Probability, for all you logicians

[FzGhouL]

Deleted

 

[FoxDemonSoavi]

zero represents nothing
any other number represents something
something is the opposite of nothing

 

[DaGuyWitBluGlasses]

The opposite of zero is every other number.

Beautiful and Pretty are both opposites of ugly.

Beautiful and Pretty are synonyms.

1 and 2 are not synonyms.

 

[Roman Candle]

Depending on how you define the opposites of numbers, anything could be the opposite of 0. For instance, 0 is a real number, so the opposite of 0 should be an imaginary number in terms of i. 0 is an integer, so the opposite of 0 should be an irrational? 0 is not nothing; it's a value, but with no magnitude.

 

[DaGuyWitBluGlasses]

Depending on how you define the opposites of numbers, anything could be the opposite of 0. For instance, 0 is a real number, so the opposite of 0 should be an imaginary number in terms of i. 0 is an integer, so the opposite of 0 should be an irrational? 0 is not nothing; it's a value, but with no magnitude.

Integers and irrational numbers are not opposing.

(Opposing invloves things that are equal/comparable. The opposite of any integer has to be an integer as well. (The opposite of the beginning of a book, is the end of a book, not something different like the end of a road.)

6i * 1 = 6i , 6i*2 = 12i, but 6i*0 = 0. So 0 can fit into the imaginary numbers just as well, there is no imaginary equivalent of 0.

 

[FzGhouL]

It depends on how you define opposite, to be honest.

So zero has infinite opposites, finite opposites, and no opposites all at the same time, varying by definition.

This is because opposites are human constructs and thus things are only opposite by construction.

 

[DaGuyWitBluGlasses]

It depends on how you define opposite, to be honest.

So zero has infinite opposites, finite opposites, and no opposites all at the same time, varying by definition.

This is because opposites are human constructs and thus things are only opposite by construction.

Practical definitions trump arbitrary ones.

There are practical reasons why you need to know a negative is the opposite of a positive (understanding how magnets work, understanding molecular structure,) There can be situations where knowing refering to opposites that are nearly identical (understanding the difference between Vitamin C your body can use, and ascorbic acid, where only half of the molecules are actually vitamin c, even tho the others have exactly the same atoms, charges, and even shape, vitamin C is specifically a left-handed).

 

[FzGhouL]

EDIT:

Actually just forget it. Waste of time. This is completely off topic. Open a new topic if you want to debate this.

 

[Shadow_Strike]

Chaosg1, I carefully didn't use the word computer, because computers can't generate real random numbers.
macchia, I see what you mean, but even if you had a number that could express the highest value in the universe in it's most fundemental unit, it wouldn't be the highest number possible. For instance, you could find that number and double it, for fun.

Anyway, thanks for your comments.

I suggest you reread your original post.

 

[Murcu]

Hang on. Guys, this is just a failing of the English language. Its unable to rectify the obvious contradiction in this math equation. This can happen before, in which words cause a problem to be impossible.

Just as much as there is an infinite chance one of the numbers won't be called, there is an equally infinite chance that it WILL be called. When you have infinite in a set, your possibility of getting something is incalcuable.

 

[Roman Candle]

ShadowStrike: A mechanism is not the same as a computer. I don't catch your meaning there.

 

[Murcu]

I think you're expected to try and see passed the obviously impossible situation of this question, and just try to imagine a computer which CAN count to infinity.
I expect he called it a mechanism in order to shy away from the more limited perception of a 'computer'

 

[Shadow_Strike]

You have two mechanisms which, based on some kind of totally random event, atomic decay or some kind of quamtum event or summat, continusly generate random numbers. Any rational real number can be found. As there are an infinite number of such values which could be found, and they are all equally likely to be found, it seems that the chance of both computers coming up with the same number is infinitely unlikely. In fact, the probability of each value is 1/(1/0), making each one infinitely unlikely. In that case, it seems infinitely unlikely that a number will be chosen at all. However, it is still entirely possible that both computers will choose a number, and entirely possible that they will both choose the same number. Now, I'm sure I've gone wrong somewhere here. Any comments?

Sure you didn't put computer in there? Thats why the first guy said computers don't generate numbers.
Because you put the word computers in there, twice.
Just sayin', because I have nothing else to say, everything I would have said has been said.

 

[Roman Candle]

Don't know how I missed that ^_^
Anyway, I think what I've already said shows what I mean. Sticks and stones may break my bones, but pedantics are just boring XD

 

[Nidhogg]

It is necessary to abandon finite number intuition to work with infinite values, so propability wouldn't exactly have the same application here. you can't say the propability is 1/infinite, simply because by doing this you are transforming infinite in a number.

By working with infinite values, all propabilities become infinite. That is, the mechanisms have infinite possibilities of generating the numbers(for example, the propability of it to generate the number 142857 is infinite). But the propability of have two of the same number is 1/2.

