A Collision at the Speed of Light

[Thalestris24]

7 minutes ago, saac said:

Louise I remember watching that Horizon video some time ago, really enjoyable and just shows how challenging the concept is. The little companion video with  Prof Gowers, for me at least, sums up the dangers of a literal interpretation of infinity. With some very basic algebraic operations Gowers demonstrates how infinity can lead to a nonsensical conclusion that 1 + 1 = 1. 

∞= 1/0    ( a commonly accepted definition) 

0 x ∞ = 1

(0+0) ∞ = 1

0 x ∞  + 0 x ∞ = 1 (line 2) but  0 x ∞ = 1  

therefore  1 + 1 = 1

The outcome is clearly "real world" nonsense; however,  as Gowers explains, one cannot treat infinity as a normal number and expect all the usual laws of arithmetic to hold.  Infinity, aye there be monsters, but also beauty as Blake found:

"To see a World in a grain of sand

And Heaven in a wild flower,

Hold infinity in the palm of your hand,

And eternity in an hour"        

William Blake fragments from "Auguries of Innocence , To See A World" 

 

Jim - still beneath an infinite cloud cover  

Yeah, infinity certainly isn't amenable to normal arithmetic since it's not a number or a quantity. Perhaps a set of sets, or just a concept. I watched both the linked videos again after I posted them. I love the mathematician/hoarder (in the Horizon prog) surrounded by the complete jumble of books and papers! Infinite mess rapidly approaching!

Louise

[andrew s]

Sorry but 1/0 is not a number and not an allowed division. While it may be commonly equated to infinity it is not done so in the mathematics of number theory. 

Regards Andrew 

[IanL]

7 minutes ago, andrew s said:

Sorry but 1/0 is not a number and not an allowed division. While it may be commonly equated to infinity it is not done so in the mathematics of number theory. 

Regards Andrew 

Exactly. I give you one slice of cake and ask you to cut it in to no slices of cake. If anyone can explain how that is done I will accept 1 + 1 = 1

[Stub Mandrel]

3 minutes ago, IanL said:

Exactly. I give you one slice of cake and ask you to cut it in to no slices of cake. If anyone can explain how that is done I will accept 1 + 1 = 1

Here's an infinite number of plates with 1/0 of a cake on each one, total one cake...

But here's the original plate and it still has an intact cake on it. Total two cakes.

>Munch<. >Burp<. Total zero cakes.

See, Biology always trumps Maths.

 

[SilverAstro]

19 minutes ago, Stub Mandrel said:

>Munch<. >Burp<. Total zero cakes.

Oh ! You greedy thing you !!

I would have settled for a few real decimal fractions of your cake, I did not want the whole of it, in fact I do not want an integral number of them

because if I had an infinite number of real decimal fractions of cake my cake pile would have been a greater infinity of cake than your infinite pile of integer number of cakes.

I think !  but that is where my head begins to hurt and need another cup of coffee, and maybe a bacon sarnie

 

[andrew s]

I think you can tell when a thread has reached the end of it's useful life... ...when the cake has all gone?

Thinking about infinity sent Cantor mad.

Regards Andrew

[Stub Mandrel]

35 minutes ago, andrew s said:

I think you can tell when a thread has reached the end of it's useful life... ...when the cake has all gone?

Thinking about infinity sent Cantor mad.

Regards Andrew

<ref>Total Perspective Vortex</ref>

[andrew s]

1 hour ago, Stub Mandrel said:

Total Perspective Vortex

I will improbably have a nice warm cup of tea.

Regards Andrew

[SilverAstro]

3 hours ago, andrew s said:

end of it's useful life... ...when the cake has all gone?

Thinking about infinity sent Cantor mad.

