In the beginning ...

[Andrew D]

Fact: thestoat wrote: "999999999999999999999999 (which we could approximate to an infinite time)"

Fact: Andrew D responded with an article on the subject which says, in pertinent part: "While there is no difference between almost surely and surely (that is, entirely certain to happen) in many basic probability experiments, the distinction is important in more complex cases relating to some sort of infinity."

Fact: In responding, thestoat conveniently ignored and omitted the phrase "in many basic probability experiments".

Fact: In responding, thestoat conveniently ignored and omitted the clause "the distinction is important in more complex cases relating to some sort of infinity."

Fact: In responding, thestoat wrote: "This is key: 'While there is no difference between almost surely and surely'".

Fact: thestoat has not even attempted to explain, in English, how the infinite number of minutes in an infinite number of hours can be the same as that infinite number of hours without a minute's being equal to an hour.

Fact: The facts speak for themselves, and others will have to draw their own conclusions.

[Andrew D]

All you need to do, Andrew, is simply say "I simply do not understand any of the maths behind infinity" ....

Except that I understand them far better than you do. Why? Because unlike you, I actually think about them.

... rather than harping on about how maths is wrong and you are right.

Who may be right and who may be wrong matters only to small minds obsessed with trivia. A proposition either does or does not withstand logical scrutiny. Yours do not.

Each of your complaints above ha been answered sidestepped by me several times (the most recent leaving me [to] state [that because I have no coherent answers,] I am bored by left with nothing but repeating myself).

Fixed that for you.

If you I don't understand[,] - sayso - I find it pointless have no choice but to keep repeating a non-answer to a question that has I have already been answered dodged several times. Repeat a question and you'll get a repeat of the answer I'll keep repeating myself - unless you say I finally admit th[a]t you I don't understand the answer you are given question I have been asked.

That too.

If you were able to explain, in English, how it is (as you claim) that the infinite number of minutes in an infinite number of hours is the same as that infinite number of hours, you would surely have done so by now.

But you have not. Because you cannot. Go ahead; prove me wrong. Or just keep running away. The choice is yours.

And again, others will have to draw their own conclusions.

[thestoat]

It's possible it may be beneath you

There's always something new to learn, or a new angle on something already learned. And I have kids who'll love it too

[thestoat]

Fact: thestoat wrote: "999999999999999999999999 (which we could approximate to an infinite time)"

Another fact that you are ignoring. I also wrote - obviously MORE than once:

"Yes, I did write that, and once again (for the last time because repeating myself is becoming boring) the approximations hold true when a large number may be considered infinity to aid in simple calculations. This sort of thing happens a great deal in physics and mathematics. I have given real examples where it is appropriate to do so."

Don't you get it?

Fact: In responding, thestoat conveniently ignored and omitted the phrase "in many basic probability experiments".

Fact: In responding, thestoat conveniently ignored and omitted the clause "the distinction is important in more complex cases relating to some sort of infinity."

Another fact that you are ignoring. I also wrote:

"I read that to mean discussing other sortS of infinity (I have read there are several), not the simple infinity we are discussing. My mistake." You see? I can admit to my mistakes!

Fact: thestoat has not even attempted to explain, in English, how the infinite number of minutes in an infinite number of hours can be the same as that infinite number of hours without a minute's being equal to an hour.

Infinity is a mathematical concept - not a physical thing (something you clearly have yet to grasp). It cannot be explained in layman's terms and more than the square root of minus 1 (which, as noted previously, is also very important in mathematics). All we can do with infinity is set some rules around how to use it - which I did and you then completely ignored by trying to apply an invalid subtraction operation to it!

How would you define 1 / 0? The answer is infinity. But how can you describe that? I suspect that, were I to produce a mathematical proof, you would not understand it (and I am pretty sure I wouldn't) so there is little point looking for one, if one exists. It may be that this is simply how infinity is *defined* - don't forget that it does not exist in reality: it is a mathematical construct. If you like, I can repeat that statement a few more times until it sinks in.

[thestoat]

Except that I understand them far better than you do. Why? Because unlike you, I actually think about them.

That is brilliant, Andrew. I forgot you are such a mathematical genius that you believe you are right and maths is wrong. Obviously others will draw their own conclusions about the level of your understanding of infinity.

Who may be right and who may be wrong matters only to small minds obsessed with trivia. A proposition either does or does not withstand logical scrutiny. Yours do not.

