Coolest comment award in #philosophical today goes to Niniane, who remarked on her interest in the peculiar properties of that most famous of all transcendental numbers: Pi. Technically, pi is the name of one specific point — and only one — in the real-number continuum. Pi is not a rational number, and so it has no exact representation in the decimal number system — nor in any other number system. The fact that it can’t be fully written out is what makes it ‘transcendental.’
I’ve heard it said of pi that, since its decimal expansion consists of an infinite number of digits, and those digits are random [odd concept of random, since the digits of pi never vary, but are always the same] you can find any conceivable sequence of digits buried somewhere in the expansion of pi. If you allow substrings of digits to be assembled into a raster pattern, you can even find pictures in it, and somewhere in Pi is a TV picture of Hitler making his opening speech to the 1934 Olympics.
I got into an argument with somebody once, whether Pi has an aleph-null or an aleph-one cardinality of digits. We know it has an infinite number of digits, but the question is, is it the same order of infinity as the integer number set, or of the same order as the reals? I argued for the former, but I wasn’t able to convince my friend. What do you think?
Does the value of Pi have anything to do with the actual universe? Was its value set at the time of the Big Bang? Or is it merely an abstract mental invention of some kind? I don’t know.
The Ivory Tower 
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The amount of digits in pi has the cardinality of the integers, this can be seen easily by constructing a bijection (surjective and injective function) between the natual numbers and the digits in pi. i.e. let 1 got to 1st digit, let 2 go to 2cnd digit, let 3 got to 3rd digit etc. Speaking as a human if you can start counting something in an orderly and constructive fashion then it’s cardinality is either a natural number or equal to the cardinality of the integers, rational, etc.
Absolutely right. One of (I can’t say the only) imporant differences between countable sets (integer-denumerable sets) and uncountable (real, aleph-1, cantorian) sets is that the former has a Next() function. That is, given any item in the set, there is one and only one set member that comes next. There is no such member for the reals, because there are an infinite (aleph-1) number of members between any two.
It has the cardinality of the integers since there is always the next digit. But you can never say which the next real is.
Its also possible that all of existence is one simple law whose natural playing out of creates the “island” of existence.
I also think that black holes are the way for the universe so recycle its matter and to “reset itself”.
Try doing some dimensional valuism.
I think pi is related to pythagorean theorem, especially the first linearly sequential integer representation of PT – 3-4-5.
Pis first few digits make reference to this,
31415 – 345
92=11
65=11
358=16
979=25
I actually have made a list of patterns in pi and I think the number of patterns and their overlap increases exponentially as pi continues.
There are more patterns.
I believe there is a more fundamental form of mathematics that is like hex/binary but more complex yet simpler.
Its also interesting the 6th digit, 9, is also the 1st digit to be repeated 6x in a row (Feynman point), whose probability of occuring is sooooo small that it occuring when it does is an antiprobability.
Also, 9^6 is 531441, a composition of the first 5 digits in pi, in addition to having an approximation of pi in it.
What is more interesting to me is why pi is equal to the reliationship between a circle’s area and its radius squared. A=pi * r squared.
Why is an irrational numbers this constant, and also why is the radius squared. What is it about the 2nd root. Why not cubed or to the 4th power? There is something also very interesting about the relationship of squared, or as I think of it compounding. Acceleration has this. 32 feet per second suared. So does finance, with compounding interest.
I don’t think it is a coincidence that the constant pi expresses the relationship of one number (area) to the square of another number (radius). I believe there is 1 math formula that will solve all math problems, including engineering, economic, and financial. Help me figure it out.
You can reach me at http://www.commercialmortgagebroker.org or at http://www.financialcompound.com
Michael
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Once upon a time in the actual universe, an imaginary object was invented using a geometric/mathematical definition:
circle: a plane curve everywhere equidistant from a given fixed point.
”
What is a plane, circle, or sphere but a mere POINT if one does not change one’s perspective or proximity ? The POINT has the potential to occupy the entirety of one’s field of vision as well.
All of geometry/mathematics is an estimate of imaginary points and there relationships… which sadly are imaginary too.
I love this stuff.
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Pingback: Mathematical Inventions « The Ivory Tower
Does the value of Pi have anything to do with the actual universe? Was its value set at the time of the Big Bang? Or is it merely an abstract mental invention of some kind? I don’t know.
Once upon a time in the actual universe, an imaginary object was invented using a geometric/mathematical definition:
circle: a plane curve everywhere equidistant from a given fixed point.
“plane” was also an invention, (“a surface containing all the straight lines connecting any two points on it”)(most “surfaces” are “surfaces of an object”. We invent the planar “surface” by metaphor, apparently.)
“straight” was also an invention (“extending continously in the same direction without curving”)(obviously, nothing material is being extended, just an idea)
“point” was also an invention (my dictionary says, somewhat idiotically, “a dimensionless geometric object having no property but location”)(how one can locate something or call it an object when it has no dimension, other than by imaginary invention, is my point)
Having defined the circle by passing through so many levels of pure invention, and then not being sure whether the ratio between a circle’s diameter and its circumference qualifies as an invention, is rather curious to me. That we would be careful so that our constructions were internally consistent so as to be more useful than if they werent consistent, does make sense however. You can’t have your pi before you bake it.
If it is a mental abstraction, how is it not an invention?
Is it a discovery? If so, does that not give the suggestion that they are independent of our mental abstraction? If the are independent do the not negate the abstract mental?
Interesting point about numbers lying between things. Is there a suggestion of a third “reality” (for desperate want of a better word) here? Something that lies between the physical and the mental? The connecting point between the two, perhaps. Something that allows the relationship between physical and mental.
Numbers are relational. As I discuss in “More Dots,” numbers are like the line between two points A and B. The line isn’t really there, although there is an important relationship between A and B.
Similarly, for two marbles, the being as two marbles is not in either one, and that’s what frustrates some thinkers. The being as two lies between the marbles. Like distance.
So.. I would say you’re half right. It’s an abstract mental. But it’s not an invention.
Could it not be said that all numbers and, infact, mathematics are an abstract mental invention of some kind?
When we count marbles, is there some universal constant that imbues the marbles with their designated number? Does the physical obey the laws of physics or do the laws of physics obey the physical? Is the distance between here and there so many miles or is it merely a way to describe a concept of “between here and there”?
I think numbers are not really an “invention” of the mind, but rather, an expression. Like the distance between two points could be referenced by a simple observation of those points but to communicate to someone else what you want them to know about said points, requires an expression to the other observer. Communication was “invented” to convey that expression, yet the distance was observable before the communication was invented to describe it. likewise, numbers are an “invention” (or rationalization) of humans to convey teir expression of the linear observance of two or more points. Yet,the distance was there all along and is constant, unchanging and infinite.We think we create new math and discover new anomolous (mathematical figures) thus being called a creation of the founder of the equation, but really it is just another word in the dictionary of infinitesimal and immeasureable constants that GOD designed from the beginning and we name as we discover. I say infinitesimal and immeasureable because in the advent of nano-technology and the desire to populate and traverse the uncharted desert of space, it seems that the constant numerical equations persist for infinity.As for going “inward” (or nano), it seems man is trying to search smaller and smaller for what reason? Not Ipods. Maybe he is trying to unconciously trying to find GOD or heaven.Luke 17:21 “Neither shall they say, Lo here! or, lo there! for, behold, the kingdom of God is within you”.Not a lot of room in the human body for a kingdom. Yet the depths somehow swallow the depths and keep revealing the light.So I say this was all here and had no name…we just gave it names as we do newborn children that come to the world as a constant, yet have no name until we seal it as fact.