Probability, for all you logicians

[Nidhogg]

Yeah, but a mechanism isn't supposed to understand the concept of cardinality. To understand this concept, it had to understand the concept of a number first, since all machines work with numbers stored in memory. But if it "knew" this concept, it wouldn't need to understand cardinality, it just could create numbers using the concept.

I still think a mechanism can do it, but cardinality can

 

[Rhazdel]

Yes, I know what cardinality is.

But, the point is that it can't actually draw any number at all. If it can't draw one number, how can it draw the same second number?

There really is no such thing as "infinite range", because a number is only a number when you define it. Therefore, if you can't define all the numbers in a range, then the range doesn't exist.

If you pick a number, you have defined a FINITE range by putting a cap on what possible numbers you can draw from. Therefore, you didn't actually draw it from an infinite range.

 

[Me(tm)]

I think I'm, repeating myself. Infinite is not a defined number. Look at this:

a - a = 0
a/0 = infinity
1/0 - 1/0 = 0
2/0 != a
2/0 - 1/0 = ?

Meaning, even defining the inifinite (an infinite from 1 or 2) will not solve the question, simply because inifite has no cardinality.

If you pick a number, you have defined a FINITE range by putting a cap on what possible numbers you can draw from. Therefore, you didn't actually draw it from an infinite range.

By drawing I hope you mean picking, because you can pick a number form a 0..googolplex range, but by no means you can write that number down. (you would be 'printing' it with powers of).

 

[Rhazdel]

Yes, I meant the "picking" definition of "draw".

 

[Roman Candle]

Here's a similar sort of problem I've been thinking about recently. √2 is a non-terminating irrational number. That is to say, its written magnitude is infinitely long, in any rational base. However, if √2 is accurate, then it expresses all that information. But this doesn't seem possible - how can a finite symbol encode infinite information, and with total accuracy?

My first thought is that (√2) is very different from other numbers, real and imaginary, because irrationals are numbers that have defined properties, (ie that (√2)^2 = 2), but undefined magnitude. In fact, √n is rather like a/n. There are certainly numbers such that a/n = m, where a, n and m are all rational. However, it does not follow from that that for any number m there is a number n such that a/n = m (ie when m = 0). In the same way, although there are certainly rational solutions for √n = m, this does not mean that for every number m there is a number n such that √n = m. My reasoning here is that by using √n you are saying "√n is a number such that √n√n = n", and so are defining a property of the number that has no defined magnitude. This puts irrational numbers in the same situation as ∞: There is no such number, but it is a concept that can be used in equations.

However, as one of my friends pointed out, it this holds true for surds, then it is also true for non-terminating rationals: If there is a number such that 1 = 3n, then n has a defined property but an undefined magnitude. But fractions are definitely not considered in the same was as irrationals. On the other hand, the information of their magnitude is finite - that is, it can be written with the 'recurring dot' notation, (I don't think I can type it :p). So perhaps this counts as having defined magnitude.

Has anyone got any comments on this particular consequence of irrationals? I'm only just starting to think about this sort of thing, and I'm not exactly sure where to begin.

 

[Rhazdel]

Here's a similar sort of problem I've been thinking about recently. √2 is a non-terminating irrational number. That is to say, its written magnitude is infinitely long, in any rational base. However, if √2 is accurate, then it expresses all that information. But this doesn't seem possible - how can a finite symbol encode infinite information, and with total accuracy?

No one said that every number must have a perfect square root. In the case of, for example, 2, no number, multiplied by itself (n^2) equals 2 exactly. But, however, the number can come as close as possible (1.99 repeating, for example, will never reach 2, but for all intensive purposes, does, in fact, equal 2). The programs hard-coded into calculators are set to pick up on such instances and round off, which is why you can enter (√2)^2 into any calculator and get 2 as a result.

So, I don't believe that it encodes the information with total accuracy, simply almost total accuracy.

However, it does not follow from that that for any number m there is a number n such that a/n = m (ie when m = 0).

One could argue that 0 is no exception. For example, √0 = 0, or a = 0, n = 0 and m = 0, while all remain rational numbers (sort of, because 0 is a strange number).

This puts irrational numbers in the same situation as ∞: There is no such number, but it is a concept that can be used in equations.

Like I said before, the concept can be used symbolically in equations. They may not work out to the exact number, but comes closer than any other number. So close, in fact, that it may be considered equal to that number.

 

[Roman Candle]

I think the best way to answer the problem is to say that √2 does not actually represent √2 in the way that 2 represents 2. Instead of actually standing in for a number, it defines a set of ways in which it can used. For instance, √2√2 doesn't equal 2 because it's mathmatically correct; we simply observe that when we see √2√2 we can equate it with 2 because that's what √2 means. It's not, however, a numerical representation.

(PS I didn't mean that √2 on a calculator or worked out has total accuracy. I meant that √2√2 doesn't aproximately equal 2; it equals exactly 2).

 

[Rhazdel]

But it does not equal exactly 2, just close to.

For example:

1/3 = 0.33 repeating.

