Johnny5

I will go back and fix it right now, thank you Dapthar. Regards

The pendulum itself precesses' date=' here is a video I was looking at recently of it: Precession of a Foucalt Pendulum Follow the explanation sequentially. The explanation begins by saying that the pendulum has just been released from rest. Now, the camera is attached rigidly to the earth, and offset slightly from the unique plane which the pendulum first swings through. Let us begin to define some frames. First, since the pendulum is released from rest, there is an angle between the wire, and the pivot point, and this angle lies in the original plane which the pendulum swings in. Let us define this angle to be theta. Now, we will define at least two reference frames, during this analysis. One will be attached to the pendulum, the other to the earth. Now, the amplitude of the pendulum decreases in time, and thus the intial theta is greater than later theta values. Let us use the subscript i, to denote the initial theta value. So we have... [math] \text{initial sweep angle} = \Theta_i [/math] Now, they don't specify this angle, but we can make a rough approximation. It looks to me like the diameter of that circle, above which the pendulum is swinging, is about three meters long. Now, the length of the wire is constant in time, and they don't specify that either, but... we can approximate the time of one swing, from the video. Now, the center of inertia of that ball traces out a curved path, in the rest frame of the earth. But, the swing is only a tiny fraction of a full circle, so lets approximate this tiny arc, using a straight line, to keep the math simple. So after one swing of the pendulum, we can analyse an isosceles triangle. The base of this triangle is 3 meters. The sides of this triangle are the unknown string length L. And the apex angle of this isosceles triangle is: [math] 2 \Theta_i [/math] Let there be an axis of one of the frames containing the center of inertia of the earth, and the pivot point. Thus, these two points are at rest in this frame, let us call this frame, frame S. So frame S is an earth frame, a frame of reference permanently attached to the earth. Now, this experiment is taking place in Sydney Australia, this says. Now, Sydney is in the southern hemisphere, that means that it lies below the earth's equator. I'm no cartographer, but I have used maps to set up communications links, so I have looked at a map of the earth more than once. I would venture to guess that the coordinates of Sydney Australia are... Well I know where 0 degrees latitude, 0 longitude is, Its right off the coast of Nigeria, about six degrees southerly, in the Atlantic. Actually, is south of Ghana, I just looked, and I eyeballed it from the map as about six degrees of latitude. So, the African country Ghana, lies six degrees north latitude above the point (0,0). In other words, the coordinates of Accra Ghana are (6,0). The line of 0 degrees longitude is called the prime meridian, and actually passes through Greenwich England. Which is about 54 degrees north latitude, something like that The capital of Ghana is Map with Accra Ghana on it by the way. Now the Antarctic circle runs east/west, at 66.5 degrees south latitude. Since it's extremely warm in Australia, this is too low. I am going to approximate Sydney at... 30 degrees south latitude, and 100 degrees east longitude. So now, we just need a world map, to check my guess. World Atlas Map showing Sydney Australia bottom right From the map, we see that Sydney Australia, is located at 34 degrees south latitude, and approximately 145 degrees east longitude. Keep in mind that the point (0,0) zero lat, zero long, is located six degrees south of the Capital of Ghana, which is Accra, in the Atlantic ocean. Now through that point (0,0) there are two lines, one is called the equator, and the other is called the prime meridian, which meet at right angles. The line that runs through the north pole, and south pole of the planet, is called the prime meridian, and this line, actually a circle really, passes through Greenwich England, and from memory, Greenwich England is as far north as New York, and New York is around 55 degrees north latitude, which means that the coordinates of England are around 54 degrees north latitude, and zero degrees east longitude. To give you some perspective, the Arctic circle is located at 66.5 degrees north latitude, and the Antarctic circle is located at 66.5 degrees south latitude. There are also the tropics. The tropic of Cancer is located at 23 degrees, 26 minutes, 22 seconds north latitude, and this circle runs parallel to the equator. The equator is a great circle with zero latitude, and runs along the buldge of the earth. The reason there is a buldge there, is because the earth is spinning, and the axis of rotation passes through the earth's poles. Ok so... There is also the tropic of Capricorn, which is a great circle which runs east/west, at 23 degrees 26 minutes 22 seconds south latitude. Now, from memory, the tropic of cancer crosses the border of three countries in North Africa. I remember them using NAM, like in vietnam. The three countries are: Niger Algeria Mali There is a point where these three countries meet, and there is a mountain range there known as, the Haggar mountains. What is significant about this location is the following: On June 21st of every year, at 12:00 noon, if you look straight up into the air, then you will see the sun. Six months later, if you do the same thing, anywhere on the great circle called the tropic of Capricorn, located at 23 degrees 26 minutes 22 seconds south latitude, you will likewise see the sun directly overhead, at 12:00 in the afternoon. Ok so... Back to the Foucalt pendulum. We now know where Sydney Australia is located. Its coordinates are: 34 degrees south of the equator 144 degrees east of the prime meridian So this is where the experiment on Foucalt's pendulum is taking place. Now, in the video, you can see that the pendulum swings roughly 3 meters, and it does so in approximately 3 seconds. So a good guess for its initial speed is one meter per second. Now, I have no idea how much the pendulum bob weighs. But ok, so there are two frames to focus on, one which is fixed to the earth, and a second one which