Draft 3
Assertion
The Jordan curve theorem holds for a set of non-visualizable, "wild" curves that can be posited in accord with the Well-Ordering Principle.
Prefatory remarks
An intuitive idea of the well-ordering of the reals can be had by considering the following:
We have that any infinite digit string of 0s and 1s may represent a real. Now a string of length n digits may be ordered from least to greatest in 2n ways, with the awareness that any digit after n is a 0.
This denumerable ordering holds for any finite n. However, denumerable ordering does not hold for the entire set 2N of course. But the Well-Ordering Principle can be interpreted to mean that the set of string permutations is precisely ordered at 2N.
We can obtain the set of all curves, including wild non-differentiable and non-fractal-like curves thus:
Orienting ourselves on the x-y grid, using Quad I for convenience, we can arbitrarily assign any y height to any x value in the interval, say, [1,2]. By this, two neighboring x points can have wildly different heights, though there would still exist a slope for them. But, by the Well-Ordering Principle, there must exist two points that are precisely ordered and that have 0 distance between them. These two points will have no slope and constitute a wild, non-visualizable curve "section." That is, we have a situation where there is a y height for every real but no slope for the curve anywhere, even though y does not equal 0.
Though the area under such a curve must be less than 1*ymax, we may find it difficult to evaluate this integral or even give a good approximation by numerical methods.
To complete the set of all planar curves, we mention fractal and fractal-like curves, which are discussed briefly below.
Proof
We form what I call a molecule, or bubble, with the understanding that such an entity may not be visualizable with a drawing.
We may define an origin-based molecule as r = cosx where, with a well-ordering of the set of radians, r is any finite length. Additionally, r(x) = cosx is a relation with 2n-1 values, accounting for "fingers" -- any of which may be infinitely short -- and "interior molecules" beyond the neighborhood of origin. An interior bubble is defined in the same way as a basis bubble, except that its relation r' = cosx requires that for every value of x, r'(x) is less than r(x) and falls between origin and r(x). (Note: an interior bubble is defined by the relation and is not considered an a priori figure here.)
This will suffice to describe any n-loop; i.e., simple loop or pretzel, though we must shortly consider some sets.
[To help one's mental picture, we can proceed thus after forming a basis molecule whereby there are no fingers or holes. We form a second molecule, which may or may not be congruent, and map it onto the first by arranging a translation of coordinates, the effect of which is to intersect the two bubbles such that they share at least two points. All points other than the join points are then erased. The join points are those which do not intersect only a bubble's points.
[We can orient a bubble any way we like and add bubbles to bubbles to our heart's content. We may get a simple loop or an n-loop pretzel.
[A pretzel may appear when two or more bubbles intersect and the intersection set is construed to be "empty" or "background." If a pretzel hole appears, then the intersection obeys the relation requirements of a molecule. A single-hole pretzel is defined by the relation s(1) and s(2) each have one value and there is a subset for which there are exactly four distinct values of s(x).]
Now the interval [r(x)low,r(x)high] is (ignoring fractals), a finite line segment. So we regard those two values as the end points of the line segment. We then require a Dedekind cut between such an end point and the end point of the corresponding, coinciding half-line. In the case of fingers and pretzel holes, we have rather than a half-line, another line segment, of course.
Now the set of such Dedekind cuts maps as a continuous curve about a finite area with no end points. Suppose the boundary curve had two end points. In that case the relation r would have 0 or 2n values, a contradiction.
So, with respect to a molecule, we have that a point on the plane is either an element of the figure's Dedekind boundary set, an element of a line segment r = cos(x) such that there are 2n-1 values of r(x) (not including origin), or neither. So then, the Jordan curve theorem is proved for molecules -- n-loops -- with continuous or wild curves. (We have not bothered with the matter of nested sets of interior loops, which clearly follows from the preceding.)
In the matter of fractal, or fractal-like, curves, it is plain that a fractal construction at any finite step n is composed of a set of self-similar bubbles of diminishing size. Clearly our proof holds for any such step. By the way, we can see immediately that we can apply, at least notionally, a wild curve to a fractal, giving us a whole new set of fractals.
In the case of a fractal, the non-trivial fractal "slopes," though non-differentiable, take on infinitesimal form in the infinite limit, but what form does a wild fractal curve take? The wild part has 0 slope. So whether the curve can be said to exist in the algorithmic limit must be determined from consideration of axioms. At this point, I am content to point out such a situation.
