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On July 19 2013 16:11 Tobberoth wrote:Show nested quote +On July 19 2013 13:29 DoubleReed wrote:
No. I'm saying I have no idea what 1/infinity means. It may be intuitive but it doesn't actually mean anything.
[Edit: If you mean Lim 1/x as x -> infinity then this equals zero. Try writing it down. It doesn't approach zero. It equals zero. Limits don't approach things. Limits equal things. The x approaches infinity in the limit, but the limits themselves don't approach stuff.]
But I forgot something. There's actually a super duper easy way to see that the probability is exactly zero.
Let's look at the probability of picking a random number on [0,1] that it lands on the interval [0.47,0.53]. Well it's 6%, right? Because the length of the interval is 0.06. So let's forget that whole measure thing and just look at lengths of intervals. What's the generalized way to find the length of an interval?
Length[a,b] = b - a. Simple.
Okay. How does this relate to the probability of picking a single number? Well, a single number can be expressed as a closed interval! What's the probability of picking a number on the interval [0.47,0.47]? Well it's just the length of the interval. Which is 0.47 - 0.47 = 0.
No calculus. No infinity. No countability. Just subtraction. That's how we like it. So if you have a theoretical one-sided dice, the odds of getting 1 is 0, because 1-1 = 0? I don't think this closed interval thing works.
You're mixing up discrete and continuous probability... His stuff works, it's called uniform distribution, it's just that you can't deal with more complex questions with only that.
Also Myrddraal, have you ever seen constructions with limits ? Like constructed Riemann integral or something ? That would help you not getting confused over that sort of stuff...
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On July 18 2013 17:28 Tobberoth wrote:Show nested quote +On July 18 2013 17:12 xM(Z wrote:i can not but equate the believe in determinism with the believe in a god. faith - 1. Confident belief in the truth, value, or trustworthiness of a person, idea, or thing.i can not accept deterministic claims because i can not accept the existence of a single god. there has to be a dualism for evolution/motion/change/vibration to occur and a validation of self, as determinism claims to be, can not exist. the mind is a stereoscopic like effect, generated by a dualism. certainty vs uncertainty. if certainty = physicalism then uncertainty = faithism (it exists by means of observation  ) if form is an expression of physicalism and believes an expression of faith, then the stereoscopic effect between form and believe is reality, your reality. I don't know why I bother clicking reply on your posts any more, but determinism is nothing like the belief in god. Everything we know, except some really fringe science where our understanding is limited at best, everything has causes and is dependant on causes. It's perfectly valid a priori knowledge that if everything depends on causes, everything is predetermined. Hell, even if some quantum events are causeless, they are still regular, which leads to determinism as well. We can't tell which atom will decay from a certain amount of radioactive materia, but we know exactly how many atoms will decay within a certain timespan, so it's still deterministic on a macro level. Non-belief in determinism is far more like the belief in god, since you believe in something supernatural which we have no reason to believe in other than finding it comforting that we are constantly in 100% control.
if there wouldn't be humans that would find comforting that they are 100% not in control, then there wouldn't be humans that would find comforting that they are 100% in control. again dualism, supply and demand.
see now, what you did is attribute proprieties to those ones and zeroes, to those believes, and that is everything there is to them.
think of a future in which all our (current) physical laws would've been broken, in which you'll have infinite effects as outcome of a single cause or infinite causes leading to a single effect and where antigravity would roam freely just because we could,
then know, that in that future some people will still believe in determinism and some others will still believe in free will. those believes transcend knowledge and transcend the environment.
even when you look at it mathematically in the context of a mathematical universe hypothesis or Ultimate Ensemble, there isn't and never will be a ultimate and anything. there will always be at least a duality.
https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis
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Tobberoth, what are you doing? Is the closed interval from [0,1] the same thing as the set {1,2,3,4,5,6}? Last I checked they weren't.
Sorry if I am being difficult, but it seems like you are getting caught up on the language I am using rather than what I am actually intending. I am assuming we can agree on the language used in the first statement on limits from Wikipedia : In mathematics, a limit is the value that a function or sequence "approaches" as the input or index approaches some value. I will do my best not to differentiate from this language in any way to describe what I am trying to say.
Yes, a limit equals something, but when calculating the limit of 1/x as the input x approaches infinity, the value of the limit L will approach 0. Now, my understanding (perhaps here is where you can fill me in and it will make sense to me) is that it is technically impossible for x to actually "reach" infinity, so it is technically impossible for the limit to "reach" 0, though instead it gets so close so as to make practically no difference. I would argue that while it makes practically no difference, and mathematically we don't run into problems treating them as equal and they are mathematically provable to be equal, if we were to define the difference, the clearest way to define the difference would be infinitesimal.
I think what I am trying to say pretty much is that to me, the concept of 1/infinity or something infinitesimal is effectively equal to 0 in almost every way. Except that I think the separate definition would be useful in terms of theoretical probability to be able to effectively describe the difference between something that is impossible and almost impossible.
