Copied from facebook.
> From Scott Usenko Browne
> There’s obviously problems with language, such as “I broke my arm” sounds like you intentionally broke your arm.
> So my question is, are there relatable or similar problems with mathematics?
mollwollfumble said:
Copied from facebook.> From Scott Usenko Browne
> There’s obviously problems with language, such as “I broke my arm” sounds like you intentionally broke your arm.
> So my question is, are there relatable or similar problems with mathematics?
Ambiguity, ambiguous. Anything like that in maths?
I’d be surprised if there weren’t.
Michael V said:
mollwollfumble said:
Copied from facebook.> From Scott Usenko Browne
> There’s obviously problems with language, such as “I broke my arm” sounds like you intentionally broke your arm.
> So my question is, are there relatable or similar problems with mathematics?
Ambiguity, ambiguous. Anything like that in maths?
I’d be surprised if there weren’t.
Gödel’s two incompleteness theorems spring to mind.
sibeen said:
Michael V said:
mollwollfumble said:
Copied from facebook.> From Scott Usenko Browne
> There’s obviously problems with language, such as “I broke my arm” sounds like you intentionally broke your arm.
> So my question is, are there relatable or similar problems with mathematics?
Ambiguity, ambiguous. Anything like that in maths?
I’d be surprised if there weren’t.
Gödel’s two incompleteness theorems spring to mind.
Which remind me of Hofstadter’s law…
And now we need to do a shout-out for the Rev.
:)
Michael V said:
sibeen said:
mollwollfumble said:Copied from facebook.
> From Scott Usenko Browne
> There’s obviously problems with language, such as “I broke my arm” sounds like you intentionally broke your arm.
> So my question is, are there relatable or similar problems with mathematics?
Gödel’s two incompleteness theorems spring to mind.
Which remind me of Hofstadter’s law…
Had to look that one up.
> Hofstadter’s Law: It always takes longer than you expect, even when you take into account Hofstadter’s Law.
I would have put that down as a version of Murphy’s Law. But it does apply in this case. Proofs are getting so long that the entire corpus of the world’s mathematicians is insufficient for confirming or denying whether the proof is correct or not.
> Kurzweil’s Law of Accelerating Returns may take effect when the task is repeated, thus counteracting – and in some cases overpowering – Hofstadter’s law.
Let’s consider Hilbert’s problems. David Hilbert’s stated twenty-three problems in mathematics in 1900. “There are two problems that are not only unresolved but may in fact be unresolvable by modern standards.” and “the status of the first and second problems is even more complicated: there is not any clear mathematical consensus as to whether the results of Gödel give definitive negative solutions or not, since these solutions apply to a certain formalization of the problems, which is not necessarily the only possible one” and “Hilbert’s 9th problem is not precise enough to specify a particular problem.”
Given that the original ‘“I broke my arm” sounds like you intentionally broke your arm’ is a question of language, the whole question puts me in mind of: https://en.wikipedia.org/wiki/Bertrand_paradox_
Simply stated, Bertrand’s paradox is:
“Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of an inscribed equilateral triangle?”
The problem has THREE different solutions, all easy to find and all equally correct, because of ambiguity in the phrase “random chord”.
On the other hand. Given a set of axioms, the set of solutions possible from those axioms ought to be completely free from ambiguity.
>> There’s obviously problems with language, such as “I broke my arm” sounds like you intentionally broke your arm.
clearly that, as an example, is intended with some humour, so there’s no obvious problem that can be extracted from the example.
Someone called?
I recently read a discussion about whether something with a probability of zero could actually happen.
The consensus seemed to be that since there are an infinite number of numbers between any two specified limits, and you can select a specific number with a random selection between two limits, then events with a probability of zero can happen.
I think that’s wrong, because if a defined number is selected, then the probability was not zero. To actually select a number at random would take an infinite time.
From which I conclude that maths can be ambiguous because in any non-trivial system there will always be potential differences of interpretation of the words used.
… and if the words used are made non-ambiguous with further words, then the further words will be ambiguous.
And so on.
transition said:
>> There’s obviously problems with language, such as “I broke my arm” sounds like you intentionally broke your arm.clearly that, as an example, is intended with some humour, so there’s no obvious problem that can be extracted from the example.
Except obviously transition is joking, so there is an obvious problem.
The previous statement was a joke.
sibeen said:
Michael V said:
mollwollfumble said:
Copied from facebook.> From Scott Usenko Browne
> There’s obviously problems with language, such as “I broke my arm” sounds like you intentionally broke your arm.
> So my question is, are there relatable or similar problems with mathematics?
Ambiguity, ambiguous. Anything like that in maths?
I’d be surprised if there weren’t.