The Jordan curve theorem also applies to any curve of infinite length that is found at, above or below the x axis. We have the relation r(x) which may have 2n-1 distinct values. We then follow the arguments above and have that a point is found in a Dedekind boundary, between a Dedekind boundary point and r(a)2n-1 or not in the Dedekind boundary but above r(a)max or below r(a)min.
Saturday, October 27, 2007
Wednesday, September 5, 2007
Detecting design
I haven't read William Dembski's books and so am not attempting to refute them. However, I have scanned his online writings, including his rebuttal of critic Richard Wein. In these intelligent design posts, my purpose is simply to think about what might constitute a strong suggestion of design, and related issues.
Dembski's essential point is that some biotic phenomena are so fantastically improbable as to constitute evidence of a designer's forethought.
One example attributed to Dembski is the case of a county clerk who "randomly" placed either Democrat or Republican on the top line. In 41 elections, the Democratic clerk placed a Republican on the top line once. The probability of the exact sequence is 2-41 = 4.54 x 10-13 and the probability that exactly one R would appear anywhere in the sequence is 41C1(2-41) = 1.86 x 10-11. Had 20 R's appeared, the probability would have been 41C20(2-41) = 0.12, which would be at the center of the distribution curve. Either of the other probabilities would lie outside a "confidence interval" of 99 percent, meaning we would have strong grounds to believe that a nonrandom force had been at work. Considering other information, such as the fact that the clerk had a motive to bias the choice, we feel quite sure that the sequence is nonrandom, though mathematically we do not have absolute proof.
(I realize that Dembski doesn't fully accept some commonly held ideas concerning statistical inference, but I will not address those matters here.)
Another example attributed to Dembski is a sequence of coin flips (perhaps 100) that, when recorded as 0s and 1s, reveal a sequence of consecutive binary integers.
In this case, the probability is 2-100 = 7.88 x 10-31, a preposterously small number that would immediately make us conclude that a nonrandom force was at work. But why so? In the case of a sequence containing say 10 heads and five tails, we want to know the probability associated with a subset of sequences. That is, we do not narrow down the probability to one element. So the probability of one specific sequence is 2-15 = .0000305, but the probability of any element of that subset is .09.
In the case of the binary digit series, we have reason to specify one element and so we conclude that there is reason to suspect a nonrandom force. That is, the bias in each individual toss appears to be very strong and the conjecture that the tosses are independent appears to be false. But, if we simply find that the 10 heads and 5 tails are not obviously 1-to-1 with a known sequence, then we have no reason to specify further down than the subset level. We could not easily reject the conjecture that the tosses are independent events, though we would have grounds to believe that the coin was biased. We could not rightly say that a nonrandom force was at work beyond simple weight bias.
That is, if we have reason to select one and only one element from the set of sequences (with n sufficiently large), then we have reason to suspect that the events are not independent and that there is at work some highly delineated nonrandom force.
So now we return to the issue of detecting design, or, that is, intelligent design (see previous ID post below).
A designer of a machine or network system actually designs an algorithm, which we might view as a sequence of logic gates. We are able to express this circuit using some logic language L. An algorithm would be 1-to-1 with any grammatical sentence in L. If we have a large enough sample of expressions, we might detect the presence of L in some sequence by checking the frequency ratios of pairs of symbols in L. If the sequence is indicative of L, the scatter plots of the symbols will show strong correlations.
For example, we could have 30 characters in E (English) appearing randomly over 10,000 spaces. For x,y element of EXE, the scatter plot correlation will be weak. But, if an excerpt from an English-language novel appears in those 10,000 spaces, the scatter plot for the pair x,y will be far more correlated.
An ungrammatical 10,000-character statement would be noise, but a grammatical statement would show pair correlations. This holds even if we can't even read English or L.
Still, a physical system must be modeled by a grammatical statement. An ungrammatical statement implies a bogus physical system (or noise).
But there is one more issue here. Machines, even relatively inefficient ones, have little internal noise in their design. So perhaps we should consider pattern recognition.
XXJXOXXXEXXX
produces a readable pattern despite the noise. As noise increases, readability decreases, so signal-to-noise ratio may be something to consider.
1AJMOLNYEV4MM
is so noisy that the word Joe may take far longer to discern.
So suppose we have a grammatical statement embedded in noise whereby the noise is represented by dummies. A pair with a dummy will show low correlation. Now, if we are painstaking and have a large enough sample, we might be able to distinguish the signal from the noise, even when the SNR is unfavorable for routine pattern recognition.
