Dec 14, 2021
27
0
50
The definition of a circle has widely been accepted as a shape in which every point on the perimeter is equidistant to the center.
However, if this is to be the accepted definition of a circle, then logic dictates that no such thing as a circle can exist.
Pi is a mathematical constant, obtained by the division of a circle's circumference by its diameter using consistent units.
Why does this calculation always yield a non-repeating infinite decimal?
Most infinite decimals exist as a reflection of the limited mathematics used to describe them.
For example, while 1/3 is an infinite decimal in a base ten system, it can be represented as a perfect decimal in a base nine system. In other words, the status of most infinite decimals is strictly relative.
So what makes Pi different?
One way of looking at Pi, is as a calculation of the average distance of any given point on the circumference of a circle from a point directly opposing it, the center of the circle being halfway between them.
So if all circles are different sizes, why is it that Pi, a mathematical constant, is a non-repeating infinite decimal?
The answer is that the distance between two points can never be objectively measured in discrete finite terms to begin with.
When a shape is constructed or observed that appears to be a circle, discrete finite terms are used to describe its dimensions.
The reality is that those "discrete finite terms" are infinitely divisible.
So when you take a shape in which all points appear to be equidistant to the center, and divide the perimeter by its width, you get an infinite non-repeating decimal regardless of what units you use, assuming they are applied relatively consistently.
What this indicates, is that the average distance to the center can only ever be calculated in infinite terms. In other words, at the heart of every circle is a singularity. Every point on the perimeter is a unique distance from the center, but when you take them as a whole you get a persistent average.
As a matter of fact, any "point" on the perimeter can itself be described as its own circle, with its own center.
So what we call a "point" is an infinitely divisible point of reference, not a discrete finite mathematical tool.
In a way, Plato was right. The circle is the most perfect of shapes. Too bad they don't actually exist.
 
Mar 4, 2020
498
70
1,780
I am certainly not a mathematician. But from what I understand about ratios.......if one describes a circle using base nine numbers.......the C/D relationship will have a different value.......but that value also.....should be a repeater.

If that is wrong, someone please correct me. If one can find a base where pi(C/D) is non repeating....then indeed it is the math we use.

You might very well be correct about a perfect circle in nature. The apparent ones we do see, always seem to have a perpendicular velocity component to the circumference. For instance, a round water ripple. It looks like a perfect circle moving out and expanding......but there is a down and up motion, as it moves out. Plus, the circle is not rotating. In 2D. Also, if the origin moves(up and down, or sideways), while in rotation or expansion, pi would be invalid. Pi only works in 2D.

And when we try to rotate disks at high speeds, they start to wobble, up and down. Maybe, mass itself, can't move in 2D, at high speeds. It might have to roll at high speed to conserve momentum.
 
Jan 27, 2020
319
96
1,780
Is a line segment divisible without limit? Take a segment of finite length, call the first “point” on it zero and the last “point” one. The distance from the origin (zero) to any point on the segment is given by a (standard) real number of the form: zero-point (0.), followed by an infinite expansion (sequence) of digits (for instance, 0.2854618326580009276492651648206517848...). This is the mathematical version of Zeno’s axiom, inaccessible to the Greek who did not know the zero.

Accepting the “existence” of the real numbers is a strong hypothesis and, consequently, their study has become a key element in the search for a set of axioms or fundamental assumptions on which elementary number theory, and by extension, the whole of mathematics, might be firmly based. It is generally accepted that the set of 10 or so statements supporting most mathematical systems is the Zermelo-Fraenkel set theory. To these statements the nonstandard approach adds three additional axioms. They are based on the definition of a new nonstandard number: the infinitesimal. An infinitesimal nonstandard number is a new type of number: by definition, it is greater than zero but always less than any standard real number, however small.

These infinitesimals possess a thoroughly elusive character because they can never be captured through any possible measurement. The reason: measurements have always as result a standard real number. Furthermore, the difference between two standard real numbers can never be a nonstandard number, which is by definition always less than any standard number. Thus the interval between two nonstandard points on the line, or two nonstandard intervals of time, can never be measured, and so these intervals are forever beyond the range of observation. They exist only by axiomatic (Platonic) definition but can never become actual in Aristotle’s sense.

The nonstandard theory adds two more nonstandard numbers as axioms. The nonstandard unlimited number, which is the inverse of an infinitesimal number, is greater than any standard number but nevertheless smaller than infinity. The nonstandard unlimited numbers are thus very large, larger than any standard number, but always finite, that is, always less than the truly infinite numbers. The nonstandard mixed numbers are so to speak in between: around each standard number, on both sides of it on the line segment, a particular set of nonstandard numbers is, on the left, greater than any other standard number but less than this particular standard number; on the right side, it is smaller than any standard number greater than this particular number, but it is still greater than this number. In summary, between zero and infinity, a new infinity of nonstandard numbers has been added by axiom.

How does this nonstandard mathematics “solve” Zeno’s paradox? Achilles, as he gets closer and closer to the tortoise, will be traversing an infinite series of ever smaller space segments until eventually he will be at nonstandard infinitesimal distance from the tortoise. From this point on, his progress until he catches his prey escapes all possibility of measurement: all the final segments being nonstandard infinitesimal distances. In other words, what “really” happens when Achilles catches the tortoise can never be known, by definition, and so the case rests.

In the early twentieth century some philosophers have been particularly intrigued by the time continuum: if the instant is zero, when do we exist? The past is already gone, the future not yet here, and the present instant zero: when can we claim that we are? Martin Heidegger’s attempt to crack this problem in Being and Time, first published in 1927, may in a certain way be considered to be one more “solution” to Zeno ́s paradox. This philosopher suggests that humans are never authentically “being"; instead, from the very moment one is born, one is already dying, i.e., not-being. “The moment you are born you are old enough to die”. He furthermore claims that the only “time” that has a sense is the unknown period a human still has before he dies, that is, only the yet non-existent future is real. If one is asked: Will you die? The answer is: Yes, of course, but not yet.

