פרסומים אחרונים במדור "מדע"

if a ≤ b and b ≤ a then a = b (antisymmetry)
if a ≤ b and b ≤ c then a ≤ c (transitivity)
a ≤ b or b ≤ a (totalness)

if a≤b and not a=b then a≤c and not a=c and c≤b and not b=c

b≤a and not b=a and a≤c and not a=c

"Platonism ... claims that mathematics is descriptive of abstract entities, of numbers and sets, that exist separately from our attempt to understand them through our mathematical systems.

Platonism has always had a great appeal for mathematicians, because it grounds their sense that they're discovering rather than inventing truths. When Gödel fell in love with Platonism, it became, I think, the core of his life. He happened to have been married, but the real love of his life was Platonism, and he fell in love, like so many of us, when he was an undergraduate.

Platonism was an unpopular position in his day. Most mathematicians, such as David Hilbert, the towering figure of the previous generation of mathematicians, and still alive when Gödel was a young man, were formalists. To say that something is mathematically true is to say that it's provable in a formal system. Hilbert's Program was to formalize all branches of mathematics. Hilbert himself had already formalized geometry, contingent on arithmetic's being formalized. And what Gödel's famous proof shows is that arithmetic can't be formalized. Any formal system of arithmetic is either going to be inconsistent or incomplete.

Gödel had intended to show that our knowledge of mathematics exceeds our formal proofs. He hadn't meant to subvert the notion that we have objective mathematical knowledge or claim that there is no mathematical proof—quite the contrary. He believed that we do have access to an independent mathematical reality. Our formal systems are incomplete because there's more to mathematical reality than can be contained in any of our formal systems".

http://www.edge.org/3rd_culture/goldstein05/goldstei...

"Gödel made it harder not to be a Platonist. He proved that there are true but unprovable propositions of arithmetic. That sounds at least close to Platonism. That sounds close to the claim that arithmetical truths are independent of any human activity".

>Gödel made it harder not to be a Platonist. He proved that there are true but unprovable propositions of arithmetic.

Mathematicians and physicists are just as guided by principles of elegance and beauty as novelists and musicians are.