What the hell? you say. Simple. You actually never saw infinite number in action, neither I did. The possibilities we have are:

-Having two diferent numbers
-Having two of the same

we can call these numbers X and Y, just to explain better. Since we're working with two mechanisms we have infinite pairs of (X,Y) where X and Y are the same, and we have, too, infinite pairs of (X,Y) where X and Y are diferent, so the two possibilities have the same number of returns!

Now the question if it is possible to generate numbers in an infinite array... well... It's propably possible, since we can undestand the composotion of numbers, and the concept of unit, hundred, thousand, and such, we could make an algorithm for a machine to understand it, too. It would be a lot more complicated, but not impossible. Sorry 'cause I don't have an answer involving math for this one, but I'll think a lot and try to answer later.

that's it...
thanks!

 

[Rhazdel]

By working with infinite values, all propabilities become infinite. That is, the mechanisms have infinite possibilities of generating the numbers(for example, the propability of it to generate the number 142857 is infinite). But the propability of have two of the same number is 1/2.

What the hell? you say. Simple. You actually never saw infinite number in action, neither I did. The possibilities we have are:

-Having two diferent numbers
-Having two of the same

we can call these numbers X and Y, just to explain better. Since we're working with two mechanisms we have infinite pairs of (X,Y) where X and Y are the same, and we have, too, infinite pairs of (X,Y) where X and Y are diferent, so the two possibilities have the same number of returns!

Not quite. You say that two mechanisms drawing a random number from an infinite array will have a 1/2 chance to produce the same number just because there are two possible outcomes:

A) The numbers drawn by each mechanism can match
B) The numbers drawn by each mechanism do not match

Your probability is off.

By flipping a coin you can get either heads or tails. What is the probability of getting 2 heads?

Probability of 1 head = 1/2

So, the probability of getting two heads is 1/2 x 1/2 = 1/4.

You say that it has a 1/2 chance. That means that if you were to run the experiment twice, the two mechanisms, on average, would choose the same number 1 out of every 2 draws from the array.

With infinite possibilities for each number, if you were to quantify infinity, would look like this:

1/infinity x 1/infinity = 1/infinity.

Because there is no limit on the array size, there can be no calculable probability it is theoretically not possible to draw any number from an endless array, because, by definition, you have to have an array in order to define what you are drawing from.

A mechanism could potentially draw a random number from a seemingly infinite array, but really all that would happen is it would subconsciously create a maximum value to complete the array before drawing the number.

Anyone can start writing down numbers and stop after reaching an certain number of digits, but it is impossible to have an infinite array, because theoretically you can have a number so large that it could not be written in font the size of a single proton across the entire universe (unless you argue that the universe is infinite, but I won't go into that). Because these numbers cannot be defined in any real way, a cap must be applied to the array in order to draw any real number from it, even if the cap is subconscious.

 

[Nidhogg]

you right.. except by this:

Anyone can start writing down numbers and stop after reaching an certain number of digits, but it is impossible to have an infinite array, because theoretically you can have a number so large that it could not be written in font the size of a single proton across the entire universe (unless you argue that the universe is infinite, but I won't go into that). Because these numbers cannot be defined in any real way, a cap must be applied to the array in order to draw any real number from it, even if the cap is subconscious.

I'll not discuss who is right or wrong about the size of universe... But I have a question for you: If it was finite, when you got to the end of it, what would be next? nothing? well, I myself believe not. Something must be after there... there's aways something after...

About a number who couldn't possibly be writen down... There's no such number... every number can be writen in a notation who could subtract the size of it to a few numbers below 100.

Possibilities are infinite. First, almost everyone is stuck by the "normal" way to write a number, that is, when it reaches ten, a new digit is add. but you could use hexadecimal witch adds a digit only when it reaches 16, or even use a notation witch only adds a digit when reaches 100.

second, there's a lot of notations to make large numbers smaller... The most famous is the scientific notation, but there's the Graham's number notation, too, witch uses tetration and knuth's up arrow notation:

http://upload.wikimedia.org/math/3/5/f/ ... 4f4b38.png[/img]

there's lots of other notations there, and you could easily create a new one.
as you said, a number too big to be writen is just theory, so, there's not much to add here.

as to the propabilities, yeah, I was equivocated. I myself said to abandon finite numbers intuition, and finished using it... XD

As I said, when working with infinite numbers, all propabilities become infinite. so you don't even have to write infinite*infinite = infinite (simply because there's no such calculation), you just have to know that the possibility is infinite itself.

As for the impossibility of taking a random number in a infinite array, I still doubt that. The computer, witch I'm using and you're using and I'm using right now, just can't do that because it don't understands the concept of a number, it just have an array of them in it's memory. but a mechanism advanced enought could get close to "learn" this concept, close enough to create it's own numbers. If it can create a number, randomizing any number from 0 to infinite would be piece of cake.

and, to talk a bit out of the topic's subject: In the medieval era, no one tough there would be more earth after the ocean... But now we already got to the moon... History only show that our mind is aways a bit closed comparing to the mind of future generations...

