Relationship between the abstract world of mathematics and the material universe.

[Budapest]

Mr GG

It gets weirder as it can also be shown that -1 = e ^( i pi). Note that both e and pi are both irrational and transcendental.

http://www.google.com/search?q=e%20is%20transcendental

I'm getting out of my league here as far as mathematics is concerned, but will add that irrational numbers are the basic flaw in maths. Transcendental is semantics, because if you try hard enough, and the drugs work, you can express anything as algebra (the quantum stuff is a wonderful example of this - yes, objects can exist in two places at the same time, and there are multiple universes, etc, but at this point in time - ha! - it's all just mathematical theory). Bottom line is, and forgive me for dumbing down, the numbers don't add up.

[Sydney Grew]

. . . because infinity cannot be explained.

Well the word "infinity" itself can; all it means is "keep on going."

And what can one say about the fact or potentiality itself of infinity? I do not think there is such an entity. Infinite past; infinite future; infinite space; infinite speed; infinite or indefinite divisibility - all mathematics - or rather, "mythi-matics."

And every one lived happily ever afterwards. That's language is it not.

In conclusion, let us just ponder the pregnant possibilities of a few Indo-European roots: middle, medium, measure, metre, muthic, Mozart, mathematics, mythos, mother . . .

In fact if one looks a little further back it all started with dem- or dom- which occurs for example in domus (house) and English timber.

[MrGongGong]

I'm getting out of my league here as far as mathematics is concerned, but will add that irrational numbers are the basic flaw in maths. Transcendental is semantics, because if you try hard enough, and the drugs work, you can express anything as algebra (the quantum stuff is a wonderful example of this - yes, objects can exist in two places at the same time, and there are multiple universes, etc, but at this point in time - ha! - it's all just mathematical theory). Bottom line is, and forgive me for dumbing down, the numbers don't add up.

They only "don't add up" if you keep thinking that they are numbers of things

[David Underdown]

Irrationals (of which transcendentals are a subset) can be expressed as series and various other means. It's not really a flaw, unexpected and caused the Greeks some grief sure. Even many rational numbers have their problems as decimal or binary representations - 1/3 takes an infinite number of decimal places to represent "accurately"

There are of course different orders of infinity as Cantor showed

[Dave2002]

Irrationals (of which transcendentals are a subset) can be expressed as series and various other means. It's not really a flaw, unexpected and caused the Greeks some grief sure. Even many rational numbers have their problems as decimal or binary representations - 1/3 takes an infinite number of decimal places to represent "accurately"

But it's really just a sequence of 3s after the decimal point - 0.3333333 ...

1/7 is a bit more fun. - 0.142857 142857 142857 ... With a longer repeating pattern.

Pi and e have AFAIK no consecutive repeating patterns anywhere.

[vinteuil]

.

Pi and e have AFAIK no consecutive repeating patterns anywhere.

... au contraire! - given that pi and e are infinite, they will have an infinite number of repeating patterns inside their numbers! (altho' we haven't yet got far along enough to find interestingly long repeating patterns)

[Nick Armstrong]

... au contraire! - given that pi and e are infinite, they will have an infinite number of repeating patterns inside their numbers! (altho' we haven't yet got far along enough to find interestingly long repeating patterns)

So, analogous to the Alphabet Associations thread in fact...

[Budapest]

Sydney, the biggest problem with infinity is not on a perception/language level. It's at a science level, which of course most of us don't properly understand, including me.

I will try to use a very brutal example of this: why can't 'they' cure cancer, which in theory is a very, very (and I'll add another 'very') simple disease. We can't cure it because our science is still very primitive. Our sums still don't add-up. Mathematicians find all kinds of curtesy ways around this. It still doesn't prevent people dying from cancer (mathematics is the baseline of all the sciences).

David Underdown, I would probably disagree with what you say. There's only really irrationals. The subsets seem to just qualify irrationality in human terms (ie, we can't figure it out and so make-up our own language to sort of explain what we don't understand).

Dave2002, maths is great fun. It's even greater fun when you realise how completely mad it is (it's a bit like a 1960s rock 'n roller waking-up in a hotel room in Novosibirsk).

MrGongGong, you're probably right.

Where's Warwick?