Now suppose we have a sufficient sample which is high on noise but low on signal. Yet the signal (grammatical statement) is there -- that is, a physical system is functioning. Would we be correct to conclude design? I think possibly not. In a large enough environment, low entropy sequences are at least plausibly the result of independent events. That is, we may find a "grammatical" pattern in a long enough sequence of garble (remember The Bible Code?).
However, if the sequence contains a grammatical statement describing the system's algorithm is low on noise, we might have grounds to suspect that the events are not independent and that some powerful nonrandom force is at work. Whether we can assume that the powerful nonrandom force has consciousness is another matter.
But, let us consider computer system and internet bugs. Bugs are like noise in a system. They are usually unintentional consequences of complexity though they might seem to be the work of a malicious hacker. But, normally, though a bug might have a cascade effect (butterfly effect), it does not replicate and does not transmit itself.
However, computer worms, viruses and Trojan Horse parasite programs are an obnoxious internet presence. I would be eager to know whether there has been one worm, virus or Trojan Horse that has arisen spontaneously from a peculiar confluence of bugs or other computer oddities. So how does a computer bug differ from a malware program? It's in the code. The malware code has a much higher information quantity than does the bug. The code sequence does not arise spontaneously, though it might I suppose if there were sufficient time and energy.
Dembski's essential point is that some biotic phenomena are so fantastically improbable as to constitute evidence of a designer's forethought.
One example attributed to Dembski is the case of a county clerk who "randomly" placed either Democrat or Republican on the top line. In 41 elections, the Democratic clerk placed a Republican on the top line once. The probability of the exact sequence is 2-41 = 4.54 x 10-13 and the probability that exactly one R would appear anywhere in the sequence is 41C1(2-41) = 1.86 x 10-11. Had 20 R's appeared, the probability would have been 41C20(2-41) = 0.12, which would be at the center of the distribution curve. Either of the other probabilities would lie outside a "confidence interval" of 99 percent, meaning we would have strong grounds to believe that a nonrandom force had been at work. Considering other information, such as the fact that the clerk had a motive to bias the choice, we feel quite sure that the sequence is nonrandom, though mathematically we do not have absolute proof.
(I realize that Dembski doesn't fully accept some commonly held ideas concerning statistical inference, but I will not address those matters here.)
Another example attributed to Dembski is a sequence of coin flips (perhaps 100) that, when recorded as 0s and 1s, reveal a sequence of consecutive binary integers.
In this case, the probability is 2-100 = 7.88 x 10-31, a preposterously small number that would immediately make us conclude that a nonrandom force was at work. But why so? In the case of a sequence containing say 10 heads and five tails, we want to know the probability associated with a subset of sequences. That is, we do not narrow down the probability to one element. So the probability of one specific sequence is 2-15 = .0000305, but the probability of any element of that subset is .09.
In the case of the binary digit series, we have reason to specify one element and so we conclude that there is reason to suspect a nonrandom force. That is, the bias in each individual toss appears to be very strong and the conjecture that the tosses are independent appears to be false. But, if we simply find that the 10 heads and 5 tails are not obviously 1-to-1 with a known sequence, then we have no reason to specify further down than the subset level. We could not easily reject the conjecture that the tosses are independent events, though we would have grounds to believe that the coin was biased. We could not rightly say that a nonrandom force was at work beyond simple weight bias.
That is, if we have reason to select one and only one element from the set of sequences (with n sufficiently large), then we have reason to suspect that the events are not independent and that there is at work some highly delineated nonrandom force.
So now we return to the issue of detecting design, or, that is, intelligent design (see previous ID post below).
A designer of a machine or network system actually designs an algorithm, which we might view as a sequence of logic gates. We are able to express this circuit using some logic language L. An algorithm would be 1-to-1 with any grammatical sentence in L. If we have a large enough sample of expressions, we might detect the presence of L in some sequence by checking the frequency ratios of pairs of symbols in L. If the sequence is indicative of L, the scatter plots of the symbols will show strong correlations.
For example, we could have 30 characters in E (English) appearing randomly over 10,000 spaces. For x,y element of EXE, the scatter plot correlation will be weak. But, if an excerpt from an English-language novel appears in those 10,000 spaces, the scatter plot for the pair x,y will be far more correlated.
An ungrammatical 10,000-character statement would be noise, but a grammatical statement would show pair correlations. This holds even if we can't even read English or L.
Still, a physical system must be modeled by a grammatical statement. An ungrammatical statement implies a bogus physical system (or noise).
But there is one more issue here. Machines, even relatively inefficient ones, have little internal noise in their design. So perhaps we should consider pattern recognition.
