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Funny, in pure mathematics the opposite is true. Examples of intuition defying theorems include the Banach–Tarski paradox, cantor sets, uncountability of the re
by blackle 11y ago
Funny, in pure mathematics the opposite is true. Examples of intuition defying theorems include the Banach–Tarski paradox, cantor sets, uncountability of the reals, gabriel's horn, the list goes on.
- hyperpape 11y agoThat said, when you prove 1 = 0, it's still often wrong.
- sytelus 11y agoPretty much every great result in Physics had been counter-intuitive: Gravity, relativity, quantum mech, nuclear physics, semi-conductors, super conductors... I'd say every major advance is science destroyed some conventional wisdom and proved previous intuition wrong.
- SideburnsOfDoom 11y ago> every major advance is science destroyed some conventional wisdom and proved previous intuition wrong. true, but not in contradiction to the original statement that counter-intuitive claims are very likely to be wrong. Carl Sagan said it well: "But the fact that some geniuses were laughed at does not imply that all who are laughed at are geniuses. They laughed at Columbus, they laughed at Fulton, they laughed at the Wright brothers. But they also laughed at Bozo the Clown." http://rationalwiki.org/wiki/Galileo_gambit http://rationalwiki.org/wiki/Galileo_gambit
- davidgerard 11y agoYes, but "counter-intuitive" means "probably wrong." There are many great results that are counter-intuitive, but there are vastly more completely stupid ideas that are counter-intuitive. Ask any academic about their green ink file. Most new ideas are bad. That doesn't mean "give them a serious look", but it does mean "though you probably won't get far with them."
- pluma 11y agoWhat is a "green ink file"? Google isn't helping.
- davidgerard 11y agoA British journalistic term for the frothing of lunatics; may or may not actually be written in green. http://rationalwiki.org/wiki/Green_ink http://rationalwiki.org/wiki/Green_ink Academics tend to get a lot of correspondence from cranks, often enough to accumulate a collection.
- tbrownaw 11y agoBanach–Tarski paradox This relies on declaring that 2x == x for sufficiently large x. Which is clearly bullshit: they're not the same, you just can't tell the difference. cantor sets The wikipedia page doesn't make clear what's supposed to be contrary to intuition about this. Unless you mean the 2x == x issue wrt cardinality? gabriel's horn This one isn't counterintuitive at all; smaller things always have more surface area. . Calculus with infinitesimals works by discarding (symbolic) terms known to be (numerically) below epsilon. Adding things to infinity works the same way. Both of these will be intimately familiar to anyone who works with algorithmic complexity ("big-O notation") or the IEEE 'float' or 'double' data types.
- steven2012 11y agoI'm pretty sure it's not just "sufficiently large x" but rather as x approaches infinity. The properties as you approach infinity are completely different, and counterintuitive, to simply a very very large number.
- gus_massa 11y agoFor the Banach–Tarski paradox it's not enough to have an x that is a \HUGE number of elements, you need an actual infinite. For this and most (all?) of the unintuitive results that use the Axiom of Choice, it is even not enough to have a countable number of elements, like the integer numbers or the rational numbers. You need a bigger infinite like the number of elements in the real number. In this case the number of points in a sphere of real numbers in R^3. (More info about the different types of infinites: http://en.wikipedia.org/wiki/Cardinal_number http://en.wikipedia.org/wiki/Cardinal_number )
- tbrownaw 11y agoThe "odd" or "counterintuitive" properties as you approach infinity, are a perfectly reasonable consequence of the general agreement to throw away information and treat infinity as all the same. But they're not all the same, they're simply large enough that any non-infinite numbers are below epsilon. Much like how infinitesimals are all equal to zero (below epsilon for any non-infinitesimal), but if they were all the same calculus wouldn't work.