The language is that the function approaches y as x approaches infinity. The Limit does not approach anything.
The reason I'm trying to be to stubborn about this is because the whole point of limits is so you don't fuss around with things like this. Limits are not some vague thing. The whole point of having the concept of a limit is that you can say "This EQUALS zero." Lim 1/x as x -> infinity = 0. It's not sorta-kinda-maybe zero. It equals zero. That's why we developed limits in the first place, to exactly answer such questions as "what does 1/infinity equal?" and the answer is zero. It's not sorta-kinda zero.
I don't know what you mean by "theoretically impossible" and "theoretically almost impossible." We are talking about numbers here. We are talking about purely theoretical constructs of randomness. We're talking about trying to find a rational number in an interval of real numbers. Think about it: We can't even properly write down an irrational number. Theoretical and not theoretical are just not useful distinctions here.
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On July 19 2013 21:02 DoubleReed wrote:
Tobberoth, what are you doing? Is the closed interval from [0,1] the same thing as the set {1,2,3,4,5,6}? Last I checked they weren't.
Maybe I misunderstood your post completely, but from I gathered, you proved that the probability of picking a single number between 0 and 1 is zero by making an interval between that single number and since a number minus that same number is 0, the probability is 0. I was just pointing out that this will obviously always be true. What's the probability of getting a 5 in the set you posted? 0, because 5-5 = 0. What I'm not getting is where you make the distinction between 0.47 - 0.47 compared to this 5-5 comparison.
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On July 19 2013 00:23 paralleluniverse wrote:Show nested quote +On July 18 2013 21:07 Rassy wrote:On July 18 2013 19:42 paralleluniverse wrote:On July 18 2013 09:10 DoubleReed wrote:On July 18 2013 08:43 yOngKIN wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. measurable only in terms of our limited huma knowledge of math and physics Uhh... no. Measurable as in Lebesgue Measure. Yep. You are completely correct. One thing I've learned is that people like to debate math on the internet and get it all wrong (like this guy), when in fact math is almost never up for debate. Especially, when it's well-established, hundreds of years old math, like measure theory. Math debates are often the most futile because people substitute their intuition with unrelenting zeal in place of mathematical rigor. Particularly in topics like measure theory, which produces counter-intuitive results to those who haven't learned the subject. In a math "debate", a general rule that I observe is the following: If you're debating about technical details, then you're talking to a crank (and probably getting nowhere). If you're debating about philosophy, it's not necessarily apparent that you're talking to a crank. Appeals to ignorance are also very common, as you've just experience. For example. people love saying that we don't understand infinity. There are some things in math that we don't understand, infinity is not one of them. Infinity is a rigorously defined and well-understood concept. With knowledge from a high school or 1st or 2nd year math course, pretty much any perceived problems or hole in our human knowledge of math that one would think of (other than famous unsolved problems), isn't actually a problem nor a hole, but rather a personal lack of knowledge in math. On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. This is correct. The distinction is between countable and uncountable. As an example to back up your fact that "Sum(Measure(An)) = Measure(Union(An))" doesn't work when n is an element of an uncountable set, we can use the uncountable set [0,infinity) and set An as the independent events "a Brownian motion hits 3 at time n". Then the LHS = 0 and the RHS = 1. Also, no I don't understand what they're arguing about either. But in a probably futile attempt to resolve it, let me state the following fact: If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly. Not "approximately 0", or "infinitesimally close to 0", or "1/infinity", or "approaches 0", or "0 in the limit", or whatever. It's simply 0. Now i have a question to parralel universe who seems to be verry sure in his statements. If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly. Not "approximately 0", or "infinitesimally close to 0", or "1/infinity", or "approaches 0", or "0 in the limit", or whatever. It's simply 0.[/QUOTE Now you pick a number between 0 and 1 an infinite amount of times, what are the odds to pick 0.548 exactly at least once? If it is 0 exactly as he say, then the answer should be 0 However if it is infinitesimally close to 0, then the odds of picking this number at least once would be 1 No? For those of you quick enough, you may have notice that I edited out my first response to this question. That's because it was incorrect. The correct solution is 0. If U_n are independent uniform random variables on [0,1] for n = 1, 2, ..., infinity, then the probability that U_n is eventually 0.548 is equal to 0. This is because P(U_n = 0.548 eventually) = 1 - P(U_n != 0.548 infinitely often), and by the 2nd Borel-Cantelli lemma, P(U_n != 0.548 infinitely often) = 1. So even if you select (countably many) infinite random numbers that are uniform on [0,1], you still won't get 0.548 exactly. Interestingly, if you have a Brownian motion, which is in some sense like randomly selecting a normal random variable at every time instance in [0,1], then the probability that it's equal to 0.548 at any particular time t, is 0. But the probability that it will equal 0.548 exactly, infinitely often, has probability 1. Note that this doesn't contradict the solution above, because here we are on an uncountable set. The Borel-Cantelli lemma also says that if we let E_n = "getting all heads in infinite flips on trial n" (a trial is one string of infinite flips), then P(E_n infinitely often) = 0. This is contrary to claims above saying that if you have infinite trials of infinite flips, then you'll get infinitely many trials of all heads. Those claims are wrong.
