Print

Please login or register.
1 Hour 1 Day 1 Week 1 Month Forever
Login with username, password and session length

[Vector]

Oh, from Pandemic? >_>

No, more like "YOU BROKE THE UNIVERSE AGAIN?  Come back when you're more worthy!"

[Sowelu]

Man, I want to take a class where I can break the universe.

[Tellemurius]

Man, I want to take a class where I can break the universe.

Its called Physics Theories of Real Life >_>

[ed boy]

This is more of a logic question than a maths question, but I don't want to clutter up the forums with another thread when this one will do.

There are some statements that cannot be true or false. Consider, for example, the statement:
-This statement is false
Trying to classify it as true or false results in a contradiction. There are other such staments, but all the ones that I can think of are recursive (that is, the statement refers to itself).

Are there any statments which are neither true not false, and are not recursive?

[Vector]

Yes, there's other sorts of issues similar to that one.


"The person who shaves the barber, in a town where the barber shaves every man who does not shave himself."

"Call a 'really big number' a number that cannot be described in 1000 words or less.

Now consider the least really big number."

(5 words: "the least really big number.")

"The statement below this one is false.

"The statement above this one is true."

The statement of the continuum hypothesis cannot be proven true or false in ZFC.


Is that what you were thinking about?

[Darvi]

Actually one of those uses a double moebius reacharound recursion.

"the cat is alive"

That one is both true and false, and thus neither. If the cat is in a box, that is.

[ed boy]

Well, those can be considered recursive.

The first example you gave was "The person who shaves the barber, in a town where the barber shaves every man who does not shave himself."
There is the possibility that the barber is female, in which case there is no paradox, but that is probably not what you meant by it.
If the barber is male, then the part of the statement "The barber shaves every man that who does not shave himself" is recursive.


The second example you gave was:
"Call a 'really big number' a number that cannot be described in 1000 words or less.
Now consider the least really big number."
(5 words: "the least really big number.")
The description of the least really big number is related to the definition of the least really big number. Since the description is related to the defnition, the definition is dependent upon the definition, and it is therefore recursive.


The third example you gave was:
"The statement below this one is false."
"The statement above this one is true."
In this case, the truth of the first statement is dependent upon the truth of the second. The second is dependunt upon the truth of the first. Finding the truth of the first requires you to find the truth of the second. Similarly, trying to find the truth of the second requires you to find the truth of the first. It is still recursive.


The fourth example you gave was that the continuum hypothesis cannot be proven true or false with ZFC. That doesn't mean that it cannot be true or false, just that it cannot be proven using the axioms of ZFC.

"the cat is alive"

That one is both true and false, and thus neither. If the cat is in a box, that is.

That is not quite the same: As far as the limited observations of the people outside the box go, the cat is both alive and dead. However, that is not saying that the cat has no dead/alive status.

[Vector]

Ah, well, every statement is either true or false within a context.  Without a context, there is no "true" or "false."  There are no statements that are both true and false outside of a logical framework that defines what "true" and "false" mean--nor even any statements that are just true or just false.

Otherwise, we wouldn't need an axiomatic system to express mathematics.  We'd just assume this context that allows us to make non-recursive statements.

I don't have much formal training in logic, but this is, at the very least, my understanding of the situation.  Hope this helps.

[Leafsnail]

There are no statements that are both true and false outside of a logical framework that defines what "true" and "false" mean--nor even any statements that are just true or just false.

Arguably, there are some things you can say which are true by their very nature.  "I promise that I will...", "I name this ship...", or "I aknowledge that..." could be examples - if I say that I am promising someone something, then I am promising that person something.  Even if I were to say "Haha, only kidding" straight afterwards that doesn't change the fact that I made a promise with my ealier statement.

Or, for always false... "This idea cannot be expressed"?

That is not quite the same: As far as the limited observations of the people outside the box go, the cat is both alive and dead. However, that is not saying that the cat has no dead/alive status.

Pretty sure quantum theory says that the subatomic particle is in literally in neither state/ both states until you measure it, at which point it changes to one or the other.  The cat was added to ridicule this idea.

