WC Archive

sciam had a nice conversation with brian greene recently!

http://www.sciam.com/print_version.cfm?articleID=000073A5-C100-1F80-B57583414B7F0103

a little more in depth is lee smolin's edge talk on LQG :D

http://www.edge.org/3rd_culture/smolin03/smolin03_print.html

Cool leads. Thanks!

Please send your comments to subhendu.das@excite.com

The article “Atoms of Space and Time,” by Lee Smolin, in the January 2004 issue of Scientific American, attempts to answer a very fundamental question: is nature continuous or discrete? However, it appears that the author is already biased toward his hypothesis that nature is discrete.

Is matter discrete?
Smolin says on page 66, “the granularity of matter is old news.” Is that really correct? Has anyone proven that electrons cannot be broken into smaller particles? We have already seen that neutrons can be broken. It is only a matter of time when our technology will break electrons.

On same page Smolin asks, “…does the world evolve in series of tiny steps, acting more like a digital computer?” The digital computer is not a discrete device. Often we casually say that it uses only zeros and ones, but in reality it does not. If you put an electronic probe of an oscilloscope at any pin of a microprocessor you will find a continuous time electrical signal. All electronic circuits are continuous. They are made of capacitors, inductors, and resistors. They all operate like continuous components.

Ironically, while the author was trying to prove discreteness of nature at Harvard, just across the Charles River, MIT was trying to generate discrete signals of digital computers using continuous time dynamical systems.

Are real numbers discrete?
On page 69 Smolin writes, “…the volume could be any positive real number.” He used such statements in several places. It seems that the author thinks real numbers are discrete. In fact hey are not. In all mathematics we use finite precision representation of numbers. But that is not the true representation. The integer 2 is really the real number 2.0000…000… with an infinite number of zeros. When we say 2.00124 we really mean 2.0012413568…9457… with an infinite number of decimal places, but our number is accurate to only five decimal places, so we do not write the rest of the digits.

When Smolin treats the discreteness of nature, which leads to very small numbers, he should really treat all numbers in all of his algebraic and differential equations as numbers with an infinite number of digits. Or he should treat all finite precision numbers of his equations as open intervals around the specified point. For example he should treat the integer 2 as an open interval (2-, 2+) where  is a very small number.

In the sense discussed above, all numbers are open intervals. You can break them into smaller and smaller intervals and you will never find the atom. Thus our mathematical concept of the real line is continuous and not discrete as the author thinks. Please send your comments to subhendu.das@excite.com

Is calculus continuous?
Surely Smolin has used lot of calculus to justify his theory. But is calculus continuous or discrete? Consider the Derivative operator D[xn]. It produces nxn-1. Is this a continuous operation? No. If you plot the graphs of xn and nxn-1, you will see a large gap between them. The Derivative operator D peels off a function in discrete steps. This Derivative cannot be used to analyze the discreteness of nature. It is too gross for the author’s subject.

One would like to have a Derivative that would behave like a continuous operator. As an example, consider the -Derivative with the definition D[xn] = nx(n-). Here  is a real number between 0 and 1. This -Derivative will fill up the gap between the two functions. We can now smoothly change  to smoothly peel the original function. If we can create capacitors and inductors that will follow the -Derivative, then we may see a completely different technology. Smolin needs a continuous calculus to find atoms, if any, of nature. Please send your comments to subhendu.das@excite.com

Are limits correct?
The concept of limits was visualized using a figure drawn on paper. In describing this figure we have used two contradictory notions. On one hand we said we are approaching a limit and on other we still used the same gross level view of the figure. When we approach a limit we must also magnify the figure to see exactly what is happening. If we approach 100 times closer to a point we should magnify the figure by 100 times or a thousand times to see the result. To observe microscopic things we need a microscope. It is the same thing as saying that we are moving closer and closer to a distant star and still thinking that the star will maintain the same small size. So we see that the fundamental notion of calculus, the limit, has failed to treat continuous and discrete behavior of nature in a consistent way. Please send your comments to subhendu.das@excite.com

Is Derivative a tangent?
Consider how we proved that the derivative is a tangent. Imagine the figure, a circle, and a line OP, starting at O on the circle, intersecting at Q and extending beyond to P. If we keep the point O fixed and move the point Q closer and closer to O along the circumference, then the line OP will move and become a tangent at O. We see the same fallacy here; we are not using a microscope. As the point Q moves towards O we must magnify the figure. If you do that then you will see that the figure is basically not changing, only becoming bigger and bigger, and the line will never become tangent. Thus to analyze atoms of nature we cannot use this derivative to represent rate of change of variables or tangent to a function. That will lead to inconsistent results. Please send your comments to subhendu.das@excite.com

Is there equality?
Consider Newton’s formula, F = ma, force equals mass times acceleration, which is not true. We will never be able to measure the quantities at the same time; there will be always a time gap of nanoseconds between measurements of all variables. According to Newton, the equality happens only when you take measurements at the same time. Thus we cannot verify the equation using our modern technology. Moreover every time we measure a quantity, like the mass, using an analog to digital converter with 32-bit resolution, our results will be always different. Because of that reason F-ma will never produce zero. Thus we do not have zero nor equality in our measurements. As a consequence of the above property of real numbers, there are no equations today. We only have inequalities. Did Smolin use equations or inequalities in his theory? Please send your comments to subhendu.das@excite.com

Is mathematics obsolete?
We cannot use existing mathematics to analyze today’s technology. Our technology is way ahead of this mathematics. It was invented over 400 years ago. When this mathematics was invented there were no microprocessors and there were no nanoseconds. In fact, whenever we have used mathematics in our embedded engineering software we had to violate all mathematical theories and assumptions to make them work.

To prove Smolin’s hypothesis we must use a continuous mathematics. A discrete mathematics will always produce discrete results. The above observations prove that our calculus is not correct and not continuous. Using this kind of mathematics, Smolin cannot treat his subject properly. His subject is too advanced for our mathematics. We need to rediscover our mathematics first. In fact Smolin has talked about it on page 68, “Perhaps with the right additions or a new mathematical structure….” Please send your comments to subhendu.das@excite.com