Unfortunately I can not claim the credit for the proposition - far cleverer mathematical minds than mine did that. I have merely learned and agree with them. But, again (I do seem to have to repeat myself a lot), Quantum mechanics does not stand logical scrutiny. Is that also false?

Your attempts of "fix" my statements clearly indicate your lack of understand of them.

If you were able to explain, in English, how it is (as you claim) that the infinite number of minutes in an infinite number of hours is the same as that infinite number of hours, you would surely have done so by now. But you have not. Because you cannot. Go ahead; prove me wrong. Or just keep running away. The choice is yours.

I cannot. I am not running away - I just can't explain it. There are many things I cannot explain: that doesn't make them false. I can prove it mathematically (and have done so), but you even dispute mathematics (where you state "The breakdown is in the mathematics by which people claim to have shown that the number of minutes in an infinite number of hours is equal to that infinite number of hours") . I thus conclude you either had no clue what the (very simple) equations meant or your genius/arrogance is such that you know better than fundamental mathematics.

[Crackpot]

http://scidiv.bellevuecollege.edu/Math/ ... Hotel.html

For those who think muliplying infinity makes a bigger number.

[thestoat]

CP - Excellent. Love the statement :

I had taken an infinite hotel that was full, added an infinite number of infinities, and when all was done, I still had an infinite number of vacancies!

There may be some who won't get it, but I thought it was a nice piece

[Crackpot]

that's one of the stories gone over in that book.

[MajGenl.Meade]

Andrew

I was guilty of sloppy language. The hypothetical is not the “infinite set of moments” but the “infinite set of all moments” – whether past or future. That should have been understood since I was not identifying “past” moments but an actual infinity which includes all actual past and all actual future. This means it includes all actual events.

From one of those discussions of Hilbert’s infinite hotel: “Infinity × 2 is still infinity. There are just as many even numbers as there are natural numbers, and there are just as many odd numbers as there are natural numbers. If you combine the set of the evens with the set of the odds, you get a set that’s no larger than either of the sets you started with”

A = C and B = C therefore A = B. There are the same number of odd and even numbers. Of course, if you define a set as consisting only of A, then one can hardly be surprised that B does not appear in that set. It certainly says nothing about a set that includes all A and all B.

Now consider the hypothetically infinite set of all past moments. That set does not contain any future moments

Of course not – given that “future” refers to that which has not yet happened. It is not contended that the past includes future moments. The contention is that there cannot be an infinite past for if there were then there would be no future moments at all. Since there are future moments, an infinite regress is not possible.

What’s odd (to me) is that we agree that there is a future – that not all possible things have occurred – but the conclusion that therefore an actual infinity of time cannot precede us is not acceptable to you. The assumption I make - and it is an assumption - is that your intent to preserve disbelief (or non-believ or an inability to know - I will not insist upon a term - you choose) ... your intent to preserve disbelief causes you to argue anything rather than admit the possibility that the universe had a beginning in finite time. Obviously you will disagree

The absolute equality of infinity as a set – that the infinite number of moments in the “past” is the same “amount” as the infinite number of moments in the “future” – is critical. In infinite time, all potentials will be actualized – otherwise they were not potentials. If the “amount” of time “past” is infinite, it is exactly the same “size” or “amount” as all time. Therefore in that identical infinity of time (“past” for clarity) all potentialities must have been actualized. Since they have not acualized, time “past” is not infinite.

Saying that (a) because a possible thing has not occurred within the hypothetically infinite set of all past moments, it cannot occur within the set of all moments, past (and present?) and future

But I have never said such a thing! You have reversed the argument entirely and the bulk of yours is misdirected. The non-occurrence (yet) of even one possible thing is exactly the proof that there is no actual infinity of time past. That’s all. Even I’m not dumb enough to say that because something hasn’t happened in the past it cannot happen in the future.

I think looking back we’ll find that I said a non-believer either must end up positing an infinite regress (that the universe has “always existed”) or there’s a spaghetti monster out there. Maybe I didn’t say that. Maybe it was “or that nothing can cause something”. Whatever

God’s eternal existence; not the same as infinite regress since He is beyond time – He created time by creating matter.

If God is uncaused why can’t the universe be uncaused? (That’s a potential fallacy by the way – assuming identity). But yes you can argue that the universe is uncaused. But then are you not an infinite regress supporter – right? (Bet you find a way to say ‘no’!). However, you know the drill. ‘Whatever has the possibility for non-existence is currently caused by another’.

