, I'll find that this figure times itself will equal the sum total of the other two sides times themselves added together. This will always be the case, no matter the size of the triangle. Why should I believe this. Because it can be proven by a series of incontrovertible steps that cannot be refuted.

How do I know that like Newtonian mechanics this pi ratio will only hold true for so long; how do I know that if the measurements were made, that at the infinite reaches of the figure, small deviations might not begin to creep in. Is pi true by axiom only then? This is math: a straight line is the shortest distance between two points (of the surface is flat). What can't be made true by definition must be proven. How can it be otherwise?

[Hashi Lebwohl]

This is the really good part--you are not required to believe any of it. Despite your disbelief, if a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then a^2 + b^2 = c^2 will remain true. This is why mathematics is how we will begin communicating with non-terrestrial species of sufficient technological advancement, should we encounter any of them.

It sounds like you want what Hilbert wanted--a way to prove that mathematics is true based on a set of axioms. Sadly, we know from later researchers that this is simply not possible. That being said, for all practical purposes we can indeed begin with a set of axioms such as "on plane P, the shortest distance between point A and point B is a straight line segment connecting A and B"...as long as we first define "point", "line", and "plane".

As far as measuring or calculating pi...well, physically there are limitations because we can't really make instruments which could measure something on the order of 10^-15 m, so if we begin with a circle having a diameter of 1m then the best we could calculate for the circumference would probably be 3.14159, where the last digit is .01mm--you need to have a really well-calibrated micrometer for such measurements.

Never forget--there is a decided difference between "internally consistent" and "true". Much of mathematics is both internally consistent and true, but sometimes "internally consistent" is as close as we can get to "true".

[Zarathustra]

The first derivation of pi was a geometric proof by Archimedes, using the Pythagorean theorem, I believe. If Peter accepts a geometric demonstration of the Pythagorean Thereom, he should have no problem with this geometric derivation of pi.

Now, as whether it's irrational or not,here's various proofs that pi is irrational and transcendental. As for why he should believe them, it would depend on how well he understands them, I assume. I'm not an expert in math, but the proof is in the proving. If you really want to be convinced, I'd say dive in and check their math!

[peter]

Hoisted on my own patard! (Well - not exactly but you get my drift). I do get the proof of Pythagoras - my math will stretch that far but that's about it. To it it in a nutshell it seems to me (and I'm quite possibly mistaken here), that in essence the Pythagoras relationship is ratio based in the same way that pi is: if this is correct then why would one be provable - the other not? It would have to be reflective of some fundamental difference in the applicability of math to straight lines as opposed to curved ones?

[Zarathustra]

It seems, Peter, that your mind is balking at the idea that we can prove something that is infinite with finite means. How do we know that the decimal representation of pi doesn't ever repeat without checking every possible digit?

Mathematicians have developed many clever ways to deal with infinity, infinite series, and even orders of infinity. These methods may seem counter-intuitive to novices like us, but the are proven to work.

On a side note, I think these constitute some of the reasons why we'll never build conscious computers. I think that the leap from finite processes to infinite proofs requires a kind of understanding that computers will never have. The computer will need to run the infinite derivation, whereas we can see "above" this process toward which it logically tends. A computer can never escape or transcend its algorithmic function, whereas our consciousness/understanding was never algorithmic to begin with.

[peter]

Well - I did struggle with Hibert's infinity hotel at it's weirder end; you could make bags of rubbish disapear and stuff (I even remember something about puppies creeping in somewhere!), but no - I do accept these things; I just wonder sometimes is all.

Here's one for example;

Say we take a 3, 4, 5 right angled triangle: we take its sides and pull them into a circle (ie of twelve units circumference). Now if the squares that had been sitting on the sides of the triangle remained in situ (ie with their vertical lines of 3, 4, and 5 along the radial lines and their outer edges formed into concentric arcs with the inner (ie 12 unit circle) ........ would the resultant relationship between their areas still maintain the Pythagorean one of the original equilateral triangle?

Here's another: I am astonished, but believe it to be true that, were you to run a piece of string around the equator one inch above the ground all the way around, it would only need to be a little over three inches longer than if it were run at ground level?

Math is weird - I just gotta get used to it! [/i]