On Apr 1, 9:58 pm, david petry <david_lawrence_pe...@yahoo.com> wrote: > On Monday, April 1, 2013 11:38:05 AM UTC-7, Dan wrote: > > Falsifiability only comes into consideration when you have something > > apparently 'external' against which to test your mathematical models . > > Precisely. The "external" reality underlying mathematics is computation. > > A very good way of thinking about this stuff is to think of the computer as the mathematicians' microscope which helps mathematicians peer into the world of computation, and then mathematics itself is the science that studies the phenomena observed in that world of computation. The "science of phenomena observable in the world of computation" includes all of the mathematics that helps us reason about the real world. > > > What's so hard to understand? > > What's hard to understand is why you ... oh, never mind.
You've never actually seen a "real computer" , nor will you ever do so . The Turing machine requires an infinite tape . At most , you have pathetic, finite automata. What about if a "real computer" makes a floating point error when calculating pi? Does that change pi? All 'models of computation' are abstract, as 'internal to Mathematics' as they could be .I build a computer not because I can't think my task out, but because it helps me execute a 'mechanized part of my thinking' faster. Historically , computability theory relies and is intertwined heavily with other mathematics, (beginning with arithmetic , going recently to the high abstractions of Category Theory), and could not have developed without the existence of said mathematics. Was Mathematics before Turing invalid or useless?
"What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing" -Godel
"It is perhaps just dawning on five or six minds that physics, too, is only an interpretation and exegesis of the world (to suit us, if I may say so!) and not a world-explanation; but insofar as it is based on belief in the senses, it is regarded as more, and for a long time to come must be regarded as more - namely, as an explanation. Eyes and fingers speak in its favor, visual evidence and palpableness do, too: this strikes an age with fundamentally plebian tastes as fascinating, persuasive, and convincing - after all, it follows instinctively the canon of truth of eternally popular sensualism. What is clear, what is "explained"? Only what can be seen and felt - every problem has to be pursued to that point. Conversely, the charm of the Platonic way of thinking, which was a noble way of thinking, consisted precisely in resistance to obvious sense-evidence - perhaps among men who enjoyed even stronger and more demanding senses than our contemporaries, but who knew how to find a higher triumph in remaining masters of their senses - and this by means of pale, cold, gray concept nets which they threw over the motley whirl of the senses - the mob of the senses, as Plato said. In this overcoming of the world and interpreting of the world in the manner of Plato, there was an enjoyment different from that which the physicists of today offer us - and also the Darwinists and anti-teleologists among the workers in physiology, with their principle of the "smallest possible force" and the greatest possible stupidity. "Where man cannot find anything to see or to grasp, he has no further business" - that is certainly an imperative different from the Platonic one, but it may be the right imperative for a tough, industrious race of machinists and bridge-builders of the future, who have nothing but rough work to do." -Friedrich Nietzsche