Is the universe “necessary” and not “contingent” – is that going to be at all profitable?

Groaning!
Meade

[Andrew D]

The increasingly peripatetic nature of thestoat’s postings should come as no surprise. It avoids the essential challenge: Explain, in English, without making a minute equal to an hour, how it is that the number of minutes in an infinite number of hours is equal to that infinite number of hours.

To his credit, thestoat concedes that he “can’t explain it.” One might well think that that should be the end of the matter.

This:

Fact: thestoat wrote: "999999999999999999999999 (which we could approximate to an infinite time)"

Another fact that you are ignoring. I also wrote - obviously MORE than once:

"Yes, I did write that, and once again (for the last time because repeating myself is becoming boring) the approximations hold true when a large number may be considered infinity to aid in simple calculations. This sort of thing happens a great deal in physics and mathematics. I have given real examples where it is appropriate to do so."

Don't you get it?

Fact: In responding, thestoat conveniently ignored and omitted the phrase "in many basic probability experiments".

Fact: In responding, thestoat conveniently ignored and omitted the clause "the distinction is important in more complex cases relating to some sort of infinity."

Another fact that you are ignoring. I also wrote:

"I read that to mean discussing other sortS of infinity (I have read there are several), not the simple infinity we are discussing. My mistake." You see? I can admit to my mistakes!

is simply weird. Once it is acknowledged that the statement “the distinction is important in more complex cases relating to some sort of infinity” does apply to “the simple infinity we are discussing,” the contrary statement that “999999999999999999999999 (which we could approximate to an infinite time)” should be withdrawn.

But thestoat continues to insist that “a large [finite] number may be considered infinity to aid in simple calculations.”[1] If the sorts of things with which we are concerned are “simple calculations,” then thestoat has conceded that he is in error. If the sorts of things with which we are concerned are not “simple calculations,” then the fact (if such it be) that finite numbers can be approximated to infinite numbers in simple calculations is irrelevant. Ah, well; logical consistency is not among thestoat’s strong points.

Speaking of weird, thestoat concedes that he “can’t explain” how the number of minutes in an infinite number of hours can be the same as that infinite number of hours without a minute’s being equal to an hour. But he says, nonetheless, that he has “merely learned and agree with” it.

What does that even mean? How can one rationally agree with a proposition which one finds inexplicable?

(Of course, one can non-rationally agree with a proposition which one finds inexplicable. That is common with matters of faith. And that is no disparagement of faith; it is simply an observation that there is more than one kind of ground for agreement with a proposition.)
-------------------------

1. thestoat claims that he has “given real examples where it is appropriate to do so”; i.e., to approximate a finite number to an infinite number. I recall no such examples, but this is a rather long thread, and I may have missed them. Perhaps he will be good enough to bring them to my attention.

[Andrew D]

This:

How would you define 1 / 0? The answer is infinity. But how can you describe that?

presents the same issue as was presented by the fact, discussed above, that any number, finite or infinite, multiplied by zero equals zero. In both cases, the answer is perfectly sensible in English, because “zero” means “nothing”.

If I have one (empty) basket, and I put into it zero apples, the number of apples in the basket is still zero. And if I put into it zero apples again and again and again, the number of apples in the basket is still zero. And no matter how many times I put into it zero apples, the number of apples in the basket is still zero.

But that is not true if I actually put apples into the basket. If I actually put apples into the basket, whether the number of apples I put into the basket is finite or infinite, the number of apples in the basket after I have done so is necessarily greater than zero. In non-mathematical terms, if I put something into the basket, the basket contains something, not nothing. And an infinite number of apples, regardless of whether that number is or is not equal to some other infinite number, is no zero apples. In mathematical terms, the statement ∞ ≠ 0 should be noncontroversial.

It appears that what thestoat is actually asserting is that “infinity” (or, better, “an infinite number”) is merely a convenient fiction which mathematicians find useful. (He says that infinity “cannot be explained in layman’s terms,” which appears to mean that it cannot be explained in a natural human language.) I have no objection to that assertion: What fictions mathematicians find useful is for mathematicians to determine, and I am not one of them.

As a lawyer, I am familiar with useful fictions. Perhaps the most famous legal fiction is that a corporation is a person. Everyone knows that a corporation is not a person in the conventional sense (which is why the phrases “natural person” and “artificial person” are so common in the law). But the fiction is useful, because it obviates the need to start from scratch in determining what “rights” a corporation possesses.