[Dave2002]

... au contraire! - given that pi and e are infinite, they will have an infinite number of repeating patterns inside their numbers! (altho' we haven't yet got far along enough to find interestingly long repeating patterns)

vinteuil

I am not convinced that simply basing an argument on infinity as you have done here constitutes a proof. I think there may be deep waters here, relating to the nature of infinities - whether they are countable or not. A casual argument based on infinities might allow you to state that from some point in the enumeration of pi, pi would include a complete enumeration of e, and vice-versa, but this is very likely not to be provably true. You might be able to argue that from some point in the enumeration of pi, that pi would include an arbitrarily large subsequence which is an enumeration of e.

Of course something can be true without being provable.

I have wondered if there is a constructive proof that a countably infinite sequence of digits exists which does not contain any consecutive repeating sequences. If so, then it is possible, that pi and e are such sequences.

[Dave2002]

Dave2002, maths is great fun. It's even greater fun when you realise how completely mad it is (it's a bit like a 1960s rock 'n roller waking-up in a hotel room in Novosibirsk).

No, definitely not mad!

Try convincing some people about 4, 5, 6 dimensional spaces. Some physicists apparently want to use 11 dimensions.

You can drive them even further insane by pointing out the possibility of fractional dimensions - say 1.5 dimensions, or even pi dimensions. Think this is impossible?

What about space filling curves. A line is clearly one dimensional, but it is possible to construct lines which will cover a two dimensional space. Check out Hausdorff dimensions!

[David Underdown]

David Underdown, I would probably disagree with what you say. There's only really irrationals. The subsets seem to just qualify irrationality in human terms (ie, we can't figure it out and so make-up our own language to sort of explain what we don't understand).

I think there are important differences - I can construct a line segment that's root 2 units long easily enough. I can't do the same for pi or e. the set of algebraic numbers (ie excluding transcendentals) is countable, the reals (including transcendentals) are not

Dave2002, I deliberately chose one third as the simplest case. A computer cannot use an infinite number of decimal places, so any decimal representation is an approximation and introduces the potential for rounding errors

[Dave2002]

Dave2002, I deliberately chose one third as the simplest case. A computer cannot use an infinite number of decimal places, so any decimal representation is an approximation and introduces the potential for rounding errors

Quite.

And how many people would accept 0.999999999999 ... (recurring) is exactly the same as 1!

[Vile Consort]

vinteuil

I am not convinced that simply basing an argument on infinity as you have done here constitutes a proof. I think there may be deep waters here, relating to the nature of infinities - whether they are countable or not. A casual argument based on infinities might allow you to state that from some point in the enumeration of pi, pi would include a complete enumeration of e, and vice-versa, but this is very likely not to be provably true. You might be able to argue that from some point in the enumeration of pi, that pi would include an arbitrarily large subsequence which is an enumeration of e.

Of course something can be true without being provable.

I have wondered if there is a constructive proof that a countably infinite sequence of digits exists which does not contain any consecutive repeating sequences. If so, then it is possible, that pi and e are such sequences.

What vinteuil says has been shown to be false. There is a theorem by Axel Thue (1906) that says you can construct an infinite sequence out of three symbols (e.g. the digits 0, 1 and 2) that has no consecutive repeats of any length. There is a proof at http://www.cs.umb.edu/~offner/files/thue.pdf.

Questions that come to my mind are (a) are there infinitely many such numbers, (b) if so, are they countable, (c) are they dense - i.e. is that one arbitrarily close to any given real number, and (d) are they the same numbers no matter what base you are working in?

[Vile Consort]

On reflection, Vinteuil wasn't talking about consecutive repeats, so the theorem doesn't disprove what he said.

I think what he might have been working towards was the question, given any infinite sequence on the digits 0-9, is it the case that for all N > 0, there is a sub-sequence of N digits that occurs twice? Well, yes, it is. In fact, for all N > 0, there is a sub-sequence of N digits that occurs infinitely often!

[Quarky]

What is the relationship between the abstract world of mathematics and the material universe?

Any thoughts?

My thought is that this question is doing the rounds of several messageboards at the moment, see http://www.google.co.uk/search?q=Rel...M4_8sgaIvYzKBw

But no one has come up with a convincing answer as yet.