Intuition alone cannot break down what both intuition and rigor have built up.
And in any case, the stance "everyone else is wrong but me" is actual insanity, which is to say I don't mean that in a pejorative sense, it's a statement of fact: it's insane.
It can be insanity, it can be incorrect and it can be correct. Do try to be accurate! 
Intuition alone cannot break down what both intuition and rigor have built up.
And in any case, the stance "everyone else is wrong but me" is actual insanity, which is to say I don't mean that in a pejorative sense, it's a statement of fact: it's insane.
It can be insanity, it can be incorrect and it can be correct. Do try to be accurate!
It's strange to me how you take enormous offense at certain bits of casual conversation, but when I express the very real sentiment that I think you're mentally ill, you don't seem to notice.
<<Actually, the clearest way might be to prove that the variance of the sum of two independent random variables is the sum of their variances: it is a very simple result from that. Anyone want to prove that? (Looking up the proof will suffice, if you like).>>
I think that was basic error analysis which we learned at school in the 6th form, mainly for physics.
Intuition alone cannot break down what both intuition and rigor have built up.
And in any case, the stance "everyone else is wrong but me" is actual insanity, which is to say I don't mean that in a pejorative sense, it's a statement of fact: it's insane.
It can be insanity, it can be incorrect and it can be correct. Do try to be accurate!
It's strange to me how you take enormous offense at certain bits of casual conversation, but when I express the very real sentiment that I think you're mentally ill, you don't seem to notice.
You know something? I DON'T think it strange that there are those who take great delight in interpreting others' responses as the result of offence having been taken. All that demonstrates is that you are both assumptive and manipulative. If you were to refer to your elemenary troll's handbook, it will quite possibly suggest it as a way to deliberately get under people's skin. However, that's for beginners only. Best not to do it except at school. It really is kids' stuff.
I mean... there are multiple things wrong with what I'm saying...
First of all it's obviously very rude.
Also, if I really thought you were so far gone, why would I try to reason with you?
Why would I be puzzled by contradictory behavior?
Yeah, I guess I'm still talking to you, so I have some hope for a resolution.
I guess the point of this post is to say, this is very rude of me, but it's not meant as an attack. I'm trying to prompt an explanation.
<<Actually, the clearest way might be to prove that the variance of the sum of two independent random variables is the sum of their variances: it is a very simple result from that. Anyone want to prove that? (Looking up the proof will suffice, if you like).>>
I think that was basic error analysis which we learned at school in the 6th form, mainly for physics.
That sounds entirely plausible. Well done for remembering.
Intuition alone cannot break down what both intuition and rigor have built up.
And in any case, the stance "everyone else is wrong but me" is actual insanity, which is to say I don't mean that in a pejorative sense, it's a statement of fact: it's insane.
It can be insanity, it can be incorrect and it can be correct. Do try to be accurate!
It's strange to me how you take enormous offense at certain bits of casual conversation, but when I express the very real sentiment that I think you're mentally ill, you don't seem to notice.
Regarding the latter half of it, you have absolutely no idea whether I'm writing things to wind you up or not. You don't know what I really believe and you do have a noticeable desire to think negatively about others and I suppose I do instinctively ignore that in others. I know why certain people congregate together online.
I might well be mentally ill in your judgement and I'm not drawn to you in such a way as to wish to congregate with you but maybe, types attract similar types. I reserve my opinion about the accuracy of your judgement and wonder why you don't pick on harder targets too. Might give you more satisfaction. I'm giving you the chance to show yourself though, am I not? Are you not happy with that?
For the record, your "proof" will probably be an ontological argument, since you are trying to use it to define the ontology of infinity or of the concept thereof. As you will understand, ontology is concerned with the real nature of things but where those things can be conceptual, at least primarily, problems obviously arise, regarding what we consider to be that reality. In the case of infinity, I personally think there is probably no reality behind the concept, whereas I assume that for @MikeKalish, for instance, that reality is genuine. It seems to be a rather more complex example than the usual subject of an ontological argument.
