On July 20 2013 04:06 Shiori wrote: strikes me as impossible for any human being to grasp, on any deep level, a reductive account of their own mind.
Strikes me like that too. Fortunately a lot of progress can be made without requiring a human to understand its own mind. Many people can look at one brain from another person in order to understand that mind better. With enough brains examined trends emerge and generalizations are inferred
On July 20 2013 06:53 DoubleReed wrote: Infinity/Infinity is indeterminate, not undefined. Undefined generally means infinity, because there is no infinity in the Real Numbers or Natural Numbers. It hasn't been defined yet. Generally you just need more information to figure out what Infinity/Infinity is. It's like Day[9] said: it depends. Sometimes it's a constant. Sometimes it's infinity. Sometimes it's zero.
Indeterminacy and being undefined aren't necessarily mutually exclusive. It depends entirely on what is meant by the term "infinity," because, as you pointed out, there isn't really any well-defined explanation for it in common number systems. Undefined is really imprecise as an adjective, whereas indeterminacy emphasizes that, given certain conditions, there could be different solutions. But over the reals, strictly, there's no such thing as asking what "infinity/infinity" is, because no manipulation of an R --> R function gives infinity afaik. Technically, lim x-->0 1/x strictly over the reals is "unbounded" rather than actually equal to infinity. But yeah, you're pretty much right. There's no good reason not to use indeterminate, if I think about it. Ty.
On July 20 2013 02:33 Darkwhite wrote: Shiori: Read Myrdraals last post in this topic. There are ample amounts of confusion and disagreement, so something has obviously gone wrong somewhere. I think different explanations involving less formalism would have avoided these misunderstandings, but I wouldn't mind hearing your take on this.
The good thing about p=1/inf is that it preserves the intuition that, with a large enough sample size N, N*p != 0.
Say we are making a function from all integers (Z) to the positive integers (N), by mapping each Z to a random number in N - let's not concern ourselves too much with picking something at random from an infinite set.
For this function, what is the probability that any given Z maps to itself? This must obviously be something along the lines of 1/(2*inf) or zero - for any given Z, there is sort of a 50% chance that it is in N, and then a 1/(size(N)) chance that it maps to itself, and size(N) is sort of inf.
The slight problem with denoting this probability as zero, is that it invites the (false) intuition that the function will have no such identity mappings at all - denoting it as zero disguises the non-impossibility of an identity mapping, if you aren't rigorously enough trained to know that infinities are difficult beasts. 1/inf keeps you alert that, with infinite candidates, depending on the sizes of the infinities in question, you might get either none or some or even an infinite amount of them.
I think this is more or less why Myrdraal wanted to write the probability as 1/inf rather than 0, which I personally think is more than okay. I think the problem was everyone spent much more time proclaiming 1/inf as heresy than trying to understand what he was actually saying.
I get what you're saying. I think the problem is that, while intuition is great, permitting writing 1/inf rather than 0 encourages a sort of informal way of thinking about probabilities, which leads people to equivocate between nonzero probability and how much of a "chance" there is of something. I mean, suppose we wrote 1/inf instead of probability zero for "almost never." Then, on the face of it, we'd have people saying that "well the chance is bigger than nothing so it's at least a chance" which is kinda fine, in a way, but it misrepresents what we actually mean when we say p(x)= 0 or p(x) = 1.
I know we have moved on from this topic but I just wanted to thank Darkwhite for interpreting what I was actually trying to say, rather than what it implies mathematically, and I'd like to try to clear up why there was so much confusion and why your attempts to explain were not very effective.
The easy part to clear up is my initial disagreement where you stated "It doesn't just converge to zero at a limit.", I took this to mean "the probability does not converge to 0 a limit". This didn't make sense to me since as far as I could tell the probability was found using a limit and this was definitely a comprehension fail on my part, but it wasn't cleared up until later.
When I read the real number scenario the way I understood it was that the probability is mathematically 0 but conceptually 1/infinite, since 0 as a concept means "nothing" and the idea that something that has "no" chance of happening must happen is logically impossible.