(And it bears noting that the fiction has its limits: A corporation’s “rights” are not coextensive with a “natural”) person’s rights. For example, corporations cannot refuse to produce information (documents, live testimony, whatever) on the ground that such information might tend to incriminate them.)

But an essential attribute of a useful fiction is that it is, well, fictional. The legal fiction that a corporation is a person is useful, but it is, nonetheless, a fiction. In other words, it is not true.

If the claim that even though a minute is not equal to an hour, the number of minutes in an infinite number of hours is equal to that infinite number of hours is a useful fiction in mathematics, it is still a fiction. In other words, it is still not true.

[Big RR]

Andrew--it is either a fiction or a consequence of mathematics and how it is structured. Face it, another mathematical concept is that when one multiplies two negative numbers they become a positive number--no tie in to any real concept, but it exists nonetheless. Likewise imaginary numbers.

[Scooter]

How would you define 1 / 0? The answer is infinity.

Incorrect. The value of 1/0 is not defined. The limit of 1/0 is infinity, but that is not at all the same thing.

[Andrew D]

And one could plausibly argue that the whole of mathematics is but a useful fiction. For centuries, the fiction of Newtonian physics was useful. (And for purposes not requiring extreme precision, it still is.) But things progressed, and we encountered things for which Newtonian physics is not useful. (According to Hawking, if GPS were based on Newtonian physics, people relying on their GPS systems would commonly end up lost.)

The assertion that when one multiplies two negative numbers, the result is a positive number may well be only a useful fiction. (For that matter, negative numbers themselves may well be only useful fictions: I have seen one apple in a basket, and I have seen twenty apples in a basket, and I can at least begin to imagine an infinite number of apples in a basket; but I have not seen, nor can I even begin two imagine a negative number of apples in a basket.)

But in what Meade and I are discussing, such things actually matter. If Meade is correct that a universe that has existed throughout an infinite past is impossible, that is at least a step toward demonstrating the existence of something which is not a part of the universe. (There remains, at least as an intellectual possibility, the idea that even a universe which has not existed throughout an infinite past was not caused by anything outside itself.) If I am correct that it has not been demonstrated that a universe that has existed throughout an infinite past is impossible, then the existence of something which is not a part of the universe has not been shown to be necessary.

[thestoat]

To his credit, thestoat concedes that he “can’t explain it.” One might well think that that should be the end of the matter.

Hardly an interesting point. There are vast tracts of maths and physics (probably other sciences too) that the finest minds on the planet cannot explain. In fact, Richard Feynman once said "If you think you understand quantum mechanics, you don't understand quantum mechanics". Conceptually I can understand infinity, when to use it and when to approximate. I think it is fairly obvious you do not. I have given you a mathematical definition of it (which you ignored).

Once it is acknowledged that the statement “the distinction is important in more complex cases relating to some sort of infinity” does apply to “the simple infinity we are discussing,” the contrary statement that “999999999999999999999999 (which we could approximate to an infinite time)” should be withdrawn.

Not at all, and I find it truly bizarre that you keep coming back to this point that has been explained so many times. Here I shall repeat it ...

Approximations are used all the time in many branches of physics and maths. They have to be since no measurement is perfect. Now, 9999999999999999999999999 millennia might not be a good approximation of infinity when dealing with the age of the universe, but it would be a good approximation when dealing with the lifespan of a ladybird.

Or consider rays of light from the sun. We consider them to be parallel. They are not parallel, but near as dammit when making calculations. They would only be parallel if the sun was infinitely far away, which of course it isn't.

Let's think of 2 of these rays of light falling on a book you are reading (let's just consider 2 rays, one on one extremity of the book, the other on the other). The rays of light can be considered to form an isosceles triangle. One side of this triangle is about 6 inches long. The other 2 sides are 93 million miles long. Question - what are the internal angles of the triangle?

Any normal person with a reasonable understanding of maths would say "hmmm. Let's pretend that this 93 million miles is infinity." When you do that, the answer becomes 90 degrees, 90 degrees and 0 degrees. That answer is normally good enough. But if you insist that you can never approximate infinity they you will say

"Ah ha - wrong answer. The actual answer is ...".

I wonder - can you tell me? My calculator can't calculate such accurate numbers. It will be something like
0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 degrees and
2 lots of
89.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998

Most people would, I suspect, be happy to say "well, call it two lots of 90 and 1 lot of zero degrees", which is what you will get if you assume, in this case, 93 million miles is infinity.