[Vector]

There are no statements that are both true and false outside of a logical framework that defines what "true" and "false" mean--nor even any statements that are just true or just false.

Arguably, there are some things you can say which are true by their very nature.  "I promise that I will...", "I name this ship...", or "I aknowledge that..." could be examples - if I say that I am promising someone something, then I am promising that person something.  Even if I were to say "Haha, only kidding" straight afterwards that doesn't change the fact that I made a promise with my ealier statement.

Or, for always false... "This idea cannot be expressed"?

Ah, yes, Austin and his performatives.  But then they are neither true nor false, just complete or incomplete.

That's a very good example, but it is more rhetorical/semiological than mathematical.  I felt that the interaction was between statements that carried both truth values simultaneously, rather than having neither applicable in the first place.

[Virex]

This is more of a logic question than a maths question, but I don't want to clutter up the forums with another thread when this one will do.

There are some statements that cannot be true or false. Consider, for example, the statement:
-This statement is false
Trying to classify it as true or false results in a contradiction. There are other such staments, but all the ones that I can think of are recursive (that is, the statement refers to itself).

Are there any statments which are neither true not false, and are not recursive?

Let's see what our options are, because I have no clue myself.
Let S be a statement. If S is not descriptive, that means evaluating S does not result in truth or falsehood, then S is neither false nor true. This is of course the trivial case, a statement that cannot be true or false by virtue of not evaluating to truth or falsehood in the first place. An example of this is "Wait for me!".


The other option is that S evaluates to truth and falsehood at the same time. This is of course possible if the statement contains a truth and a falsehood at the same time, such as, "That blue car is red". That would however mean that S is not atomic, but can be divided into a set of atomic statements S_i. Contradictions between any set (S_i,S_j) with j =/= i just implies the statement as a whole must be false, because falsehood supersedes truth (The sky is cloudy and green tonight would be an example).


However, if any atomic statement S_c contradicts itself, then we've got ourselves a situation where a non-trivial statement is both true and false, violating the principle that truth and falsehood must exclude each other. Now to determine the truth of S_c we usually compare it according to meta-statements (statements about statements) to other statements. If our reference statements and meta-statements are true and consistent then the statement is true. (for example, meta-statement: Truth is found by comparing a statement to statements about direct observations. I see that apple is red, so the apple must be red). If either set contains falsehood, then it's a logical fallacy. If the set of statements and meta-statements is contradictory, then our evaluation of S can be contradictory as well.


Now if we assume that the set of comparison statements and meta-statements used to evaluate S_c, S_c can both be contradictory and atomic if and only if S_c is at the same time the statement under consideration and part of the set of comparison statements or the meta-statements (it might be that such an expression is always a meta-expression, but I'm not sure of that). The example of "This statement is false" is a nice example. It is the statement under consideration, but at the same time it is part of the meta-statements that is used to determine the truth or falsehood of the statement itself. If we remove it from that set, it becomes non-descriptive, because it is not applied to the only statement it could be applied to.
We can expand this to include compound expressions where no atomic expression is contradictory but the whole expression is nevertheless contradictory. This can be only if the statement adds an inconsistency to the statements used to consider the expression. Since we assume that each atomic subexpression is well-defined this can only be if two subexpressions on which another subexpression depends are contradictory. (Assume for the moment that apple-shaped objects are green and that cars are red. Is an apple-shaped car green?). Note that these kinds of expressions need not be self-referential although they can be. It is just necessary for one subexpression to depend on a contradictory set of other expressions.

[Vattic]

Hopefully this is a suitable place to post this. Given the questions about logic above I'd assume so.

A friend of the family who professes an interest in mathematics mentioned something odd and I hope someone here can help clear things up. He said that if in roulette the ball lands on red that it is more likely to land on red the next spin adding "why would it change". He also said the same applies to tossing a fair coin. This sounded wrong to me and seems like an example of the gambler's fallacy. I mentioned this and he said he meant mathematical likelihood and in support of his claim said it was a "proper formula" and refused to explain further. I said I'd look it up but have to admit to being stumped. I'd be willing to bet that at least his examples are invalid. Can anyone explain likelihood and how it would apply, if at all, to the mentioned examples? I'm ashamedly inexperienced when it comes to mathematics, something I should fix but am unsure how, so layman's terms would be appreciated.