In the first post of yours I replied to, you said that "the probability of randomly selecting any particular real number in a trial is actually zero.", because of the emphasis due to the underline I thought you were trying to imply more than just mathematically 0, but I have come to realise that it was probably just a symptom of trying to get your point across to Wherebugsgo, who has a tendency to be.. difficult.
I tried to query you on the language but you instead explained to me how infinitesimal is equal to 0 with regards to real numbers, which definitely helped, until paralleluniverse came along and stated explicitly that the probability is not infinitesimal. Logically this is impossible, assuming the probability is 0 and infinitesimal is defined in real numbers as 0, so the only way this could make sense would be if he disagreed with your definition and he had some way to prove that the answer was 0 and not infinitesimal. Since language seemed to be a problem, I tried to use my limited (heh) understanding of limits to show how it might be shown to be infinitesimal (while also pointing out that I thought they were mathematically equal) and even you disagreed despite previously giving a definition of the only infinitesimal number being 0.
Well, how do you want us to talk? We said something in terms of mathematically defined terms when we said that the probabilities we were discussing were exactly zero and exactly one. And that's incontrovertibly true from the point of view of mathematics. If we explain it without "formalism," then how do we even put the question? What does probability mean if it's divorced from the mathematical definition of probability? What does probability 0 mean? What does probability 1 mean? If we're using concepts instead of numbers in the case of infinitesimals, then why are we using numbers like 0 and 1 for probability otherwise? Why not just say things like "Must happen," "Will happen," "Can't happen," and "Won't happen" rather than bringing probability into it at all?
I mean, we literally have said several times that probability = 0 doesn't mean "can't happen," so I'm not sure where the confusion is.
In short here is why there was confusion, you made an informal statement: "the probability is actually 0", I questioned the language of the statement and instead of explicitly stating "yes, formally it is 0, informally you can think of it as infinitesimal" you gave me the information to link my understanding with the formal definition, which was fine until someone else made another informal statement stating: "the probability is not infinitesimal".
The thing is understanding concepts and how they relate to maths is important, especially to me as a programmer, I need to be able to make sure I understand concepts correctly in order to model them as code. I don't need to understand exactly how all the maths that I use works, just that it does work and in what context, any additional understanding is just a bonus. What I don't like about using the value 0 to describe both "impossible" and "almost impossible" is that it is ambiguous, and ambiguity causes problems with logic. A computer can't tell if we mean 0 "impossible" or 0 "almost impossible" and Darkwhite seems to understand what I mean that in this context 1/inf seems to do a better job of explaining what we mean. It may seem like I am being finicky, but you guys were being finicky with the maths so it goes both ways .
Anyway, this whole topic is a little bit off track of the OP, but it does mirror quite closely the discussions that people have with regards to the mind (conceptual) and the brain (rigorously defined structure) and it shows quite clearly how the two can be confused.
Is there a brain area for mind wandering? For religious experience? For re-orienting attention? A recent study casts serious doubt on the evidence for these ideas, and rewrites the rules for neuroimaging.
Brain-mapping experiments attempt to identify the cognitive functions associated with discrete cortical regions. They generally rely on a method known as “cognitive subtraction.” However, recent research reveals a basic assumption underlying this approach—that brain activation is due to the additional processes triggered by the experimental task—is wrong.
----
Dual-process theory is outlined in the recent book Thinking Fast and Slow by the Nobel Prize winner Daniel Kahneman. Classic dual-process theory postulates a fight between deliberate reasoning and primitive automatic processes. But the fight that is most obvious in the brain is between two types of deliberate and evolutionarily advanced reasoning—one for empathetic, the other for analytic thought, the researchers said.
On July 20 2013 02:33 Darkwhite wrote: Shiori: Read Myrdraals last post in this topic. There are ample amounts of confusion and disagreement, so something has obviously gone wrong somewhere. I think different explanations involving less formalism would have avoided these misunderstandings, but I wouldn't mind hearing your take on this.
The good thing about p=1/inf is that it preserves the intuition that, with a large enough sample size N, N*p != 0.
