Looking for an interesting challenge
- Thread starter Aldebaran
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I've read with interest the posts by Aldebaran and Kizmet. I wasn't sure if I could say much more, so I left it. But maybe I can say more, its just I probably shouldn't - explaining things too much can make them foggy. But anyway see what you think.
How does this apply to the Goldbach Conjecture, though? What is it that is imperfect? I take it is the answer to the question, “what is the probability that the conjecture is true?”
The question is very dependent on what we know. If you were to ask this question of somebody who knows nothing about this then they might say one thing, if you ask the same question of somebody who has studied it then they would say something else. If you ask it of somebody who happens to have a proof (or a disproof) then the answer would be very different indeed. These are different likelihoods (and perhaps also different senses of “likelihood”). Questions of probability are inherently (but perhaps not entirely) subjective – they inherently depend on what you know.
Is it the case that I only know what I know imperfectly? If I work out what I know haven’t I just learnt something?
The above ‘paradox’ is not really a problem, but I introduce it to emphasise that we are learning something when we find a probability. We are dealing with a question about an infinite set of numbers. If we can prove that there are only a finite number of counter-examples (I think this has been done), what does this say of the probability? If done with proper asymptotic care then this says that the density of counter-solutions is zero. I would like to say that the probability of any particular even number being a counter example is zero. But here we move into something very hand-wavy. What does it mean to select randomly over an infinite domain? Is it possible to pick an even number at random – in such a way that you would pick any other even number with equal likelihood? You can use a random number generator if you think it helps, but it won’t. The problem of likelihood is beyond computation.
Not only that, I have an apparent possibility with a probability of zero. This is not quite as absurd as it sounds [but I still need to point out that this is a very informal way of talking, the formality is important here] – but how does it help me? The possibility of disproving the conjecture false is zero regardless of whether or not it is true.
What more could I learn about the likelihood of the conjecture being true short of a proof of the conjecture? If there is a real probability (expressing my imperfect sense of uncertainty) that it is true then not only do I not know it but I cannot even interpret it either.
The talk of probability standing on itself, with interpretations resulting in actual calculations in no way allows us to talk about probability in the way we want to. There is nothing to be gained by such talk. However, I am not advancing a result here - as there is nothing to be lost either, we just need to be careful not to let such talk trip us up.
If perfection is impossible, what does it mean to talk about imperfections existing in themselves? Is there any content here? I would like to counter, "perfection is a product of the mind, reality is not the opposite of this - it is indifferent to this." But even that isn't right. Its more simply that I cannot put the concept of imperfection to use without the contrast of a perfect conception and the perfect conception is entirely of my own creation.
If you do not attribute practical use and value it isn't visible but still a probability (that occurs without being noticed by you).
Like I said : I would see probability as standing on itself, but interpretations and applications inevitably are of human construct and the moment you give it a practical use it is no longer a pure probability, because caught within a frame of likely occuranec (which can be correct or not).
If you give it value and interpretation it becomes objective feature in the frame of your observation, if you don't it is a feature inherent to everything that isn't perfect.
In my view perfection does not exist or if it exists we are not able to perceive it, but I would think evolution and/or change would come to a halt once perfection is reached.
The fact there is a probability that the coin falls on either side shows that probability exists on itself as a likelihood unlimited in its likeliness with the freedom to quantify itself or not. It occurs within and from within when you toss the coin.
Like I said, I can be totally wrong... I'm just thinking about it as we go
salaam.
How does this apply to the Goldbach Conjecture, though? What is it that is imperfect? I take it is the answer to the question, “what is the probability that the conjecture is true?”
The question is very dependent on what we know. If you were to ask this question of somebody who knows nothing about this then they might say one thing, if you ask the same question of somebody who has studied it then they would say something else. If you ask it of somebody who happens to have a proof (or a disproof) then the answer would be very different indeed. These are different likelihoods (and perhaps also different senses of “likelihood”). Questions of probability are inherently (but perhaps not entirely) subjective – they inherently depend on what you know.
