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From: Andrzej Kozlowski on 6 Jan 2010 05:56 On 6 Jan 2010, at 11:13, DrMajorBob wrote: > I completely understand that Mathematica considers 1.2 Real, not Rational... but that's a software design decision, not an objective fact. I think we are talking cross purposes. You seem to believe (correct me if I am wrong) that numbers somehow "exist". Well, I have never seen one - and that applies equally to irrational and rationals and even (contrary to Kronecker) integers. I do not know what the number 3 looks like, nor what 1/3 looks like (I know how we denote them, but that's not the sam thing). So I do not think that the notion of "computer numbers" makes any sense and hence to say that all computer numbers are rational also does not make sense. There are only certain things that we interpret as numbers and when we interpret them as rationals they are rationals and when we interpret them as non-computable reals than they are just that. Of course we know that a computer can only store a finite number of such objects at a given time, but that fact in no sense makes them "rational". Andrzej Kozlowski
From: DrMajorBob on 6 Jan 2010 05:57 Obviously, it DOES make them rational "in a sense"... the sense in which I mean it, for example. Bobby On Tue, 05 Jan 2010 20:41:34 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > > On 6 Jan 2010, at 11:13, DrMajorBob wrote: > >> I completely understand that Mathematica considers 1.2 Real, not >> Rational... but that's a software design decision, not an objective >> fact. > > I think we are talking cross purposes. You seem to believe (correct me > if I am wrong) that numbers somehow "exist". Well, I have never seen one > - and that applies equally to irrational and rationals and even > (contrary to Kronecker) integers. I do not know what the number 3 looks > like, nor what 1/3 looks like (I know how we denote them, but that's not > the sam thing). So I do not think that the notion of "computer numbers" > makes any sense and hence to say that all computer numbers are rational > also does not make sense. There are only certain things that we > interpret as numbers and when we interpret them as rationals they are > rationals and when we interpret them as non-computable reals than they > are just that. > Of course we know that a computer can only store a finite number of such > objects at a given time, but that fact in no sense makes them "rational". > > Andrzej Kozlowski -- DrMajorBob(a)yahoo.com
From: Andrzej Kozlowski on 6 Jan 2010 05:57 Well, I think when you are using Mathematica it is the designers of Mathematica who decide what is rational and what is not. And when you are not using Mathematica (or other similar software which interprets certain computer data as numbers), than I can't imagine what you could possibly mean by a "computer number". Andrzej On 6 Jan 2010, at 11:45, DrMajorBob wrote: > Obviously, it DOES make them rational "in a sense"... the sense in which I mean it, for example. > > Bobby > > On Tue, 05 Jan 2010 20:41:34 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > >> >> On 6 Jan 2010, at 11:13, DrMajorBob wrote: >> >>> I completely understand that Mathematica considers 1.2 Real, not Rational... but that's a software design decision, not an objective fact. >> >> I think we are talking cross purposes. You seem to believe (correct me if I am wrong) that numbers somehow "exist". Well, I have never seen one - and that applies equally to irrational and rationals and even (contrary to Kronecker) integers. I do not know what the number 3 looks like, nor what 1/3 looks like (I know how we denote them, but that's not the sam thing). So I do not think that the notion of "computer numbers" makes any sense and hence to say that all computer numbers are rational also does not make sense. There are only certain things that we interpret as numbers and when we interpret them as rationals they are rationals and when we interpret them as non-computable reals than they are just that. >> Of course we know that a computer can only store a finite number of such objects at a given time, but that fact in no sense makes them "rational". >> >> Andrzej Kozlowski > > > -- > DrMajorBob(a)yahoo.com