is fixed to the pendulum bob. We can switch back and forth between reference frames if necessary. The origin of frame S, is the center of inertia of the earth. The origin of frame S`, is the center of inertia of the pendulum bob. So lets continue to follow the explanation given along with the video. Ok, let me define the XY plane of frame S, as follows: The first swing of the pendulum lies in the XY plane of frame S. Now the camera is fixed in frame S, and you can see the handrail on the stairs in the background, at the same location in S, exactly one hour after the pendulum was released. Now, the camera isn't actually lying in the XY plane of frame S, it is slightly offset. That is why you can see the pendulum swing at a slight angle in the first video, the moment the pendulum is released, but this angle is practically zero, relative to the XY plane. If the camera were moved over just a few inches, then the pendulum wire would be seen to go directly away from the camera, and come directly back. This is not exactly the case, because the camera is slightly outside of the XY plane of frame S. But the important phenomenon to be noticed in this experiment, is that exactly one hour later, the center of inertia of the pendulum bob can now be clearly seen swinging outside of the XY plane. On the first swing of the pendulum, the center of inertia of the pendulum bob was in the XY plane. The bob goes back and forth every six seconds, which means that after one hour has passed, that the pendulum swung back and forth 3600/6 = 600 times. To understand this calculation, it helps to understand units. There are sixty seconds per minute, and there are sixty minutes per hour, so the number of seconds in one hour is given by: [math] \frac{60s}{m} \cdot \frac{60m}{h} = \frac{3600s}{h} [/math] Units of minutes cancel out, and so you are just left with units of seconds per hour. Now, the period of the pendulum T, is the time taken for the pendulum to go all the way back and forth. And by analysis of the video, it takes the pendulum 3 seconds to move away, and 3 seconds to come back, so the period T of the pendulum is 6 seconds. So there are six seconds per swing. Let n denote the number of swings per hour, n is what we want to solve for. n swings per hour = x swings per second*3600 seconds per hour. [math] n = \frac{1 swing}{6 seconds} \cdot \frac{3600s}{h} = \frac{3600}{6} = 600 \frac{\text{swings}}{\text{hour}} [/math] Now, the second video shows the state of the universe one hour after the pendulum was released. You can clearly see from the video, that the pendulum is swinging outside the original plane it swung through. And this is the phenomenon under study. The question to be answered is,"Why didn't the pendulum stay in the XY plane of reference frame S?" Now, in the explanation given at the site, it says that one hour after the pendulum was released, the plane precessed about 9 degrees. Now, one guess as to why this happens, comes from Galileo's law of inertia. Galileo's law of inertia, says that an object at rest will remain at rest, or move in a straight line at a constant speed, unless acted upon by an external force. So let us focus on the pendulum bob. It is hung from a very long wire, which i will just guess is 10 meters long, thats about 30 feet. So the ceiling at the University of New South Wales, has been assumed to be approximately 30 feet high, in the room the camera is in. It's just a guess. Correction: I just checked, and New York is at 47 degrees north latitude, and Greenwich England is at 51 degrees north latitude. Back to the Foucalt's Pendulum. At the site, they say that the pendulum is swinging north/south. So let us focus upon this for a moment. The coordinates of Sydney are (-34,144); that is 34 degrees south latitude, 144 degrees east longitude. So let there be a meridian running through the University of New South Wales, a great circle, containing the earth's poles. Now, here is diagram which relates the S frame to the rest of the universe. Celestial Sphere Here is an interesting article on the north star (labelled Polaris in the previous link). Here is a sky chart, containing Ursa Major Here is a picture which shows how to find polaris (the pole star, the north star) given that you were able to locate Ursa Major (The big dipper): Polaris and Ursa Major Suppose you go outside of your house, on a starry night . I just included a link to that famous painting by Van Gogh, because I have much appreciation for the arts, as well as the sciences. But back to business, understanding the Foucalt pendulum. Suppose you go outside your house, on a starry night, with a telescope, and a magnetic compass. The magnetic compass will tell you which direction the north pole is in. And that knowledge will come in handy, since you are trying to locate the North star in the night sky. Now, in a moment i will discuss the physics of how a compass works, but right now, I want to focus on how you can locate the north star in the night sky. Since you have a compass needle which points magnetic north no matter how you turn your body, you can imagine a great circle, a meridian, running through whatever location you are at on the surface of earth, all the way around the earth, and this imaginary great circle, will contain both the North pole, and the south pole. Very cold at both places. Oh, and by the way, the first human being to set foot upon the North pole was: I believe Amundsen. Yes, it was Amundsen, Roald Amundsen, a Norwegian explorer of both poles. Amundsen had actually already been to the south pole, before he journeyed to the North pole. An earlier navy explorer tried to reach the north pole, but he missed it by 20 nautical miles. Here is a link to a wikipedia article on Roald Amundsen Ok so, once you know the direction north is, just begin lookin upwards, tilting your neck back along the imaginary great circle, so that the back of your head is moving towards the south pole... and look for the big dipper. Once you find it, you can spot the north star from the following diagram: Polaris relative to Ursa Major As you can see, the north star is part of the little dipper, seen here. And if you click on the interactive star chart, you can see the location of a few other constellations, relative to the north star.