Like Achilles: will he ever catch the tortoise? Yes, of course, but not yet. So, Heidegger states, there are two manner of being: the inauthentic and the authentic. In the former, which is where most of us choose to be, our allotted time-span is this not yet, this unknown future which allows us to escape from, to conceal, the unbearably displeasing fact: we are mortal. In this manner of being, we never actually die, we are always alive; death is “only” a potentiality. And we can say such strange things as: I have no time, don’t waste your time, etc. Whereas being authentically is equivalent, in a sense, to dying! That is, to fully accept human mortality. Put differently, inauthentic being is equivalent to live, to be, in the standard part of the time scale: there we can “measure” time with watches in standard numbers. Whereas authentic being is traversing our life-span in the nonstandard numeration, which is beyond measure, and where, in a sense, as soon as we are born we are already dying .

Recently a new, fascinating twist to this old conundrum appeared: Cris Calude’s Lexicon. Here is succinctly how it applies. Zeno’s paradox, and its possible “solutions”, must be somehow explicitly stated in a communicable language. This implies some linear sequence of symbols; the set of allowed symbols constituting the pertinent alphabet. Any finite sequence can be unambiguously coded in binary (or decimal) and thus corresponds exactly to some rational number. This paper, for example, corresponds to the rational number “w”. On the other hand, real numbers are infinite sequences of digits (in whatever chosen code or base). Question: Is there a real number that with certainty contains the word w (i.e.: exactly this paper)? Answer, Yes, and furthermore, there exists a real number that contains every possible “word”. That is, that contains everything that can be explicitly stated, coded, communicated. Here is how that number is constructed, in binary: simply add one after the other every possible binary sequence of 1,2,3,4,... bits:
0,1,00,01,10,11,000,001,011,111,110,100, 010,101, 0000,0001,...
all the way to infinity. By construction, absolutely everything that can be explicitly stated is represented, at least once, in this sequence.

Now it can be shown that this special real number not only contains, by construction, every possible finite linear sequence, say William Shakespeare’s complete works, but also that it contains every possible linear sequence infinitely many times! This is easily proved. Again, call some sequence, say this paper, w. Now construct these sequences:
w0
w00
w000 w0000 .................
all the way to infinity. Since by construction, each of them is already on our specially constructed binary real number, all of these “words” or binary sequences must also appear, at least once. But in each of them w appears, hence w appears infinitely many times. And this is the case for every possible w, QED.

It has been shown in 1998 by Calude and Zamfirescu that there exist real
numbers that present this remarkable property independent of the employed code or alphabet (binary, decimal, or, for instance, all the symbols on a computer keyboard). These are the Lexicons. Thus a Lexicon contains infinitely many times anything imaginable and not imaginable, everything ever written, or that will ever be written, any description of anything, of any phenomenon, real or imaginary, etc., etc.

So what is the relation of this surprising result with Zeno’s paradox? We humans are limited and mortal. We can only name, we can only put our fingers on, we can only actually exhibit, the rational (finite) numbers. But underlying all our mathematics, and Zeno’s paradox, are the real numbers. These, at least the immense majority of them, contain potentially everything, and infinitely many times. What we can actually exhibit is only a vanishing small subset of the underlying potentially possible number set. Now, we may call some “fact”corresponding to some rational numbera novelty. But this is just a Zeno-type illusion: everything is already contained (or expressed) infinitely many times in the infinite set of the reals almost all being Lexicons.

But what is a fact? Something that may have happened and that is amenable to a communicable description (measurement). For instance: the result of the sense- perception by which we observe Achilles catching the tortoise. Thus, something that corresponds to some finite sequence, to some rational number. But this “fact”, this sequence, this change, movement, kinêsis, is infinitely many times recorded in infinitely many real numbers ... if they exist, of course. Then it follows from the admittance of the existence of the real numbers, of Zeno’s infinite divisibility axiom of space and time, that there can never be any novelty. Everything is, always, and nothing ever becomes: Parmenides was right after all!

“The thing that has been, it is that which shall be; and that which is done is that which shall be done : and there is no new thing under the sun” ; so says the Ecclesiastes 1.9 (composed after 250 BC). Everything, the theory tells us, is right there under our nose, so to speak, potentially, but inaccessible. Whereas anything that actually happens, happened, or will happen, is an illusion, since it has been “there”, already, always. A final question remains, however: has anybody anytime laid his fingers on such a Lexicon? Surprisingly, yes, Greg Chaitin’s marvellous and mysterious real number page8image189738176is a Lexicon. But that is another story.

See: https://www.cs.auckland.ac.nz/research/groups/CDMTCS/researchreports/089walter.pdf

After twenty-five centuries Zeno’s paradox* is still with us: if we admit the existence of the real numbers we run into trouble; we deny it, and we find a different set of equally intractable problems since now mathematics, and thus physics describing “change”, become problematic. We are left with admiration for those early Greek philosophers, who unveiled the fundamental limits of human reason (and of mathematics ).
Hartmann352

* Zeno's paradox see: https://iep.utm.edu/zeno-par/
 

Attachments

  • Like
Reactions: Onemetertwometer
Dec 14, 2021
27
0
50
I am certainly not a mathematician. But from what I understand about ratios.......if one describes a circle using base nine numbers.......the C/D relationship will have a different value.......but that value also.....should be a repeater.

If that is wrong, someone please correct me. If one can find a base where pi(C/D) is non repeating....then indeed it is the math we use.

You might very well be correct about a perfect circle in nature. The apparent ones we do see, always seem to have a perpendicular velocity component to the circumference. For instance, a round water ripple. It looks like a perfect circle moving out and expanding......but there is a down and up motion, as it moves out. Plus, the circle is not rotating. In 2D. Also, if the origin moves(up and down, or sideways), while in rotation or expansion, pi would be invalid. Pi only works in 2D.

And when we try to rotate disks at high speeds, they start to wobble, up and down. Maybe, mass itself, can't move in 2D, at high speeds. It might have to roll at high speed to conserve momentum.
Thanks for taking time to reply!
I fear I may have articulated myself poorly.
I was not suggesting that Pi is an artifice of base-10 mathematics, I was indicating the exact opposite. No matter what system of math you use (from binary code to base 3, base 4, base 5 ... ∞) or what units you use, any calculation of Pi made with the use of consistently applied units will yield an infinite decimal that never repeats. However, any calculated value for Pi can never be presumed accurate.
Take a circle with a circumference of 10 centimeters. Use this measurement to calculate Pi.
Now, measure the circle to the nearest femtometer. Will your calculation of Pi remain the same? If you simply convert your initial measurement to femtometers, the answer is yes. However, if you measure your circle to the nearest femtometer and find that its dimensions deviate in any way from your initial measurements, then your calculated value for Pi may be altered as well.
Since all units are infinitely divisible, you can rest assured that your calculated value of Pi can never be conclusively accurate to any arbitrarily contrived significant digits.
What is 1 million digits of Pi in a base 100 system?
500,000 digits of Pi.
PS: if a supercomputer could be taught base 1,000 math, would it be easier to calculate Pi, or determine newly discovered Prime numbers?
 
