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Author
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Topic: d* The Littlest Number
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Melvin H. Fox
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Member # 1684
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posted 03. December 2005 09:58
Once upon a time there was a real number line. It was a continuous string of numbers rational and irrational and it was the framework around which the physical universe was built. At least that is what they led me to believe during my years of formal education.
I am a teacher now and until very recently I have been instructing my students along the same real line. One day a calculus student of mine brought me an article “Atoms of Space and Time” [Scientific American: January 2004]. In it Lee Smolin reported on how space time is now thought to be made up of tiny indivisible sections called quanta. Further it was reported that a central prediction of the loop quantum gravity theory, LQG, was that there is a nonzero absolute minimum volume of 10^-99 cubic centimeters. My immediate reaction was one of concern for the job security of the calculus. After all it holds the position of tour guide to the universe. If LQG or any similar theory which establishes the universe as a discrete space is true, then limits as displacement goes to zero are merely practical approximations.
Only in the back of my mind did I start to question the validity of R^3 as a model for three dimensional space. Once I read some of Dembski’s work I made a solid connection and asked myself this question: What if there is a smallest positive number in the real physical universe? This would not only apply to displacements but to any variable of a physical nature. Even time itself would be a discrete variable. LQG predicts that time ticks along at 10^-43 second jumps. Time does not exist between the ticks. In other words, if the solution to some differential equation yields dt < 10^-43, then we must conclude dt = 0.
Would it not follow then that there would exist some smallest probability p such that if the probability of event E occurring by chance were less than p, then we would conclude event E could not occur by chance?
Theorem: The smallest positive real number is 1.
Proof: Let d* be the smallest positive number to show that d* = 1. One and only one of the following must hold: d* = 1, d* < 1, or d* > 1. By our assumption we cannot have d* > 1. If d* < 1 then [d*]^2 < d* and d* would not be the smallest. Therefore d* = 1 and 1 is the smallest positive number.
What is wrong with this proof? As E.E. Escultura points out is his paper “Countably infinite counterexamples to Fermat’s Last theorem” [http://www.users.bigpond.com]:
quote:
The main culprit here is the use of necessary condition without existence theory…
The existence of d* is a vacuous statement, a true nonsense about anything that does not exist. Why did I bring this up, I thought I was trying to argue that there is a smallest positive number. I give the proof as evidence for the following conjecture: If there is a smallest positive number d* in the physical universe, then the real number system is not a good model for solving problems involving large or small numbers. This is clearly so because it has been shown that positive numbers of arbitrary smallness exist in R. This leaves the real number system as an inadequate tool to use when explaining the probabilities of rare events. Again from Escultura:
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A fundamental defect of mathematics we are raising for the first time is the lack of distinction between the subjective universe of thought and concepts, on the one hand, and the objective universe of their representation, on the other.
Apart from observation, there is no way to verify that the existence of a mathematical construct [i.e. circle or an arbitrarily small number] exists in the physical world. Even if we had a perfect circle in the palm of our hand we could not verify it because there is always error when we measure. Would it not be vacuous then to say, that any measured quantity of the physical universe is less than d*?
In his paper, “How real are real numbers?” [arXiv:math.HO/0411418v3 29 Nov 2004] Gregory Chaitin takes a different tact which involves the concept of irreducible complexity and arrives at a similar conclusion:
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Why should we believe in real numbers, if most of them are uncomputable? Why should we believe in real numbers, if most of them, it turns out, are maximally unknowable, like the halting probability?
Physics has been married to R for some time now. Divorce is a possibility but the honeymoon is definitely over. At this time I have no better estimate of d* than 10^-150 [Dembski’s UPB]. I believe that for any calculated value derived from measured data in the physical universe [this would include probabilities of physical events] which is less than 10^-150, the burden of proof has now switched. One must show the existence of this small number apart from its existence in R.
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Melvin H. Fox
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Member # 1684
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posted 07. December 2005 11:23
The set of real numbers is a mathematical construct. In most analysis texts R is constructed from Q [the set of rational numbers]. A real number is defined as a cut. A cut is any non empty proper subset of Q with the following properties:
Let C be a cut.
[I] If p is an element of C, q is an element of Q, and q[II] If p is an element of C, then p
Note: p, q, and r are all rational numbers.
In other words, a real number is the set of all rational numbers less than some boundary. The boundary can be a rational number or some undenumerated entity between two rational numbers. How is it known that these entities represent some physical attribute, without the assumption that the physical universe is a continuous space? It might be said that we need these numbers, the square root of 2 for example, because the set of rational numbers is unable to give the displacement for an object that moves one kilometer due north followed by a one kilometer movement due east. Perhaps this exact path is not physically possible and therefore no exact calculation of the displacement is needed? It is true that some displacements would not be possible in a discrete universe. Could it be the case that most real numbers are irrelevant in the physical universe? If so, it is then reasonable to assert that there is a smallest relevant number [a number which actually represents some quality of the physical universe] that is greater than zero.
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TheLeftReverend
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Member # 1835
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posted 08. December 2005 11:23
I am an amateur scientist and mathematician (I hope in the good sense). It has always amazed me that most science is now predicated on the belief that mathematics describes the observable world. Experiments are rarely done by observation; the senses of touch, smell, taste are obsolete. Information is gathered, manipulated and presented in mathematical form.
When experiments produce unexpected outcomes, we tweak the mathematics; it is inconceivable to us that the inverse square law, for example, or the Pythagorean theorem simply does not apply at all times and places.
