Phi

[webolife]

I have no real disagreement with you here... it's largely a matter of perspective.

[klypp]

It's not nit-picking. Many people use irrational numbers as evidence that "science has proven God" or other such claims. Many lay-people don't know how to interpret an irrational number and will believe a mathematician if s/he states that it has direct physical significance. We must always keep concepts and abstractions in their proper context/place. As conveniences. It's not nit-picking when books have been written about "pi philosophy" and other such crap. This is a widespread misconception in almost every facet of society. Kids are taught this stuff as early as third grade and are practically never taught about the physical significance.

Look at this image:


If there is a "physical significance" to the number 1, there is also a "physical significance" to the square root of 2.

What you are teaching, altonhare, is no less crap than the crap you're trying to debunk.

[altonhare]

Klypp you have completely missed the message. The triangle you have shown is an idealized geometric figure. You have to keep in mind what numbers actually refer to, what you're actually doing when you use numbers/measurements. If you measure the hypotenuse of that triangle you will inevitably end up with a rational number. The length/width/height of any shape is rational whether you measure it or not. If not, when does the shape "decide" it has reached the "infinitieth" decimal place and stop expanding? Irrational numbers are abstractions used to describe *idealized* shapes. How can the length from here to there be pi or 2.5? Think about it! Just because we can write an equation or get a result doesn't mean it is physically significant.

[altonhare]

The physical interpretation of "1" in the image you showed is that we can lay down 1 standard-object along the triangle's width and height. The sqrt(2) is the result of an equation you have applied to this figure to determine the distance along what we call the "hypotenuse". Instead of actually measuring this distance you have assumed it follows some standardized mathematical relation. The reason we do this is because it standardizes idealized shapes, i.e. it's the same reason we have pi. Instead of every physicist in the world measuring their own circular objects we have a "standard circle" that obeys specific mathematical rules. This makes calculations and sharing work easier. We have a standard "right triangle" also that obeys specific mathematical rules. Nature doesn't care about our mathematical rules. The distance from A to B is what it is, She scoffs at our abstractions.

In fact, if you measured the hypotenuse you would always find that it's rational. Even down to the scale which humans cannot see, the distance is rational. sqrt(2) is an abstract mathematical concept with no physical significance.

[webolife]

...because nature has a finite smallest "unit" of distance, whether it is Planck's distance or something else,
therefore the number of units is countable... is that what you're getting at?
What I disagree [with Alton] about is the insistence that irrational relationships are of no physical significance. Since when does the relationship between two numbers have to fit our mathematical constructs in order to be valid? I said before, and stick by, my disallowance of complex/imaginary numbers, by definition, but irrational number relationships occur everywhere in the universe, and calculations with them are useful for predicting and generalizing. Of course, numbers are not physical objects, per se, rather "adjectives" good for describing observed patterns. How is this not physically significant?

[altonhare]

What I disagree [with Alton] about is the insistence that irrational relationships are of no physical significance.

Let's keep the discussion/debate simple, I have gotten off onto horrible tangents with people in the past. The question really is simple:

"Can the distance from A to B be irrational?"

While distance is not the only relevant measurement, lets just concentrate on one little chunk at a time. Ideally I'd like to hear a definite "yes" or "no", if no how do you support such a position?

If I am measuring the distance from A to be with my brick I may have to break my brick in half to get it right. I may have to do it again etc. An irrational number implies that I can never break my brick in half enough times to measure the distance from A to B. Ever. I will be breaking my brick in half incessantly but never getting the final answer. I concede that I may need to break my brick in half many gazillions of times, but surely I will eventually get to an answer. The distance from A to B is what it is, it's not forever expanding as I add more decimal points to my calculation is it? Surely that's a ludicrous proposition.

but irrational number relationships occur everywhere in the universe, and calculations with them are useful for predicting and generalizing.

Name a single physical relationship that IS irrational, not one that is accurately described as irrational. You cannot by definition because nobody has measured something indefinitely.

Of course, numbers are not physical objects, per se, rather "adjectives" good for describing observed patterns. How is this not physically significant?

Obviously numbers are physically significant when we use them in reference to objects. The question of physical significance is not whether we can use something to correlate data, but whether it is physical. Just because we can correlate observations with equations does not magically imbue the equations with physical significance! We do not need an experiment to resolve these issues. Is the distance from A to B a definite value or is it perpetually expanding?