To be honest, it doesn't matter who invented it, whether or not it was Euclid in particular, is irrelevent to me. Although, what would matter, if knowable, was the year. As for something he defined, no he didnt define it. The person who found it, deduced it. Actually, I am interested in the history of the "Euclidean algorithm" if you happen to know about it. I think investigating the inventor of a mathematical idea is part of the process of understanding that idea, but that's just my opinion. Oh and here is something for you... Algorithm [math] \alpha \lambda \gamma o \sigma = \text{painful}[/math] [math] \alpha \rho \eta \theta \mu o \sigma = \text{calculation or number} [/math] [math] \alpha \lambda \gamma o \rho \eta \theta \mu o \sigma = \text{painful calculation}[/math] Regards PS: What about my question on whether or not you should use Z+ as opposed to Z? Now, I don't currently remember the algorithm, but I do remember it was very simple. And i know that it connects to the fundamental axiom of arithmetic, namely that any positive integer can be written as a product of prime numbers. And this all connects to greatest common multiple, and some other things. I remember how to break a natural number down into the product of primes. Take 486 for example. First, factor out the lowest prime number, which is 2, as follows: 486=2*243 Stick with 2, until you cannot factor it out anymore. You cannot factor out 2 from 243, so move on to the next prime number, which is 3. Can you factor out 3 from 243? Here is a quick way to check... 2+4+3=9, and 9 is evenly divisible by 3, so yes. So factor out 3 from 243 as follows: 486=2*243=2*3*81 Stick with 3. 486=2*243=2*3*81=2*3*3*27 Stick with 3. 486=2*243=2*3*81=2*3*3*3*9 Stick with 3. 486=2*243=2*3*81=2*3*3*3*3*3 And the algorithm ends. Now using the transitive property of equality, we have: 486=2*3*3*3*3*3 This is the one, and only way, to write 486 as a product of prime numbers.