Dec 14, 2021
27
0
50
Is a line segment divisible without limit? Take a segment of finite length, call the first “point” on it zero and the last “point” one. The distance from the origin (zero) to any point on the segment is given by a (standard) real number of the form: zero-point (0.), followed by an infinite expansion (sequence) of digits (for instance, 0.2854618326580009276492651648206517848...). This is the mathematical version of Zeno’s axiom, inaccessible to the Greek who did not know the zero.

Accepting the “existence” of the real numbers is a strong hypothesis and, consequently, their study has become a key element in the search for a set of axioms or fundamental assumptions on which elementary number theory, and by extension, the whole of mathematics, might be firmly based. It is generally accepted that the set of 10 or so statements supporting most mathematical systems is the Zermelo-Fraenkel set theory. To these statements the nonstandard approach adds three additional axioms. They are based on the definition of a new nonstandard number: the infinitesimal. An infinitesimal nonstandard number is a new type of number: by definition, it is greater than zero but always less than any standard real number, however small.

These infinitesimals possess a thoroughly elusive character because they can never be captured through any possible measurement. The reason: measurements have always as result a standard real number. Furthermore, the difference between two standard real numbers can never be a nonstandard number, which is by definition always less than any standard number. Thus the interval between two nonstandard points on the line, or two nonstandard intervals of time, can never be measured, and so these intervals are forever beyond the range of observation. They exist only by axiomatic (Platonic) definition but can never become actual in Aristotle’s sense.

The nonstandard theory adds two more nonstandard numbers as axioms. The nonstandard unlimited number, which is the inverse of an infinitesimal number, is greater than any standard number but nevertheless smaller than infinity. The nonstandard unlimited numbers are thus very large, larger than any standard number, but always finite, that is, always less than the truly infinite numbers. The nonstandard mixed numbers are so to speak in between: around each standard number, on both sides of it on the line segment, a particular set of nonstandard numbers is, on the left, greater than any other standard number but less than this particular standard number; on the right side, it is smaller than any standard number greater than this particular number, but it is still greater than this number. In summary, between zero and infinity, a new infinity of nonstandard numbers has been added by axiom.

How does this nonstandard mathematics “solve” Zeno’s paradox? Achilles, as he gets closer and closer to the tortoise, will be traversing an infinite series of ever smaller space segments until eventually he will be at nonstandard infinitesimal distance from the tortoise. From this point on, his progress until he catches his prey escapes all possibility of measurement: all the final segments being nonstandard infinitesimal distances. In other words, what “really” happens when Achilles catches the tortoise can never be known, by definition, and so the case rests.

In the early twentieth century some philosophers have been particularly intrigued by the time continuum: if the instant is zero, when do we exist? The past is already gone, the future not yet here, and the present instant zero: when can we claim that we are? Martin Heidegger’s attempt to crack this problem in Being and Time, first published in 1927, may in a certain way be considered to be one more “solution” to Zeno ́s paradox. This philosopher suggests that humans are never authentically “being"; instead, from the very moment one is born, one is already dying, i.e., not-being. “The moment you are born you are old enough to die”. He furthermore claims that the only “time” that has a sense is the unknown period a human still has before he dies, that is, only the yet non-existent future is real. If one is asked: Will you die? The answer is: Yes, of course, but not yet.

Like Achilles: will he ever catch the tortoise? Yes, of course, but not yet. So, Heidegger states, there are two manner of being: the inauthentic and the authentic. In the former, which is where most of us choose to be, our allotted time-span is this not yet, this unknown future which allows us to escape from, to conceal, the unbearably displeasing fact: we are mortal. In this manner of being, we never actually die, we are always alive; death is “only” a potentiality. And we can say such strange things as: I have no time, don’t waste your time, etc. Whereas being authentically is equivalent, in a sense, to dying! That is, to fully accept human mortality. Put differently, inauthentic being is equivalent to live, to be, in the standard part of the time scale: there we can “measure” time with watches in standard numbers. Whereas authentic being is traversing our life-span in the nonstandard numeration, which is beyond measure, and where, in a sense, as soon as we are born we are already dying .

Recently a new, fascinating twist to this old conundrum appeared: Cris Calude’s Lexicon. Here is succinctly how it applies. Zeno’s paradox, and its possible “solutions”, must be somehow explicitly stated in a communicable language. This implies some linear sequence of symbols; the set of allowed symbols constituting the pertinent alphabet. Any finite sequence can be unambiguously coded in binary (or decimal) and thus corresponds exactly to some rational number. This paper, for example, corresponds to the rational number “w”. On the other hand, real numbers are infinite sequences of digits (in whatever chosen code or base). Question: Is there a real number that with certainty contains the word w (i.e.: exactly this paper)? Answer, Yes, and furthermore, there exists a real number that contains every possible “word”. That is, that contains everything that can be explicitly stated, coded, communicated. Here is how that number is constructed, in binary: simply add one after the other every possible binary sequence of 1,2,3,4,... bits:
0,1,00,01,10,11,000,001,011,111,110,100, 010,101, 0000,0001,...
all the way to infinity. By construction, absolutely everything that can be explicitly stated is represented, at least once, in this sequence.

Now it can be shown that this special real number not only contains, by construction, every possible finite linear sequence, say William Shakespeare’s complete works, but also that it contains every possible linear sequence infinitely many times! This is easily proved. Again, call some sequence, say this paper, w. Now construct these sequences:
w0
w00
w000 w0000 .................
all the way to infinity. Since by construction, each of them is already on our specially constructed binary real number, all of these “words” or binary sequences must also appear, at least once. But in each of them w appears, hence w appears infinitely many times. And this is the case for every possible w, QED.

It has been shown in 1998 by Calude and Zamfirescu that there exist real
numbers that present this remarkable property independent of the employed code or alphabet (binary, decimal, or, for instance, all the symbols on a computer keyboard). These are the Lexicons. Thus a Lexicon contains infinitely many times anything imaginable and not imaginable, everything ever written, or that will ever be written, any description of anything, of any phenomenon, real or imaginary, etc., etc.

