Infinitely maddening

Originally posted by DeepThought
The proof I reproduced above is the standard one due to Cantor. If that is the one from the YouTube video then it is correct, but other proofs are possible. I'd regard a decimal number as a number written in base 10. Any decimal which can be written down with a finite number of digits is automatically rational. 0.123 for example is 123÷1000. I would ...[text shortened]... ht present the proof better than I did:
http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

"Talking about a decimal number with an infinite number of digits is perfectly sound. Writing out such a number is a supertask but that in itself isn't any great objection."

I beg to differ.

A question: Is the exact value of pi a decimal number? Is it possible to write down all decimals, infinitely many?

"You could just have a Turing machine calculate pi and have it write the first digit after ½ a second, the next digit ¼ of a second later and so on."

Sorry, such a Turing machine does not exist, and cannot ever be constructed. Time in itself is quantizised and a digit cannot be printed in an interval shorter than this smallest time interval, so the printer will stop there, well before you reach the very last digit in this infinite string.

Originally posted by FabianFnas
"Talking about a decimal number with an infinite number of digits is perfectly sound. Writing out such a number is a supertask but that in itself isn't any great objection."

I beg to differ.

A question: Is the exact value of pi a decimal number? Is it possible to write down all decimals, infinitely many?

"You could just have a Turing machine cal ...[text shortened]... the printer will stop there, well before you reach the very last digit in this infinite string.

pi is a number, decimal is a representation. Conceptually there is no problem with an infinite string of digits. If you are going to insist that there is no such thing as an infinitely long string of symbols as an abstract concept then you throw out Cantor's uncountability proof. In fact you throw out the notion that any infinite set can be mapped to another set. This might be difficult to justify to mathematicians.

Part of the definition of Turing machines is an infinitely long tape. They are conceptual devices, there is a variant of Turing machines which can copy themselves so that NP problems can be analysed.

Where did you get the idea that time is quantised? Standard model physics doesn't say that and General Relativity certainly doesn't. There are speculative theories that it might be, but none of them have any experimental authority. You are simply not justified in making that claim. The existence of a Planck time is not the same as time being quantised.

Originally posted by DeepThought
pi is a number, decimal is a representation. Conceptually there is no problem with an infinite string of digits. If you are going to insist that there is no such thing as an infinitely long string of symbols as an abstract concept then you throw out Cantor's uncountability proof. In fact you throw out the notion that any infinite set can be mapped to ...[text shortened]... d in making that claim. The existence of a Planck time is not the same as time being quantised.

First, half a Planck time doesn't exist. Like half a photon. Also non-existent.

Secondly, whenever you imply that it is possible to write down any infinite number of digits, then I will object.

Originally posted by FabianFnas
Secondly, whenever you imply that it is possible to write down any infinite number of digits, then I will object.

Does the number "Pi" exists as an infinite number of digits even if it's not possiblee to write it down in full?

Originally posted by adam warlock
Does the number "Pi" exists as an infinite number of digits even if it's not possiblee to write it down in full?

Yes.

"October 17, 2011: The record has been improved to 10 trillion digits."
I suppose that this record is beaten now. But at that time 10 trillion numbers were known. (If this is the exact truth then...) does this mean that the 10 trillion first digit was totally unknown? Yes. And if you are the first to compute n digits, the next one (the n+1th one), is there, but still unknown.

So, yes, the number Pi does exist as an infinite number of digits, even if it's not possible to know them all. In fact, the last infinite digits of the decimal representation of pi are still unknown. (!)

Originally posted by FabianFnas
Yes.

So you only object when one says that it is possible to write down Pi in full but don't object to the notion of its existence. That's interesting...

Originally posted by adam warlock
So you only object when one says that it is possible to write down Pi in full but don't object to the notion of its existence. That's interesting...

Yes. There are practical reasons that you cannot write down an infinite number of characters of any kind. And a limit of time too. And material. And ink. And a lot of other reasons too that this is completely impossible.

Sometimes I hear people ask "Fabian, if you go by the speed of light, then..." And I interrupt him and say, "No, you cannot." Everything based of an false fact that you can go by the speed of light can lead anywhere and doesn't have anything with reality to do. When I was young I philosophized about what would happen if you divide by zero. I had some theories. But it was completely a waste of time. I put that thinking aside when my teacher proved to me that it was completely impossible and against the very foundation of mathematics.

Originally posted by FabianFnas
When I was young I philosophized about what would happen if you divide by zero. I had some theories. But it was completely a waste of time. I put that thinking aside when my teacher proved to me that it was completely impossible and against the very foundation of mathematics.

Do you remember your teacher's proof of that? And can you tell me why it goes against the very foundations of mathematics?

Originally posted by FabianFnas
First, half a Planck time doesn't exist. Like half a photon. Also non-existent.

Secondly, whenever you imply that it is possible to write down any infinite number of digits, then I will object.

Maths may control Physics but Physics will never control Maths.

Why should Maths thought experiments be subject to constraints of the real world?

Originally posted by adam warlock
Does the number "Pi" exists as an infinite number of digits even if it's not possiblee to write it down in full?

Adam - you are normally so precise!
You would cringe at anyone else talking about "an infinite number"
(although we all know what you mean)

Originally posted by wolfgang59
Adam - you are normally so precise!
You would cringe at anyone else talking about "an infinite number"
(although we all know what you mean)

😳😳😳😳😳😳😳😳😳😳

I couldn't think of a better way to say it in english. Is "infinite numbers" (instead of infinite number of digits) a better way to say it?

PS: in my defense I'll say that in Portuguese the equivalent sentence: "um número infinito de dígitos" sounds, and is, perfectly alright.

Originally posted by adam warlock
PS: in my defense I'll say that in Portuguese the equivalent sentence: "um número infinito de dígitos" sounds, and is, perfectly alright.

I believe 'an infinite number of digits' is perfectly correct in English too.

Originally posted by FabianFnas
First, half a Planck time doesn't exist. Like half a photon. Also non-existent.

Secondly, whenever you imply that it is possible to write down any infinite number of digits, then I will object.

The way the Planck time is obtained is a game of playing with numbers. One takes Newton's constant, Planck's constant, the speed of light and mixes them together so something with the units of time come out. This gives the scale at which one would expect Quantum Gravity effects to dominate. There are good reasons to believe that it is impossible to measure time intervals shorter than this, but that is a different statement to the one you made which is that there are no time intervals shorter than this.

There are speculative theories where there are shortest distance scales and hence shortest time scales, but none of them have any experimental justification. These shortest possible distances are not necessarily the same as the Planck distance. There are also theories, equally speculative, that there is no such shortest distance scale, we just can't probe them.

Physics is an empirical subject. If a theory does not agree with experiment it doesn't matter who has proposed it it is just not true. Until an experiment has been done a theory does not deserve the name. You are treating quantization of time as a theory, and treating it as if it had experimental authority, when it is only a hypothesis - a sort of glorified guess. Anyone in the physics community who claims that time is quantized is engaging in a P.R. exercise for their research. There are some reasons to think it might be, as other things are quantized, but that question is still open.

Originally posted by twhitehead
I believe 'an infinite number of digits' is perfectly correct in English too.

I was being pedantic about "an infinite number".

I don't know any. Do you? 😉

Originally posted by wolfgang59
I was being pedantic about "an infinite number".

I don't know any. Do you? 😉

That is not what it means. If I say there were 'a large number of digits', I am not talking about a specific large number, but rather that the number of digits is large. The word 'number' in this instance is synonymous with 'quantity'. So think of 'an infinite quantity of digits' or a 'large quantity of digits'.