Gödel’s two incompleteness theorems spring to mind.
I didn’t know / had forgotten that Gödel had two incompleteness theorems.
But since the second is merely the application of the first to itself, surely he had an infinite number of incompleteness theorems.
Someone should write a book about that.
The Rev Dodgson said:
transition said:
>> There’s obviously problems with language, such as “I broke my arm” sounds like you intentionally broke your arm.clearly that, as an example, is intended with some humour, so there’s no obvious problem that can be extracted from the example.
Except obviously transition is joking, so there is an obvious problem.
The previous statement was a joke.
yeah, there are people out there intentionally breaking their own arms, so regularly it features as a serious idea, a serious possibility when someone says I broke my arm.
transition said:
The Rev Dodgson said:
transition said:
>> There’s obviously problems with language, such as “I broke my arm” sounds like you intentionally broke your arm.clearly that, as an example, is intended with some humour, so there’s no obvious problem that can be extracted from the example.
Except obviously transition is joking, so there is an obvious problem.
The previous statement was a joke.
yeah, there are people out there intentionally breaking their own arms, so regularly it features as a serious idea, a serious possibility when someone says I broke my arm.
me – how are you rev?
rev – I broke my arm.
me – that’s not good, how’d you do that?
rev – I rested my arm on a railway track in front of a train.
me – everything else is good, though?
rev – yes i’m fine.
transition said:
transition said:
The Rev Dodgson said:Except obviously transition is joking, so there is an obvious problem.
The previous statement was a joke.
yeah, there are people out there intentionally breaking their own arms, so regularly it features as a serious idea, a serious possibility when someone says I broke my arm.
me – how are you rev?
rev – I broke my arm.
me – that’s not good, how’d you do that?
rev – I rested my arm on a railway track in front of a train.
me – everything else is good, though?
rev – yes i’m fine.
Thing is. However it happened, it was you and your arm.
transition said:
transition said:
The Rev Dodgson said:Except obviously transition is joking, so there is an obvious problem.
The previous statement was a joke.
yeah, there are people out there intentionally breaking their own arms, so regularly it features as a serious idea, a serious possibility when someone says I broke my arm.
me – how are you rev?
rev – I broke my arm.
me – that’s not good, how’d you do that?
rev – I rested my arm on a railway track in front of a train.
me – everything else is good, though?
rev – yes i’m fine.
OK, so maybe the statement “I broke my arm” is not obviously ambiguous, but obviously the statement “I broke my arm is obviously ambiguous” is obviously ambiguous.
The Rev Dodgson said:
transition said:
transition said:yeah, there are people out there intentionally breaking their own arms, so regularly it features as a serious idea, a serious possibility when someone says I broke my arm.
me – how are you rev?
rev – I broke my arm.
me – that’s not good, how’d you do that?
rev – I rested my arm on a railway track in front of a train.
me – everything else is good, though?
rev – yes i’m fine.OK, so maybe the statement “I broke my arm” is not obviously ambiguous, but obviously the statement “I broke my arm is obviously ambiguous” is obviously ambiguous.
anything that thinks starts with safe assumptions. Tries anyway, tries and tests that way.
generally you wouldn’t get I intentionally broke my own arm out of I broke my arm.
most people aren’t going to jump to self-harm as an explanation, given how often arms (bones) are known to get broken unintentionally, commonly known as accidents.
reminds me of those road signs drowsy drivers die, sounds like an instruction, and too bad if you had some neurological condition that results in objects compelling behaviors related. Probably everyone has some form of that. Inclines the receptivity to culture, and tool use.
and there’s this alphabet business.
The Rev Dodgson said:
sibeen said:
Michael V said:Ambiguity, ambiguous. Anything like that in maths?
I’d be surprised if there weren’t.
Gödel’s two incompleteness theorems spring to mind.
I didn’t know / had forgotten that Gödel had two incompleteness theorems.
But since the second is merely the application of the first to itself, surely he had an infinite number of incompleteness theorems.
Someone should write a book about that.
It is rumoured that in his final days, Godel had found another incompleteness proof, but he never wrote it down. True story.
The Rev Dodgson said:
Someone called?I recently read a discussion about whether something with a probability of zero could actually happen.
The consensus seemed to be that since there are an infinite number of numbers between any two specified limits, and you can select a specific number with a random selection between two limits, then events with a probability of zero can happen.
I think that’s wrong, because if a defined number is selected, then the probability was not zero. To actually select a number at random would take an infinite time.
From which I conclude that maths can be ambiguous because in any non-trivial system there will always be potential differences of interpretation of the words used.
… and if the words used are made non-ambiguous with further words, then the further words will be ambiguous.