Ty for explanation it is much apreciated.
It still feels like a trick to make it a countably infinite number of trails and not uncountably infinite.
Are there countably infinite numbers between 0 and 1 or uncountably infinite numbers?
If its uncountably infinite numbers between 0 and 1 then it makes no sense to use countably infinite for the number of trials,and i think you should use uncountably infinite then, wich would give probability 1 like you said in a later post.
If there are countably infinite numbers between 0 and 1 then i have to agree with you.
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On July 19 2013 21:02 DoubleReed wrote:Tobberoth, what are you doing? Is the closed interval from [0,1] the same thing as the set {1,2,3,4,5,6}? Last I checked they weren't. Show nested quote +Sorry if I am being difficult, but it seems like you are getting caught up on the language I am using rather than what I am actually intending. I am assuming we can agree on the language used in the first statement on limits from Wikipedia : In mathematics, a limit is the value that a function or sequence "approaches" as the input or index approaches some value. I will do my best not to differentiate from this language in any way to describe what I am trying to say.
Yes, a limit equals something, but when calculating the limit of 1/x as the input x approaches infinity, the value of the limit L will approach 0. Now, my understanding (perhaps here is where you can fill me in and it will make sense to me) is that it is technically impossible for x to actually "reach" infinity, so it is technically impossible for the limit to "reach" 0, though instead it gets so close so as to make practically no difference. I would argue that while it makes practically no difference, and mathematically we don't run into problems treating them as equal and they are mathematically provable to be equal, if we were to define the difference, the clearest way to define the difference would be infinitesimal.
I think what I am trying to say pretty much is that to me, the concept of 1/infinity or something infinitesimal is effectively equal to 0 in almost every way. Except that I think the separate definition would be useful in terms of theoretical probability to be able to effectively describe the difference between something that is impossible and almost impossible. The language is that the function approaches y as x approaches infinity. The Limit does not approach anything. The reason I'm trying to be to stubborn about this is because the whole point of limits is so you don't fuss around with things like this. Limits are not some vague thing. The whole point of having the concept of a limit is that you can say "This EQUALS zero." Lim 1/x as x -> infinity = 0. It's not sorta-kinda-maybe zero. It equals zero.
Now you're just arguing against things that I am not saying. I think you just lack the capability to answer my question how the value reaches 0 if x can not reach infinity, that is why you are being stubborn, surely if you knew how to answer it you would (this is not to say that it is not answerable). You clearly didn't read my post very well because I said it mathematically EQUALS 0, so you repeating this is useless, I did not say sorta-kinda-maybe 0 I said equals. The definition that I am using of infinitesimally small is such that 0 + 1/infinity = 0 as shown here ( http://en.wikipedia.org/wiki/0.999...) where 0.999... is equal to 1. You even admit this in the next part where you say 1/infinity is 0 as 0 + 0 = 0.
That's why we developed limits in the first place, to exactly answer such questions as "what does 1/infinity equal?" and the answer is zero. It's not sorta-kinda zero.
You are saying that 1/infinty is equal to 0? If this is the case then you agree with me that the probability is 0 and 1/infinity.
I don't know what you mean by "theoretically impossible" and "theoretically almost impossible." We are talking about numbers here. We are talking about purely theoretical constructs of randomness. We're talking about trying to find a rational number in an interval of real numbers. Think about it: We can't even properly write down an irrational number. Theoretical and not theoretical are just not useful distinctions here.
Conceptually they are very clearly different, if you can't see this you are too blinded by the maths to actually look at the concepts. Here is a really simple example. Since there are uncountably infinite real numbers between 0 and 1 the probability that 0.5 is selected is almost impossible, however a number must be selected and 0.5 could be the number that is selected so it is not actually impossible. However it is impossible for 1.5 to be selected, because it does not lie within the set [0,1].
I don't know why you are harping on my use of the word theoretical, I just used the word theoretical to show that I am discussing theory of probability rather than its practical application. You are talking about numbers I am talking about concepts and attempting to use math to show how these concepts could be represented or possibly proved.
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Myrddraal, your confusion stems from treating infinity as a number when it is not. Using the real numbers, there's no such thing as 1/infinity, just like there is no such thing as 1/cat or 1/chair. You are right when you say that the probability is infinitesimal, but the real numbers contain no infinitesimal except 0. Ergo, it is 0 and only 0.
These two things are together known as the archimedean property if you want to look it up.