So what is the relation of this surprising result with Zeno’s paradox? We humans are limited and mortal. We can only name, we can only put our fingers on, we can only actually exhibit, the rational (finite) numbers. But underlying all our mathematics, and Zeno’s paradox, are the real numbers. These, at least the immense majority of them, contain potentially everything, and infinitely many times. What we can actually exhibit is only a vanishing small subset of the underlying potentially possible number set. Now, we may call some “fact”corresponding to some rational numbera novelty. But this is just a Zeno-type illusion: everything is already contained (or expressed) infinitely many times in the infinite set of the reals almost all being Lexicons.

But what is a fact? Something that may have happened and that is amenable to a communicable description (measurement). For instance: the result of the sense- perception by which we observe Achilles catching the tortoise. Thus, something that corresponds to some finite sequence, to some rational number. But this “fact”, this sequence, this change, movement, kinêsis, is infinitely many times recorded in infinitely many real numbers ... if they exist, of course. Then it follows from the admittance of the existence of the real numbers, of Zeno’s infinite divisibility axiom of space and time, that there can never be any novelty. Everything is, always, and nothing ever becomes: Parmenides was right after all!

“The thing that has been, it is that which shall be; and that which is done is that which shall be done : and there is no new thing under the sun” ; so says the Ecclesiastes 1.9 (composed after 250 BC). Everything, the theory tells us, is right there under our nose, so to speak, potentially, but inaccessible. Whereas anything that actually happens, happened, or will happen, is an illusion, since it has been “there”, already, always. A final question remains, however: has anybody anytime laid his fingers on such a Lexicon? Surprisingly, yes, Greg Chaitin’s marvellous and mysterious real number View attachment 1703is a Lexicon. But that is another story.

See: https://www.cs.auckland.ac.nz/research/groups/CDMTCS/researchreports/089walter.pdf

After twenty-five centuries Zeno’s paradox* is still with us: if we admit the existence of the real numbers we run into trouble; we deny it, and we find a different set of equally intractable problems since now mathematics, and thus physics describing “change”, become problematic. We are left with admiration for those early Greek philosophers, who unveiled the fundamental limits of human reason (and of mathematics ).
Hartmann352

* Zeno's paradox see: https://iep.utm.edu/zeno-par/
Doesn't Zeno's paradox align precisely with the observation of "singularities" in physics?
Why do so many physicists seem to treat the existence of the Singularity as if it is irrational, when precisely the opposite is true?
When we describe Achilles catching the tortoise, we are using words as a symbolic representation of an experiential reality. It can hardly be considered a "fact" that "apple" is actually an apple.
So when I say "the apple fell on Newton's head" I am using symbolism to convey a widely accepted interpretation of past events. Even if I personally saw the apple fall on his head, any claim that I am stating an objective fact is baseless.
Commensurately, infinitesimals are the only real numbers.
Intervals only exist in the abstract, or as they relate to a symbolic representation of experience.
 
Dec 14, 2021
27
0
50
Is a line segment divisible without limit? Take a segment of finite length, call the first “point” on it zero and the last “point” one. The distance from the origin (zero) to any point on the segment is given by a (standard) real number of the form: zero-point (0.), followed by an infinite expansion (sequence) of digits (for instance, 0.2854618326580009276492651648206517848...). This is the mathematical version of Zeno’s axiom, inaccessible to the Greek who did not know the zero.

Accepting the “existence” of the real numbers is a strong hypothesis and, consequently, their study has become a key element in the search for a set of axioms or fundamental assumptions on which elementary number theory, and by extension, the whole of mathematics, might be firmly based. It is generally accepted that the set of 10 or so statements supporting most mathematical systems is the Zermelo-Fraenkel set theory. To these statements the nonstandard approach adds three additional axioms. They are based on the definition of a new nonstandard number: the infinitesimal. An infinitesimal nonstandard number is a new type of number: by definition, it is greater than zero but always less than any standard real number, however small.

These infinitesimals possess a thoroughly elusive character because they can never be captured through any possible measurement. The reason: measurements have always as result a standard real number. Furthermore, the difference between two standard real numbers can never be a nonstandard number, which is by definition always less than any standard number. Thus the interval between two nonstandard points on the line, or two nonstandard intervals of time, can never be measured, and so these intervals are forever beyond the range of observation. They exist only by axiomatic (Platonic) definition but can never become actual in Aristotle’s sense.

The nonstandard theory adds two more nonstandard numbers as axioms. The nonstandard unlimited number, which is the inverse of an infinitesimal number, is greater than any standard number but nevertheless smaller than infinity. The nonstandard unlimited numbers are thus very large, larger than any standard number, but always finite, that is, always less than the truly infinite numbers. The nonstandard mixed numbers are so to speak in between: around each standard number, on both sides of it on the line segment, a particular set of nonstandard numbers is, on the left, greater than any other standard number but less than this particular standard number; on the right side, it is smaller than any standard number greater than this particular number, but it is still greater than this number. In summary, between zero and infinity, a new infinity of nonstandard numbers has been added by axiom.

How does this nonstandard mathematics “solve” Zeno’s paradox? Achilles, as he gets closer and closer to the tortoise, will be traversing an infinite series of ever smaller space segments until eventually he will be at nonstandard infinitesimal distance from the tortoise. From this point on, his progress until he catches his prey escapes all possibility of measurement: all the final segments being nonstandard infinitesimal distances. In other words, what “really” happens when Achilles catches the tortoise can never be known, by definition, and so the case rests.

In the early twentieth century some philosophers have been particularly intrigued by the time continuum: if the instant is zero, when do we exist? The past is already gone, the future not yet here, and the present instant zero: when can we claim that we are? Martin Heidegger’s attempt to crack this problem in Being and Time, first published in 1927, may in a certain way be considered to be one more “solution” to Zeno ́s paradox. This philosopher suggests that humans are never authentically “being"; instead, from the very moment one is born, one is already dying, i.e., not-being. “The moment you are born you are old enough to die”. He furthermore claims that the only “time” that has a sense is the unknown period a human still has before he dies, that is, only the yet non-existent future is real. If one is asked: Will you die? The answer is: Yes, of course, but not yet.