Is it the case that I only know what I know imperfectly? If I work out what I know haven’t I just learnt something?
The above ‘paradox’ is not really a problem, but I introduce it to emphasise that we are learning something when we find a probability. We are dealing with a question about an infinite set of numbers. If we can prove that there are only a finite number of counter-examples (I think this has been done), what does this say of the probability? If done with proper asymptotic care then this says that the density of counter-solutions is zero. I would like to say that the probability of any particular even number being a counter example is zero. But here we move into something very hand-wavy. What does it mean to select randomly over an infinite domain? Is it possible to pick an even number at random – in such a way that you would pick any other even number with equal likelihood? You can use a random number generator if you think it helps, but it won’t. The problem of likelihood is beyond computation.
Not only that, I have an apparent possibility with a probability of zero. This is not quite as absurd as it sounds [but I still need to point out that this is a very informal way of talking, the formality is important here] – but how does it help me? The possibility of disproving the conjecture false is zero regardless of whether or not it is true.
What more could I learn about the likelihood of the conjecture being true short of a proof of the conjecture? If there is a real probability (expressing my imperfect sense of uncertainty) that it is true then not only do I not know it but I cannot even interpret it either.
The talk of probability standing on itself, with interpretations resulting in actual calculations in no way allows us to talk about probability in the way we want to. There is nothing to be gained by such talk. However, I am not advancing a result here - as there is nothing to be lost either, we just need to be careful not to let such talk trip us up.
If perfection is impossible, what does it mean to talk about imperfections existing in themselves? Is there any content here? I would like to counter, "perfection is a product of the mind, reality is not the opposite of this - it is indifferent to this." But even that isn't right. Its more simply that I cannot put the concept of imperfection to use without the contrast of a perfect conception and the perfect conception is entirely of my own creation.
Learing something does not lead to knowledge, but it can lead to the suggestion thereof (see my posts on "knowlegde" in other threads)
That is not what I meant when I said
"It occurs from within the imperfect so I'm thinking that it is the imperfection itself"
I meant that if a probability occurs inevitably from within the imperfect, probability and imperfection are one. You can have more then one imperfection within one and the same hence more than one probability is possible.
This reasoning requires to look beyond the practical implementation of what is named "probability" and see it as inherent to everything imperfect.
It is because perfection is out of reach that everything inevitably is imperfect, if we can perceive that imperfect thing or not.
You can perceive something as being perfect, or think about it as if it is perfect, but that is indeed only imagination.
I'm going to translate back for myself what I posted here to check if I reasoned somewhat coherently and from there translate into English whatever I can come up with as follow up or correction of my idea.
Could be complete nonsense, but at the moment I don't think so. Looks very logical to me.
salaam.
What I am saying - to put it another way - is that it does not usually make sense to quantify our certainty eg. "I am 99.9% certain that such and such is so."
This presents a false picture of how we base our convictions.
It may be the case that there is a level of certainty we have over something - but it is not simply that we cannot calculate it, the problem is that we cannot make sense of it at all. It makes no difference to us whether we recognise it.
This presents a false picture of how we base our convictions.
It may be the case that there is a level of certainty we have over something - but it is not simply that we cannot calculate it, the problem is that we cannot make sense of it at all. It makes no difference to us whether we recognise it.
I have no certainty about anything. Every perception is subjective and incomplete. That perfection isn't possible is however not difficult to witness, even with our limited capabilities: There would be absolute stagnation of everything as soon as perfection was reached.
salaam.
I don't think you were talking about imperfection existing in itself. I do, however, think that you use "imperfection" as a quality in itself. I want to say that I don't understand that until there is a contrast with something perfect.
Hope that helps.
Well OK. I would say that probability is intimately linked to knowledge - certainly knowledge in an informatic sense. (Shannon entropy)
Perhaps instead of saying "knowledge", I could say "suggestion of knowledge" in the above. Perhaps I could say this all the time. What purpose would this serve?