1/3 x 3 = 1

0.33 x 3 = 0.99 repeating.

It doesn't work out, because 1/3 never ends. It comes infinitesimally close to the answer, but never actually equals 1. We can, however, consider 0.99 repeating as equal to 1.

It's just the way the numbers work out. 1/3 is an infinite answer (much like √2), and the remainder is lost somewhere in infinity. It's the way that the mathematical construct is designed. Not everything works out perfectly (or rather, the way you want it to, because, math actually does work out perfectly).

The number 0.33 repeating does not perfectly represent 1/3 (much as √2 does not actually represent √2, symbolically, anyway), but at the same time, it does. It works out to an infinite degree, and therefore, we can say that mathematically, √2√2 does, in fact, equal 2 in much the same way we can say that 0.33 repeating times 3 does equal 1.

It's merely a simplification of repeating decimals.

P.S. And the only reason I mentioned calculators is because if you punch in √2√2 it will give you 2 as your answer, even though it is not exact.

 

[Nidhogg]

Actually I think it is.
The problem about numbers that repeat infinitely, is that they aren't accurate at all. That means:

1/3 is almost equal to 0.333 repeating

1/3 * 3 is equal to 1

0.333 * 3 is equal to 0.999 repeating(since here it doesn't tell where 0.333 came from, i think)

√2 is accurate. this is a symbol made to help with the use of the actual number, witch is irrational. think about complex numbers. √-1 become the imaginary "i", so we can work with it properly. working with the origin of the irrational numbers is fairly simpler than working with themselves. If I saw a "1/3" somewere, I wouldn't change it until I find the final answer to the problem. taht serves to the √2 aswell.

So yeah, √2√2 equals exactly 2. At least that's what I believe.

Recently I thought about a "neverending mechanism". This may sound dumb(maybe it IS dumb), but it is actually a mechanism that does a loop in itself. it counts a certain amount of numbers at each loop, and have a counter for the number of loops it does. each loop, it has a chance of stoping counting, at a number pick from a finite range(I was thinking about 360, the number of degrees of a circle)

Okay, I can stop writing bullshit now.

Peace.

 

[The Clap]

There's still a chance you can get the same number. Yeah it's pretty big and unlikely but unlikely doesn't mean it won't happen.

 

[Roman Candle]

√2 can't possibly be accurate, because it would be symbolising an infinitely complex, (as in complicated, not real vs imaginary), value. I think the same goes for imaginary numbers. When you use i, you aren't really refering to a number, to which the square is -1; it's simply an indicator that you can use certain rules and techniques. Likewise, when you use √2, you aren't really refering to a number or value; it just carries a set of rules about how you can manipulate it.

Also, if you made a number generator like the one you described, each number would have a lower chance of being picked, since the probability of the loops having ended before it was chosen would always be increasing, so it wouldn't be a random number generator.

 

[Rhazdel]

I see what you mean, Roman Candle.

Yes, the problem with repeating decimals is that they are not exact, as you said. Numbers in math do not always work out the way we would like them to. It's just kind of the nature of the beast.

In my example (1/3), 3 does not go evenly into 1. You get a fraction of a fraction as a remainder, and that remainder gets lost in infinity. Think of 1/3 as 0.33 and 1/3. Because there is no way to write that mathematically, the remainder is dropped off and lost in the repeating decimal. Another way to think of 1/3 is 0.33 with 0.33 as a remainder, with 0.33 as a remainder, with....etc., because each fraction of the fraction has...well, another fraction. There is a remainer, because 0.33 times 3 does not equal 1. 0.33 repeating times 3 gives you 0.99 repeating. The remainder is split among the repeating decimal across the 3 times you multiplied it. This infinitely small remainder is lost, and hence, you don't ever get to 1.

The same applies for √2. No number, times itself equals 2. Therefore, the remainder of the actual square root is lost in infinity.

I think the way Roman Candle put it does sum it up nicely, that √2 is more of a conceptual way of thinking of a value, rather than an actual value, because the actual value is not perfectly accurate.

 

[___]

I think the very simple conclusion to your logical experiment is that 1/infinity != 0. That should be self evident. Being non-quantifiable isn't the same as being non-existent. Or did I miss something in here? It's not impossible to draw a random number from infinity using your system, just infinitely improbable. It sounds like a non-answer unless you consider the two distinct concepts, but if anything your logic experiment proves that they must be distinct concepts.

How do you symbolize infinity without special characters, btw, if anyone knows?

 

[Rhazdel]

That was the whole crux of this debate. Numbers are only infinite in theory. A number isn't really a number until it is defined by people (or man-made mechanisms), so is there really such a thing as inifinite range? In theory, yes. In practice, no, because you will never be done defining your range and therefore will never know what you are drawing from. So, how can you draw from a range that doesn't exist?

That's my theory, anyway. I discussed it in a little more detail in earlier posts.

 

[Me(tm)]

How do you symbolize infinity without special characters, btw, if anyone knows?

If x is allowed... x / 0. Otherwise, I don't know...