How Math Explains the World.pdf
tice that 1(2n1) 2 n 3 (2n3), and so on. Adding the right sides, we get n sums of 2n, or 2n^2. So 2S 2 n^2 , and the resu ...
If A denotes the set of all positive integers n such that the sum of the first n odd integers is n^2 , we have shown that A cont ...
cause of a course he took from a charismatic instructor who was confined to a wheelchair. Gödel was highly conscious of his own ...
(he was once attacked by a gang of thugs who thought he was Jewish), and when one of his inf luential teachers was murdered by a ...
The Halting Problem At approximately the same time that Gödel came up with his incomplete- ness theorem (which perhaps should be ...
H and does the opposite; if H outputs “halt,” then N loops, and if H out- puts “loop,” then N halts. Since H presumably is able ...
viruses is equivalent to the halting problem; no such program can be written.^7 What Is, or Might Be, Undecidable I’m not sure w ...
If I denotes the identity of the group (the symmetry that leaves all the vertices in their original positions), then we have the ...
erators that are subject to a number of relationships, as in the above ex- ample. The word problem for such a group is to find a ...
turning coffee into theorems,” as Erdos drank prodigious quantities.^11 Goodstein’s theorem^12 is reminiscent of this problem; i ...
http://www.miskatonic.org/godel.html. The text box from Rudy Rucker’s Inf inity and the Mind contains Gödel’s argument in comput ...
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8 i Al Space and T me: Is That l There Is? The Second Solution High-school algebra is more than five decades in my rearview mirr ...
substituting this into the second equation gives W5. It’s easy to check that these numbers solve the problem; a garden with a l ...
weights increased from left to right in each row, and from top to bottom in each column. When Mendeleyev began his work, not all ...
cloud chamber. It isn’t recorded whether, after the discovery of the posi- tron, Dirac turned to Kapitsa and said, “There!” If D ...
end in the universe we know—or have we just not discovered complex cookies? Heisenberg said something about the role of mathemat ...
Electrical storms in the center of the Milky Way do not affect us, but the gravity that emanates from the black hole at its cent ...
equations?”^4 Unsaid, but implied, was “Why not other equations?” Why does the universe in which we live support Einstein’s equa ...
our sector of it) abhors infinities much the same way as it was thought that nature abhorred a vacuum. As a result, no particle ...
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