Like Achilles: will he ever catch the tortoise? Yes, of course, but not yet. So, Heidegger states, there are two manner of being: the inauthentic and the authentic. In the former, which is where most of us choose to be, our allotted time-span is this not yet, this unknown future which allows us to escape from, to conceal, the unbearably displeasing fact: we are mortal. In this manner of being, we never actually die, we are always alive; death is “only” a potentiality. And we can say such strange things as: I have no time, don’t waste your time, etc. Whereas being authentically is equivalent, in a sense, to dying! That is, to fully accept human mortality. Put differently, inauthentic being is equivalent to live, to be, in the standard part of the time scale: there we can “measure” time with watches in standard numbers. Whereas authentic being is traversing our life-span in the nonstandard numeration, which is beyond measure, and where, in a sense, as soon as we are born we are already dying .

Recently a new, fascinating twist to this old conundrum appeared: Cris Calude’s Lexicon. Here is succinctly how it applies. Zeno’s paradox, and its possible “solutions”, must be somehow explicitly stated in a communicable language. This implies some linear sequence of symbols; the set of allowed symbols constituting the pertinent alphabet. Any finite sequence can be unambiguously coded in binary (or decimal) and thus corresponds exactly to some rational number. This paper, for example, corresponds to the rational number “w”. On the other hand, real numbers are infinite sequences of digits (in whatever chosen code or base). Question: Is there a real number that with certainty contains the word w (i.e.: exactly this paper)? Answer, Yes, and furthermore, there exists a real number that contains every possible “word”. That is, that contains everything that can be explicitly stated, coded, communicated. Here is how that number is constructed, in binary: simply add one after the other every possible binary sequence of 1,2,3,4,... bits:
0,1,00,01,10,11,000,001,011,111,110,100, 010,101, 0000,0001,...
all the way to infinity. By construction, absolutely everything that can be explicitly stated is represented, at least once, in this sequence.

Now it can be shown that this special real number not only contains, by construction, every possible finite linear sequence, say William Shakespeare’s complete works, but also that it contains every possible linear sequence infinitely many times! This is easily proved. Again, call some sequence, say this paper, w. Now construct these sequences:
w0
w00
w000 w0000 .................
all the way to infinity. Since by construction, each of them is already on our specially constructed binary real number, all of these “words” or binary sequences must also appear, at least once. But in each of them w appears, hence w appears infinitely many times. And this is the case for every possible w, QED.

It has been shown in 1998 by Calude and Zamfirescu that there exist real
numbers that present this remarkable property independent of the employed code or alphabet (binary, decimal, or, for instance, all the symbols on a computer keyboard). These are the Lexicons. Thus a Lexicon contains infinitely many times anything imaginable and not imaginable, everything ever written, or that will ever be written, any description of anything, of any phenomenon, real or imaginary, etc., etc.

So what is the relation of this surprising result with Zeno’s paradox? We humans are limited and mortal. We can only name, we can only put our fingers on, we can only actually exhibit, the rational (finite) numbers. But underlying all our mathematics, and Zeno’s paradox, are the real numbers. These, at least the immense majority of them, contain potentially everything, and infinitely many times. What we can actually exhibit is only a vanishing small subset of the underlying potentially possible number set. Now, we may call some “fact”corresponding to some rational numbera novelty. But this is just a Zeno-type illusion: everything is already contained (or expressed) infinitely many times in the infinite set of the reals almost all being Lexicons.

But what is a fact? Something that may have happened and that is amenable to a communicable description (measurement). For instance: the result of the sense- perception by which we observe Achilles catching the tortoise. Thus, something that corresponds to some finite sequence, to some rational number. But this “fact”, this sequence, this change, movement, kinêsis, is infinitely many times recorded in infinitely many real numbers ... if they exist, of course. Then it follows from the admittance of the existence of the real numbers, of Zeno’s infinite divisibility axiom of space and time, that there can never be any novelty. Everything is, always, and nothing ever becomes: Parmenides was right after all!

“The thing that has been, it is that which shall be; and that which is done is that which shall be done : and there is no new thing under the sun” ; so says the Ecclesiastes 1.9 (composed after 250 BC). Everything, the theory tells us, is right there under our nose, so to speak, potentially, but inaccessible. Whereas anything that actually happens, happened, or will happen, is an illusion, since it has been “there”, already, always. A final question remains, however: has anybody anytime laid his fingers on such a Lexicon? Surprisingly, yes, Greg Chaitin’s marvellous and mysterious real number View attachment 1703is a Lexicon. But that is another story.

See: https://www.cs.auckland.ac.nz/research/groups/CDMTCS/researchreports/089walter.pdf

After twenty-five centuries Zeno’s paradox* is still with us: if we admit the existence of the real numbers we run into trouble; we deny it, and we find a different set of equally intractable problems since now mathematics, and thus physics describing “change”, become problematic. We are left with admiration for those early Greek philosophers, who unveiled the fundamental limits of human reason (and of mathematics ).
Hartmann352

* Zeno's paradox see: https://iep.utm.edu/zeno-par/
I read a book called "The Case Against Reality" by Donald Hoffman that covered some of these topics. I found it fascinating.
So I guess it simply boils down to the fact that cosmological physicists don't want to accept the obvious conclusion resulting from their observations, right?
There is no beginning of the Universe, because any conceivable unit of time is infinitely divisible, which can be demonstrated with basic logic. The Singularity is a single all-encompassing force at the heart of all symbolic representations of reality. The symbolic representation can never reflect the objective reality, because the objective reality defies subjective representation. A particle is an infinitely divisible construct of spacetime just like a black hole or a baseball.
Dark matter and dark energy can be understood as a reflection of our attempts to describe symbolic representations of reality in terms of discrete units of spacetime as they relate to rational numbers.
 
Dec 14, 2021
27
0
50
Is a line segment divisible without limit? Take a segment of finite length, call the first “point” on it zero and the last “point” one. The distance from the origin (zero) to any point on the segment is given by a (standard) real number of the form: zero-point (0.), followed by an infinite expansion (sequence) of digits (for instance, 0.2854618326580009276492651648206517848...). This is the mathematical version of Zeno’s axiom, inaccessible to the Greek who did not know the zero.

Accepting the “existence” of the real numbers is a strong hypothesis and, consequently, their study has become a key element in the search for a set of axioms or fundamental assumptions on which elementary number theory, and by extension, the whole of mathematics, might be firmly based. It is generally accepted that the set of 10 or so statements supporting most mathematical systems is the Zermelo-Fraenkel set theory. To these statements the nonstandard approach adds three additional axioms. They are based on the definition of a new nonstandard number: the infinitesimal. An infinitesimal nonstandard number is a new type of number: by definition, it is greater than zero but always less than any standard real number, however small.