Say we are making a function from all integers (Z) to the positive integers (N), by mapping each Z to a random number in N - let's not concern ourselves too much with picking something at random from an infinite set.
For this function, what is the probability that any given Z maps to itself? This must obviously be something along the lines of 1/(2*inf) or zero - for any given Z, there is sort of a 50% chance that it is in N, and then a 1/(size(N)) chance that it maps to itself, and size(N) is sort of inf.
The slight problem with denoting this probability as zero, is that it invites the (false) intuition that the function will have no such identity mappings at all - denoting it as zero disguises the non-impossibility of an identity mapping, if you aren't rigorously enough trained to know that infinities are difficult beasts. 1/inf keeps you alert that, with infinite candidates, depending on the sizes of the infinities in question, you might get either none or some or even an infinite amount of them.
I think this is more or less why Myrdraal wanted to write the probability as 1/inf rather than 0, which I personally think is more than okay. I think the problem was everyone spent much more time proclaiming 1/inf as heresy than trying to understand what he was actually saying.
I get what you're saying. I think the problem is that, while intuition is great, permitting writing 1/inf rather than 0 encourages a sort of informal way of thinking about probabilities, which leads people to equivocate between nonzero probability and how much of a "chance" there is of something. I mean, suppose we wrote 1/inf instead of probability zero for "almost never." Then, on the face of it, we'd have people saying that "well the chance is bigger than nothing so it's at least a chance" which is kinda fine, in a way, but it misrepresents what we actually mean when we say p(x)= 0 or p(x) = 1.
I know we have moved on from this topic but I just wanted to thank Darkwhite for interpreting what I was actually trying to say, rather than what it implies mathematically, and I'd like to try to clear up why there was so much confusion and why your attempts to explain were not very effective.
The easy part to clear up is my initial disagreement where you stated "It doesn't just converge to zero at a limit.", I took this to mean "the probability does not converge to 0 a limit". This didn't make sense to me since as far as I could tell the probability was found using a limit and this was definitely a comprehension fail on my part, but it wasn't cleared up until later.
When I read the real number scenario the way I understood it was that the probability is mathematically 0 but conceptually 1/infinite, since 0 as a concept means "nothing" and the idea that something that has "no" chance of happening must happen is logically impossible.
In the first post of yours I replied to, you said that "the probability of randomly selecting any particular real number in a trial is actually zero.", because of the emphasis due to the underline I thought you were trying to imply more than just mathematically 0, but I have come to realise that it was probably just a symptom of trying to get your point across to Wherebugsgo, who has a tendency to be.. difficult.
I tried to query you on the language but you instead explained to me how infinitesimal is equal to 0 with regards to real numbers, which definitely helped, until paralleluniverse came along and stated explicitly that the probability is not infinitesimal. Logically this is impossible, assuming the probability is 0 and infinitesimal is defined in real numbers as 0, so the only way this could make sense would be if he disagreed with your definition and he had some way to prove that the answer was 0 and not infinitesimal. Since language seemed to be a problem, I tried to use my limited (heh) understanding of limits to show how it might be shown to be infinitesimal (while also pointing out that I thought they were mathematically equal) and even you disagreed despite previously giving a definition of the only infinitesimal number being 0.
Well, how do you want us to talk? We said something in terms of mathematically defined terms when we said that the probabilities we were discussing were exactly zero and exactly one. And that's incontrovertibly true from the point of view of mathematics. If we explain it without "formalism," then how do we even put the question? What does probability mean if it's divorced from the mathematical definition of probability? What does probability 0 mean? What does probability 1 mean? If we're using concepts instead of numbers in the case of infinitesimals, then why are we using numbers like 0 and 1 for probability otherwise? Why not just say things like "Must happen," "Will happen," "Can't happen," and "Won't happen" rather than bringing probability into it at all?
I mean, we literally have said several times that probability = 0 doesn't mean "can't happen," so I'm not sure where the confusion is.