[webolife]

I'll bite. If you keep breaking your brick in half, or any fraction of pieces, you will always get a rational result, obviously.
But I look right there on my slide rule and see the number pi sitting there along with the other numbers on the scale.
I have no problem visualizing the exact length root-2, just as Klypp offered. Planck's distance is so small as to far exceed any abilty to measure or detect, yet it is an alleged finite unit of theoretically countable [rational] distance. Yet you would say that even it can be cut in half, to the utter dismay of Planck So if this is all you mean by irrational #'s having no physical significance, what's the point? Don't just repeat yourself here, it's not doing you or me or anyone any good. Is Avogadro's number actually counting something [have you counted a mole lately?], or is it a representational concept, based on certain assumptions of finitude? So, there's a rational number that likely has no real physical significance. I have read a pretty good argument that c = 1. There's a nice rational number for you, but how does it help to describe what you actually observe when light "happens"? At least pi, phi, root-3 and root-2 are useful for describing observed patterns in nature, and how can that be insignificant? OK, so you conceded that these numbers may be physically significant, but just not "physical"?
So what is the argument really all about? (It's been fun anywho.)

[klypp]

Let's keep the discussion/debate simple, I have gotten off onto horrible tangents with people in the past. The question really is simple:

"Can the distance from A to B be irrational?"

and

Name a single physical relationship that IS irrational, not one that is accurately described as irrational. You cannot by definition because nobody has measured something indefinitely.

A rational number is a number which can be expressed as a ratio of two integers. Ratio -> rational. Do you get it???

When the ratio is divided out, it becomes a terminating or repeating decimal.
An irrational number can not be expressed as a ratio. Irrational numbers cannot be represented as terminating or repeating decimals. They can, however, be represented in other ways.
If you make yourself a measure stick and mark 1, 2, 3 and so on, you may just as well mark phi on the same stick - with exactly the same precision as your 1, 2 and 3 marks. The length phi meters, is a precisely described length.

This is kids stuff.

[webolife]

Whoa, Klypp, how'd we get on the same side of an issue???!!! Yea for open discussion!

[klypp]

Yeah, I know...

This sure is a strange universe...

[rcglinsk]

Mathematics is a language like any other, therefore its symbols must refer to something just as words do.

What, then, does a number represent? First off, does one refer to an object or a concept? Is it like a rock or is it like "up"? Can I hand you or throw you a one? No, I cannot, because it is a concept, a relationship among objects.

What, then, is the relationship of the object(s) referred to by a number? What is the relationship that distinguishes 1 from 2 from 3 etc.?

The answer is that each number refers to a set of criteria that distinguish it uniquely. A number indicates that there are X objects within a distance d of each other, each of which is at least a distance D from every object in some set Z (which could be every other object in the universe).

1: I
2: II
3: III
4: IIII
5: IIIII

The operations such as addition and subtraction refer to motion. The "=" symbol is just a separator to indicate that the criteria met by the objects indicated on the left are different from the criteria met by the objects indicated on the right.

1+1=2: I I = II
1+2=3: I II = III

A negative number has no physical significance. Zero only has physical significance when it proceeds the "=" symbol (i.e. the result of an action). Nothing cannot be the cause of an action, only something can perform an action. Therefore 0 has no physical meaning on the left side of the "=" symbol.

Decimals in our numerology are just a standardized scaling factor. The smallest number we use can be set to 1 and everything else scaled up from there so that we work only with whole numbers. Without this caveat a fraction has no physical significance. When you cut a pizza in half you don't have two 0.5 pizzas, you have two portions of pizza. Cutting a pizza in half is represented mathematically:

II = I I
or
2 portions of pizza separated by 10 angstroms = 1 portion of pizza + 1 portion of pizza separated by 1 inch

All you've done is move two objects further away from each other and identified them separately. What you identified previously as a "whole pizza" no longer exists. There are now two portions of pizza.

Carrying this to its logical extreme the smallest number would refer to a single fundamental constituent (the building block of everything, a continuous object). The pizza may contain 10E50 of these fundamental constituents. When we separate two portions of the pizza we would say:

10E50 Fund. Consts. =5E50 Fund. Consts. + 5E50 Fund. Consts

It's okay to not know what the most fundamental continuous objects of the universe are, we can apply this to whatever the most fundamental object we can quantify is. If it is an atom then we talk about the motion of atoms at specific distances. If it is an electron and a quark then we talk about the motion of electrons and quarks.

