sqrt(2) is rational?

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English: Foto of blackboard picture of a const...

logic is a funny thing. there’s syntax and there’s its meaning. totally independent.
it is quite frequent in maths that while syntax stays the same, meaning changes. the reason why syntax stays the same is: so you can make the same claims. write the same thing over and over again, only the meaning changes. this way the deep insight is shared that abstractly it’s the same thing.
for example numbers. we’re used to a decimal system, 10 digits. abstractly it’s the same as a binary system though. instead of denoting which number-system is meant, be sloppy. adding the b to every number over and over again is redundant. write once you’re talking of binary numbers, and everybody will understand.
only problem is when quoting someone. you’d need to repeat the definitions. so a non-mathematician might write 1+1=10 because someone wrote that. more correct would be to first define what the numbers mean. i.e. say: a number a_na_{n-1}\ldots a_1a_0 means \sum_{i=0}^n 2^i\cdot a_i

 this will be the 1+1=10 kind of proof. i.e. change the definition to get something unexpected.
the definition of real numbers based on rational ones is to look at limits of open rational intervals. every real number is a (non-unique) sequence of open intervals, each containing the next. and every such sequence which converges to an empty set represents some real number.
this real number then will be contained in every single interval of that sequence.

more exactly, every real number is a whole lot of such sequences, a whole equivalence class. two sequences are in the same equivalence class if eventually the intervals mutually contain eachother.
i.e. there exists a sub-sequence of one such that they’re inside the intervals of the other, at same index. and also the other way around, as both are open intervals, by definition containing points other than their border.

in a way real numbers are infinitesimal intervals, while rational numbers are actual points you can draw.
but then, \sqrt 2 is also a point you can draw! just take a  1×1 square, it will have diameter \sqrt{1^2+1^2}.
well, ability to draw means it’s constructible on paper with a ruler and compass. so, no, this can’t be the proof.
we need to show, \left(\frac pq\right)^2==2 or p^2==2\cdot q^2 for whole numbers p,q:
obviously p must be an even number, but then squaring it makes it divisible by 4. hence q must be an even number, and p^2 must be divisible by 8. and so on.
in the end both, p and q must contain all powers of 2. i.e. they must be infinite in size.

is that allowed? in the axioms of ZFC it isn’t. actually it isn’t the axioms on their own. what isn’t allowed is to create an expression that is infinite in length. in the language of logic this is clearly forbidden, you can’t have logic-expressions “converge” to something.
but why not? in fact, there are several approaches to create a set \sqrt 2 is contained in, as a point!

i.e. you can actually turn some real numbers into actual constructible points. apart from circle also other shapes could be used for construction. naturally construction of those other tools require some kind of approximation.
but then, if you restrict yourself to the 1-dimensional line, rational numbers are an approximation of their own already!
it’s possible to check if a rational number is what it claims to be. just use the compass to copy the rational number q times along the straight line. in the same way also other approximations can be verified for accuracy.

math isn’t just geometry. it’s also about formulas and functions. most prominent example is derivatives and their inverse, integration.
every formula has a formally definitive derivative. but formal integration is not always possible. so, while derivatives are constructible by formulas, integration is an approximation.
hence, real numbers created by integrals of formulas can be verified by derivative. but as language allows only finite expressions, these real numbers are countable.

English: Illustration of how the rational numb...

English: Illustration of how the rational numbers may be ordered and counted. Illustration contains a grid and an arrow, and two Swedish words (the words for “nominator” and “denominator”). (Photo credit: Wikipedia)

the notion of countable means: there exists a mapping onto the natural numbers.
for example whole numbers are countable: map every positive number to the odd numbers. negative numbers and zero are mapped to even numbers including zero.
a similar method works also for rational numbers. just interpret them as integer-coordinates on a half-plane.
then use some distance-function mapping them on natural numbers. and then add some ordering to points with the same distance.
(well, maybe this will skip some natural numbers. but it shouldn’t be a problem to re-index.)

no such strategy will work for infinite-length expressions. polynomials are countable, but power-series \sum_{i=0}^{\infty} a_i\cdot x^i aren’t. for every method to count them, one can create an element that isn’t counted.
i.e. a map from natural numbers to real numbers has no inverse! isn’t surjective!

thanks to this limitation of logic, only countably many reals are constructible. an uncountable amount remains as intervals.
in logic the constuction of a point cannot take up infinitely many steps. if it would, maths would be an intangible mess,
you cannot prove consistency by finite-size proofs then.