In short here is why there was confusion, you made an informal statement: "the probability is actually 0", I questioned the language of the statement and instead of explicitly stating "yes, formally it is 0, informally you can think of it as infinitesimal" you gave me the information to link my understanding with the formal definition, which was fine until someone else made another informal statement stating: "the probability is not infinitesimal".
The thing is understanding concepts and how they relate to maths is important, especially to me as a programmer, I need to be able to make sure I understand concepts correctly in order to model them as code. I don't need to understand exactly how all the maths that I use works, just that it does work and in what context, any additional understanding is just a bonus. What I don't like about using the value 0 to describe both "impossible" and "almost impossible" is that it is ambiguous, and ambiguity causes problems with logic. A computer can't tell if we mean 0 "impossible" or 0 "almost impossible" and Darkwhite seems to understand what I mean that in this context 1/inf seems to do a better job of explaining what we mean. It may seem like I am being finicky, but you guys were being finicky with the maths so it goes both ways .
Anyway, this whole topic is a little bit off track of the OP, but it does mirror quite closely the discussions that people have with regards to the mind (conceptual) and the brain (rigorously defined structure) and it shows quite clearly how the two can be confused.
0 and 1/infinite are not the same wich becomes clear when you multiply them with an order of infinity, so you are completely right and the fault lies with the people who tried to explain it to you from different point of vieuws. Math is full of abstract concepts,there is nothing wrong with them. It is only physicians who use math that often have a problem with abstract concepts as it is difficult to aply concepts to the real world. For physicians 0 and 1/infinity are the same, Just like 1.3 for physicians does mean all numbers between 1.250 up to 1.349999999999999999999
On August 04 2013 04:11 Mothra wrote: Apologies if this is bad place to put this. Didn't really want to make new thread. Anyway, interesting article about neuroimaging:
Is there a brain area for mind wandering? For religious experience? For re-orienting attention? A recent study casts serious doubt on the evidence for these ideas, and rewrites the rules for neuroimaging.
Brain-mapping experiments attempt to identify the cognitive functions associated with discrete cortical regions. They generally rely on a method known as “cognitive subtraction.” However, recent research reveals a basic assumption underlying this approach—that brain activation is due to the additional processes triggered by the experimental task—is wrong.
----
Dual-process theory is outlined in the recent book Thinking Fast and Slow by the Nobel Prize winner Daniel Kahneman. Classic dual-process theory postulates a fight between deliberate reasoning and primitive automatic processes. But the fight that is most obvious in the brain is between two types of deliberate and evolutionarily advanced reasoning—one for empathetic, the other for analytic thought, the researchers said.
I always have problems with scientific research without proper limitations on their claims. How does it follow that thought is associated with brain processes that their content must be determined by them? Epiphenomalism seems to me a philosophical and not a scientific stance.
On July 20 2013 02:33 Darkwhite wrote: Shiori: Read Myrdraals last post in this topic. There are ample amounts of confusion and disagreement, so something has obviously gone wrong somewhere. I think different explanations involving less formalism would have avoided these misunderstandings, but I wouldn't mind hearing your take on this.
The good thing about p=1/inf is that it preserves the intuition that, with a large enough sample size N, N*p != 0.
Say we are making a function from all integers (Z) to the positive integers (N), by mapping each Z to a random number in N - let's not concern ourselves too much with picking something at random from an infinite set.
For this function, what is the probability that any given Z maps to itself? This must obviously be something along the lines of 1/(2*inf) or zero - for any given Z, there is sort of a 50% chance that it is in N, and then a 1/(size(N)) chance that it maps to itself, and size(N) is sort of inf.
The slight problem with denoting this probability as zero, is that it invites the (false) intuition that the function will have no such identity mappings at all - denoting it as zero disguises the non-impossibility of an identity mapping, if you aren't rigorously enough trained to know that infinities are difficult beasts. 1/inf keeps you alert that, with infinite candidates, depending on the sizes of the infinities in question, you might get either none or some or even an infinite amount of them.
I think this is more or less why Myrdraal wanted to write the probability as 1/inf rather than 0, which I personally think is more than okay. I think the problem was everyone spent much more time proclaiming 1/inf as heresy than trying to understand what he was actually saying.