As such we never actually have to resort to fractions and decimals, although we use these as conveniences. Dealing with factors of 10^50 gets old very fast. These conveniences work well as long as we do not forget where they actually come from, as long as we do not forget the physical significance.

So, an irrational number has no physical significance because it indicates an infinite number of objects. It implies there is no fundamental constituent/component. We cannot scale an irrational number to a whole number via a physical interpretation.

Mathematics is an excellent tool, but we get into trouble when we forget that it is merely a tool.

Hiya, if I might try to add a bit.

Imagine a series of sticks arranged in a spiral. The largest sticks are on the outside and the shortest toward the inside. The way to arrange these sticks in the smallest area is for them to have lengths in series that correspond to the Fibonacci sequence. The number phi corresponds physically to an infinite series of sticks. Now, that's not possible. Hence your point about the smallest fundamental unit. The number phi, out to a finite but very large ratio in the Fibonacci sequence, is clearly built into the structure of biology. Since biology has the characteristic of layer upon layer at almost any scale, the Fibonacci sequence ratios become something of an evolutionary driving force.

[junglelord]

Clearly PHI is a fundamental component of reality.
APM quantifies it as the fundamental ratio of aether cycles to matters quantum spin.
This is derived via SI units from Plancks length, speed of light, Coulombs Constant, Permittivity and Permeablity of Vacuum, Comptons wavelength. It is the product of aether cycles (quantum 2 spin rotating magnetic field) related to the primary angular momentum (quantum spin number) which it encapsulated to from matter via tensegrity.

This matter must also obey the Fuller Synergetics rules of existance based on Harmonic Resonance and Pi and experiments carried out by Fuller with spheres and harmonic nodes via resonance.

Since Charge is derived from Pi, (as expressed in Coulombs Constant) then it is essential to understand both relationships, PHI and Pi. Charge is only properly understood if you have both the structure (Pi) and the function (Synergetics/Harmonic Resonance)

Add e to the list of fundamental attributes of reality and you have re-organized math around a unify principle.
This will solve all problems.

[rcglinsk]

I'll bite. If you keep breaking your brick in half, or any fraction of pieces, you will always get a rational result, obviously.
But I look right there on my slide rule and see the number pi sitting there along with the other numbers on the scale.
I have no problem visualizing the exact length root-2, just as Klypp offered. Planck's distance is so small as to far exceed any abilty to measure or detect, yet it is an alleged finite unit of theoretically countable [rational] distance. Yet you would say that even it can be cut in half, to the utter dismay of Planck So if this is all you mean by irrational #'s having no physical significance, what's the point? Don't just repeat yourself here, it's not doing you or me or anyone any good. Is Avogadro's number actually counting something [have you counted a mole lately?], or is it a representational concept, based on certain assumptions of finitude? So, there's a rational number that likely has no real physical significance. I have read a pretty good argument that c = 1. There's a nice rational number for you, but how does it help to describe what you actually observe when light "happens"? At least pi, phi, root-3 and root-2 are useful for describing observed patterns in nature, and how can that be insignificant? OK, so you conceded that these numbers may be physically significant, but just not "physical"?
So what is the argument really all about? (It's been fun anywho.)

Avagadro's number of protons and electrons is one gram. It's nice and circular.

[rcglinsk]

Let's keep the discussion/debate simple, I have gotten off onto horrible tangents with people in the past. The question really is simple:

"Can the distance from A to B be irrational?"

and

Name a single physical relationship that IS irrational, not one that is accurately described as irrational. You cannot by definition because nobody has measured something indefinitely.

A rational number is a number which can be expressed as a ratio of two integers. Ratio -> rational. Do you get it???

When the ratio is divided out, it becomes a terminating or repeating decimal.
An irrational number can not be expressed as a ratio. Irrational numbers cannot be represented as terminating or repeating decimals. They can, however, be represented in other ways.
If you make yourself a measure stick and mark 1, 2, 3 and so on, you may just as well mark phi on the same stick - with exactly the same precision as your 1, 2 and 3 marks. The length phi meters, is a precisely described length.

This is kids stuff.

I don't think you can mark phi. Phi is the ratio of numbers in the Fibonacci sequence. There is no infinite ratio out there, only particular very large ones. You'd have to pick a ratio to actually mark it.

[altonhare]

I'll bite. If you keep breaking your brick in half, or any fraction of pieces, you will always get a rational result, obviously.

Great, case closed.