In the following post, I am going to do a few examples of Euclid's algorithm. It appears in Book seven, as proposition 2. Greatest Common Divisor

No, I feel I need to fully understand not only the Michelson Morely experiment, which took place around 1879, but also the General theory of relativity. I would be interested in any version of it, which incorporates particle exchange. Thank you, and regards

Let the theory be that magnetic monopoles exist. Therefore, it is not the case that: [math] \nabla \bullet \vec B = 0 [/math] Where B denotes magnetic field. That is exactly what I was thinking of doing. Writing gravity as electromagnetic force. Change the constants. Something like that. I think this because of the Van Allen belt, and the precession of this planet. Regards I will come up with something more mathematical in a moment. The earth is already tilted. Here is the angle: Earth is tilted in the following RANGE: [math] 22^{\circ} \ \ \text{to} \ \ 24.5^{\circ} [/math] Here is my source. Notice the following statement: This requires a lot of thought to understand, a starchart will help to understand it. Here is an article on the tropic of cancer Paraphrased slightly from wikipedia: The tropic of cancer (latin for crab), is one of five circles of latitude that mark maps of the earth. The exact positition of this line of latitude that runs parallel to the earth's equator, is: 23 degrees, 26 minutes, and 22 seconds. What is special about this line of latitude is this: It is the farthest northern latitude at which the sun can appear directly overhead. Regions of this earth, which lie above the tropic of cancer, are in what is called the "Northern temperate zone." This zone runs from the tropic of cancer up to the arctic circle, located at 66.5 degrees north latitude. The southern temperate zone, runs from the tropic of capricorn (located at 23 degrees, 26 minutes, 22 seconds south latitude of this planet, all the way to the Antarctic circle. So, now here is the thing: In order to understand this, it helps to imagine yourself in one of the countries that the line of caner runs through. I know some geography. Consider Africa. Along the cusp, there is Ivory Coast, there is Ghana, Nigeria, I think in that order. And from memory, the point (0,0) is just below Nigeria, i think six degress south or something like that, its just off the coast of Africa, in the water. It might be 6 degrees below Ghana actually. So we need one of the countries further north, from a map... There is a mountain range lying along the tropic of cancer, called the Ahaggar mountains. This mountain range lies right along the border of three different African countries, Niger, Algeria, and Mali.'' So suppose we are situatued way high up in those mountains, on June 21st. here is what will happen. At 12:00 noon, the sun will be directly over our heads. All we have to do, is look straight up into the air, and we will see the sun. Afterwards, in order to have the same effect happen, we have to begin walking directly towards the south pole. Now, after some period of time passes, the same effect will occur at the tropic of capricorn. I will have the date in a moment. Ok, here are the facts: Definition: The summer solstice is defined to be the day at which the sun reaches its highest point above the equator, as the earth wobbles in its slightly elliptical orbit about the sun. The exact line of latititude is 23 degrees, 26 minutes, 22 seconds north latitude, and the exact day of this is June 21st (though this varies slightly). Definition: The winter solstice is defined to be the day at which the sun reaches its lowest point below the equator, as the earth wobbles in its slightly elliptical orbit about the sun. The exact line of latidude is 23 degrees, 26 minutes, 22 seconds south latitude, and the exact day of this is December 21st. (though this varies slightly). So the two dates to remember are: June 21st, and December 21st. Count the months between them July 21st August 21st September 21st October 21st November 21st December 21st Exactly six.