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On July 19 2013 16:11 Myrddraal wrote:
Yes, a limit equals something, but when calculating the limit of 1/x as the input x approaches infinity, the value of the limit L will approach 0. Now, my understanding (perhaps here is where you can fill me in and it will make sense to me) is that it is technically impossible for x to actually "reach" infinity, so it is technically impossible for the limit to "reach" 0, though instead it gets so close so as to make practically no difference. I would argue that while it makes practically no difference, and mathematically we don't run into problems treating them as equal and they are mathematically provable to be equal, if we were to define the difference, the clearest way to define the difference would be infinitesimal.
If f(x) = 1/x then lim x-->infinity f(x) = 0. The value of the limit doesn't "approach" zero; it is zero exactly.
Your definition of x + 1/infinity = x for all x is problematic structurally for algebra because it means that there are infinitely many additive identities. But that doesn't make any sense, because you can prove that additive identities are always unique in a group.
See here.
You are saying that 1/infinty is equal to 0? If this is the case then you agree with me that the probability is 0 and 1/infinity.
Only in the same sense that the probability is zero and 0/1...
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gedatsu: If you cannot imagine such a thing as 1 divided by infinity, that's a failure of your imagination. You might as well say there is no such thing as the imaginary unit = sqrt(-1). The imaginary unit is very useful because you can use it in calculations to get sensible, non-imaginary answers further down the line. In the same way, 1/infinity is an sort-of-okay way to denote the ratio the of all the positive integers being exactly five. If you select a random, positive integer once for each positive integer which exists, the expected number of fives is one - the infinities in the denominator and the one you multiply by are of exactly the same size and cancel out. If you replace 1/infinity by 0, you lose the distinction between an infinitesimal and an exact zero, which does matter if you are going to do further calculations which happen to involve infinities.
You wouldn't really go around writing 1/infinity much when doing rigorous mathematics, because it fails to make an important distinction of cardinality. Do note that the dx which shows up in integrals is an infinitesimal, which you cannot just replace with 0 - even though that is indeed its size. In some formulations of integration, is represents a length along the x-axis when the interval is partitioned into N pieces as N grows towards infinity - i.e. it represents the length [total width of interval * 1/infinity].
Myrdraal: Most of what you are writing makes sense if interpreted charitably. The people answering you seem to be too busy pointing out that your notation isn't permissible in formal mathematics - which is, as far as I can see, completely missing the point.
Shiori: You cannot possibly be serious. He explicitly said he was a programmer and wanted to understand something with his limited mathematical background - do you really, honestly think he's concerned about preserving the additive identity? He is using 1/inf as a concept, not a number - and I have pointed out, above, why such a concept is somewhat useful. Nobody is pretending 1/inf is a number, and none of us are going to place it anywhere in a coordinate system as something distinct from zero.
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On July 20 2013 00:25 Darkwhite wrote:
gedatsu: If you cannot imagine such a thing as 1 divided by infinity, that's a failure of your imagination. You might as well say there is no such thing as the imaginary unit = sqrt(-1). The imaginary unit is very useful because you can use it in calculations to get sensible, non-imaginary answers further down the line. In the same way, 1/infinity is an sort-of-okay way to denote the ratio the of all the positive integers being exactly five. If you select a random, positive integer once for each positive integer which exists, the expected number of fives is one - the infinities in the denominator and the one you multiply by are of exactly the same size and cancel out. If you replace 1/infinity by 0, you lose the distinction between an infinitesimal and an exact zero, which does matter if you are going to do further calculations which happen to involve infinities.
You wouldn't really go around writing 1/infinity much when doing rigorous mathematics, because it fails to make an important distinction of cardinality. Do note that the dx which shows up in integrals is an infinitesimal, which you cannot just replace with 0 - even though that is indeed its size. In some formulations of integration, is represents a length along the x-axis when the interval is partitioned into N pieces as N grows towards infinity - i.e. it represents the length [total width of interval * 1/infinity].
Myrdraal: Most of what you are writing makes sense if interpreted charitably. The people answering you seem to be too busy pointing out that your notation isn't permissible in formal mathematics - which is, as far as I can see, completely missing the point.
Shiori: You cannot possibly be serious. He explicitly said he was a programmer and wanted to understand something with his limited mathematical background - do you really, honestly think he's concerned about preserving the additive identity? He is using 1/inf as a concept, not a number - and I have pointed out, above, why such a concept is somewhat useful. Nobody is pretending 1/inf is a number, and none of us are going to place it anywhere in a coordinate system as something distinct from zero.
I can imagine it just fine. I can also work in a number system that does include it, such as the hyperreals. I've no problem with people thinking about 1/infinity as shorthand for the limit of 1/x as x tends to infinity. But it is a very important point to know that this is just a mental shortcut and formally incorrect. Myrddraal did not seem to understand this.
I'm the same vein, dx being an infinitesimal is another simplification that is not formally correct. Limits are usually taught in high school using that concept, but it is not how they are actually defined.
