A little conceptual snippet from GEB:

Everything in mathematics can be seen from a "meaningless" perspective. In some sense everything from algebra to complex analysis can be stripped of meaning by thinking of them as just "a rearrangement of symbols according to a set of arbitrary rules".

And, somewhere, that's really all they are.

It's only once we apply an interpretation to these operations that they become "math". Same with numerical rearrangement. Just because it seems like it's just a algorithmic manipulation doesn't mean there isn't an interesting interpretation of the meaning somewhere.

To be pathological, take the phrase X2YZ to mean "X + Y = Z if X < Y and Y - X = Z if Y > X" and both 1234 and 3241 are true. Now that manipulation suddenly has a whole lot of (silly) meaning.

Okay I'm only replying to this one reply but my point is relevant to all discussion re meaning.

Always be suspicious of any statement about numbers that relies on the base. This property of 6174 is really a property of the number's representation in base 10. In another base, it's the same number, but doesn't have the same property. That's why it seems suspiciously meaningless.

Oh, sure.

My reply was more of a meta argument than really a defense of the meaning of 6174. There's no reason to say that there isn't some property of its representation in base N that isn't useful, but it's pretty unlikely to be general in any interpretation we're used to dealing with (thus the re-representation of "2").

A nice example is the divisibility of numbers, by rearranging the digits and performing simple operations on them you can quickly figure out if a number is divisible by a certain other number without actually doing the division.

for an example see:

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

or google for 'eleven check'.

A snippet from a yahoo answers page:

"a number is divisible by 11 if and only if when you add the digits that are an odd number of places from the right, and then you add the digits which are an even number of places from the right, the difference of these is divisible by 11."

Who ever thought that x^3 transformed into 2x^2 would be useful one day? Of course I'm being facetious in this example, but the simple rearrangement of the mathematical expression above has profound implications for how we interact with the world.

Yes I know that there is a whole derivation to the simple derivative example I've given, but who is to say that the 6174 transformation might have preceded an actual use?

Who's to say that this odd mathematical transformation actually leads to something useful one day?

Did you mean 3x^2?

I don't really see the connection. Your example is the derivative of a function, and what I am saying is that rearranging 1234 to 3421 doesn't mean anything to me.

Now you have some new number who's only relationship to the old number is that it's one of (432)-1 possible permutations of it. Other than that, what's the significance?

The numbers in the article aren't rearranged arbitrarily, they're sorted to ascending and descending order.

I don't really see the connection. Your example is the derivative of a function, and what I am saying is that rearranging 1234 to 3421 doesn't mean anything to me.

Now you have some new number who's only relationship to the old number is that it's one of (432)-1 possible permutations of it. Other than that, what's the significance?

Just because there is no name for this operation does not mean it is not valid (or meaningful). There are many mathematical constructs which do not have any physical meaning.

6174 it's the only real solution of this system of equations: 9 ≥ a ≥ b ≥ c ≥ d ≥ 0 abcd - dcba = ABCD

Are you positive that's true for all bases?

The statement as juanpaulo gives it has no hidden dependence on bases. It is all spelled out by the conditions. The symbols a b c d stand for any values, not the "actual digits" 0-9, which are also merely placeholders for values with no "true meaning" of their own.

He is using both digits and the nubers that sequence of digits represents in that equation.

In base 10, 7641 - 1467 = 6174.

In base 11, 7641 - 1467 = 4808.

Finding tricks that work in base 10 is interesting, but it's a much more fun to look for patterns that work across several number systems. A simple example is for any base > 3, (base - 1) times x where base > x > 1; you get a 2 digit number [x - 1],[base - x]

No, they are not. The symbols in the equation are simply numbers, and "9 ≥ a ≥ b ≥ c ≥ d ≥ 0" constrains them to 0-9.

You are mistaking the map for the territory, sort of. The positional aspects of the number system get encoded the equations and then the mere "numbers" don't matter anymore.

I think we are talking past each other. In base ten if you start with 1234 and cycle though asdf - fdsa > new number you hit a single stable number 6174 .

In base 11 using 6174 you cycle:

4808, 87A3, 7094, 9272, 3098, 9452, 7094 (cycle)

I was thinking of a http://en.wikipedia.org/wiki/Logistic_map for this equation abcd - dcba > might be interesting across different bases.

Yes they are. Where he meant 1000a+100b+10c+d he wrote abcd.