Earth has a magnetic field. The evidence is my compass. The earth precesses. The evidence is the Foucalt pendulum. So the sun is emitting particles, we feel its warmth. If they interact with the earth's magnetic field, somehow, then umm well then umm they can rock the earth. Make it wobble. Make it precess, just like ummm in an hydrogen atom... The Zeeman effect. How is that for evidence?

hmmm... What is an isomorphism again? Also, what is Euclid's algorithm again. Also, is 1 a prime number? Thanks Matt

You start out, let p be a prime. A prime number is a number which is only divisible by one and itself. The first few primes are in the following set: {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53...} Now, focus on the following statement: In the expression above, let Z denote the integers? Shouldn't it be the natural numbers Matt? The standard symbol for the integers is: [math] \mathbb{Z} [/math] The standard symbol for the natural numbers is: [math] \mathbb{N} [/math] I'm going to assume you meant the natural numbers. Now, there is a question as to what x,y denote. They are just arbitrary natural numbers. So... [math] \forall x \in \mathbb{N} \forall y \in \mathbb{N} \exists k \in \mathbb{N} [ x=y+kp ][/math] Where p denotes a prime. But the x,y are used to define an equivalence relation on the set.

My bad. One of the field axioms is that there is an element 0, of any field distinct from 1. [math] \text{Existence of an Additive Identity} [/math] [math] \exists 0 \forall x \in \mathbb{R} [0+x=x] [/math] Hence... if the randomly chosen field F was {1,2,3} then one of the symbols 1,2,3 must denote that which 0 denotes, in the field axioms. So I should have written: [math] \mathbb{F} = \mathcal{f} 0,1,2 \mathcal{g} [/math]

I'll get right on on it. [math] \text{Non-triviality field axiom} [/math] [math] \text{not(0=1)} [/math]

I am trying to recall modulo arithmetic Matt, and then express the logic behind it correctly is all. Can you sort of give me a push in the right direction please? What I mean by that is this, how would you start off a lecture on the subject? There will be a branch from what is well known, to modulo arithmetic. I want to see it. Thanks 2=1+1 regardless... Yeah but then you can reach a contradiction this way... Suppose you are working with the following field: [math] \mathbb{F} [/math] So the field axioms are all true statements. Now, one of the axioms of a field, is that the symbol 1 denotes an element of any field conceivable. Therefore, regardless of whether the field chosen was the reals, the rationals, the complex numbers, whatever... the following expression is guaranteed to denote a true statement: [math] 1 \in \mathbb{F} [/math] Now, the following expression denotes a true statement, regardless of the field chosen. The field could be chosen at random, and the following expression would still denote a true statement: [math] 1+1 \in \mathbb{F} [/math] And this is so, because a field is closed under addition. Now, the closure axiom of a field can be stated logically, as follows: [math] \text{Axiom Of Closure of a randomly chosen field} \ \ \mathbb{F} [/math] [math] \forall x \in \mathbb{F} \forall y \in \mathbb{F} [x+y \in \mathbb{F}][/math] The possibility that x=y has been left open. Consider the case where x=y, the axiom of closure reduces to: [math] \forall y \in \mathbb{F} [y+y \in \mathbb{F}][/math] Now, one of the field axioms is the following: [math] \text{Existence of a multiplicative identity 1} [/math] [math] \exists 1 \forall x \in \mathbb{R} [1 \cdot x = x] [/math] Uniqueness is left as a theorem. But the point is, that the previous axiom ensures that any field has at least one point in it, being denoted by the symbol 1. And so now, using the special case of the closure axiom stated earlier, together with existential instantiation, the following expression denotes a true statement, regardless of what field was randomly chosen: [math] 1+1 \in \mathbb{F} [/math] And repeating the process, it follows that the following expression denotes a true statement, for any field F: [math] 1+1+1 \in \mathbb{F} [/math] And so on. Now, is the time to cover the issue of whether or not the field is finite, or infinite, and introduce the definition of: [math] \text{Char} \ \mathbb{F} [/math] ... the characteristic of a field F. So now, to modulo n arithmetic... Let it be stipulated(that means that both you and I agree) that the randomly chosen field has three and only three elements in it. It only makes sense that we should (deontic logic word) choose different symbols to denote each of the three elements. Let the following be stipulated to be true: [math] \mathbb{F} = \mathcal{f} 1,2,3 \mathcal{g} [/math] So the elements of F are distinct by design. And saying that the elements of the set are distinct, is just a quick way to say this: [math] \text{not(1=2) and not(1=3) and not(2=3)} [/math]