From: DrMajorBob on 6 Jan 2010 05:58 RandomReal[] returns numbers from a countable set of rationals. Or call them reals, if you must; it still selects from a countable set of possibilities... not from the uncountable unit interval in the reals. The range of RandomReal[] is a set of measure zero, just like the algebraic numbers. Bobby On Tue, 05 Jan 2010 02:08:24 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > > On 5 Jan 2010, at 15:47, DrMajorBob wrote: > >> If computer reals are THE reals, why is it that RandomReal[{3,4}] can >> never return Pi, Sqrt[11], or ANY irrational? > > It can't possibly do that because these are computable real numbers the > set of computable real numbers if countable and has measure 0. > Computable numbers can never be the outcome of any distribution that > selects numbers randomly from a real interval. > > The most common mistake people make about real numbers is to think that > numbers such as Sqrt[2] or Pi as being in some sense typical examples of > an irrational number or a transcendental number but they are not. They > are very untypical because they are computable: that is, there exists a > formula for computing as many of their digits as you like. But we can > prove that the set of all reals with this property is countable and of > measure 0. So Sqrt[2] is a very untypical irrational and Pi a very > untypical transcendental. So what do typical real look like? Well, I > think since a "typical" real is not computable we cannot know all of its > digits and we cannot know any formula for computing them. But we can > know a finite number of these digits. So this looks to me very much like > the Mathematica concept of Real - you know a specified number of > significant digits and you know that there are infinitely many more than > you do not know. It seems to me the most natural way to think about > non-computable reals. > > Roger Penrose, by the way, is famous for arguing that our brain is > somehow able to work with non-computable quantities, although of course > not by using digital expansions. But this involves quantum physics and > has been the object of a heated dispute since the appearance of "The > emperor's New Mind". > -- DrMajorBob(a)yahoo.com
From: Andrzej Kozlowski on 6 Jan 2010 05:58
Well, you are obviously misunderstanding what I am trying to explain but I have no desire to spend any more time on it. I give up. Perhaps you should try to explain yourself why Mathematica gives In[1]:= Element[1.2, Rationals] Out[1]= False In[2]:= Element[1.2, Reals] Out[2]= True and you might also read http://en.wikipedia.org/wiki/Computable_number (but that's the last time I posting anything to do with any logic or mathematics here.) Andrzej Kozlowski On 5 Jan 2010, at 22:31, DrMajorBob wrote: > RandomReal[] returns numbers from a countable set of rationals. > > Or call them reals, if you must; it still selects from a countable set of possibilities... not from the uncountable unit interval in the reals. > > The range of RandomReal[] is a set of measure zero, just like the algebraic numbers. > > Bobby > > On Tue, 05 Jan 2010 02:08:24 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > >> >> On 5 Jan 2010, at 15:47, DrMajorBob wrote: >> >>> If computer reals are THE reals, why is it that RandomReal[{3,4}] can >>> never return Pi, Sqrt[11], or ANY irrational? >> >> It can't possibly do that because these are computable real numbers the set of computable real numbers if countable and has measure 0. Computable numbers can never be the outcome of any distribution that selects numbers randomly from a real interval. >> >> The most common mistake people make about real numbers is to think that numbers such as Sqrt[2] or Pi as being in some sense typical examples of an irrational number or a transcendental number but they are not. They are very untypical because they are computable: that is, there exists a formula for computing as many of their digits as you like. But we can prove that the set of all reals with this property is countable and of measure 0. So Sqrt[2] is a very untypical irrational and Pi a very untypical transcendental. So what do typical real look like? Well, I think since a "typical" real is not computable we cannot know all of its digits and we cannot know any formula for computing them. But we can know a finite number of these digits. So this looks to me very much like the Mathematica concept of Real - you know a specified number of significant digits and you know that there are infinitely many more than you do not know. It seems to me the most natural way to think about non-computable reals. >> >> Roger Penrose, by the way, is famous for arguing that our brain is somehow able to work with non-computable quantities, although of course not by using digital expansions. But this involves quantum physics and has been the object of a heated dispute since the appearance of "The emperor's New Mind". >> > > > -- > DrMajorBob(a)yahoo.com |