Btw, it is equally false to say that i = sqrt - 1. It is however true that i ^2 = - 1. Squaring is not strictly an invertible function. Again an example of an ok mental shortcut, that can lead to serious problems if treated without care.
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On July 20 2013 00:25 Darkwhite wrote:
gedatsu: If you cannot imagine such a thing as 1 divided by infinity, that's a failure of your imagination. You might as well say there is no such thing as the imaginary unit = sqrt(-1). The imaginary unit is very useful because you can use it in calculations to get sensible, non-imaginary answers further down the line. In the same way, 1/infinity is an sort-of-okay way to denote the ratio the of all the positive integers being exactly five. If you select a random, positive integer once for each positive integer which exists, the expected number of fives is one - the infinities in the denominator and the one you multiply by are of exactly the same size and cancel out. If you replace 1/infinity by 0, you lose the distinction between an infinitesimal and an exact zero, which does matter if you are going to do further calculations which happen to involve infinities.
The imaginary unit isn't a number in the real numbers. That's the point. I dislike the equivocation between "sensible" and "non-imaginary." The reason we have the complex numbers is because the complex numbers are useful. In any problem with a real solution, the existence or not existence of sqrt(-1) doesn't matter because it is necessarily simplified into something that does exist in the real numbers. That's why, when we solve the quadratic equation, we say there are "no real solutions" if the discriminant is negative; giving complex solutions is only relevant if we are actually working with complex numbers.
The idea here is that there just aren't any infinitesimal elements in the real numbers except for 0. Yes, there are other constructions where there are different infinitesimals, but we're talking about the real numbers here. I'm not sure if you're disagreeing with me, but if you are, could you define what an infinitesimal actually is in the real numbers?
You wouldn't really go around writing 1/infinity much when doing rigorous mathematics, because it fails to make an important distinction of cardinality. Do note that the dx which shows up in integrals is an infinitesimal, which you cannot just replace with 0 - even though that is indeed its size. In some formulations of integration, is represents a length along the x-axis when the interval is partitioned into N pieces as N grows towards infinity - i.e. it represents the length [total width of interval * 1/infinity].
The dx in integrals is actually not an infinitesimal in modern calculus. It is a remnant of Leibniz's notation for calculus, which we largely use, but doesn't retain the meaning he associated with it. Modern calculus is based on Weierstrass's formalization of it, namely through the use of limits + Cauchy's (incredibly annoying) epsilon-delta characterization of said limits. While it's true that there are infinitesimals used in nonstandard analysis to define calculus (with the hyperreals) that's not really what's being discussed right now, and just because we use the dx doesn't mean that we're calling it an infinitesimal. What the dx actually is depends entirely on what kind of integral definition you're working from (Riemann, Lebesgue, etc). For Lebesgue integrals, for example, the dx refers to measure, which is absolutely not the same thing as "an infinitesimal" because such a thing isn't necessarily defined.
I'm not totally sure what you're saying with the number five and the positive integers, so I won't comment much on that. However, since you are using infinitesimals, which are rigorously defined in the hyperreal field, have a different sort of hierarchy than the kind talked about by Cantor with his transfinite numbers i.e. 1+infinity refers to a different thing than (5)(infinity) assuming that the infinities in question are identical. This isn't exactly the case over other fields. To illustrate the problems with doing infinite arithmetic, think about Hilbert's Hotel. We have a countably infinite number of rooms, all empty, just like the integers. So then let's say I give you all the prime ones, i.e. an infinite number with N_0 cardinality. What is the value of my number of rooms divided by your number of rooms?
Over the extended reals (https://en.wikipedia.org/wiki/Extended_real_number) infinity/a = infinity for any a, including infinity, and a/infinity = 0 for any a in the reals.
Myrdraal: Most of what you are writing makes sense if interpreted charitably. The people answering you seem to be too busy pointing out that your notation isn't permissible in formal mathematics - which is, as far as I can see, completely missing the point.
You act like the notation is irrelevant. The reason we're emphasizing notation is because we're trying to point out the difference between what we're saying and what we aren't saying. The problem is that the word "infinitesimal" has different definitions depending on the context of how you're using it.
Shiori: You cannot possibly be serious. He explicitly said he was a programmer and wanted to understand something with his limited mathematical background - do you really, honestly think he's concerned about preserving the additive identity? He is using 1/inf as a concept, not a number - and I have pointed out, above, why such a concept is somewhat useful. Nobody is pretending 1/inf is a number, and none of us are going to place it anywhere in a coordinate system as something distinct from zero.
Considering the additive identity is a fundamental property of groups, yes, he should be concerned about it, given that without it nothing would make any sense at all.
1/inf as a concept isn't necessarily wrong, but it has to be conceptually consistent with whatever system you're working over. Say you're working over the field with two elements; then asking what 1/inf is means absolutely nothing because one of its elements doesn't even exist in the field.