I didn't say the sun was a magnetic monopole at all. I just said, what if the sun emits them. Well what if there were magnetic monopoles, what would Maxwell's equations look like then?

What if the sun emits umm magnetic monopoles?

Yes they show the same thing, but the second postulate of relativity is not an empirical fact. Regards

I don't believe gravity curves space at all. In fact, I am starting to think that gravity might be related to magnetism. Do you know what the Van Allen belt is? Kind regards

Are you basing this conclusion on Maxwell's equations?

I think I get this, but lets just check ok? In trying to answer this question, I have had to address several issues in set theory. Specifically, I am wondering how to handle a case where x=y, in a discussion which involves set theory in some way, either explicitely or implicitely. Generally, when I use the universal quantifier, I leave open the possibility that x=y, because I can close it easily by writing.... [math] \forall x \forall y [ \text{if not(X=Y) then such and such} ] [/math] I have no reason to change this practice. But, when using the roster method, i commonly employ the following axioms: Axiom I: For any x an element of A, and any y an element of A: If A={x,y} and x=y then A={x,x} (by substitution). Axiom II: For any x an element of A, If A={x,x} then A={x}. Now, I started this post, trying to prove the following theorem (which I worded)... [math] \text{Theorem: In any infinite field F, with 1,2 } \in \text{F, not (1=2). } [/math] And in the course of trying to prove it, I had to wonder whether or not by writing [math] \text{with 1,2 } \in \text{F} [/math] That contained the information on whether or not 1=2. Because, if the answer is that it contains the information that not (1=2), then there can be constructed a rapid proof. But, it cannot contain the information that 1=2, since one of the axioms of a set, is that the elements of a set are distinct. So at most, it leaves open as possible that 1=2. So that's where I am at now. Suppose you see the following expression: [math] 1 \in \mathbf{F} \ \& \ 2 \in \mathbf{F} [/math] The serious question arises as to at what point a reasoning agent is to conclude that not(1=2). What those symbols are to denote, has to be decided by the specific individual who formulated the expression using those symbols. No other conclusion is rational, so... with that in mind... One of the tasks of another individual who reads the expression is, to first decide whether or not the expression denotes a statement. If the answer is yes, then that reasoning agent can go on to try to decide whether or not the statement is true. On the other hand, if the answer is no, then the individual should not bother to try and determine the truth value of a statement, since that individual would have already decided that the expression does not denote a statement. But, a quite common practice by reasoning agents, is to try and look for meaning in someone else's expression. In other words, the first stage of their reasoning process, when analyzing someone else's expression, is to try and find the other individual's meaning. Once the meaning is determined, then it follows that the expression denotes a statement, and then the person can go on to try and determine the truth value of the statement. So this is a complex problem. Proof: Suppose that 1=2, in an infinite field where 1,2 denote elements of the field. Suppose that you are a reasoning agent who understands that the elements of a set must be distinct by a set theoretic axiom. Then you can use set theory to immediately draw the conclusion that since 1,2 were stipulated to be elements (plural not singular) of the field F, they must be distinct, wherefore you can immediately see that the hypothesis (1=2) contradicts a set theoretic axiom. To understand the basic axiom that I am referring to, consider this: In the case where {x,y} denotes a set, and x=y (meaning that whatever symbol x denotes, has also been denoted by symbol y), then {x,y}={x,x} using the substitution principle. But, a set is suppose to be just a collection of distinct objects, where the order of elements is irrelevent. So, for any set {x,x} it must be the case that: {x,x}={x} Which is just a singleton set, a set containing only one element. If {x,y} is a set then not (x=y). On the other hand, suppose you are a reasoning agent who does not know the set theoretic axioms, but you do know the field axioms of algebra. Then you can use the following proof to prove the theorem: Suppose that 1=2, in an infinite field where 1,2 denote elements of the field. Now, let the definition of 2 be given by [math] 2 \equiv 1+1 [/math] Using the axiom of reason, known as conjunction, the following statement "should" be true (where I am using a bit of Deontic logic as well): [math] 1 = 2 \& 2 = 1+1 [/math] Now, using the transitive axiom of equality, the following statement "should" be true: 1=1+1 Then 1=1+1 Then, by the seldom referred to field axiom known as the axiom of addition, which is: [math] \forall x,y,z \in \mathbb{F}[ x=y \Rightarrow x+a=y+a ][/math] We have as a special case: 1+(-1)=(1+1)+(-1) And we know that -1 is an element of the field, because The smallest P, for which [math] 1_1+1_2+1_3+... 1_p = 0 [/math] Is called the "characteristic of the field."