These infinitesimals possess a thoroughly elusive character because they can never be captured through any possible measurement. The reason: measurements have always as result a standard real number. Furthermore, the difference between two standard real numbers can never be a nonstandard number, which is by definition always less than any standard number. Thus the interval between two nonstandard points on the line, or two nonstandard intervals of time, can never be measured, and so these intervals are forever beyond the range of observation. They exist only by axiomatic (Platonic) definition but can never become actual in Aristotle’s sense.

The nonstandard theory adds two more nonstandard numbers as axioms. The nonstandard unlimited number, which is the inverse of an infinitesimal number, is greater than any standard number but nevertheless smaller than infinity. The nonstandard unlimited numbers are thus very large, larger than any standard number, but always finite, that is, always less than the truly infinite numbers. The nonstandard mixed numbers are so to speak in between: around each standard number, on both sides of it on the line segment, a particular set of nonstandard numbers is, on the left, greater than any other standard number but less than this particular standard number; on the right side, it is smaller than any standard number greater than this particular number, but it is still greater than this number. In summary, between zero and infinity, a new infinity of nonstandard numbers has been added by axiom.

How does this nonstandard mathematics “solve” Zeno’s paradox? Achilles, as he gets closer and closer to the tortoise, will be traversing an infinite series of ever smaller space segments until eventually he will be at nonstandard infinitesimal distance from the tortoise. From this point on, his progress until he catches his prey escapes all possibility of measurement: all the final segments being nonstandard infinitesimal distances. In other words, what “really” happens when Achilles catches the tortoise can never be known, by definition, and so the case rests.

In the early twentieth century some philosophers have been particularly intrigued by the time continuum: if the instant is zero, when do we exist? The past is already gone, the future not yet here, and the present instant zero: when can we claim that we are? Martin Heidegger’s attempt to crack this problem in Being and Time, first published in 1927, may in a certain way be considered to be one more “solution” to Zeno ́s paradox. This philosopher suggests that humans are never authentically “being"; instead, from the very moment one is born, one is already dying, i.e., not-being. “The moment you are born you are old enough to die”. He furthermore claims that the only “time” that has a sense is the unknown period a human still has before he dies, that is, only the yet non-existent future is real. If one is asked: Will you die? The answer is: Yes, of course, but not yet.

Like Achilles: will he ever catch the tortoise? Yes, of course, but not yet. So, Heidegger states, there are two manner of being: the inauthentic and the authentic. In the former, which is where most of us choose to be, our allotted time-span is this not yet, this unknown future which allows us to escape from, to conceal, the unbearably displeasing fact: we are mortal. In this manner of being, we never actually die, we are always alive; death is “only” a potentiality. And we can say such strange things as: I have no time, don’t waste your time, etc. Whereas being authentically is equivalent, in a sense, to dying! That is, to fully accept human mortality. Put differently, inauthentic being is equivalent to live, to be, in the standard part of the time scale: there we can “measure” time with watches in standard numbers. Whereas authentic being is traversing our life-span in the nonstandard numeration, which is beyond measure, and where, in a sense, as soon as we are born we are already dying .

Recently a new, fascinating twist to this old conundrum appeared: Cris Calude’s Lexicon. Here is succinctly how it applies. Zeno’s paradox, and its possible “solutions”, must be somehow explicitly stated in a communicable language. This implies some linear sequence of symbols; the set of allowed symbols constituting the pertinent alphabet. Any finite sequence can be unambiguously coded in binary (or decimal) and thus corresponds exactly to some rational number. This paper, for example, corresponds to the rational number “w”. On the other hand, real numbers are infinite sequences of digits (in whatever chosen code or base). Question: Is there a real number that with certainty contains the word w (i.e.: exactly this paper)? Answer, Yes, and furthermore, there exists a real number that contains every possible “word”. That is, that contains everything that can be explicitly stated, coded, communicated. Here is how that number is constructed, in binary: simply add one after the other every possible binary sequence of 1,2,3,4,... bits:
0,1,00,01,10,11,000,001,011,111,110,100, 010,101, 0000,0001,...
all the way to infinity. By construction, absolutely everything that can be explicitly stated is represented, at least once, in this sequence.

Now it can be shown that this special real number not only contains, by construction, every possible finite linear sequence, say William Shakespeare’s complete works, but also that it contains every possible linear sequence infinitely many times! This is easily proved. Again, call some sequence, say this paper, w. Now construct these sequences:
w0
w00
w000 w0000 .................
all the way to infinity. Since by construction, each of them is already on our specially constructed binary real number, all of these “words” or binary sequences must also appear, at least once. But in each of them w appears, hence w appears infinitely many times. And this is the case for every possible w, QED.

It has been shown in 1998 by Calude and Zamfirescu that there exist real
numbers that present this remarkable property independent of the employed code or alphabet (binary, decimal, or, for instance, all the symbols on a computer keyboard). These are the Lexicons. Thus a Lexicon contains infinitely many times anything imaginable and not imaginable, everything ever written, or that will ever be written, any description of anything, of any phenomenon, real or imaginary, etc., etc.

So what is the relation of this surprising result with Zeno’s paradox? We humans are limited and mortal. We can only name, we can only put our fingers on, we can only actually exhibit, the rational (finite) numbers. But underlying all our mathematics, and Zeno’s paradox, are the real numbers. These, at least the immense majority of them, contain potentially everything, and infinitely many times. What we can actually exhibit is only a vanishing small subset of the underlying potentially possible number set. Now, we may call some “fact”corresponding to some rational numbera novelty. But this is just a Zeno-type illusion: everything is already contained (or expressed) infinitely many times in the infinite set of the reals almost all being Lexicons.

But what is a fact? Something that may have happened and that is amenable to a communicable description (measurement). For instance: the result of the sense- perception by which we observe Achilles catching the tortoise. Thus, something that corresponds to some finite sequence, to some rational number. But this “fact”, this sequence, this change, movement, kinêsis, is infinitely many times recorded in infinitely many real numbers ... if they exist, of course. Then it follows from the admittance of the existence of the real numbers, of Zeno’s infinite divisibility axiom of space and time, that there can never be any novelty. Everything is, always, and nothing ever becomes: Parmenides was right after all!