Gödel’s incompleteness theorem says in our mathematical logic you cannot prove consistency.
the proof starts out with the observation that expressions can be counted. same with proofs. then it goes on showing how this counting can be implemented in the language of logic.
the problem arises when you start proving stuff about these numbers. then you get into the same mess as with my proof above. some things require expressions of infinite length. but as history has shown, then inconsistency easily would creep in.
when dealing with infinity, a lot of additional information is required. literally, an infinite amount of information, infinite sets of axioms and such. talking of infinity is always alike to leaning out of a window: easy to lose your grip on the ground.



la petite morte is what dreams have in common with reality

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some say “life is a big dream”. I disagree with that.

“life is just a big dream” however sounds somewhat more true.
the word “just” signifies a kind of limitation. here it means that life is at most as limited as dreams are.
maybe dreams have more limitations than reality.
but obviously the author of this saying is convinced life has some important limitations in common with dreams.

I guess I have to explain in more detail why I used the notion “at most” instead of “at least”.
let’s start with set-theory: a set is a collection of elements.
however at the same time a set is also a collection of elements that fulfil certain properties.
a finite set can have each of its objects individually listed in such a description, an infinite set can not.
the reason is that for defining some properties only finite-length expressions are allowed.
so when dealing with sets of infinite size, one really is dealing with limitations of some predefined infinite set.
to define an infinite set you take another infinite set and add limitations to it, you throw out elements.

now lets say life is a set of things that could happen, particularly my life is a set of experiences I could potentially have.
dreams are part of my life too, for me it is possible to re-experience all the same things I already experienced in dreams.
those things happened in my mind while I had these dreams. wouldn’t be surprising if my mind could repeat that.
even when I am awake and perceiving my surroundings, I can still additionally experience these things.
this is called hallucinations or day-dreams. therefore the set called “life” additionally contains what otherwise isn’t part of the set called “dreams”.

life has less limitations than dreams, in terms of what one could possibly experience.
for example when awake I can experience defecating, and my creations will remain real. can’t do that in mind alone.
some things possible in reality simply are not possible in dreams. hence dreams have more limitations than life even more than waken life.
the trick here is to interpret dreams as experiences of mind, instead of taking the hallucinations at face value.
then it is natural that experiences of mind also are possible when fully awake.
leaves the question, does life have any limitations at all?
but it is for sure, the limitations we have in life also apply to dreams, they merely are irrelevant there since dream-experiences aren’t bodily experiences.

I have been told, the limitation life has in common with dreams is that both are completely pointless.
you live your life, and eventually you die, there is nothing you gain. similarly a nice dream eventually ends — in disappointment.
also the other way around: all your life long you had those fears, as death comes they all are pointless, a gigantic relief.
at least this is the current time’s interpretation of the quote I made at the beginning.
we can’t really know how the words were originally meant.

dreams are so much more than life could ever offer. a dream is like an experiment in a laboratory!
all the things that encumber us in our waken state, they all are gone in our dreams.
we even frequently lose our memory of life’s hardships, when we dream.
in these ideal conditions we can experiment with our mind’s potentialities, explore mind’s limitations.
can’t do that in the mess we call “life”. in dreams we are responsible for the order, when awake this responsibility is shared.
who ever tried to re-live experiences from a dream also in waken reality, quickly will get disappointed.
some people are capable of experiencing hallucinations, alike to the ones experienced as dreams.
but even for these people such experience is more limited than within dreams.
the biggest difference is that when sleeping we have our eyes closed.
so while waken hallucinations  must adapt to our surroundings, dreams only adapt to our 4 facial senses.
light falling on the eyelids, smells, sounds, tastes, all find their way into dreams.
but when hallucinating awake there further is the sense of touch, and actual shapes we see with our eyes.
these two make it quite impossible to hallucinate for example about sexual experience. but wet dreams we can have anytime.