If it's not distinct from zero over a field, then it is zero in that field. The problem here is that intuitive concepts do not transcend the definition of fields, because if you don't define a field, then what you're talking about is utterly ambiguous. If you're asking whether 1/infinity is 0 or an infinitesimal, and whether these are "different," the answer is exactly that it depends what field you're talking about.
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This thread is now completely about math!
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I think we might have wondered off topic a bit here, folks. Math is neat, but only a small part of the brain.
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On July 20 2013 00:41 gedatsu wrote:Show nested quote +On July 20 2013 00:25 Darkwhite wrote:
gedatsu: If you cannot imagine such a thing as 1 divided by infinity, that's a failure of your imagination. You might as well say there is no such thing as the imaginary unit = sqrt(-1). The imaginary unit is very useful because you can use it in calculations to get sensible, non-imaginary answers further down the line. In the same way, 1/infinity is an sort-of-okay way to denote the ratio the of all the positive integers being exactly five. If you select a random, positive integer once for each positive integer which exists, the expected number of fives is one - the infinities in the denominator and the one you multiply by are of exactly the same size and cancel out. If you replace 1/infinity by 0, you lose the distinction between an infinitesimal and an exact zero, which does matter if you are going to do further calculations which happen to involve infinities.
You wouldn't really go around writing 1/infinity much when doing rigorous mathematics, because it fails to make an important distinction of cardinality. Do note that the dx which shows up in integrals is an infinitesimal, which you cannot just replace with 0 - even though that is indeed its size. In some formulations of integration, is represents a length along the x-axis when the interval is partitioned into N pieces as N grows towards infinity - i.e. it represents the length [total width of interval * 1/infinity].
Myrdraal: Most of what you are writing makes sense if interpreted charitably. The people answering you seem to be too busy pointing out that your notation isn't permissible in formal mathematics - which is, as far as I can see, completely missing the point.
Shiori: You cannot possibly be serious. He explicitly said he was a programmer and wanted to understand something with his limited mathematical background - do you really, honestly think he's concerned about preserving the additive identity? He is using 1/inf as a concept, not a number - and I have pointed out, above, why such a concept is somewhat useful. Nobody is pretending 1/inf is a number, and none of us are going to place it anywhere in a coordinate system as something distinct from zero. I've no problem with people thinking about 1/infinity as shorthand for the limit of 1/x as x tends to infinity. But it is a very important point to know that this is just a mental shortcut and formally incorrect. Myrddraal did not seem to understand this.
I agree with everything not snipped from the quote.
I disagree that this is a very important point - he isn't here to be formally trained in mathematics or to pass an algebra exam. I think it is a very confusing point which is not at all relevant to what he is trying to understand. The straight answer to the question he asked is that 0 and 1/inf are both acceptable ways to think about the probability in this context, even if 1/inf doesn't cut it in rigorous mathematics. I would even say that, conceptually, 1/inf is more accurate, even though it numerically evaluates to 0.
I am sure he would agree that, if he for some reason had to represent 1/inf as a float, it would be indistinguishable from 0. What I really don't get is why the people here who know the mathematics and have read him explicitly explaining his background keep obsessing about irrelevant formalities and nuances. Being rigorous is important when writing academic papers, but it isn't the best way to explain these things to someone without high-level, formal mathematical education.
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"If the brain were so simple we could understand it, we would be so simple we couldn't."
I guess this quote reflects the age in which it was uttered?? Lots of people now seem to think the brain can be fully deciphered, and I think it'll be done eventually. By this I don't mean that we could fully simulate a brain with all its structures, chemical reactions and electrical phenomena. That's an unrealistic expection and it wouldn't be practical anyway.
What's gonna happen is that the emergent properties of the brain that cause its functions, such as an emotion or a state of conciousness, will be characterized, and by putting those together, it'll be possible to emulate what a brain does in vitro. And whether those emergent properties are really just chemistry and electricity...is a question of semantics and philosophy, but it seems like the best theory for me! And I'm satisfied with that, just like I'm satisfied with our knowledge of gravity, even if the particles which cause it have not been identified.