“The thing that has been, it is that which shall be; and that which is done is that which shall be done : and there is no new thing under the sun” ; so says the Ecclesiastes 1.9 (composed after 250 BC). Everything, the theory tells us, is right there under our nose, so to speak, potentially, but inaccessible. Whereas anything that actually happens, happened, or will happen, is an illusion, since it has been “there”, already, always. A final question remains, however: has anybody anytime laid his fingers on such a Lexicon? Surprisingly, yes, Greg Chaitin’s marvellous and mysterious real number View attachment 1703is a Lexicon. But that is another story.

See: https://www.cs.auckland.ac.nz/research/groups/CDMTCS/researchreports/089walter.pdf

After twenty-five centuries Zeno’s paradox* is still with us: if we admit the existence of the real numbers we run into trouble; we deny it, and we find a different set of equally intractable problems since now mathematics, and thus physics describing “change”, become problematic. We are left with admiration for those early Greek philosophers, who unveiled the fundamental limits of human reason (and of mathematics ).
Hartmann352

* Zeno's paradox see: https://iep.utm.edu/zeno-par/
Thanks for taking time to respond. This is the kind of thing that keeps me up at night, and clearly you know your stuff 😀
 
Dec 14, 2021
27
0
50
Is a line segment divisible without limit? Take a segment of finite length, call the first “point” on it zero and the last “point” one. The distance from the origin (zero) to any point on the segment is given by a (standard) real number of the form: zero-point (0.), followed by an infinite expansion (sequence) of digits (for instance, 0.2854618326580009276492651648206517848...). This is the mathematical version of Zeno’s axiom, inaccessible to the Greek who did not know the zero.

Accepting the “existence” of the real numbers is a strong hypothesis and, consequently, their study has become a key element in the search for a set of axioms or fundamental assumptions on which elementary number theory, and by extension, the whole of mathematics, might be firmly based. It is generally accepted that the set of 10 or so statements supporting most mathematical systems is the Zermelo-Fraenkel set theory. To these statements the nonstandard approach adds three additional axioms. They are based on the definition of a new nonstandard number: the infinitesimal. An infinitesimal nonstandard number is a new type of number: by definition, it is greater than zero but always less than any standard real number, however small.

These infinitesimals possess a thoroughly elusive character because they can never be captured through any possible measurement. The reason: measurements have always as result a standard real number. Furthermore, the difference between two standard real numbers can never be a nonstandard number, which is by definition always less than any standard number. Thus the interval between two nonstandard points on the line, or two nonstandard intervals of time, can never be measured, and so these intervals are forever beyond the range of observation. They exist only by axiomatic (Platonic) definition but can never become actual in Aristotle’s sense.

The nonstandard theory adds two more nonstandard numbers as axioms. The nonstandard unlimited number, which is the inverse of an infinitesimal number, is greater than any standard number but nevertheless smaller than infinity. The nonstandard unlimited numbers are thus very large, larger than any standard number, but always finite, that is, always less than the truly infinite numbers. The nonstandard mixed numbers are so to speak in between: around each standard number, on both sides of it on the line segment, a particular set of nonstandard numbers is, on the left, greater than any other standard number but less than this particular standard number; on the right side, it is smaller than any standard number greater than this particular number, but it is still greater than this number. In summary, between zero and infinity, a new infinity of nonstandard numbers has been added by axiom.

How does this nonstandard mathematics “solve” Zeno’s paradox? Achilles, as he gets closer and closer to the tortoise, will be traversing an infinite series of ever smaller space segments until eventually he will be at nonstandard infinitesimal distance from the tortoise. From this point on, his progress until he catches his prey escapes all possibility of measurement: all the final segments being nonstandard infinitesimal distances. In other words, what “really” happens when Achilles catches the tortoise can never be known, by definition, and so the case rests.

In the early twentieth century some philosophers have been particularly intrigued by the time continuum: if the instant is zero, when do we exist? The past is already gone, the future not yet here, and the present instant zero: when can we claim that we are? Martin Heidegger’s attempt to crack this problem in Being and Time, first published in 1927, may in a certain way be considered to be one more “solution” to Zeno ́s paradox. This philosopher suggests that humans are never authentically “being"; instead, from the very moment one is born, one is already dying, i.e., not-being. “The moment you are born you are old enough to die”. He furthermore claims that the only “time” that has a sense is the unknown period a human still has before he dies, that is, only the yet non-existent future is real. If one is asked: Will you die? The answer is: Yes, of course, but not yet.

Like Achilles: will he ever catch the tortoise? Yes, of course, but not yet. So, Heidegger states, there are two manner of being: the inauthentic and the authentic. In the former, which is where most of us choose to be, our allotted time-span is this not yet, this unknown future which allows us to escape from, to conceal, the unbearably displeasing fact: we are mortal. In this manner of being, we never actually die, we are always alive; death is “only” a potentiality. And we can say such strange things as: I have no time, don’t waste your time, etc. Whereas being authentically is equivalent, in a sense, to dying! That is, to fully accept human mortality. Put differently, inauthentic being is equivalent to live, to be, in the standard part of the time scale: there we can “measure” time with watches in standard numbers. Whereas authentic being is traversing our life-span in the nonstandard numeration, which is beyond measure, and where, in a sense, as soon as we are born we are already dying .

Recently a new, fascinating twist to this old conundrum appeared: Cris Calude’s Lexicon. Here is succinctly how it applies. Zeno’s paradox, and its possible “solutions”, must be somehow explicitly stated in a communicable language. This implies some linear sequence of symbols; the set of allowed symbols constituting the pertinent alphabet. Any finite sequence can be unambiguously coded in binary (or decimal) and thus corresponds exactly to some rational number. This paper, for example, corresponds to the rational number “w”. On the other hand, real numbers are infinite sequences of digits (in whatever chosen code or base). Question: Is there a real number that with certainty contains the word w (i.e.: exactly this paper)? Answer, Yes, and furthermore, there exists a real number that contains every possible “word”. That is, that contains everything that can be explicitly stated, coded, communicated. Here is how that number is constructed, in binary: simply add one after the other every possible binary sequence of 1,2,3,4,... bits:
0,1,00,01,10,11,000,001,011,111,110,100, 010,101, 0000,0001,...
all the way to infinity. By construction, absolutely everything that can be explicitly stated is represented, at least once, in this sequence.