Oh yes ! I had forgotten that, I expect they will be here any minute in their white coats

(I half expected you to ask if I was trying to make a cardinal point, shucks ,  , out caked and out guessed  )

[saac]

9 hours ago, IanL said:

Exactly. I give you one slice of cake and ask you to cut it in to no slices of cake. If anyone can explain how that is done I will accept 1 + 1 = 1

Exactly the point, 1 + 1 does not equal 1, infinity cannot be handled by normal arithmetical operators else it leads to nonsensical results. Effectively what the exercise demonstrates as mentioned by Prof Gower is that our initial boundary conditions are in error namely 1/0 does not equal infinity. 

 

Andrew, what if we approximate 0 to be a really small number tending towards the infinitely small and then divide it in to 1.  To all practical intents and purpose we have divided 1 by 0.   Would the result not tend towards infinity?  By approximation then 1/0 could be considered, for the purpose of the proof at least, to equal infinity. You need to humour me here I've stacked the boundary conditions   Of course the more we have to make these concessions the more the alarm bells should be ringing that infinity is not what it pretends to be.

I don't accept that infinity is a number/property of any practical or corporeal concern. I've always considered infinity to be a process, rooted firmly in mathematical curiosity/flimflam!  Like entropy as seen by the engineer, a useful tool to derive something that is more useful but of little value in its own right. 

 

Jim

[saac]

18 hours ago, SilverAstro said:

Only in 2 dimensions ? Straight up it is quite thin, probably 

It may as well be infinite straight up, my pier is only 1.75m  

Jim

[andrew s]

19 hours ago, saac said:

Andrew, what if we approximate 0 to be a really small number tending towards the infinitely small and then divide it in to 1.  To all practical intents and purpose we have divided 1 by 0.   Would the result not tend towards infinity?  By approximation then 1/0 could be considered, for the purpose of the proof at least, to equal infinity. You need to humour me here I've stacked the boundary conditions   Of course the more we have to make these concessions the more the alarm bells should be ringing that infinity is not what it pretends to be.

Hi Jim, you get into the theory of taking the limits of functions. This is an area I am not expert in. As you propose you can use the idea that taking a limit as your small number approaches zero but (and it is a big but) this can't be used in all circumstances and has a number of strict conditions attached to it.

As an example in geometry is where you can use co-ordinates of 1/x rather than x where 0 -> infinity and infinity -> 0 (-> is mapped to.)  

Other example is where a finite section semi infinite section of a  curve can generate an infinite area when rotated about, say, the x axis  but enclose a finite volume! It's all about taking limits. This would mean that if it were a paint pot it would not have enough paint to paint it's interior!!!!! My brain hurts.

But, in the end division by 0 is not a valid number period.

Regards Andrew

PS I have corrected a mistake I made above. The curve in question is y = 1/x on the line 1 to  +∞ (you can go as close as you like to zero but not include it. I apologies for the error but I was recalling this from about 1971 or 2. I can provide the proof (I worked it out last night) but for now leave it to the reader.

[saac]

31 minutes ago, andrew s said:

This would mean that if it were a paint pot it would not have enough paint to paint it's interior!!!!! My brain hurts.

But, in the end division by 0 is not a valid number period.

Regards Andrew

Andrew thanks, that is about as close I've seen to a practical analogy that makes sense.  

Jim

[Tiki]

 

1 hour ago, saac said:

Exactly the point, 1 + 1 does not equal 1, infinity cannot be handled by normal arithmetical operators else it leads to nonsensical results. Effectively what the exercise demonstrates as mentioned by Prof Gower is that our initial boundary conditions are in error namely 1/0 does not equal infinity. 

 

Jim

There is also a second error in Prof Gower's exercise....

 

0x1=0

0x(1-1+1)=0

1-1+1=0      [oops, just cancelled the zeros. line 2 in Prof Gower's exercise]

1+1=1

whilst 1/0 is not a number , 0/0 is undefined and can be manipulated in an equation to provide any value.