XXJXOXXXEXXX
produces a readable pattern despite the noise. As noise increases, readability decreases, so signal-to-noise ratio may be something to consider.
1AJMOLNYEV4MM
is so noisy that the word Joe may take far longer to discern.
So suppose we have a grammatical statement embedded in noise whereby the noise is represented by dummies. A pair with a dummy will show low correlation. Now, if we are painstaking and have a large enough sample, we might be able to distinguish the signal from the noise, even when the SNR is unfavorable for routine pattern recognition.
Now suppose we have a sufficient sample which is high on noise but low on signal. Yet the signal (grammatical statement) is there -- that is, a physical system is functioning. Would we be correct to conclude design? I think possibly not. In a large enough environment, low entropy sequences are at least plausibly the result of independent events. That is, we may find a "grammatical" pattern in a long enough sequence of garble (remember The Bible Code?).
However, if the sequence contains a grammatical statement describing the system's algorithm is low on noise, we might have grounds to suspect that the events are not independent and that some powerful nonrandom force is at work. Whether we can assume that the powerful nonrandom force has consciousness is another matter.
But, let us consider computer system and internet bugs. Bugs are like noise in a system. They are usually unintentional consequences of complexity though they might seem to be the work of a malicious hacker. But, normally, though a bug might have a cascade effect (butterfly effect), it does not replicate and does not transmit itself.
However, computer worms, viruses and Trojan Horse parasite programs are an obnoxious internet presence. I would be eager to know whether there has been one worm, virus or Trojan Horse that has arisen spontaneously from a peculiar confluence of bugs or other computer oddities. So how does a computer bug differ from a malware program? It's in the code. The malware code has a much higher information quantity than does the bug. The code sequence does not arise spontaneously, though it might I suppose if there were sufficient time and energy.
Monday, September 3, 2007
Weeding out dummies
Below we discuss a statistical method for discriminating between signal and noise. We might be able to tell whether a transmission from deep space is a message of some kind simply by doing scatter plots of presumed symbol pairs.
This leads to a cryptological point: the use of dummies is not neceesarily a protection against frequency analysis. When a dummy is paired with a symbol, the scatter plot may well show low correlation (no football shape). The code-makers must make sure in advance that every dummy pairs with every symbol in a likely ratio.
Otherwise, code crackers can check correlations and weed out all the dummies. They are then left with the simple task of doing a routine frequency analysis on the remaining symbols.
Yet I can't help thinking that for dummies to be useful that they will either correlate poorly with symbols, or that, if they are designed to correlate strongly, the correlation will not be characteristic of letter/number correlations.
Computer codes don't ordinarily rely on dummies. But, secret messages are sent via all sorts of means. Low tech messages might easily rely on dummies. If you use such a system, beware.
This leads to a cryptological point: the use of dummies is not neceesarily a protection against frequency analysis. When a dummy is paired with a symbol, the scatter plot may well show low correlation (no football shape). The code-makers must make sure in advance that every dummy pairs with every symbol in a likely ratio.
Otherwise, code crackers can check correlations and weed out all the dummies. They are then left with the simple task of doing a routine frequency analysis on the remaining symbols.
Yet I can't help thinking that for dummies to be useful that they will either correlate poorly with symbols, or that, if they are designed to correlate strongly, the correlation will not be characteristic of letter/number correlations.
Computer codes don't ordinarily rely on dummies. But, secret messages are sent via all sorts of means. Low tech messages might easily rely on dummies. If you use such a system, beware.
Thursday, August 30, 2007
Is there a God code?
In previous posts, I discussed means of assessing whether a transmission from outer space implied intelligence. The intelligent design theorists use as an example a transmission from space that follows a pattern of the prime numbers in consecutive order. Such a transmission would be immediately recognizable as a communication, or sign of intelligence. Similarly, the ID backers say, a proto-cell is astronomically unlikely to have fallen together spontaneously.
I have argued that we can't necessarily accept that a prime number transmission implies intelligence. We first have to know about primes. But anyway, it occurs to me that cryptographers and communications engineers would have little problem with such a challenge -- if by intelligence we mean something to which humans can relate.
As is well known, every English letter occurs with a specific frequency. Hence each pair in the alphabet A (each element of AXA) also has a specific ratio. This also holds for words and for symbols that are part of some language L in general. That is, if a symbol is used in some form of communication of ideas, it bears a stable relationship with every other symbol in L.