It really would be a good idea to be clear about what the proposition is before you refer to a "proof". I bet you couldn't clarify this example.
I have previously pointed out that mathematicians have discovered a great deal about things that are inherently infinite. Some of these are very far from common intuition (such as the hierarchy of cardinals and its relationship to provability). There was no simple proposition here, except to observe that thinking of a single thing called "infinity" is not at all close to the reality.
As I also pointed out, there are different flavours of even the simplest notions of the infinite.
The infinity of the natural numbers is a simple theorem (proven by Euclid, I think) and is about counting. Mathematicians call the generalisation of this notion "cardinality". It behaves differently with infinite sets than finite ones.
A different notion is when you say that the sum of an infinite series (say 1/n for all natural numbers n) is infinite. This is better viewed as being about magnitude, for which we use real numbers. While the infinite number of natural numbers is like an addition to the finite natural numbers, the infinite sum of a series is like an addition to the real numbers.
As I mentioned, there is a rather esoteric mathematical object called the hyperreal numbers which includes the latter infinity and infinitessimals, and a large range of different examples. This is a more complete picture of the infinities of magnitude, while the hierarchy of cardinal numbers gives the right way to deal with infinities of counting.
<<Actually, the clearest way might be to prove that the variance of the sum of two independent random variables is the sum of their variances: it is a very simple result from that. Anyone want to prove that? (Looking up the proof will suffice, if you like).>>
I think that was basic error analysis which we learned at school in the 6th form, mainly for physics.
That sounds entirely plausible. Well done for remembering.
It's little things such as that, that I really appreciate. It isn't so much the compliment as the recognition of humanity in others. Thankyou. I'll do the same for you, when I can.
there are those who take great delight in interpreting others' responses as the result of offence having been taken
It's a very reasonable interpretation, considering your responses to me over the last 24 hours have variously characterized me as a bully, deliberately provocative, as well as angry and childish.
So as to state my own version of events, over the last 24 hours I haven't made a single post on these forums while feeling any sort of emotion. A few times I've been frank to the point of being rude with you, but I don't particularly feel any emotion and don't have a goal of provoking.
I am sure I can appear patronising, supercilious and various other things, but really all I am trying to do is draw people's attention to knowledge that exists (and which I have no personal credit for!) in a neutral way.
Like @llama36, I am not emotional, unless enthusiasm for knowledge counts.
For the record, your "proof" will probably be an ontological argument, since you are trying to use it to define the ontology of infinity or of the concept thereof. As you will understand, ontology is concerned with the real nature of things but where those things can be conceptual, at least primarily, problems obviously arise, regarding what we consider to be that reality. In the case of infinity, I personally think there is probably no reality behind the concept, whereas I assume that for @Mike_Kalish, for instance, that reality is genuine. It seems to be a rather more complex example than the usual subject of an ontological argument.
It really would be a good idea to be clear about what the proposition is before you refer to a "proof". I bet you couldn't even clarify this example.
I have previously pointed out that mathematicians have discovered a great deal about things that are inherently infinite. Some of these are very far from common intuition (such as the hierarchy of cardinals and its relationship to provability). There was no simple proposition here, except to observe that thinking of a single thing called "infinity" is not at all close to the reality.
As I also pointed out, there are different flavours of even the simplest notions of the infinite.
The infinity of the natural numbers is a simple theorem (proven by Euclid, I think) and is about counting. Mathematicians call the generalisation of this notion "cardinality". It behaves differently with infinite sets than finite ones.
A different notion is when you say that the sum of an infinite series (say 1/n for all natural numbers n) is infinite. This is better viewed as being about magnitude, for which we use real numbers. While the infinite number of natural numbers is like an addition to the finite natural numbers, the infinite sum of a series is like an addition to the real numbers.
As I mentioned, there is a rather esoteric mathematical object called the hyperreal numbers which includes the latter infinity and infinitessimals, and a large range of different examples.
I'm not in a state of mind where I'm thinking clearly. I've had a busy day and am rather too tired to focus. I can function well in some areas without focussing but wouldn't want to be playing chess.
