common measure

[AntonioLao]

Pythagoras (548-495 BC) discovered irrational numbers. This discovery became the secret of a religious society formed by his followers known as the Order of the Pythagoreans. The exposure of this mathematical secret punishable by death is now known as the Pythagorean Crisis. From number theory of sets it is well known that the set of irrational numbers is infinitely larger than the set of whole numbers which is also known to be infinitely large. This crisis emerges in relation to the problem of comparing a line segment expressible in whole numbers with that of another.

If one compares a short segment R with a longer segment S, it is possible R can fit exactly a whole number of times into S. This possibility implies that R is a measure of S and S is a multiple of R. Moreover, if R does not fit exactly it is still possible to find a smaller segment T that fits a whole number of times into both R and S. In this case, T is known as a common measure of both R and S and the ratio S/R is a rational number called an improper fraction or a mixed number as the sum of a whole number and a proper fraction where the numerator is always lesser than the denominator. However, a simple proof can be shown that the side R and diagonal S of a square do not have a common measure. By the Pythogorean Theorem S²=2R². If a common measure exists then S=M/N and M and N are both whole numbers. Furthermore, M/N is given in lowest terms which imply that M and N cannot be both even numbers but if M is even then M² is also even. Consequently, there exists a whole number K such that M=2K then M²=4K²=2N² where S=MT and R=NT. That is N and N² are also both even thus showing a proof by contradiction known as reductio ad absurdum.

[SteveA]

...From number theory of sets it is well known that the set of irrational numbers is infinitely larger than the set of whole numbers which is also known to be infinitely large. This crisis emerges in relation to the problem of comparing a line segment expressible in whole numbers with that of another.

If one compares a short segment R with a longer segment S, it is possible R can fit exactly a whole number of times into S. This possibility implies that R is a measure of S and S is a multiple of R. Moreover, if R does not fit exactly it is still possible to find a smaller segment T that fits a whole number of times into both R and S.
...

A way to potentially unify all these is to look at the information and forms it takes to construct various quantities.

If there is said to exist 1/100th of something, that means there must be 100 of something else, so you have at least two quantities, a 1 and 100. If 100 was determined via 10*10, then you similarly have a geometric relationship of an area (or specifically a square) for which 2 independent properties must exist in order to make measurements of those quantities and a 3rd property would have to be present to isolate the single unit (as well as other properties to be able to distinguish between 10 different numbers in the other 2 dimensions) etc.

So when someone writes 0.01, or 1/100, there already exists implicit structures, properties, dimensions, geometric forms etc.

If we're making measurements of very "small" quantities of energy or very "precise" positions in space etc., this implies a reciprocally larger quantity of information exists as a measurement platform or assumed set of physical laws/properties already exists.

Look at how this applies to measurements in physics. Subatomic particles cannot be directly detected by physical perceptions but instead are derived indirectly by an accumulation of information on very large scales. The "smaller" a particle is, the more precise the measurements need to be in various respects and this requires a reciprocally larger quantity of information to determine.

Notice how much matter, energy, time and information it takes to create some of the very "smallest particles"





How's that for creating phenomenon on larger and larger scales of number theory? Are "subatomic" particles truly small?

Actually, the Uncertainty Principle says they aren't specifically small. They need to be at least as large as a quantum unit of perception in some respect, otherwise we shouldn't know they exist at all!

So imagine particles and motions in space being similar to various quantities and geometric operations - given any finite set of these, only a finite set of quantities can be constructed ... if there's a fundamental way the universe can "count", then we'll always find new "particles" on larger and larger scales of observation (with any finite limits of perception or comprehension) because there are always forms on larger scales with a greater density of information that can't be described by collections of previously existing particles.

[AntonioLao]

Unfortunately, the common measure of quantum mechanics is the irrational Planck's constant of action which can be rationalized by dividing it by 2pi to give the quantum of spin.

[Bogie]

Pythagoras (548-495 BC) discovered irrational numbers. This discovery became the secret of a religious society formed by his followers known as the Order of the Pythagoreans. The exposure of this mathematical secret punishable by death is now known as the Pythagorean Crisis. From number theory of sets it is well known that the set of irrational numbers is infinitely larger than the set of whole numbers which is also known to be infinitely large. This crisis emerges in relation to the problem of comparing a line segment expressible in whole numbers with that of another.

If one compares a short segment R with a longer segment S, it is possible R can fit exactly a whole number of times into S. This possibility implies that R is a measure of S and S is a multiple of R. Moreover, if R does not fit exactly it is still possible to find a smaller segment T that fits a whole number of times into both R and S. In this case, T is known as a common measure of both R and S and the ratio S/R is a rational number called an improper fraction or a mixed number as the sum of a whole number and a proper fraction where the numerator is always lesser than the denominator. However, a simple proof can be shown that the side R and diagonal S of a square do not have a common measure. By the Pythogorean Theorem S²=2R². If a common measure exists then S=M/N and M and N are both whole numbers. Furthermore, M/N is given in lowest terms which imply that M and N cannot be both even numbers but if M is even then M² is also even. Consequently, there exists a whole number K such that M=2K then M²=4K²=2N² where S=MT and R=NT. That is N and N² are also both even thus showing a proof by contradiction known as reductio ad absurdum.

Will you get into any trouble with the Pythagoreans buy leaking this information? .

Acutally that all makes perfect math sense but what I latch on to in it is the infinitely large sets of whole numbers and irrational numbers. I love to see people acknowledge infinity; I wish everyone was as comfortable with physical infinities as you and I are with mathetical infinities.

[SteveA]

Will you get into any trouble with the Pythagoreans but leaking this information? .

Acutally that all makes perfect math sense but what I latch on to in it is the infinitely large sets of whole numbers and irrational numbers. I love to see people acknowledge infinity; I wish everyone was as comfortable with physical infinities as you and I are with mathetical infinities.

A comment I'd like to add is that if two infinite things don't share a common reference, then it wouldn't seem they could exist in the same space.

Notice that we can compute something like:

1=lim(x/x) as x->infinity

Because 1/1=1, 2/2=1 3/3=1 etc.

But if we have two disconnected and unrelated infinite quantities, there aren't necessarily comparable units that can exist in the same space:

?=lim(x/y) as x and y ->infinity

On the other hand, if we have a single value, such as time, for which x and y are derived/functions of it, then these two "infinities" could remain comparable and related in a single space.

[AntonioLao]

Will you get into any trouble with the Pythagoreans

You can get into trouble if and only if you go back in time to meet them.