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On July 20 2013 01:07 Shiori wrote:Show nested quote +On July 20 2013 00:25 Darkwhite wrote:
gedatsu: If you cannot imagine such a thing as 1 divided by infinity, that's a failure of your imagination. You might as well say there is no such thing as the imaginary unit = sqrt(-1). The imaginary unit is very useful because you can use it in calculations to get sensible, non-imaginary answers further down the line. In the same way, 1/infinity is an sort-of-okay way to denote the ratio the of all the positive integers being exactly five. If you select a random, positive integer once for each positive integer which exists, the expected number of fives is one - the infinities in the denominator and the one you multiply by are of exactly the same size and cancel out. If you replace 1/infinity by 0, you lose the distinction between an infinitesimal and an exact zero, which does matter if you are going to do further calculations which happen to involve infinities. The imaginary unit isn't a number in the real numbers. That's the point. I dislike the equivocation between "sensible" and "non-imaginary." The reason we have the complex numbers is because the complex numbers are useful. In any problem with a real solution, the existence or not existence of sqrt(-1) doesn't matter because it is necessarily simplified into something that does exist in the real numbers. That's why, when we solve the quadratic equation, we say there are "no real solutions" if the discriminant is negative; giving complex solutions is only relevant if we are actually working with complex numbers. The idea here is that there just aren't any infinitesimal elements in the real numbers except for 0. Yes, there are other constructions where there are different infinitesimals, but we're talking about the real numbers here. I'm not sure if you're disagreeing with me, but if you are, could you define what an infinitesimal actually is in the real numbers? Show nested quote +You wouldn't really go around writing 1/infinity much when doing rigorous mathematics, because it fails to make an important distinction of cardinality. Do note that the dx which shows up in integrals is an infinitesimal, which you cannot just replace with 0 - even though that is indeed its size. In some formulations of integration, is represents a length along the x-axis when the interval is partitioned into N pieces as N grows towards infinity - i.e. it represents the length [total width of interval * 1/infinity]. The dx in integrals is actually not an infinitesimal in modern calculus. It is a remnant of Leibniz's notation for calculus, which we largely use, but doesn't retain the meaning he associated with it. Modern calculus is based on Weierstrass's formalization of it, namely through the use of limits + Cauchy's (incredibly annoying) epsilon-delta characterization of said limits. While it's true that there are infinitesimals used in nonstandard analysis to define calculus (with the hyperreals) that's not really what's being discussed right now, and just because we use the dx doesn't mean that we're calling it an infinitesimal. What the dx actually is depends entirely on what kind of integral definition you're working from (Riemann, Lebesgue, etc). For Lebesgue integrals, for example, the dx refers to measure, which is absolutely not the same thing as "an infinitesimal" because such a thing isn't necessarily defined. I'm not totally sure what you're saying with the number five and the positive integers, so I won't comment much on that. However, since you are using infinitesimals, which are rigorously defined in the hyperreal field, have a different sort of hierarchy than the kind talked about by Cantor with his transfinite numbers i.e. 1+infinity refers to a different thing than (5)(infinity) assuming that the infinities in question are identical. This isn't exactly the case over other fields. To illustrate the problems with doing infinite arithmetic, think about Hilbert's Hotel. We have a countably infinite number of rooms, all empty, just like the integers. So then let's say I give you all the prime ones, i.e. an infinite number with N_0 cardinality. What is the value of my number of rooms divided by your number of rooms? Over the extended reals (https://en.wikipedia.org/wiki/Extended_real_number) infinity/a = infinity for any a, including infinity, and a/infinity = 0 for any a in the reals. Show nested quote +Myrdraal: Most of what you are writing makes sense if interpreted charitably. The people answering you seem to be too busy pointing out that your notation isn't permissible in formal mathematics - which is, as far as I can see, completely missing the point. You act like the notation is irrelevant. The reason we're emphasizing notation is because we're trying to point out the difference between what we're saying and what we aren't saying. The problem is that the word "infinitesimal" has different definitions depending on the context of how you're using it. Show nested quote +Shiori: You cannot possibly be serious. He explicitly said he was a programmer and wanted to understand something with his limited mathematical background - do you really, honestly think he's concerned about preserving the additive identity? He is using 1/inf as a concept, not a number - and I have pointed out, above, why such a concept is somewhat useful. Nobody is pretending 1/inf is a number, and none of us are going to place it anywhere in a coordinate system as something distinct from zero. Considering the additive identity is a fundamental property of groups, yes, he should be concerned about it, given that without it nothing would make any sense at all. 1/inf as a concept isn't necessarily wrong, but it has to be conceptually consistent with whatever system you're working over. Say you're working over the field with two elements; then asking what 1/inf is means absolutely nothing because one of its elements doesn't even exist in the field. If it's not distinct from zero over a field, then it is zero in that field. The problem here is that intuitive concepts do not transcend the definition of fields, because if you don't define a field, then what you're talking about is utterly ambiguous. If you're asking whether 1/infinity is 0 or an infinitesimal, and whether these are "different," the answer is exactly that it depends what field you're talking about.
I am going to leave the more extreme tangents lying.
I am not saying that notation is irrelevant in all settings. I am saying that proper notation is not the way to communicate with someone who isn't familiar with said formalism. Do you expect that Mydraal knows, or can reasonably be expected to read up on, what a group and an additive identity is and why its uniqueness is a big deal? Does it seem like the sort of explanations he has been given in these two pages have cleared things up for him?
