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From: DrMajorBob on 7 Jan 2010 02:32 I don't disagree with most of that. You agree, I think, that we do certain things, even as small children, that we don't know how to program into a computer. It's interesting to watch Japanese robots as they GLACIALLY improve... none of them yet reaching what five year old humans can do. It's astonishing the physical feats a ROACH can perform, with a brain barely large enough to see, if it were exposed. I can walk, but I can't explain how. I can spell more words than it seems possible I could have memorized. I think I can recognize people I know, better than any computer (usually a hell of a lot better than iPhoto, though DoD probably has some serious contenders). I suspect I can recognize incorrect English better than any computer program. A lot of this is the result of evolutionary programming; we're born with brains wired for this stuff. It's analog computing, or virtually so... discrete on the level of DNA or neurons, but effectively analog. Chess programs numerically "score" positions based on heuristics like the total point value of pieces, how many squares each side controls, how many times pieces (and empty squares) are attacked and defended, are pieces protected by more valuable pieces (bad), is this a memorized win or loss, an opening gambit that rarely succeeds, et cetera... and I believe the rest is just dynamic programming. If you can "branch and bound" 20 layers deep, heuristics don't have to be especially good. Even human chess masters routinely analyze ten steps ahead on some lines, dismissing most paths via intuition alone. Bobby On Wed, 06 Jan 2010 20:46:46 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > I really think we are all the time talking cross purposes. Obviously > everything mathematica does is "algebra" in a very wide sense. But a > great many human topological arguments are also "algebra" in the same > sense. The same applies to arguments used in analysis. Essentially they > are just "discrete" deductive steps, based on certain statements which > are taken as facts - and all of that is perfectly reproducible by a > computer, although it may be hard. There is no reason why one should not > be able to store in a computer's memory a vast number of facts about the > topology of, say, three manifolds and then get the computer, by purely > discrete means, deduce from them new facts. The computer would then be > doing topology and the statements that it discovered would be > interpreted as statements about objects whose existence depends on the > the continuum. > > There is only one difference between this process and what actually > happens in the human mind - we have a geometric intuition to guide us in > selecting what might be true and also the steps that we need in a proof > - a computer has no geometric intuition at all. This is the crux of the > matter. This is also what all these books of Penrose are really about. > The issue is exactly this "intuition". Formal, rigorous mathematics is > not, in principle, different from what computer can do. Intuition may or > may not be. We do not know whether intuition is not simply based on > having a vast amount of knowledge stored in our memory. > Computers now can play chess on the level of the strongest grandmasters. > The way this was achieved was not by making than try to compute all the > possibilities in any given situation - this is the way to nowhere, but > by storing vast amounts of human knowledge and making computers imitate > the way humans play chess. Now, it is pretty hard, to discover just by > studying the moves in a chess game, whether one or both players are > computer programs or not. A human chess player uses above all his > intuition so that he does not need to consider the great majority of > possible moves in a given situation because his intuition tells him they > are bad moves. A computer has no intuition, but it can learn to act as > if it had one, simply by finding in its stored database of games an > identical or very similar position and choosing a move stored in its > memory. > There is no obvious reason why computers should not some day be able to > do topology as well as they can play chess. They would be making use of > human knowledge in doing that. To a large extent this is what > Mathematica already does. That's my entire point. > > Andrzej Kozlowski > > > > > > On 7 Jan 2010, at 11:29, DrMajorBob wrote: > >>> But Mathematica does or if you prefer "simulates" a lot of mathematics >>> that only makes sense under the assumption of continuity. >> >> Continuity of a function does NOT depend on completeness in the domain, >> and I suspect that >> >> Resolve[Exists[x, x^2 == 2], Reals] >> >> True >> >> succeeds based on algebra, not topology. >> >> You or I might (MIGHT) treat it as a topological problem, but I doubt >> Resolve can do so. >> >> A better example might be >> >> Reduce[Exists[x, Exp[x^7 + 3 x - 11] + x - 6/10 == 0], Reals] >> >> True >> >> A human might have great difficulty solving the equation, but he might >> easily establish that the LHS is negative for some value and positive >> for another, hence a solution exists in between. >> >> Yet, since this works: >> >> FindInstance[ Exp[x^7 + 3 x - 11] + x - 6/10 == 0, x] >> >> {{x -> Root[{-3 + 5 E^(-11 + 3 #1 + #1^7) + 5 #1 &, >> 0.599896128076431511686789719766}]}} >> >> I suspect that algebra and root-search was used in Resolve. >> >> Unless a developer can confirm that Resolve didn't find a solution, >> merely proved that one could be bracketed? >> >> To do that, Resolve would have to know the LHS is continuous on the >> real line, and haven't we found, frequently, that Mathematica CAN'T >> identify continuous functions? >> >> And what does THIS mean? >> >> 0.5998961280764315116867897197655402817356291002252018609367`30. // \ >> RootApproximant >> >> Root[1 - #1 + #1^3 - 7 #1^4 + 10 #1^5 - 16 #1^6 + 15 #1^7 - 15 #1^8 + >> 15 #1^9 - 16 #1^10 + 12 #1^11 - 10 #1^12 + 11 #1^13 - >> 2 #1^14 + #1^15 + 8 #1^16 &, 3] >> >> (Note the constant included in the output from FindInstance.) >> >> Did FindInstance (and Resolve) generate and solve a series >> approximation to -3 + 5 E^(-11 + 3 #1 + #1^7) + 5 #1 & ? >> >> Or is the RootApproximant result a pure accident? >> >> Bobby >> >> On Wed, 06 Jan 2010 19:42:51 -0600, Andrzej Kozlowski >> <akoz(a)mimuw.edu.pl> wrote: >> >>> The important word was "in principle". I have never claimed that >>> Mathematica can do topology. I work in topology and when I do that I >>> do not use Mathematica. But Mathematica does or if you prefer >>> "simulates" a lot of mathematics that only makes sense under the >>> assumption of continuity. In particular things like >>> >>> Resolve[Exists[x, x^2 == 2], Reals] >>> >>> True >>> >>> Mathematica obviously does this "discretely" (so does the human brain) >>> but this is a statement about the reals not the rationals. To think in >>> any other way just makes no sense to me. >>> >>> Andrzej >>> >>> >>> On 7 Jan 2010, at 10:28, DrMajorBob wrote: >>> >>>> Yes, this discussion is far too philosophical... but it HAS >>>> illuminated a few real-world Mathematica behaviors. >>>> >>>>> are you only claiming that "all computer reals are rationals" or are >>>>> you also claiming that "all reals are rationals"? >>>> >>>> The former. >>>> >>>>> If not, then what is the difference between the two? >>>> >>>> A great deal. >>>> >>>> I can imagine the woof and weave (the topology) of real numbers; >>>> computers can't do that. I can state four assumptions and show that >>>> every set with these properties is topologically isomorphic to what >>>> we call "the real line", with NO reference to real numbers, numeric >>>> representations, or real arithmetic. We did just that in a special >>>> topics course when I was a sophomore; none of us knew, when we >>>> started, what the end-goal would be... but that's where we arrived. >>>> >>>> The idea that a computer's mimicry of reals is equivalent to that is >>>> just... absurd. >>>> >>>> A computer can't begin to grasp the topology; it begins and ends with >>>> arithmetic. (That includes smart algorithms such as GroebnerBasis and >>>> RootApproximant, which are, root and branch, arithmetical.) >>>> >>>> Computers can do arithmetic on a finite subset of the reals, it can >>>> do symbolic algebra faster than a human, and Mathematica's >>>> arbitrary-precision arithmetic and large integers simulate >>>> nonstandard analysis in a limited way... but that's very far from >>>> understanding reals the way a topologist does or fields the way a >>>> algebraist does, or nonstandard analysis as a mathematical logician >>>> does. >>>> >>>>> Why can't a computer, in principle of course, perfectly simulate the >>>>> activity of the human brian that we call "doing mathematics"? >>>> >>>> In principle of course, human minds ARE computers... but not the kind >>>> we're likely to build, anytime soon. >>>> >>>> You're not claiming that Mathematica simulates the mind of a >>>> mathematician, I hope? >>>> >>>> Show me