Perhaps you can see that approximating large numbers to infinity can actually be very useful, if you understand the maths. Or perhaps you still don't agree?

That is what I mean by a simple calculation. Therefore I most certainly do not concede I am in error. Everything I have said has a very firm basis in logic.

Speaking of weird, thestoat concedes that he “can’t explain” how the number of minutes in an infinite number of hours can be the same as that infinite number of hours without a minute’s being equal to an hour

Hang on - didn't you just say that at the start of the post?

How can one rationally agree with a proposition which one finds inexplicable?

Ah, I see. Then please explain to me the nature of quantum mechanics - or do you disagree with QM? Unless you refute this statement in your next sentence, in which case I wonder why you bring it up.

Perhaps he will be good enough to bring them to my attention.

See above.

[thestoat]

Wow. This is brilliant. When I ask what "one divided by zero" is, you state

presents the same issue as was presented by the fact, discussed above, that any number, finite or infinite, multiplied by zero equals zero. In both cases, the answer is perfectly sensible in English, because “zero” means “nothing”.

I didn't ask about multiplication. I asked about division

If I have one (empty) basket, and I put into it zero apples, the number of apples in the basket is still zero. And if I put into it zero apples again and again and again, the number of apples in the basket is still zero. And no matter how many times I put into it zero apples, the number of apples in the basket is still zero.

I didn't ask about addition or multiplication. I asked about division

These are basic maths concepts. And different from each other.

What fictions mathematicians find useful is for mathematicians to determine, and I am not one of them.

Clearly.

It appears that what thestoat is actually asserting is that “infinity” (or, better, “an infinite number”) is merely a convenient fiction which mathematicians find useful

I have never said that infinity is fictional. Merely that it is nor a real number. And I mean "real" in a mathematical sense.

But an essential attribute of a useful fiction is that it is, well, fictional. The legal fiction that a corporation is a person is useful, but it is, nonetheless, a fiction. In other words, it is not true. It appears that what thestoat is actually asserting is that “infinity” (or, better, “an infinite number”) is merely a convenient fiction which mathematicians find useful

So if quantum mechanics is "not true", I would be very interested in your summation of how it is used in various physical instruments that rely on it. Please explain.

[thestoat]

How would you define 1 / 0? The answer is infinity.

Incorrect. The value of 1/0 is not defined. The limit of 1/0 is infinity, but that is not at all the same thing.

Errr ... that doesn't make any sense at all, Scooter. Mathematics talks about limits in certain areas of calculus, a good example being integration. That most certainly does not apply in this case where we are dividing real integers. The answer is infinity.

To understand division by zero, one must check it with multiplication: multiply the quotient by the divisor to get the original number. However, no number multiplied by zero will produce a product other than zero. To satisfy division by zero, the quotient must be bigger than all other numbers, i.e., infinity

http://en.wikipedia.org/wiki/Division_by_zero

[thestoat]

The assertion that when one multiplies two negative numbers, the result is a positive number may well be only a useful fiction. (For that matter, negative numbers themselves may well be only useful fictions: I have seen one apple in a basket, and I have seen twenty apples in a basket, and I can at least begin to imagine an infinite number of apples in a basket; but I have not seen, nor can I even begin two imagine a negative number of apples in a basket.)

You really need to move away from apples - they don't seem to be helping.

If I owe the bank money, and I don't have the money to repay it, my bank balance has a negative number in it.

If it makes you more comfortable, I could rephrase this to

If I owe the bank a load of apples, and I don't have the money to buy the apples required, my bank balance has a negative number in it.

[Andrew D]

I didn't ask about multiplication. I asked about division.

Oh, for crying out loud. Multiplication and division are correspondent operations. The number of times that two goes into six (division) is three. The number of twos one needs to get to six (multiplication) is also three. Do you really not grasp that those are different ways of saying the same thing?

[thestoat]

I didn't ask about multiplication. I asked about division.

Oh, for crying out loud. Multiplication and division are correspondent operations. The number of times that two goes into six (division) is three. The number of twos one needs to get to six (multiplication) is also three. Do you really not grasp that those are different ways of saying the same thing?

Not in this case. You can multiply 2 by 0 apples. What do you get with 2 divided by zero apples?????????? If they are so similar why didn't you use an example with division rather than multiplication?