But I look right there on my slide rule and see the number pi sitting there along with the other numbers on the scale.

I thought we resolved this issue Web. You just said we can break out brick in half and get a rational number. Why do we care what some ignorant mathematician wrote on your slide rule? Can we break our bricks in half and get a rational number or not? You have a habit of agreeing with me then disagreeing. What's your stance eh?

I have no problem visualizing the exact length root-2, just as Klypp offered.

Again, I thought your opening sentence resolved this issue. You JUST said that we will always get a rational result. Your powers of visualization far exceed mine. In fact they far exceed anyone on the planet. When I try to visualize what numbers refer to I start having trouble when I get to the hundreds. For instance to visualize 1.28 I can either imagine 128 bricks or I can take one whole brick and another (2 bricks), break one in half (1.5), then break it in half again (1.25). At this point it becomes pretty difficult to visualize getting 3/1000. Usually it's easier to just scale everything up to whole bricks and visualize 128 bricks. It takes a bit of time counting but I get there.

How do you personally visualize sqrt(2)? Do you actually visualize it, or is it an abstract concept? If you cannot actually show me sqrt(2) physical objects, or at least S*sqrt(2) where S is a finite scaling factor, how can you visualize it? What's your actual stance Web?

Just because we can conceptualize abstractions does not imbue them with physicality. This is called reification. I can conceptualize all kinds of things that don't actually make any sense if I think about it carefully.

Is Avogadro's number actually counting something [have you counted a mole lately?], or is it a representational concept, based on certain assumptions of finitude?

I have not counted a lot of things lately, the issue here is whether it is countable (rational). Last I checked Avogadro's constant was rational i.e. it reduces down to counting objects (formally atoms in carbon-12). The number is physically significant. It refers to counting something, although we may be correct or incorrect on the exact nature of that something. In fact we know we must be incorrect because we don't yet understand what a "gram" really is i.e. we don't understand gravitation. Why does Na carbon 12 atoms always have a weight of one gram? What is the physical mechanism? This question hasn't been answered, implying we do not understand either the atom or gravity yet. Discerning the architecture of the fundamental constituents of the universe is the primary task of physics. When we discover rational relationships it tells us we are "on the right track". Avogadro's constant tells us there is something significant about our observations of what we call "atoms".

Finally, I never said every rational number has physical significance. I said only rational numbers CAN have physical significance. Rationality is a necessary but insufficient criterion. I can throw out rational numbers all day long but they don't mean anything unless I reference objects.

I have read a pretty good argument that c = 1. There's a nice rational number for you, but how does it help to describe what you actually observe when light "happens"?

We don't explain nature with measurements. We explain nature with objects. Measurements come in at the last step of the scientific method. C=1 is just reporting a measurement in some units. What we are primarily interested in, is WHY C is ALWAYS C. This is a question for physics, math has no power to explain this counter intuitive phenomenon. If I ask you why the pencil fell to the floor will you answer with an equation? An equation can only describe how fast. It cannot tell you why!

At least pi, phi, root-3 and root-2 are useful for describing observed patterns in nature, and how can that be insignificant?

Web you have difficulty discerning between a mathematical description and a physical explanation. Irrational relationships are abstractions that are useful for standardization and convenience. They have no power to explain because, as you said, if we break our brick in half enough we will always get a rational result! If we always get rational results how can irrationals be significant beyond being mere conveniences!? You can't have your cake and eat it too Web, pick a side (we're at war!). That's a joke from The Colbert Report.

OK, so you conceded that these numbers may be physically significant, but just not "physical"?

Again, not physically significant. Irrationals are conveniences. RCG made some very good points. While biology is characterized by layer upon layer at "almost every scale" he (and you) concede that there cannot be an "infinite number of sticks". At some point the number terminates, indicating we have reached the "smallest scale". I am *far* more interested in where this termination occurs than I am in the irrational abstraction phi. The last decimal point represents the most fundamental constituents in the universe!

Shouldn't we be concentrating on finding where it terminates? Using the irrational abstraction phi has distracted us from the most important question, that is "how small does the "universe" get?" This is a good reason to recognize irrationals for what they are, conveniences.

I don't think you can mark phi. Phi is the ratio of numbers in the Fibonacci sequence. There is no infinite ratio out there, only particular very large ones. You'd have to pick a ratio to actually mark it.

Thank you rcg. I think that will help web see why I don't care if some ignorant mathematician marks an irrational on his slide rule.