On Fri, Oct 12, 2007 at 05:16:04PM +0100, Mike Tintner wrote: > > How is maths grounded?
Wow. Many algebraic systems ar grounded in a set of axioms, which are assumed to be true. > Our decimal number system is obviously based on the basic numbers 1 - 10 - > which are countable by hand. Digital. We have numbers like 2031, 43458, > 1,000,002 - which are all reducible to, and built up from, physically > countable tens. This argument, taken to its limit, is known as "ZFC", the zermelo-frenkel-axiom-of-choice axioms. Some mathematicians, the constructivists, try to ground everything in ZFC. > We absolutely need that. False. There are non-ZFC systems that get explored. More generally, the study of what one can derive from a set of axioms is known as "first-order logic", which taken to its abstraction, involves the category of logic, aka Grothendieck's Topos theory. Then, there's things like "model theory" which simply aren't grounded in this way. Modern algebraic geometry, e.g. schemes, are not grounded on ZFC, mostly cause its not constructivist. Non-concrete categories are, well, roughly speaking bigger than the biggest infinities, and so ZFC doesn't really address that. -------------------- As to "visualizing" math: I notice that many topologists, those working in 3D, couldn't write down an equation to save thier life (and I'm pretty much not exaggerating... I've been to lectures where only figures get drawn on the blackboard, and someone tries to ask about a formula, and you get these dumb stares...) And the opposite: a non-infrequent meme states that one must never visualize, as you will be deceived by your brain, and that only algebra can be trusted. --linas ----- This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?member_id=8660244&id_secret=53126807-461e8e