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On July 20 2013 01:21 Darkwhite wrote:Show nested quote +On July 20 2013 00:41 gedatsu wrote:On July 20 2013 00:25 Darkwhite wrote:
gedatsu: If you cannot imagine such a thing as 1 divided by infinity, that's a failure of your imagination. You might as well say there is no such thing as the imaginary unit = sqrt(-1). The imaginary unit is very useful because you can use it in calculations to get sensible, non-imaginary answers further down the line. In the same way, 1/infinity is an sort-of-okay way to denote the ratio the of all the positive integers being exactly five. If you select a random, positive integer once for each positive integer which exists, the expected number of fives is one - the infinities in the denominator and the one you multiply by are of exactly the same size and cancel out. If you replace 1/infinity by 0, you lose the distinction between an infinitesimal and an exact zero, which does matter if you are going to do further calculations which happen to involve infinities.
You wouldn't really go around writing 1/infinity much when doing rigorous mathematics, because it fails to make an important distinction of cardinality. Do note that the dx which shows up in integrals is an infinitesimal, which you cannot just replace with 0 - even though that is indeed its size. In some formulations of integration, is represents a length along the x-axis when the interval is partitioned into N pieces as N grows towards infinity - i.e. it represents the length [total width of interval * 1/infinity].
Myrdraal: Most of what you are writing makes sense if interpreted charitably. The people answering you seem to be too busy pointing out that your notation isn't permissible in formal mathematics - which is, as far as I can see, completely missing the point.
Shiori: You cannot possibly be serious. He explicitly said he was a programmer and wanted to understand something with his limited mathematical background - do you really, honestly think he's concerned about preserving the additive identity? He is using 1/inf as a concept, not a number - and I have pointed out, above, why such a concept is somewhat useful. Nobody is pretending 1/inf is a number, and none of us are going to place it anywhere in a coordinate system as something distinct from zero. I've no problem with people thinking about 1/infinity as shorthand for the limit of 1/x as x tends to infinity. But it is a very important point to know that this is just a mental shortcut and formally incorrect. Myrddraal did not seem to understand this. I agree with everything not snipped from the quote. I disagree that this is a very important point - he isn't here to be formally trained in mathematics or to pass an algebra exam. I think it is a very confusing point which is not at all relevant to what he is trying to understand. The straight answer to the question he asked is that 0 and 1/inf are both acceptable ways to think about the probability in this context, even if 1/inf doesn't cut it in rigorous mathematics. I would even say that, conceptually, 1/inf is more accurate, even though it numerically evaluates to 0.
But we're talking about "almost surely" versus "surely," which is what spawned this entire discussion. Both of those terms are defined in a particular way according to mathematics. You can't ignore the precise definition because it's only in the context of the definition that the question makes any sense at all.
I don't think anyone is deliberately trying to confuse him. We've stated over and over that there are no infinitesimals (other than zero) in the field of real numbers. That's it. There just aren't any! How is that appealing to complicated or confusing higher mathematics?
1/inf doesn't even mean anything outside of a particular context in which 1 and infinity and division are defined. Since we're not talking about really abstract mathematics, we're assuming he means the reals, or some extended version of the real line. In either case, there are no infinitesimal elements other than zero...
The probability function we were discussing outputs a real number on [0,1]. If something isn't a real number, it can't be spat out by this function. That's the point.
I am sure he would agree that, if he for some reason had to represent 1/inf as a float, it would be indistinguishable from 0. What I really don't get is why the people here who know the mathematics and have read him explicitly explaining his background keep obsessing about irrelevant formalities and nuances. Being rigorous is important when writing academic papers, but it isn't the best way to explain these things to someone without high-level, formal mathematical education.
In what ways is 1/inf not indistinguishable from zero? Do these ways have any meaning over the real numbers, extended real numbers, complex numbers, or what? Do they have any meaning in the context of probability? If 1/inf isn't a number, but is instead a "concept," then what does that mean? Probability is on [0,1]. What does it mean for p(x) = concept unless concept is in [0,1] because [0,1] is the range of p...
I am not saying that notation is irrelevant in all settings. I am saying that proper notation is not the way to communicate with someone who isn't familiar with said formalism. Do you expect that Mydraal knows, or can reasonably be expected to read up on, what a group and an additive identity is and why its uniqueness is a big deal? Does it seem like the sort of explanations he has been given in these two pages have cleared things up for him
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.
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so 1/infinite times infinite is now 0 ?
And 1 times infinite, devided by infinite is 1?
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On July 20 2013 01:54 Shiori wrote:
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.
Probability zero can mean either "can't happen" or "will almost never happen". That's why making the distinction is necessary.
Probability of picking a specific real number between 0 and 1 = 0 (almost never)
Probability of picking a card from a standard deck that is not hearts, clubs, spades or diamonds = 0 (never)
Right?
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On July 20 2013 02:09 Rassy wrote:
so 1/infinite times infinite is now 0 lol?
No. infinity/infinity is undefined. However, lim n-->infinity of n/n = 1.
However, your question is sorta vague so I'm not totally sure..
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EPT