So if a set of discrete pulses arrived at some bandwidth, we might use scatter diagrams to check each possible pair of frequencies at time interval t and see whether, for pair (x,y) a football shape emerges, and to check the level of correlation. Of course, if the correlation is too weak, either the sample is insufficient or the pair (x,y) cannot be construed as a grammatical structure in L. If the correlation is too linear, then we would have to consider mechanical rather than symbolistic cause, as with pulsars.
Noise would yield a group of scatter diagrams most of which lack a football shape. Those with the shape would be checked more closely to see whether a language is implied.
This technique should work in most cases, though I am not sure whether it would work in the specific case of a prime number sequence. More thought necessary. But, the argument may still proceed.
The question arises: can we discover such a symbolistic code at the level of the origin of life? We are not necessarily talking about the DNA or RNA codes, but about whether the makeup of the first cells might indicate an intelligence.
How might we determine whether some system has been designed by an intelligence rather than being a consequence of the "hidden hand" or regulator of chaotic processes? We might model the system's workings with a blueprint, perhaps a Boolean circuit whereby the symbols represent logic gates. If such a circuit C's symbols are all paired (CXC) and scatter plots are done, we would check for the telltale football shapes.
If they showed, we would be fairly sure that some intelligence was "speaking" through the blueprint. But a lack of such evidence would not rule out intelligent design, of course.
A study might be done comparing Boolean circuits of artificial and inorganic natural systems to see whether there are significant differences. This information might be used for comparison with organic or near-organic systems. If the symbol language of an organic system was fairly close to that of an artificial system yet different from an inorganic system, that would be big news. Still, lack of such evidence would not disprove intelligent design.
So the point here is that the odds against a fluke outlier may be extraordinarily high but that isn't necessarily analagous to an improbable signal from space. In the latter case, we can apply statistical methods to determine the likelihood of non-random activity associated with communication versus the random pulses associated with noise.
What statistical methods can be used to check for non-randomness in the "signal" of early life forms? I gave one method, though I do not think a positive result is likely.
Yet, we cannot foreclose the possibility that a strong statistical approach might yield surprising results.
I have argued that we can't necessarily accept that a prime number transmission implies intelligence. We first have to know about primes. But anyway, it occurs to me that cryptographers and communications engineers would have little problem with such a challenge -- if by intelligence we mean something to which humans can relate.
As is well known, every English letter occurs with a specific frequency. Hence each pair in the alphabet A (each element of AXA) also has a specific ratio. This also holds for words and for symbols that are part of some language L in general. That is, if a symbol is used in some form of communication of ideas, it bears a stable relationship with every other symbol in L.
So if a set of discrete pulses arrived at some bandwidth, we might use scatter diagrams to check each possible pair of frequencies at time interval t and see whether, for pair (x,y) a football shape emerges, and to check the level of correlation. Of course, if the correlation is too weak, either the sample is insufficient or the pair (x,y) cannot be construed as a grammatical structure in L. If the correlation is too linear, then we would have to consider mechanical rather than symbolistic cause, as with pulsars.
Noise would yield a group of scatter diagrams most of which lack a football shape. Those with the shape would be checked more closely to see whether a language is implied.
This technique should work in most cases, though I am not sure whether it would work in the specific case of a prime number sequence. More thought necessary. But, the argument may still proceed.
The question arises: can we discover such a symbolistic code at the level of the origin of life? We are not necessarily talking about the DNA or RNA codes, but about whether the makeup of the first cells might indicate an intelligence.
How might we determine whether some system has been designed by an intelligence rather than being a consequence of the "hidden hand" or regulator of chaotic processes? We might model the system's workings with a blueprint, perhaps a Boolean circuit whereby the symbols represent logic gates. If such a circuit C's symbols are all paired (CXC) and scatter plots are done, we would check for the telltale football shapes.
If they showed, we would be fairly sure that some intelligence was "speaking" through the blueprint. But a lack of such evidence would not rule out intelligent design, of course.
A study might be done comparing Boolean circuits of artificial and inorganic natural systems to see whether there are significant differences. This information might be used for comparison with organic or near-organic systems. If the symbol language of an organic system was fairly close to that of an artificial system yet different from an inorganic system, that would be big news. Still, lack of such evidence would not disprove intelligent design.
So the point here is that the odds against a fluke outlier may be extraordinarily high but that isn't necessarily analagous to an improbable signal from space. In the latter case, we can apply statistical methods to determine the likelihood of non-random activity associated with communication versus the random pulses associated with noise.
What statistical methods can be used to check for non-randomness in the "signal" of early life forms? I gave one method, though I do not think a positive result is likely.
Yet, we cannot foreclose the possibility that a strong statistical approach might yield surprising results.
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