Cheers.

[ed boy]

So if I understand you correctly:
-Complex statments are composed of atomic statements (statements that cannot be broken up into smaller statements)
-These atomic statements are combined by logical operators to form complex statements
-If a statement is composed of a finite number of atomic statements, then the truth of each of the atomic statements can be determined, and therefore the truth of the statement as a whole can be determined
-If an equation is self-referential, then attempting to break it down into a finite number of atomic statements it imposible
-Therefore, if an equation is self-referential, it may be impossible to determine if it is true or not.
That does not mean all self-referential statments are neither true nor false. Let us imagine that I toss a coin. We can define the statements A and B such that:
A="The coin comes up as 'heads'"
B=A OR NOT(A) OR B
The equation B is self-referential, but it appears to be true and not false. Am I right on this?

A friend of the family who professes an interest in mathematics mentioned something odd and I hope someone here can help clear things up. He said that if in roulette the ball lands on red that it is more likely to land on red the next spin adding "why would it change". He also said the same applies to tossing a fair coin. This sounded wrong to me and seems like an example of the gambler's fallacy. I mentioned this and he said he meant mathematical likelihood and in support of his claim said it was a "proper formula" and refused to explain further. I said I'd look it up but have to admit to being stumped. I'd be willing to bet that at least his examples are invalid. Can anyone explain likelihood and how it would apply, if at all, to the mentioned examples? I'm ashamedly inexperienced when it comes to mathematics, something I should fix but am unsure how, so layman's terms would be appreciated.

Let us construct a maximum likelihood estimator.

Let us call the chance that the roulette wheel comes up red as P and the chance that it does not as Q, with P+Q=1. The roulette wheel does not change probability with each spin, so P and Q are constant. If we take N results from the roulette wheel, then the number of reds, which we shall call X, follows a binomial distribution with parameters N and P. Let us similarly call the number of times the wheel comes up as something other than red Y. X+Y=N.

He is referring to likelihood, we will construct a likelihood estimator. When we do this, we are given a set of results and we use these to guess the paramaters of the distribution. In this case, we will be given X and Y and we will use these to estimate P (and hence Q). Since X follows a binomial distribution, for any value of P the likelihood of X is given by:

Prob(X)=^NC_X*P^X*Q^Y
Prob(X)=^NC_X*P^X*(1-P)^Y
We are given what X and Y are, so they are constant. We want to find the value of P that gives the largest chance of Prob(X). Note that Prob(X) is a function of P, so we can differentiate it with respect to P. At it the maximum value of Prob(X), its derivative with respect to P will be zero. However, deriving the above with respect to P is possible, but a bit awkward. Consider instead the function L=ln(Prob(X)). ln is an increasing function, so it will be at its maximum when Prob(X) is at its maximum. It is also a lot nicer to differentiate:
L=ln(^NC_X)+Xln(P)+Yln(1-P)
^dL/_dP=^X/_P-^Y/_(1-P)
When P is at its maximum value, P, then ^dL/_dP will be zero.
^X/_P-^Y/_(1-P)=0
^X/_P=^Y/_(1-P)
X*(1-P)=Y*P
X=X*P+Y*P
X=P*(X+Y)
X=P*N
P=X/N

In the example you gave, N=1 and X=1. This would give the estimate for P as 1. However, it should be noted that this is a statistically "best guess" for P given the information available.

[Virex]

So if I understand you correctly:
-Complex statments are composed of atomic statements (statements that cannot be broken up into smaller statements)
-These atomic statements are combined by logical operators to form complex statements
-If a statement is composed of a finite number of atomic statements, then the truth of each of the atomic statements can be determined, and therefore the truth of the statement as a whole can be determined
-If an equation is self-referential, then attempting to break it down into a finite number of atomic statements it imposible
-Therefore, if an equation is self-referential, it may be impossible to determine if it is true or not.
That does not mean all self-referential statments are neither true nor false. Let us imagine that I toss a coin. We can define the statements A and B such that:
A="The coin comes up as 'heads'"
B=A OR NOT(A) OR B
The equation B is self-referential, but it appears to be true and not false. Am I right on this?