And so on.
> I recently read a discussion about whether something with a probability of zero could actually happen. The consensus seemed to be that since there are an infinite number of numbers between any two specified limits, and you can select a specific number with a random selection between two limits, then events with a probability of zero can happen.
Douglas Adams infinite improbability drive, eh.
It sounds like you need to apply a little of Reymond du Bois “Infinitary Calculus” to this.
But in the meantime let’s go philosophical and ask how you know that the probability is zero. Taking Kant’s distinction between “practical reason” and “pure reason”. Practical reason says that the probability is zero because no occurrence has ever been observed. In which case the event can happen. Pure reason says that the probability is zero because theory says it can’t happen. In which case the theory could be wrong and it can happen. It’s no good claiming the theory has always worked in the past because then we’re back at practical reason, so it can still happen that the theory is wrong and the event can happen.
Or to put it another way, nothing is impossible. As which point you notice that the statement “nothing is impossible” is self contradictory because if nothing is impossible then it is always possible that the statement “nothing is impossible” is false.
But enough of the philosophy, back to the mathematics. There is a set of infinitesimal numbers. For convenience, let me write infinity as ‘w’. Then 1/w > 1/w^2 > 1/w^3 > 1/e^w > 1/e^(w^2).
The number of integers is w, that’s the definition of w.
The number of real numbers between 0 and 1 is of order e^w. There’s a little subtlety here involving the axiom of choice but I’m going to skip over that, ask me if you want details.
So. If the true probability of an event is 1/w then because 1/ w > 1/e^w then this event can occur. If the true probability is 1/e^(w^2) then because 1/e^(w^2) < 1/e^w it can’t occur.
They ought to teach this in high school. It’s not difficult to understand.
mollwollfumble said:
The Rev Dodgson said:
Someone called?I recently read a discussion about whether something with a probability of zero could actually happen.
The consensus seemed to be that since there are an infinite number of numbers between any two specified limits, and you can select a specific number with a random selection between two limits, then events with a probability of zero can happen.
I think that’s wrong, because if a defined number is selected, then the probability was not zero. To actually select a number at random would take an infinite time.
From which I conclude that maths can be ambiguous because in any non-trivial system there will always be potential differences of interpretation of the words used.
… and if the words used are made non-ambiguous with further words, then the further words will be ambiguous.
And so on.
> I recently read a discussion about whether something with a probability of zero could actually happen. The consensus seemed to be that since there are an infinite number of numbers between any two specified limits, and you can select a specific number with a random selection between two limits, then events with a probability of zero can happen.
Douglas Adams infinite improbability drive, eh.
It sounds like you need to apply a little of Reymond du Bois “Infinitary Calculus” to this.
But in the meantime let’s go philosophical and ask how you know that the probability is zero. Taking Kant’s distinction between “practical reason” and “pure reason”. Practical reason says that the probability is zero because no occurrence has ever been observed. In which case the event can happen. Pure reason says that the probability is zero because theory says it can’t happen. In which case the theory could be wrong and it can happen. It’s no good claiming the theory has always worked in the past because then we’re back at practical reason, so it can still happen that the theory is wrong and the event can happen.
Or to put it another way, nothing is impossible. As which point you notice that the statement “nothing is impossible” is self contradictory because if nothing is impossible then it is always possible that the statement “nothing is impossible” is false.
But enough of the philosophy, back to the mathematics. There is a set of infinitesimal numbers. For convenience, let me write infinity as ‘w’. Then 1/w > 1/w^2 > 1/w^3 > 1/e^w > 1/e^(w^2).
The number of integers is w, that’s the definition of w.
The number of real numbers between 0 and 1 is of order e^w. There’s a little subtlety here involving the axiom of choice but I’m going to skip over that, ask me if you want details.
So. If the true probability of an event is 1/w then because 1/ w > 1/e^w then this event can occur. If the true probability is 1/e^(w^2) then because 1/e^(w^2) < 1/e^w it can’t occur.
They ought to teach this in high school. It’s not difficult to understand.
I disagree that it is not difficult to understand. I also disagree that what you said is true.
Let’s look at the process of selecting a number from the set of all integers.
There are two possibilities.
Either the number has a finite length, in which case the selection was not random.
Or the number has infinite length, in which case it is impossible to define it.
So either the event can happen, in which case its probability is finite, or it can’t happen, it which case it has zero probability.
The fact that there are different types of numbers that have a higher order of infinity seems entirely irrelevant to me.
Pi vs pie
> Or the number has infinite length, in which case it is impossible to define it.
No. That’s where your argument fails. Every number of infinite length is defined as the limit of a series. That’s the definition of a real number.