Mathematica proving topological theorems (beyond FINITE >>>> groups and graphs)... and you might have something. >>>> >>>> Bobby >>>> >>>> On Wed, 06 Jan 2010 18:44:15 -0600, Andrzej Kozlowski >>>> <akoz(a)mimuw.edu.pl> wrote: >>>> >>>>> It seems to me that this entire discussion has turned into pure >>>>> philosophy and isn't really suitable for this forum. But to put it >>>>> all in a nutshell: I do not see any reason to think that anything >>>>> that a computer can do is in a fundamental way different to what >>>>> human brain does. So, if you claim that "all computer reals are >>>>> rational" I can't see how this is different from the claim that "all >>>>> reals are rational" - since reals surely exist only in mathematics, >>>>> which is a product of the human mind. >>>>> >>>>> Now, as I mentioned earlier, Roger Penrose has tried to argue that >>>>> the human brain is fundamentally different from a computer and that >>>>> it has some sort of access to "real numbers" that a computer cannot >>>>> achieve (he formulates this in terms of Turing machines and >>>>> computability but essentially it amounts to the same thing). This >>>>> view remains very controversial and seems to be a minority one. But >>>>> anyway, you do not seem to be referring to this sort of thing. So >>>>> put this question to you: are you only claiming that "all computer >>>>> reals are rationals" or are you also claiming that "all reals are >>>>> rationals"? If not, then what is the difference between the two? Why >>>>> can't a computer, in principle of course, perfectly simulate the >>>>> activity of the human brian that we call "doing mathematics"? >>>>> >>>>> Andrzej Kozlowski >>>>> >>>>> >>>>> On 7 Jan 2010, at 08:59, DrMajorBob wrote: >>>>> >>>>>> If I'm told that finite-precision reals are not Rational "because >>>>>> Mathematica says so", but that Mathematica success (by some >>>>>> algorithm) in finding a Root[...] representation doesn't mean the >>>>>> number is algebraic... yet I know that all finite binary expansions >>>>>> ARE both rational and algebraic as a matter of basic arithmetic... >>>>>> then I question whether Mathematica is saying anything either way. >>>>>> >>>>>> Perhaps it's just Mathematica USERS holding forth in each direction. >>>>>> >>>>>> I think the view of reals as monads (a la nonstandard analysis) >>>>>> melds with the fact that reals are irrational A.E. and >>>>>> non-algebraic A.E., while monads are, of course, consistent with >>>>>> the spirit of Mathematica's arbitrary-precision arithmetic (WHEN IT >>>>>> IS USED). The OP posted a number far beyond machine precision, so >>>>>> it's reasonable to come at this from that arbitrary-precision >>>>>> world-view... in which case you're "right" and I'm "wrong". >>>>>> >>>>>> I called all the reals rational, and you called them monads (or >>>>>> equivalent). >>>>>> >>>>>> Fine. >>>>>> >>>>>> Bobby >>>>>> >>>>>> On Wed, 06 Jan 2010 16:46:20 -0600, Andrzej Kozlowski >>>>>> <akoz(a)mimuw.edu.pl> wrote: >>>>>> >>>>>>> >>>>>>> On 7 Jan 2010, at 04:19, DrMajorBob wrote: >>>>>>> >>>>>>>>> Well, I think when you are using Mathematica it is the designers >>>>>>>>> of >>>>>>>>> Mathematica who decide what is rational and what is not. >>>>>>>> >>>>>>>> Not to repeat myself, but RootApproximant said 100 out of 100 >>>>>>>> randomly chosen machine-precision reals ARE algebraic. >>>>>>> >>>>>>> No, they are not real algebraic. RootApproximant gives algenraic >>>>>>> approximations to these numbers and in fact it uses a test for >>>>>>> what makes a good approximation. In never says that these numbers >>>>>>> themselves are algebraic. You have been completely confused about >>>>>>> this. The method RootApproximant uses is the LLL method, which >>>>>>> finds approximations. Because of this it will give you a number of >>>>>>> different approximations for the same real. For example >>>>>>> >>>>>>> In[7]:= RootApproximant[N[Pi, 10], 2] >>>>>>> >>>>>>> Out[7]= (1/490)*(71 + Sqrt[2156141]) >>>>>>> >>>>>>> In[8]:= RootApproximant[N[Pi, 10], 3] >>>>>>> >>>>>>> Out[8]= Root[37 #1^3-114 #1^2-36 #1+91&,3] >>>>>>> >>>>>>> So how come N[Pi,10] is equal to two quite different algebraic >>>>>>> numbers? >>>>>>> You should first understand what an algorithm does (e.g. >>>>>>> RootApproximant) before making weird claims about it. (In fact >>>>>>> Daniel Lichtblau already explained this but you just seem to have >>>>>>> ignored it). >>>>>>> >>>>>>> Andrzej Kozlowski >>>>>>> >>>>>>>> >>>>>>>> If your interpretation is correct and consistent with >>>>>>>> Mathematica, and