Would this mean that our current science is as much of a construct as the mathematics on which it is based?
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Melvin H. Fox
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Member # 1684
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posted 09. December 2005 15:40
All of mathematics is built from literally nothing. Weather we use the construction of the natural numbers from the Peano axioms or the standard set theory construction we start with zero and the idea there is more and go from there. Yes, all of math is made up. The famous 19th century mathematician Dedikind is quoted as saying “God made the natural numbers and man invented the rest.” We use this invention to describe reality. Chris Beling has posted a very good explanation of the process on another thread here at Brainstorms [Is 2nd Law a special case of 4th Law?]. His post was posted 09. September 2005 23:05. Is this what should be done? I don’t know. How would you study things you can’t observe with your senses? What is suggested on this thread is that the set of real numbers may not be the best framework to use to describe reality. Further, the set of rational numbers might be equally misleading. For example, if time comes in discrete packages then all real periods of time can be represented by a proper subset of the rational numbers.
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Terry Mullett
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Member # 1810
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posted 10. December 2005 23:02
It is precisely because math does describe the world that it is a construct. The same thing goes for science, theology or any intellectual endeavor. Including quantum theory. Before we get too excited about the exotica of the quantum world, it might be good to keep that in mind.
The isomorphism between a description and the thing described is always limited, which sounds a lot like what Melvin is getting at in terms of numbers. But what I think is missing in the discussion is the matter of what makes one sort of description better than another. It may be obvious, but not really put forth here that I can see. It is, basically, that some ways of stating things are more helpful than others when you are trying to manipulate the subject matter. Helpful in what way? That depends on what you're trying to accomplish.
Furthermore, the most helpful explanatory device in one situation is not necessarilly the most helpful in any other situation. For instance, if one is operating at a level of granularity that a minimum distance or a minimum time interval at these magnitudes really could make a difference, then I'd wonder about the usefulness of continuous mathematics right along with you. At the level of, say, firing a missile, planning a shot on a pool table, or thinking about where species come from or what goes on in a brain, quantum rules are more than just lost in the details. Those rules are as effectively meaningless at the macro level as continuous math may be at the quantum level.
For example, ancient Greeks, involved in a flourishing of monumental architecture, wanted the measure of the diagonals of 1x1 and 1x2 areas. These things are as real as a quantum minimum distance. The latter may get more respect as "real", in some circles, because big experiments or fancy technology are invovled, yielding exciting and mysterious results (not to mention big exponents), but the ramifications of the former are demonstrably more significant.
I'm afraid there's something systemic about the fact that atoms yield to quarks which then yield to strings... nature may be fractal, and at any level of detail we can never really say we've bottomed out. Or is it thinking that's fractal? In any event, I'm also quite confident that there are at least a few more layers of surprises below the quantum level, and the rules at the quantum level will have to yield to other forms of wierdness yet to be discovered. Of course, the real wierdness isn't in those things we can't determine with our senses: those things are exactly what they are. It is the clumsy way we project paradigms practiced in familiar realms of observation onto phenomena in unfamiliar realms that generates the wieredness.
The ancient Egyptians expressed everything between the integers as sums of unit fractions. To the earliest Greek mathematicians, fractions weren't actually numbers at all. Zero was invented by a Hindu mathematician less than 1500 years ago. The rational numbers are as much a construct as all the rest of it.
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Melvin H. Fox
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Member # 1684
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posted 12. December 2005 09:57
Hi Terry,
I agree that the real numbers are no more a construct than the rational number. In fact I agree with most of your post. I thank you for your input on this unpopular idea. Here are my thoughts on some of the points you made.
Mathematical constructions are not bound by physical realities. One is free to invent a set of axioms [even an arbitrary set] and then see what sort of structure follows. It is true that the primary use of mathematics is to describe reality. In order to minimize error we do change the rules. Euclidian geometry is used for local analysis while hyperbolic geometry is more suited for universal analysis. I am not suggesting that the set of real numbers is false; nor am I suggesting it is useless. What I am suggesting is that it may not be the best model in a variety of situations. The situations I refer to would occur if the universe is not a continuous space. Very small real numbers, such as 10^-200, become suspect. To say this number is an accurate description in a discrete physical universe just because it is a real value solution to the real valued function we are using to describe a physical situation is not enough. The more accurate description could very well be zero. This whole debate over intelligent design and evolution is about what real happened. If we can’t know for certain and are forced to accept the best description, then a discrete universe points to zero rather than a number like 10^-200 as being the best description. If we like the idea of a perfect circle, then we can ignore the fact that this construct can not exist in a discrete space or we can be content with taking home a nearly perfect circle. If as you say we are in store for infinitely more small surprises, well my argument then would amount to zero.
-Mel
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Preston
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Member # 1839
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posted 17. December 2005 15:04
I am in no way along the lines of the level of thinkers that are on this board, I actually happened upon this board looking up the definition for a quiz. I do however, have a thought in regards to your theory.
Isn't the thought of a minimum number proved by the fact a perfect circle can not be created?
In that, if there was an infinite value to create between 1 and 0 you could always form an arc. However, in this case the ability to bend the line or decrease the value to a point that is the minimum.
In this theory the circle could only by a colusion of the smallest possible measurement to form the image of a circle but could never truly become a circle. The summary would be a perfect circle is actually a polygon.
Just my thoughts on a Saturday.
[ 17. December 2005, 19:56: Message edited by: Preston ]
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