I vaguely remember this now. Ok... Characteristic of a field. Modulo arithmetic. Like a clock. You break fields down into "finite" and "infinite" A field is infinite if there is no limit to the number of distinct elements of it. The reals, the rationals, the integers, are all examples of non-finite fields. In a finite field, there are a finite number of distinct elements. Consider normal arithmetic. One of the field axioms, is that 1 is an element of the field. And another field axiom, is closure of addition. Which means that for any elements x,y in the field, x+y is also an element of the field. Therefore, since 1 is an element of any field, then for any field the following statement is true: 1+1 is an element of the field. Now, normally we associate the string "1+1" as being equivalent to 2, and we write: 1+1=2 and we say the expression above denotes a true statement, where we implicitely assume that not(1=2). This isn't really an assumption, it's a stipulation. But, in higher math, the meaning of the symbol '2' doesn't have to be associated with what we normally associate it with. When using modulo arithmetic, it is quite possible that 2=1. But it is also possible that there are 12 elements in the field, like a clock. So that what the arabic numerals denote, can be varied. Here is one fact that will not vary: [math] \mathcal{A} = \mathcal{f} 0,1,2,3,4,5,6,7,8,9 \mathcal{g} [/math] In the statement above, the symbol [math] \mathcal{A} [/math]denotes "the set of arabic numerals." In the decimal system we have: [math] 2 \equiv 1+1 [/math] [math] 3 \equiv 2+1 [/math] [math] 4 \equiv 3+1 [/math] [math] 5 \equiv 4+1 [/math] [math] 6 \equiv 5+1 [/math] [math] 7 \equiv 6+1 [/math] [math] 8 \equiv 7+1 [/math] [math] 9 \equiv 8+1 [/math] [math] 10 \equiv 9+1 [/math] [math] 11 \equiv 10+1 [/math] and so on. And each successive natural number, is distinct from the elements of the natural number system which come before it. We can say this using first order logic, as follows: [math] \forall x \in \mathbb{N}[ \neg (x+1=x) ] [/math] But, if you are doing modulo arithmetic then the expression above denotes a false statement, instead of a true statement. All that happened, is that someone changed what the arabic numerals are denoting.