From: | H. S. Teoh <hsteoh@...> |
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Date: | Tuesday, April 8, 2003, 15:29 |

On Tue, Apr 08, 2003 at 11:26:51PM +1000, Tristan wrote: [snip]> * This is an impossible set. If it doesn't belong to itself, then it > satisfies the condition and belongs to itself, which means it doesn't > satisfy the condition, and so it doesn't belong to itself, and we're > back to the initial one again. Apparently the guy who came up with this > set, whose name I've forgotten,If memory hasn't failed me, that'd be Bertrand Russell. And that impossible set is called Russell's Paradox.> went off to write a _Principia Mathematica_ and took a lot of pages to > prove that 1+1=2,This thing about proving 1+1=2 is really a moot argument, because any proof must necessarily involve defining the operation + so that it behaves like what happens when you put two collections of physical objects together. Which means you've already defined 1+1 to be 2 to begin with, although possibly in an indirect way. In other words, it's a tautology. The exercise is more of a test that a particular definition of + makes sense relative to what we think of as addition/subtraction, rather than some magical, self-establishing "proof" that 1+1=2 in the semantic sense.> which only makes me *want* to be the first person one of my Maths > Lecturers has ever met whose attempted to read the thing. Unfortunately, > I doubt I'd understand most of it...It may be easier to understand if you understand that mathematicians prove things in such obscure ways because they want to be able to derive everything from the smallest possible set of assumptions. Believe it or not, operations we take for granted like + and - are actually rather complex beasts, mathematically speaking. That's why mathematicians want to be able to reduce it to something "simpler"[1]. Therefore, they reduce everything into set theory, which is mathematically simpler (although intuitively complex, since sets don't really behave like what we think of as sets of physical objects). And even the idea of sets itself would be overly complex (mathematically speaking) if we allowed sets of arbitrary objects; therefore in formal set theory, sets don't contain anything except other sets. By the same argument, the only relational operation on sets is the "contains" relation. The most basic assumption of set theory (axiom #1) is that at least one set exists. This is usually understood to mean the empty set. Putting these together, you can think of 0 as the empty set, 1 as the set that contains the empty set, 2 as the set containing the 0 and 1 (i.e., the empty set and the set that contains the empty set), ad infinitum. In other words, the entire field of mathematics is made of empty sets; all theorems are ultimately just pronouncements about empty sets. Now we know where mathematicians dispose their cups after consuming all that coffee.[2] :-P [1] Of course, "simple" here means expressed using the minimal set of mathematical rules; such things are hardly "simple" in the intuitive sense. [2] After all, "mathematicians are just machines that turn coffee into theorems." -- YHL> Tristan (has come to the conclusion that mathematicians *must* have too > much time on their hands to come up with things like that...Nah, they are just too filled with caffeine. It does strange things to their brains. Now if only math departments would stop having a coffee machine in the kitchen... :-P> And then to prove that 1+1=2...)Like I said, it's more of a proof that the theory that mathematicians have come up with actually makes sense relative to what we intuitively understand as 1+1=2, than an actual proof that 1+1 is indeed 2. Or, to put it another way, it's the mathematicians' way of justifying the cost of all that coffee they consumed, after it made them dream up of such a wacky concept as building the entirety of mathematicians from empty sets.[3] [3] "Look, look! you stack up those empty coffee cups, and it proves that 1+1=2!" T -- Regression testing: euphemism for "we'll do one isolated test case here, another isolated test case there, and we'll call it a day."

John Cowan <jcowan@...> | |

JS Bangs <jaspax@...> | |

Christophe Grandsire <christophe.grandsire@...> |