 

[Roman Candle]

A computer could never understand a number. A computer just has information on it, in certain configurations. It's only when that affects the screen in ways we perceive as colours and shapes that we understand the number. The computer just has a mass of data; if you took away the certain piece of data which 'records' where the different types begin and end in the memory, ('record' in inverted commas because there's nothing inherent about the data which makes it a record; it could just as easily be interpretted as a number, but different hardware acts on it), then all the meaning which could have been drawn before is lost. In a way, the syntax of the information has been lost. There's nothing inherent about the shape 1 which makes it a numeral, or indicative of a number. It only becomes a number in the mind of someone who already knows that that shape refers to that number.

I've lost my way a little here, but the point is, I don't think a computer could understand a number. But neither do I think that that makes any difference - it still has to form a physical representation of the number, and it seems impossible to do that if you want each number to have equal probability.

 

[Rhazdel]

you right.. except by this:


I'll not discuss who is right or wrong about the size of universe... But I have a question for you: If it was finite, when you got to the end of it, what would be next? nothing? well, I myself believe not. Something must be after there... there's aways something after...

About a number who couldn't possibly be writen down... There's no such number... every number can be writen in a notation who could subtract the size of it to a few numbers below 100.

Possibilities are infinite. First, almost everyone is stuck by the "normal" way to write a number, that is, when it reaches ten, a new digit is add. but you could use hexadecimal witch adds a digit only when it reaches 16, or even use a notation witch only adds a digit when reaches 100.

second, there's a lot of notations to make large numbers smaller... The most famous is the scientific notation, but there's the Graham's number notation, too, witch uses tetration and knuth's up arrow notation:

there's lots of other notations there, and you could easily create a new one.
as you said, a number too big to be writen is just theory, so, there's not much to add here.

as to the propabilities, yeah, I was equivocated. I myself said to abandon finite numbers intuition, and finished using it... XD

As I said, when working with infinite numbers, all propabilities become infinite. so you don't even have to write infinite*infinite = infinite (simply because there's no such calculation), you just have to know that the possibility is infinite itself.

As for the impossibility of taking a random number in a infinite array, I still doubt that. The computer, witch I'm using and you're using and I'm using right now, just can't do that because it don't understands the concept of a number, it just have an array of them in it's memory. but a mechanism advanced enought could get close to "learn" this concept, close enough to create it's own numbers. If it can create a number, randomizing any number from 0 to infinite would be piece of cake.

and, to talk a bit out of the topic's subject: In the medieval era, no one tough there would be more earth after the ocean... But now we already got to the moon... History only show that our mind is aways a bit closed comparing to the mind of future generations...

You are right about the universe. There are many debates about the paradoxical nature of infinite. How can the universe be infinite, but at the same time, how can it be finite? I have no opinion on the matter, because to me, both possibilities seem equally impossible (I believe in more of a combination theory, personally, but....). But, that is a little off topic, so it is totally open to interpretation.

And, yes, you could shorten numbers using notation, but, again, the theory stands. The notation would have to be mapped out to an infinite degree, because the numbers would never stop, therefore the mechanism (not necessarily a computer, mind you) would have to define all the notation.

Plus, even if you use notation, the number could still be theoretically too big to write across the universe. Imagine using a notation system in which dots, each the size of a single proton was used. Combinations of these dots gave the notation. It is still entirely possible (if you argue that the universe is finite) that the mechanism could draw a number so large that even using every possible minute proton in the universe could not provide adequate notation. Storing data in energy form, also runs into limitations, not with size, but simple supply of energy! Again, it is all theory. And if you argue that the universe is infinite, then all of this goes out the window.

And I know that I didn't have to put infinite * infinite = infinite, I did that to illustrate a point, that's all.

I just believe that if you are trying to pull a number from an array, you must define the array first. The number 16 fits into the array of 0 to infinity, but it is simply because my brain has put a subconscious cap on my array. There are numbers in that array so large, my brain would never consider them.

Numbers are constructs of the human mind, and as such, you cannot draw a real number out of an array that contains things outside the original construct. People have defined numbers, and therefore, there (arguably) is no such thing as infinite numbers because a number is nothing without human definition and interpretation. The concept of infinite numbers holds true, but actual, factual data about infinite numbers is not. Like I said, there will always be a maximum number in the array, even if it subconscious.

 

[Shadow_Strike]

If the universe is finite, then why the hell do you want to know what comes next? Does it matter? How will it affect your everyday life?
Maybe its nothing, a hard concept to grasp for people, maybe its another universe, like galaxies or something, maybe the universe is just a strand of spaghetti.
You don't know, it doesn't matter if you know or not. If you don't know whats on the other side of the street, can't see ~ can't hear ~ can't travel, nada, you can't assume its either nothing or something. Its one or the other and it honestly makes no difference.