if Mathematica is internally consistent on the >>>>>>>> topic, virtually all of those reals should NOT have been >>>>>>>> algebraic. >>>>>>>> >>>>>>>> Mathematica designers wrote RootApproximant, I assume? >>>>>>>> >>>>>>>> Hence, I'd have to say your interpretation is no better than mine. >>>>>>>> >>>>>>>> Bobby >>>>>>>> >>>>>>>> On Wed, 06 Jan 2010 04:57:26 -0600, Andrzej Kozlowski >>>>>>>> <akoz(a)mimuw.edu.pl> wrote: >>>>>>>> >>>>>>>>> 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 >>>>>>>>> >>>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> -- >>>>>>>> DrMajorBob(a)yahoo.com >>>>>>> >>>>>> >>>>>> >>>>>> -- >>>>>> DrMajorBob(a)yahoo.com >>>>> >>>>> >>>> >>>> >>>> -- >>>> DrMajorBob(a)yahoo.com >>> >> >> >> -- >> DrMajorBob(a)yahoo.com > -- DrMajorBob(a)yahoo.com
From: Richard Fateman on 7 Jan 2010 02:32 Noqsi wrote: .... > > Machine reals are not the reals or the rationals: they are themselves, > with their own special properties. Those who reason as if machine > reals are either real or rational often suffer adverse consequences. If you treat each "machine real" (that is, hardware or software FLOAT format object) as a representation of an exact rational number, then I think that you are in much better shape in terms of numerical analysis than if you treat each object as some kind of fuzz-ball. > Much of the art of numerical analysis depends upon understanding these > special properties and their consequences. > I agree. But I disagree with the assertion that you do better with fuzz-balls. You especially can do careful analysis using a computer algebra system where you can carry around symbolic "machine epsilon" data, and do arithmetic on such expressions. RJF
From: Richard Fateman on 7 Jan 2010 02:33 Andrzej Kozlowski wrote: > Just one more comment, I hope my last one on this subject. Obviously > RandomReal make it choices out of a countable set of entities. One would > have to be insane to claim otherwise and I am not that yet. > > But, Mathematica does not regard these entities as rational numbers and > so they are not that. If you call them rationals the it does not make > *mathematical* sense (because rationals have measure 0). So, if > Mathemaitca does not regard them as rationals they are not rationals. You can refer to Rationals as whatever Mathematica calls Rationals. But the rational numbers include all numbers that are represented by finite explicit binary strings in a floating-point format. They also include other numbers whose binary expansions are infinite, but repeat. Can Mathematica represent Reals that are NOT RATIONAL? Sure. Here are examples: Sqrt[2], 3*Pi, 4*E. 3*E +4*E^E + 5*E^E^E. Incidentally, it is not known if E+Pi is rational. > How could they be that ? Until they are interpreted by Mathematica, they > are not numbers at all but just some data stored in computer memory - > which are not numbers of any kind. true, but other programs can also interpret them. As numbers, as ASCII character strings, as pointers into memory. Mathematica interprets them as > non-computable irrationals No, that's not the way computer programs work. Mathematica allows some set of operations like +, *, printing. That's all. They are obviously computable and finitely representable as well, but Mathematica doesn't need to have an opinion on this, and neither do we have to attribute opinions to Mathematica. If you think that the operations that Mathematica performs are consistent with YOUR view that these numbers are non-computable irrationals, I suppose that is your view, but it is certainly unnecessary for others to hold this view. in order to make mathematical sense when > returning them while simulation a real distribution, because all other > numbers have measure 0. There is a literature on pseudo-random numbers that makes mathematical sense without any such interpretation. > > This is all about "simulating mathematics" - numbers do not live in any > sense inside computers. To say that "all computer numbers are rational" > is weird - there is no such things as "computer numbers". Numbers exist > only and (probably) exclusively in the human mind. Actually, you just said that Mathematica interprets --blah blah. Maybe you think that Mathematica has a human mind? > > To say that 1.2 is rational in Mathematica even if Mathematica says > > Element[1.2, Rationals] > > False > > does not make any sense at all. It makes perfect sense to say (in Mathematica) that 1.2 is a rational number because it is equal to a rational number. Huh?? 1.2==12/10 True. (A better example would be 1.25, since 1.2 is not representable exactly in binary. This example of 1.2 actually reveals a "misfeature of mathematica. 