MilitantModerate

Using cos(0) was my second instinct. My first was an inverse ln which is really e^0 so I was not sure it was a legit solution. To be completely truthful my zeroth instinct was to try and work down from arccos(0).

Farren

huh?

Watch:

~0 + ~0 + ~0 + ~0 = 100

100! = 100 x 011 x 010 x 001 = 11000

where problem?

Farren

The problem I was talking about was the issue of negatives and binary operators, like if you assume that a negative sign is allowed then

A | ~A
--------
0 | 1
1 | 0
-1| ?

But of course when binary operations such as NOT, AND and OR are done, we don't normally use a negative sign, we use devices such as two's compliment, with an assumed sign bit. i.e. We encode the sign in the number. So if the sign bit is the 8th bit, then

01111111 = 127
10000000 = -1
10000001 = -2

and so on. Now in this case, a ~ operation on the high bit yields a negative/positive toggle.

10000000 = -1
~10000000 = 01111111 = 127

Since a schema can exist where the low bit is the sign bit, this could screw with the argument, so the declared schema is important, hence "8bit 2's compliment".

I'm not too sure if this is simply a programming issue or a maths issue though. The necessity of encoding the sign into the bits is a universal one for Turing machines and Lambda calculus, where operators, signs and numbers all reduce to numeric values, given distinction only by their position in relation to other numbers.

The actual intent behind Turing Machines and Lambda Calculus though, was to reduce pure maths to a formal, underlying grammar, a task at which both functionally equivalent ideas succeed quite well (which is why they're the basis of computing). In theory, any math formulation can be expressed as a Turing machine, so this could be a pure math problem.

On the other hand, its equally valid to say that these conventions are arbitrary selections which don't change the nature of the operations, so we could assume the absence of such a schema, and then the bits can only represent positive, natural numbers (real numbers require a mantissa/exponent 2's compliment schema). This does not invalidate the calculation, it simply limits the representation.

hmm....

Farren

I+1:

I have to admit I don't fully understand your math (the math I've retained is what I need to code) but I think I get the thrust of what you're speculating about.

It seems to me its an epistemological issue, like "What's a real mathematical function and what's an arbitrary function?"

I've got some books lying about in layman's English with essays by Penrose and Godel on these kinds of issues. It seems to me that the reason Turing Machines work so well as an underpinning for maths is that the only definitive solution is to say any consistently encoded function or algorithm has equal validity.

In other words, any set of transformations that can be encoded using the same base symbol set has equal merit as a mathematical operation. Within an encoding schema, no one operation has merit over another.

Since theoretically any math function can be encoded using Turing machine/Lambda calculas schemas, none should be considered more arbitrary than another. This is the basis of Godel's demonstration of inconsistency in our formal mathematical system. Or have I perhaps misconstrued something?

Farren

Can't leave this alone

I've just realised that as soon as you start expressing the problem in terms of Turing machines, functions/operators become numbers, which invalidates any solution, so the terms must remain suitably vague and assumptions made for any solution to be valid.

InfinityPlusOne

This is correct.
And I believe that that's the case, mathematically. Consider some operation W. I'm going to define it such that whenever W operates on any number the result is 37. As far as I know, this is valid mathematically, but what purpose does it serve?

I was once told that you're free to define any operation you want, but if you can't come up with a context for it then you might've wasted your time.

It's a matter of seeing math as a means to an end, or an end unto itself.

Loren Pechtel

No--my point is that you only have 4 0's. Where did you get the 8 and the 2 to specify the schema?

Farren

Loren, in order to declare 24 to mean twenty four you must specify that you are working to the base 10, and thereby introduce a schema. By your reasoning you have thus used up a Zero and illegally introduced a One.

The declaration of a schema is specifying a symbolic convention, not using numbers in the system. This is what I mean. It falls outside of the problem or every other solution suffers the same fate.

Silent Acorns

Isn't this pi^pi - pi? This certainly isn't equal to 24.

We've been holding back so we don't ruin it for everyone. By the way, my solution was:

(0!+0!+0!+0!)!=24

I always thought that 0!=1 could be defended as:

n! = (n)(n-1)!
(n-1)!=n!/n
0! = (1-1)! = 1!/1 = 1/1 = 1

Kevin Dorner

Nope, 3^3-3, which is 24. Those aren't square brackets (notice the tops are missing) but are the floor function (round down to integer.) Invert for ceiling function.

But at least you checked it.