Now it can be shown that this special real number not only contains, by construction, every possible finite linear sequence, say William Shakespeare’s complete works, but also that it contains every possible linear sequence infinitely many times! This is easily proved. Again, call some sequence, say this paper, w. Now construct these sequences:
w0
w00
w000 w0000 .................
all the way to infinity. Since by construction, each of them is already on our specially constructed binary real number, all of these “words” or binary sequences must also appear, at least once. But in each of them w appears, hence w appears infinitely many times. And this is the case for every possible w, QED.

It has been shown in 1998 by Calude and Zamfirescu that there exist real
numbers that present this remarkable property independent of the employed code or alphabet (binary, decimal, or, for instance, all the symbols on a computer keyboard). These are the Lexicons. Thus a Lexicon contains infinitely many times anything imaginable and not imaginable, everything ever written, or that will ever be written, any description of anything, of any phenomenon, real or imaginary, etc., etc.

So what is the relation of this surprising result with Zeno’s paradox? We humans are limited and mortal. We can only name, we can only put our fingers on, we can only actually exhibit, the rational (finite) numbers. But underlying all our mathematics, and Zeno’s paradox, are the real numbers. These, at least the immense majority of them, contain potentially everything, and infinitely many times. What we can actually exhibit is only a vanishing small subset of the underlying potentially possible number set. Now, we may call some “fact”corresponding to some rational numbera novelty. But this is just a Zeno-type illusion: everything is already contained (or expressed) infinitely many times in the infinite set of the reals almost all being Lexicons.

But what is a fact? Something that may have happened and that is amenable to a communicable description (measurement). For instance: the result of the sense- perception by which we observe Achilles catching the tortoise. Thus, something that corresponds to some finite sequence, to some rational number. But this “fact”, this sequence, this change, movement, kinêsis, is infinitely many times recorded in infinitely many real numbers ... if they exist, of course. Then it follows from the admittance of the existence of the real numbers, of Zeno’s infinite divisibility axiom of space and time, that there can never be any novelty. Everything is, always, and nothing ever becomes: Parmenides was right after all!

“The thing that has been, it is that which shall be; and that which is done is that which shall be done : and there is no new thing under the sun” ; so says the Ecclesiastes 1.9 (composed after 250 BC). Everything, the theory tells us, is right there under our nose, so to speak, potentially, but inaccessible. Whereas anything that actually happens, happened, or will happen, is an illusion, since it has been “there”, already, always. A final question remains, however: has anybody anytime laid his fingers on such a Lexicon? Surprisingly, yes, Greg Chaitin’s marvellous and mysterious real number View attachment 1703is a Lexicon. But that is another story.

See: https://www.cs.auckland.ac.nz/research/groups/CDMTCS/researchreports/089walter.pdf

After twenty-five centuries Zeno’s paradox* is still with us: if we admit the existence of the real numbers we run into trouble; we deny it, and we find a different set of equally intractable problems since now mathematics, and thus physics describing “change”, become problematic. We are left with admiration for those early Greek philosophers, who unveiled the fundamental limits of human reason (and of mathematics ).
Hartmann352

* Zeno's paradox see: https://iep.utm.edu/zeno-par/
Imagine a square of 2 meters by 2 meters.
Draw a diagonal at 90°. Then, a line from 89° connecting at 91° at the opposite side, 88° to 92°, and so on, all the way around the square. If you divide the perimeter of the square by the average length of these lines, wouldn't you end up with a number close to Pi? The more precise you make your measurements, and the larger your dataset, the closer you will get.
The same is true for a rectangle, or an octagon, or a dodecahedron. Pi isn't just a peculiarity of circles, it is endemic to all geometry. Further evidence of the Singularity, if I might be so bold.
 
Mar 4, 2020
498
70
1,780
Discussing mathematical concepts is a nice way to spend a rainy afternoon. But these concepts do not apply to physical science. Converting from a waveform collapse.....to a physical result.......is religious dogma to me. That's not true science.

pi is a VERY STRICT conditional number. It would be extremely hard to find pi in nature. This is one reason why the motion of particles, the motion of plasma, and the motion of galaxies still puzzle us. It is also the reason science believes that orbits are elliptical. And that the forces of gravity produce such an orbit. And our gravity equations show such a dynamic.

And there is no such thing as a physical entity that can be divided forever. And other little things like mass.....that is not super-positional.

No matter how hard I try, I can not discern, why educated men, would deploy such a strategy. Mathematical concepts do not apply to physical concepts. It's so obvious.

Nature has been laughing so hard and so long at us........our only hope is for nature to expose herself.

Trying to explain physicality with our present mathematics concepts has kept us ignorant.

And the only singularity man has ever detected is ......life.
 
  • Like
Reactions: Onemetertwometer
Dec 14, 2021
27
0
50
Discussing mathematical concepts is a nice way to spend a rainy afternoon. But these concepts do not apply to physical science. Converting from a waveform collapse.....to a physical result.......is religious dogma to me. That's not true science.

pi is a VERY STRICT conditional number. It would be extremely hard to find pi in nature. This is one reason why the motion of particles, the motion of plasma, and the motion of galaxies still puzzle us. It is also the reason science believes that orbits are elliptical. And that the forces of gravity produce such an orbit. And our gravity equations show such a dynamic.

And there is no such thing as a physical entity that can be divided forever. And other little things like mass.....that is not super-positional.

No matter how hard I try, I can not discern, why educated men, would deploy such a strategy. Mathematical concepts do not apply to physical concepts. It's so obvious.

Nature has been laughing so hard and so long at us........our only hope is for nature to expose herself.

Trying to explain physicality with our present mathematics concepts has kept us ignorant.

And the only singularity man has ever detected is ......life.
"Life" is just a symbolic representation of experiential reality. There is only one Singularity, and it can be detected in every observable aspect of reality.
Pi can be found anywhere in nature, assuming you are looking for it.
Draw a hexagon in the sand. Whether it appears to be regular or irregular, in reality it is neither, just as in reality it can not be a hexagon. Nonetheless, in relative terms, if you measure its perimeter, take an average of all lines drawn at opposing angles, and divide the perimeter by this average to the best of your abilities in terms of precision, you will end up with a number resembling Pi.
You can never end up with a number that objectively represents Pi, because Pi can only be calculated in relative terms with units of arbitrary complexity that are ultimately infinitely divisible.
Right?
 

ASK THE COMMUNITY

TRENDING THREADS