1.2==5404319552844595/4503599627370496 True. So 1.2 is actually Mathematica-equal to another rational number. Many, in fact. ) If you capitalize the term and wish to say that 1.2 is not a Rational in Mathematica, that is just a convention based on the "type" of data that is input to Mathematica with a decimal point and is therefore stored in a memory format that is labeled "Real" which (in Mathematica) is a superclass of "Rational". That is, "1/2" is a Rational but is also a Real and is incidentally also a Complex. But 0.5 is not a Real. This categorization of types in Mathematica does not determine the membership (or not) of a particular numeric VALUE in a mathematical category such as "rational". From a mathematical perspective, any legal number in an ordinary floating-point format can represent only a rational value. > > Andrzej Kozlowski > > > > > On 6 Jan 2010, at 07:04, Andrzej Kozlowski wrote: > >> 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. OK. >> 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 the explanation is that Mathematica takes numbers written with a decimal point and labels them "Real". This has nothing to do with their values, which are, most assuredly, equal to rational numbers. And in particular, 1.2==12/10 in Mathematica should trouble you if you believe Mathematica speaks meaningfully on these issues. >> >> and you might also read >> >> http://en.wikipedia.org/wiki/Computable_number >> But this would be irrelevant. The Mathematica documentation says, "When domain membership cannot be decided the Element statement remain [sic] unevaluated". "cannot be decided" is not a statement about decidability in the technical "computability" sense. It is a statement about this version of Mathematica not being programmed to make a decision. Thus the fact that Mathematica 6.0 cannot decide if e+pi is rational is not a deep result, and it is referring to the mathematical literature about conjectures on the matter. It just happens that the program fails to decide. The program seems to be a jumble of some sort, since it knows that Mathematica 6.0 does not know e^e is definitely NOT Rational. (It is known not to be rational). It just hasn't been programmed. Yet. It would be nice if the documentation were clearer on this. Regards RJF ...snip..
From: DrMajorBob on 7 Jan 2010 02:33 We've already noted that RandomReal[] outputs are not only of measure zero... but also finite. Hence, arguing that computable numbers have measure zero and hence can't be RandomReal[] outputs, seems beside the point. Bobby On Wed, 06 Jan 2010 04:56:38 -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: János Löbb on 8 Jan 2010 04:13
> >> Why can't a computer, in principle of course, perfectly simulate the >> activity of the human brian that we call "doing mathematics"? > > In principle of course, human minds ARE computers... but not the kind > we're likely to build, anytime soon. > > You're not claiming that Mathematica simulates the mind of a > mathematician, I hope? > Well, well, if I remember well Penrose whole point is that mathematicians are doing mathematics, but they do not know how :) According to him the process is definitely NOT logical thinking, but a "computation" done by the microtubule immersed into the quantum world. Let me tell a non-mathematical example. Gabo - G=E1briel M=E1rquez - was traveling in Mexico on the Trans American Highway and in a curve he could glance down from the road into an exceptionally green valley. At the same moment as he glanced down into this green valley from somewhere - like the Poynting vector - the whole story of "Onehundred year solitude" CAME into his mind. He said it was so clear and vivid that he could have dictated the first three chapters from word to word to a typist. My guess is that it did not came as "text" into his mind, but rather as a "film" or "life" with all the usual emotional side effects, that make it possible to write it down so preciously as he did in the following 8 months. It is anyone guess what was the bandwidth of the "channel" to that "place" where the story came from that in a blink of an eye hundreds of peoples whole life was transfered into his brain. My guess is that there is no computer on Earth that can do the same in an hour even with the parenting hands and logical minds of highly capable mathematicians of a few hundreds/thousands/hundred- thousands. J=E1nos |