So, what is the meaning of waken life then? why not just live in dreams all the time?
truth is all I said in the previous paragraph is unimportant, except maybe for a scientist who writes books about mind.
the most important characteristic of dreams and of reality in general is the continuity, the (relative) stability.
we do something, and it has an impact, we are the cause for some effects. our actions are what makes life meaningful.
of course no effect is forever. nothing will last. but relatively to our lifespan, we can actually build up something reliable.
some of our deeds will have effects for hundreds of years.
throw a plastic bag into the ocean and you made a monument for centuries.
for your whole life you then can rely on any fish caught there to contain remnants of your contribution.
dreams have something similar too, but there time-spans are much shorter since dreams are shorter than life is.
and also relatively speaking, the effects dreams have on the future is much more limited than what we do when awake.
the difference here is that dreams depend on mind only. as soon as mind forgets, also its seemingly stable creations disappear.

one just can’t live in dreams alone, our mind isn’t meant to be used that way and our society hasn’t yet automated the gathering of food.
maybe in future this will change, maybe in future nobody will ever forget anything and eating will be a relic of the conservatives.
but till then we have to face the facts. human isn’t meant to live life inside of whatever mind.
even in terms of learning and understanding, our mind isn’t really trustworthy.

this worthlessness of mind, in terms of instability and fragility, this is the only thing that equally limits waken life and dreams alike.
any student will tell you: memory is the biggest challenge in acquiring knowledge, healthy nutrition comes second and could also become part of the same problem.
of course we have great memory, we could learn thousands of books by heart. but that’s not the problem I’m talking about.
let’s look at dreams for example: you see a wall, then you turn around, and as you look again the wall is gone. why?
obviously you noticed the wall is gone, so it cannot be a problem of memory. you remember there’s supposed to be a wall.
same with the student: after many exercises, a completely analogous problem at the exam seems unsolvable. black-out.
and even when the exam is passed, later in job and wherever applicable in private life, all the training is wasted.
we learned things in school and in reality we never even get the idea to apply them. why?
in both, dreams and school, mind is only as strong as it was a moment ago.
you didn’t perform the algebraic exercises a moment ago, then you now must relearn how to do them.
in a dream you stopped looking at the wall, so it isn’t surprising when it’s gone completely.

you must keep mind occupied with some activity, otherwise that activity will need to be re-learned anew.
as a rule of thumb, after 2 weeks without training, whatever abilities you had are lost.
an exercise must be repeated once a week to keep mind alert for that kind of situations.

there is the saying “for a man with a hammer as the only tool, every problem looks like a loose nail”.
this point of view, this seeing only loose nails all around you, I claim this point of view has a time-limit.
i.e. you put your hammer aside for a week or two, and you’ll stop seeing all those loose nails.
maybe other people have a different time-limit, but for me it is at least 1 week and at most 2 weeks.
when I was pretty good at some math stuff and I didn’t do it for that time, I am not good at it anymore.
of course I still can do it, my memory of how to do it isn’t lost, I just stop being so masterful at it.

in dreams there is no such time-limit, instead some sort of attachment is required.
in order for the wall to stay where it was as I turn my back to it, I must continue to feel its presence.

maybe I hear the wall, in the sense that sounds from behind it are muffled.
or maybe I feel how the air-currents get stopped by the wall, or I see its shadow.
maybe I feel the coldness of the wall or I feel how it is looming behind my back like a giant.
luckily this kind of continuity can be trained, now I even am able to leave a room and return to it without problems.
however, that kind of training too has the derogation factor. after about a year without training also this ability is lost.

now to summarize: human is changing all the time, nothing is forever, most things wont hold even for a lifetime.
abstractly seen, if a human had only a single ability, it would be as if that human would reincarnate every week, into the same body.
once a week it would be as if that human died, and someone else is then occupying that body.
this new host still has the same physical set-up, also the same knowledge, but character and abilities differ.
it’s as if the previous host would have left behind some book, from which to re-learn the abilities.
similarly also character can be re-acquired from the remnants of the previous person in that body.

but this kind of relearning has the same limitation as any kind of communication:
the main landmarks of the knowledge can be conveyed in detail, but it’s up to the person to connect the dots.
in mathematics one could say that only a countable subset can be conveyed, the completion must be done manually.
but this isn’t accurate, our memory can only store a finite amount of information, no communication can go beyond that.
so we have some limited description, some thoughts telling us what to do, how to do it we must figure out.
keep this in mind next time you learn something new, it really matters how you store that memory.
no matter how good you are at something after learning it, it’s important to formulate what you learnt for your future selves.