I've found by trial and error that when I can't focus, sometimes I can think very well: originally but slowly. Better leave this until another day when I'm feeling zippy. After all, this is your bread and butter and I've noticed that you have expressed the last couple of explanations very clearly. I'm capable of deciphering mathemaical code but I don't enjoy it and not when I'm tired!
However, there's a pattern which keeps being repeated. All these examples of infinity are applied to types of numeric series. But a stochastic process of tossing a coin is a real thing. It's hypothetical, as an infinitely long process must be, but it refers to an hypothetical concept about a real thing as opposed to real concepts about ideals. That must be where the conceptual problem or difference lies, I believe.
I guess I can only be clear about my side of things... sometimes your posts @optimissed are frustrating to me, and so it would be very convenient for me if I could judge you either as someone I'll enjoy replying to, or as someone I'll perpetually ignore.
It's frustrating for me but, I suppose you're somewhere in the middle.
I've meant everything I've said, so I don't retract anything, but based on your posts, I think there are times when I've offended you without meaning to, and I think it's reasonable for me to apologize for that.
I don't think you're a bad person, but sometimes it's not easy for me to talk to you.
I guess I can only be clear about my side of things... sometimes your posts @optimissed are frustrating to me, and so it would be very convenient for me if I could judge you either as someone I'll enjoy replying to, or as someone I'll perpetually ignore.
It's frustrating for me but, I suppose you're somewhere in the middle.
I've meant everything I've said, so I don't retract anything, but based on your posts, I think there are times when I've offended you without meaning to, and I think it's reasonable for me to apologize for that.
I don't think you're a bad person, but sometimes it's not easy for me to talk to you.
I did react, only for an instant, when you called me a random person the other day but really my reaction was because that was obviously untrue and also "random person" is a well-known online dismissal or put-down that kids use a lot. You just seem to have a lot to prove and it's so unnecessary. You perhaps also tend to be cautious and non-committal and maybe I'm not those things. I've probably offended you without meaning to. Unlike Elroch, I don't think you are unemotional. I think, perhaps, you're both emotional but it's kept under lock and key in public, for each of you, for your own reasons which are to do with the way you present yourselves to others in his strange environment online.
I am sure I can appear patronising, supercilious and various other things, but really all I am trying to do is draw people's attention to knowledge that exists (and which I have no personal credit for!) in a neutral way.
Like @llama36, I am not emotional, unless enthusiasm for knowledge counts.
For what little it may be worth, my view of you has changed a lot, positively, approximately over the past three or four months.
...
I did react, only for an instant, when you called me a random person the other day
But did he call you a truly random person?
More seriously, considering the infinite sequence of coin tosses, the first thing to accept is that it's not real or physical. It's impossible. But it is intuitively a meaningful thing. It might also be a useful thing (like infinite decimal numbers are).
Another useful point is that, once considered as an abstract object, time disappears. You can accurately conceive of an infinite number of coin tosses all taking place simultaneously, merely labelled by the natural numbers (this is how infinite stochastic processes are defined).
Given that, intuition about the sequence never completing can be discarded. Like Zeno's paradox doesn't help you understand real-valued functions, that doesn't help to understand the Bernoulli Process.
I am sure I can appear patronising, supercilious and various other things, but really all I am trying to do is draw people's attention to knowledge that exists (and which I have no personal credit for!) in a neutral way.
Like @llama36, I am not emotional, unless enthusiasm for knowledge counts.
You have never appeared as any of those to me. In fact, I've been impressed that you seem to have gone out of your way not to insult me when I've said things that were clearly wrong.
For the record, your "proof" will probably be an ontological argument, since you are trying to use it to define the ontology of infinity or of the concept thereof. As you will understand, ontology is concerned with the real nature of things but where those things can be conceptual, at least primarily, problems obviously arise, regarding what we consider to be that reality. In the case of infinity, I personally think there is probably no reality behind the concept, whereas I assume that for @Mike_Kalish, for instance, that reality is genuine. It seems to be a rather more complex example than the usual subject of an ontological argument.