Yes, self-referential statements need not be inconsistent ("this statement is true" springs to mind), but for an atomic statement to be inconsistent it needs to be self-referential¹. A complex statement is inconsistent if one of it's parts is itself inconsistent, but it can also be inconsistent if it contains a statement that depends on preceding statements that contradict each other.

Also I do not think that a self-referential statement is necessarily atomic. For example, in "Apples are fruit and this statement is false", there are two atomic statements. Although the statement is self-referential, "Apples are fruit" and "This statement is false" are distinct logical statements that can be used separately. On the other hand, trying to divide it into more atomic statements would not be possible².

¹This follows from the same explanation as given for complex statements, there must be a contradiction in the statements on which at least 1 statement depends. For an atomic statement, the statement that causes the inconsistency must be the same as the statement that directly or indirectly depends on it (or else the statement wouldn't be atomic), so inconsistent, atomic statements are always self-referential.

²A first clue is given by the fact that there is only 1 logical operator in the sentence, but that is not sufficient, because "Apples, pears and tomatoes are fruit" contains only one logical operator (and), but it's actually 3 atomic statements. A formal analysis would however show that there is no way to break up the 2 statements in the given sentence further so that the parts can be combined back into the original statements using only logical operators.

[Darvi]

Hopefully this is a suitable place to post this. Given the questions about logic above I'd assume so.

A friend of the family who professes an interest in mathematics mentioned something odd and I hope someone here can help clear things up. He said that if in roulette the ball lands on red that it is more likely to land on red the next spin adding "why would it change". He also said the same applies to tossing a fair coin. This sounded wrong to me and seems like an example of the gambler's fallacy. I mentioned this and he said he meant mathematical likelihood and in support of his claim said it was a "proper formula" and refused to explain further. I said I'd look it up but have to admit to being stumped. I'd be willing to bet that at least his examples are invalid. Can anyone explain likelihood and how it would apply, if at all, to the mentioned examples? I'm ashamedly inexperienced when it comes to mathematics, something I should fix but am unsure how, so layman's terms would be appreciated.

Cheers.

The thing is that previous rolls do not influence the probability of later rolls. Like, not at all. Even if the game was rigged, the chance of the next game resulting in red isn't any bigger than it was before.


But if he means the likelyhood, then he was probably implying that the probability of the game being biased towards red is bigger after the roll, meaning the average likelyhood of the next roll being red is also slightly higher.


To put it more clearly, if you had 1000 rolls, which all ended up being red, then there's a good chance that the game is rigged, and the next roll will probably end up as red too. Of course if it's unknown if it actually is rigged or not, the probability of the next roll being red will still be less than 50% (due to the 0 rolls).
People who claim that the next roll will be black "because it will even itself out eventually" are of course deluding themselves.




Anyways, had a fun, but effing hard math test today (an optional one). See if you can handle it.

Spoiler: This one should be a piece of pi. (click to show/hide)

The summets of a nonagone are labeled with the numbers from 1 to 9, in such a way that the sum of 3 following numbers is always divisible by 3.
How many possibilities are  there to arrange the numbers, assuming that any positions that can be achieved by rotating each other are the same?

Spoiler: Not too hard either. (click to show/hide)

If R_n is the rest of 2ⁿ/1000, and S is the sum of all R's, what's the rest of S/1000?

Spoiler: For experts (click to show/hide)

L is a straight line with a slope of 5/12 and which passes through the point A(24,-1), and M is a line perpendicular to L passing through B(5,6). P is a point with the coordinates (-14,27).

Now let L be the X axis and M the Y axis (with identical units, of course). In this new system, P has the coordinates (a,b).

What's a+b?

Spoiler: Not even Vector could solve this one! (click to show/hide)

Na, I'm kidding, of course she can!

A cube whose edges are 10 long is hanging over a plane(lol yeah, I laughed). The corner closest to the plane is called A, and the three corners adjacent to it are 10, 11, and 12 away from the plane.

The distance between the plane and A can be expressed as (r-sqrt(s))/t, where r,s, and t are positive integers.

Find r+s+t.