I get what you're saying. I think the problem is that, while intuition is great, permitting writing 1/inf rather than 0 encourages a sort of informal way of thinking about probabilities, which leads people to equivocate between nonzero probability and how much of a "chance" there is of something. I mean, suppose we wrote 1/inf instead of probability zero for "almost never." Then, on the face of it, we'd have people saying that "well the chance is bigger than nothing so it's at least a chance" which is kinda fine, in a way, but it misrepresents what we actually mean when we say p(x)= 0 or p(x) = 1.
I know we have moved on from this topic but I just wanted to thank Darkwhite for interpreting what I was actually trying to say, rather than what it implies mathematically, and I'd like to try to clear up why there was so much confusion and why your attempts to explain were not very effective.
The easy part to clear up is my initial disagreement where you stated "It doesn't just converge to zero at a limit.", I took this to mean "the probability does not converge to 0 a limit". This didn't make sense to me since as far as I could tell the probability was found using a limit and this was definitely a comprehension fail on my part, but it wasn't cleared up until later.
When I read the real number scenario the way I understood it was that the probability is mathematically 0 but conceptually 1/infinite, since 0 as a concept means "nothing" and the idea that something that has "no" chance of happening must happen is logically impossible.
In the first post of yours I replied to, you said that "the probability of randomly selecting any particular real number in a trial is actually zero.", because of the emphasis due to the underline I thought you were trying to imply more than just mathematically 0, but I have come to realise that it was probably just a symptom of trying to get your point across to Wherebugsgo, who has a tendency to be.. difficult.
I tried to query you on the language but you instead explained to me how infinitesimal is equal to 0 with regards to real numbers, which definitely helped, until paralleluniverse came along and stated explicitly that the probability is not infinitesimal. Logically this is impossible, assuming the probability is 0 and infinitesimal is defined in real numbers as 0, so the only way this could make sense would be if he disagreed with your definition and he had some way to prove that the answer was 0 and not infinitesimal. Since language seemed to be a problem, I tried to use my limited (heh) understanding of limits to show how it might be shown to be infinitesimal (while also pointing out that I thought they were mathematically equal) and even you disagreed despite previously giving a definition of the only infinitesimal number being 0.
Well, how do you want us to talk? We said something in terms of mathematically defined terms when we said that the probabilities we were discussing were exactly zero and exactly one. And that's incontrovertibly true from the point of view of mathematics. If we explain it without "formalism," then how do we even put the question? What does probability mean if it's divorced from the mathematical definition of probability? What does probability 0 mean? What does probability 1 mean? If we're using concepts instead of numbers in the case of infinitesimals, then why are we using numbers like 0 and 1 for probability otherwise? Why not just say things like "Must happen," "Will happen," "Can't happen," and "Won't happen" rather than bringing probability into it at all?
I mean, we literally have said several times that probability = 0 doesn't mean "can't happen," so I'm not sure where the confusion is.
In short here is why there was confusion, you made an informal statement: "the probability is actually 0", I questioned the language of the statement and instead of explicitly stating "yes, formally it is 0, informally you can think of it as infinitesimal" you gave me the information to link my understanding with the formal definition, which was fine until someone else made another informal statement stating: "the probability is not infinitesimal".
The thing is understanding concepts and how they relate to maths is important, especially to me as a programmer, I need to be able to make sure I understand concepts correctly in order to model them as code. I don't need to understand exactly how all the maths that I use works, just that it does work and in what context, any additional understanding is just a bonus. What I don't like about using the value 0 to describe both "impossible" and "almost impossible" is that it is ambiguous, and ambiguity causes problems with logic. A computer can't tell if we mean 0 "impossible" or 0 "almost impossible" and Darkwhite seems to understand what I mean that in this context 1/inf seems to do a better job of explaining what we mean. It may seem like I am being finicky, but you guys were being finicky with the maths so it goes both ways .
Anyway, this whole topic is a little bit off track of the OP, but it does mirror quite closely the discussions that people have with regards to the mind (conceptual) and the brain (rigorously defined structure) and it shows quite clearly how the two can be confused.