[saac]

Good spot Tiki. I'm busy watching University Challenge at the moment and I wonder what Wolfson - Cambridge would make of infinity; one or two mathematical boffins there although trailing Edinburgh at the moment.  To be fair to the prof though I think he was offering his little proof with tongue very much in cheek,  just trying to show how we have to be careful with our treatment and understanding of infinity.

 

Jim

[SilverAstro]

1 hour ago, saac said:

what if we approximate 0 to be a really small number tending towards the infinitely small and then divide it in to 1.  To all practical intents and purpose we have divided 1 by 0.

but if we continue   to humour you  !sorry  ,  if we continue that logic (for to be consistent throughout)  then we must do that in the next line also -  which was 0 x ∞ = 1  - 

so we get (something very very small) x ( something very very large) which is indeterminate and cannot be equated to 1 (especially if we rigorously apply  'tending to the limit'  of/on each ) so that house tumbles.

 

[SilverAstro]

Arrgh I forgot UChallenge

 

[saac]

2 minutes ago, SilverAstro said:

 

so we get (something very very small) x ( something very very large) which is indeterminate and cannot be equated to 1 (especially if we rigorously apply  'tending to the limit'  of/on each ) so the house, maybe even the universe, tumbles.

 

And maybe it has,,an infinite number of times Sorry, being mischievous here     

 

Jim

[SilverAstro]

Just now, saac said:

And maybe it has,,an infinite number of times Sorry, being mischievous here   

arnt we all !

I was in the middle of editing my post to limit it to the house of cards only collapsing  when you picked me up

 

[Stub Mandrel]

14 minutes ago, Tiki said:

 

There is also a second error in Prof Gower's exercise....

 

0x1=0

0x(1-1+1)=0

1-1+1=0      [oops, just cancelled the zeros. line 2 in Prof Gower's exercise]

1+1=1

whilst 1/0 is not a number , 0/0 is undefined and can be manipulated in an equation to provide any value.

1/0 = 2/0

Multiply both sides by zero:

1=2

I'm not entirely convinced that 0 is a quantity. Only positive integers can be actual numbers of things.

[saac]

22 minutes ago, Stub Mandrel said:

 

I'm not entirely convinced that 0 is a quantity. Only positive integers can be actual numbers of things.

Agreed, but I think that is a whole new thread and my brain hurts enough; I can just about deal with infinity but not one iota more  

Jim

[andrew s]

1 hour ago, Stub Mandrel said:

 

I'm not entirely convinced that 0 is a quantity. Only positive integers can be actual numbers of things.

While 0 was late on the block compared to the +ve integers we need it along with all the real and complex numbers to describe /model the reality we inhabit.

It is interesting that pi is transcendental and irrational was known to be very early on (you can't square the circle).

If 0 is a quantity or not depends on what you define as a quantity. 

Regards Andrew

[saac]

1 hour ago, andrew s said:

While 0 was late on the block compared to the +ve integers we need it along with all the real and complex numbers to describe /model the reality we inhabit.

It is interesting that pi is transcendental and irrational was known to be very early on (you can't square the circle).

If 0 is a quantity or not depends on what you define as a quantity. 

Regards Andrew

Now wouldn't a history of number development and the role of mathematical concepts in advancing our understanding of reality be a worthwhile read.  Calculus is an obvious example but how often was a new branch of mathematics developed specifically in response to a particular need; thinking about the likes of Dirac etc. I'd love to see how one advance in mathematics lead to another in cosmology/physics. My old maths teacher would be smiling to hear me say that; I was always mere mechanic in the subject.

Jim

 

[Thalestris24]

Maybe you guys should put infinity to bed and have a look at some quantum mechanics instead!

Louise

[SilverAstro]

2 hours ago, Thalestris24 said:

Maybe you guys should put infinity to bed and have a look at some quantum mechanics instead!

Easy-peasy, no prob, I've got a tranny radio that shows that that stuff works, without even needing a tunnel diode.

but even CERN cant (yet) create a singularity or an infinity