I think at this point I should emphasize, when I talk of death, I mean it!
in my experience it is wrong to beautify the memory-loss into something happening concurrently.
it’s really a cut, one moment I had an ability and the next moment I don’t have it anymore.
but it’s also wrong to claim the real you is dying at some point in your body’s life.
it’s always a small loss of abilities, one ability after the other is lost, till a whole bunch is gone.
also it happens at random time, mostly at a time you relax, for example during sleep.
and most importantly the loss isn’t being noticed till you need the things lost.
it’s as if part of you died and you are facing some zombie instead, some alien person.
in such moments we then say “why did I do such a stupid thing?”
of course it isn’t an alien, we just suppress the fact that nothing is forever, even our mind dies — piece by piece.

no matter if dream or reality, in a way we all die a little from time to time, repeatedly. always keep in mind that secondary mortality.
what’s the point in learning anything at all? you wont improve your abilities after these 2 weeks of training.
whatever new tricks you learn after that time, you will forget about all the old tricks which you then neglected.
if your goal is some kind of mental achievement, never waste more than those 2 weeks.
you want enlightenment? it takes only 2 weeks! if it doesn’t then you will never reach it! give up! or at least focus your life on it!

well, that really isn’t true. as I implied while talking about dreams, the ability of not-forgetting can be trained.
in dreams I have managed to do that, why not when awake?
if abilities of mind are as important for you as they are for me, do that!
keep track of how quickly you lose abilities and make that time become longer.
the secret to this it to avoid being distracted. in the dreams mind must be continuously occupied with the objects.
similarly when awake keep your mind occupied with the abilities you want to keep.
make a list of what abilities you need and create a training-schedule.
this schedule isn’t the important thing here, what one must learn is to be more systematic in the training.
if mind wanders off into unrelated fields, a lot of time is wasted that could have been used for being occupied with some new tricks.
and most importantly, always keep track of what your mind does do, at any time.
it’s important not to be controlled by circumstances, and instead keep up control of the own mind…

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Fractal Concepts

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I mentioned Jenson’s Paradox. it really is no paradox at all, when you think of it in terms of real numbers.

just to clarify, I am talking of connecting 2 points with a zig-zag path.
this path has infinitely many rectangular angles and a total length of 2.
that path looks like a diagonal line though.
also mathematically it can’t be distinguished from the diagonal line.

for every point below the diagonal, at some moment in the construction the zig-zag line will be above that point.
for example the point C, or any point between C and D.
already in the 2nd step they all are way below the constructed zig-zag shape.

however, the concept of infinity makes it clear that this isn’t a diagonal line.
the problem is that there are uncountably many points not being put onto the diagonal line.

in my construction the corners get mirrored onto the diagonal between A and B
this process is repeated only countably infinite many times
this leaves uncountably many parts that aren’t part of the diagonal.
what’s worse, you don’t even need to believe in uncountable quantities.
is \sqrt 3 an x-coordinate for a point of the zig-zag line lying on the straight line connecting  A and B?
mathematically it is. it converges there. but at which step does that point get added?
unfortunately mathematics doesn’t come up with tools for talking about such cases. the notion of “derivative” comes close though…

therefore it isn’t surprising that the length is 2, as the constructed line is below the diagonal most of the time. wouldn’t even be surprising if the length of a similar line was infinite!

and this is what fractals are about: a discrepancy between size and actual objects of that dimension.
thanks to a point of view that takes infinities into account we can investigate the fractal’s shape.
of course our actual world doesn’t contain true fractals. and the uncertainty-principles prevent that in small sizes too.
however, abstract processes can be of fractal nature, our thoughts are such a beast of infinite repetition.

of course we don’t live forever, and so also infinite thinking makes no sense.
eventually there’s a  level where thinking would become coarse-grained.
but that’s beside the point! to estimate where your finite thinking will go, you need an approximation in terms of infinities.

additionally there’s the theory that part of our thinking is happening in the way of a quantum-mechanical computer.
if that’s true, then infinities would play an important role in studying how we think.
then our thoughts would be sort of fractals created by simple rules.
and just like fractals, our thoughts most likely will never encompass everything. there always will be holes, just like in above fractal…

as a side-note, is above construction really creating a fractal?
when you scale it up evenly, the length will be doubled, just like the straight line.
neither we have self-similarities nor “rough shape”, it’s merely lacking derivatives.
however, stretch it in the x-axis, and it will be size 3. a line stretched that way would be size \sqrt 5