0 and 1/infinite are not the same wich becomes clear when you multiply them with an order of infinity, so you are completely right and the fault lies with the people who tried to explain it to you from different point of vieuws. Math is full of abstract concepts,there is nothing wrong with them. It is only physicians who use math that often have a problem with abstract concepts as it is difficult to aply concepts to the real world. For physicians 0 and 1/infinity are the same, Just like 1.3 for physicians does mean all numbers between 1.295 up to 1.349999999999999999999
(0 * infinity) and (infinity * 1/infinity) are both indeterminate, so I don't know what you're talking about.
It's physicist, not physician. Physician refers to doctors.
Math has abstract concepts with rigorous definitions. It's not flexible or interpretative or subject to different points of view. If you would like to ignore them, then fine, but don't pretend to be the authority when discussing it then.
And, if anything, it's the physicists who don't like the idea of 0 and 1/infinity being the same, because physicists have the intuition that 1/infinity is somehow greater than zero (because the numerator is not zero). Which is perfectly reasonable intuition and generally practical, but not using any rigorous definitions. Mathematically, 1/infinity is zero. If you don't like that, then just use arbitrarily large numbers instead of infinity. Everyone likes arbitrarily large numbers.
Of course, the real problem is one of notation. infinity isn't something you multiply and such, it's a limit. You have to use it in limits. You have to say things like Lim 1/x as x -> infinity = 0. That's the notation we're talking about. Trying to deal with infinity in other ways is non-rigorous, and often leads to indeterminate results.
(0 * infinity) and (infinity * 1/infinity) are both indeterminate, so I don't know what you're talking about.
0*infinity is 0 as far as i know, and infinity*1/infinity is indeterminate indeed.
"infinity isn't something you multiply and such"
No on the contrary, there are manny calculations you can do with orders of infinity.Just like there are manny calculations done with irrational numbers.
On August 04 2013 05:00 Rassy wrote: (0 * infinity) and (infinity * 1/infinity) are both indeterminate, so I don't know what you're talking about.
0*infinity is 0 as far as i know, and infinity*1/infinity is indeterminate indeed.
"infinity isn't something you multiply and such"
No on the contrary, there are manny calculations you can do with orders of infinity.Just like there are manny calculations done with irrational numbers.
0 * infinity is indeterminate, just like infinity/infinity and infinity-infinity is indeterminate. It may be convenient for some calculations to assume that 0 * infinity is zero, but that's not rigorous because infinity is not a real number and therefore you aren't supposed to use it with the operations of the real numbers.
I have no idea what irrational numbers have to do with this. Irrational numbers are real numbers, which is closed under +-*/^. Positive and negative infinity have to do with closure of the real numbers under limits and sequences (giving you the "extended real numbers"). The term "undefined" was coined for infinities because positive and negative infinity are not real numbers. Just because we have intuitions of how to use them with the real number operations, doesn't mean that is defined.
well, when i said determinism can not support itself they called me a troll or when i talked about dualism and fighting and sides and winning ...
But the fight that is most obvious in the brain is between two types of deliberate and evolutionarily advanced reasoning—one for empathetic, the other for analytic thought, the researchers said.
by evolution: we can not already be, everything we will ever be.
Imagine if we worked on a computer for millions and millions of years. It would be so incomprehensibly complex that people might start thinking there must be something else to it. But there isn't, it's still just a computer, powered by electricity like all others.
That's what I think has happened with the brain. It is so ridiculously complex that people need some other explanation for why it works, so they start thinking there is more to it than there actually is.
there no book that can answer this question for you. just think about it this way. if we have trouble overcoming our intuition that consciousness is not reducible to an arrangement of physical objects and forces, what is most likely is not that our intuitions about consciousness are wrong, but that we do not understand what we mean by one or all of the three: "arrangement" "physical" "objects and forces". can you define any of these things rigorously without begging the question (i.e. making their putative incapability of producing consciousness part of their definition)? I'm quite sure that I don't know what I'm talking about when I talk about these things.
EPT
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