Really you seem to be just saying that reality is not the same as your imagination or perception. The concept of perfection seems useless.
Why would imperfection have a temporal expression? Why not a spatial expression?
Do you really have no certainty? Is that what you think on reflection or is it how you act? There should be a difference. The thinking sceptic and the acting advocate... One is not superior to the other, but they do suit different circumstances.
Something inherently present is not a quality (which is something that can be attributed separately).
No it's not. That is the whole point. It exists even if it is not "known" by human attribution of value and frame.
It serves to stay aware of the fact that true knowledge resides out of reach of the human mind. Hence everything we perceive should be doubted on its real true form and appearance, constantly.
OK thing-in-itself not quality-in-itself. I was trying not to shoehorn you!
I didn't mean that probabilities have to be known. Just that probability takes its meaning from what we know and what we know we don't know - at least in part. We can also express knowledge in the informatic sense in terms of probabilities (quantities that is).
I was aware of what it serves in general but it does not serve any particular purpose. What I mean by "knowledge", you mean by "suggestion of knowledge". I don't know what true knowledge is, which is why I don't need a contrast.
Certainty is really an attitude relative to a supposition. Good sense can lie behind certainty - and that's what I have been exploring here. But faith can also lie behind certainty. There's nothing inherently wrong with either.
Not the thing in itself: Its imperfection, inherently present in the thing and without it the thing wouldn't exist (or be) as the thing it is.
Probability does have meaning by itself: Without it you would have perfection. Hence my thought that probability and imperfection are one and the same.
If you start talking about "knowledge" it is easy to forget it isn't knowledge you talk about since you need to contrast it constantly against the limitations of the human mind and the restrictive influence of human perception.
salaam;
I find them dangerous because they invite limitation, if not directly impose limits.
salaam.
merlin wood
mind traveller 3rd class
Not much I can say I'm afraid. I don't know what true knowledge is. I don't know what it is to know something without the limitations of the human mind and human perception. But since we're using human words I think it is reasonable to give them human meaning!
Likewise with the idea of perfection. I don't know what it is. I could come up with guesses, but they would not be quite right. For instance - perhaps its something to do with the way reality relates to our idea of reality? However it seems that you mean something more objective than that.
If imperfection and probability were the same then why do we have two words?
Have you discovered something in this identity? Have you discovered something about the nature of probability or something about the nature of imperfection? Neither perhaps? Both maybe?
We can have conceptions that seem very real but turn out to be incoherent. They seem to fragment we look at them on inspection. Sometimes I have "inspired moments" like this - usually when trying to solve a difficult problem.
But also look at dreams. Sometimes my dreams ramble on without making any sense but logic is not important for the dream. When, upon waking, I ask myself what my dream was about I cannot even say.
Does an artist have a conception of what his creation is to be? Yes but its only through the work of the artist that this conception makes sense. So there is sense but there is not a perfect conception - not even in the abstract.
Pure mental constructs are not real - even on their own terms. Yet we seem to be able to refer to them.
Put ideas to work and then they are real. Leave them idle and they could be anything and nothing.
But also look at dreams. Sometimes my dreams ramble on without making any sense but logic is not important for the dream. When, upon waking, I ask myself what my dream was about I cannot even say.
Does an artist have a conception of what his creation is to be? Yes but its only through the work of the artist that this conception makes sense. So there is sense but there is not a perfect conception - not even in the abstract.
Pure mental constructs are not real - even on their own terms. Yet we seem to be able to refer to them.
Put ideas to work and then they are real. Leave them idle and they could be anything and nothing.
kyser_soze
Hawking's Angry Eyebrow
See, I come at this from exactly the opposite route - that placing a 'God' character in charge of everything, creating everything is precisely what the limited human brain came up with, because the basic concept that we are here by chance, not design, that chaos and probability are the main features of the universe, is too alienating for many people to accept.
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