[SCoopsdk]

Chiaro?

Perfectly, but......

Let me restate my proof: since our perceptions imply change they presuppose time (change without time can't be imagined, which is why it is a contradictory concept), and therefore they can't be the cause of time.

Theodore, using terms like:

imply : when something is being suggested without being explicitly stated and by inference, association, or necessary consequence rather than by direct evidence

and

presuppose (again): when believing or supposing in advance the existence of time when that said existence hasn't been proven.......

does not help your case.

Take away the presuppositions and the implications because I don't doubt them, they just don't make time true, they simply support my assertion that Time is a construct of the mind.

Dude

You can call me Scoops

Time is motion and change

Exactly, nothing more and nothing less.

You can read more bout the concept here: Abstract Atomism

[noemon]

This man Jeff Grutt or whatever tries to argue the non-existence of past and future, and terms his theory as the present replacing itself, which if argued correctly holds credence, but what is its use? its purpose? This concept does not contradict our theory of time, it just redefines it. As i said the stream is factual, if you say the past is past, the present is now and the future is afterwards, and then go on to say the present replaces itself creating a past and therefore a future, you have not proven the theory of time wrong, you have just redefined it. This man, misquoting various philosophers he goes on trying to cancel our notion of past and future, and redefines the present as a present that changes accepts the notion of motion and change, but still remains present....because he assumes misquoting Buddha and states: that the current perception of time....."This is not the Buddhist position since it could violate the logic of Buddhist momentariness" this could, not this would cancel it for sure, which means that he has not even clarified in his head whether our notion of time actually cancels the Buddhist position, it could he says not that it actually does another LOL. He states that there is absolutely no relation between 10 o clock and 11 oclock, he sees them as different entities that each exists in itself, and that each is completely independent with the next, not that the one replaces the next, but that one is cancelled and the next is born by itself not by interactivity of matter, and that is there is no stream, no relation, while he and you, yourself accept that there is a stream, a stream like a river, is like a fluid, it is connected, each unit of water is related to the next unit of water, each unit of water replaces the next unit of water, because it touches the next unit of water, each unit of water is not in no relation with the next or the previous, they collide with each other and hence the replacement of one with the next, they do not exist themsleves independently just like the dimensions, the give way to the next they exist alltogether. He just fails to see what he discusses about....totally LOL. And i ask you and this man as well, with his totally contradictory and misquoted definition of time where he sees every moment with absolutely no relation with the previous or the next, that they do not touch each other, even though he considers the whole thing a stream whereas in a stream everything is united in fact and that is why its a thread, a stream, a line, lets assume that he is correct and that the units of time are in no relation absolutely with one another, has he also found a method to measure these units? or is he babbling non-sense that himself is unable to reason, his redefinition, does it serve any scientific purpose? Or is he simply trying to be smart?

Let us see what this man says:

The second name I give to the Buddhist theory of time is replacement presentism. I use the term “presentism” since, according to Buddhism and the doctrine of momentariness, only the present exists due to the fact that only one moment ever exists. The reason there are no moments before or after the present is because, in a theory of time and change based on momentariness, where moments are destroyed and copies of moments come into being, the destruction of one moment and the creation of another indicates that there can
only ever be one moment. I argue why this is the case in much more detail in this subsection.

I use the word “replacement” because if there is no persistence, there cannot be two moments side-by-side in time lest there be a possibility that in some pair of moments that are side-by-side somehow the same object shows up in each moment and thus persists. If there were, for example, two moments next to one another, this would mean that a given moment is not destroyed when a following moment comes into being, and instead when one moment comes into being, there is a moment before it that has not been destroyed. This is not the Buddhist position since it could violate the logic of Buddhist momentariness. But if it were instead that case that there is only one moment that ever exists, and two moments can never exist side-by-side, then it can only be the case that one present moment (p1) is completely replaced by another (p2). If only one moment ever exists it can only be a present since it is a now, and since there are no moments before or after it to make it a past or future.

LOL, IF, key word, secondly, how does he define the moment that just got cancelled and gave its place to the present moment and how does he define the next? non existant units, Basically he says that its not 1+1+1+1 moments but that each moment is cancelled and stops to exist, when the next moment comes about. He attacks the notion of measurement not just time but of everything, he attacks the notion of mathematics, the notion that 1+1=2.

This is what he argues about, and he offers no proof, only some metaphysical mumbo jumbo.

You argue that Time exists because we perceive it and hence we make it be, similar to this mans notions. False by definition.

You, me we are all units/moments and we do not exist because we are able to measure you or me or the unit, in width, height, time, space and volume terms. You exist because you do, we measure you because you exist, the fact that we measure you is the proof that you exist, you do not exist because we measure you. Unless you seriously belive that you do not actually exist, and that you are just an illusion.

Do you understand? When we measure energy, it means that energy exists, and thus we are able to measure it, it doesnot exist because we measure it. Your height is the distance between your level to the ground and your head, but because this is in relation to the ground, if you put this in relation to the Sun, or to another universe, then width and height are not enough to measure your position from the Sun to your head, The sun exists we see it everyday, we assign the sun as the point of reference, as an object as a unit and go find the distance between the sun and the moon, then we go measure more distance and more and more using objects that exist. And more dimensions to be taken into account, in relation to non-matter again more dimensions, and everytime we get stuck, we experiment and through experiment we see it happening and hence its existence, we see it we dont simply assume it, and we assign more dimensions in relation with the larger distances every/time we find in front of us, and thus energy exists. Do you seriously believe that you will disprove our scientific method, or that this mr Gutt or whatever did so, that he disproved Hopkins?

LOL.

Do you believe that without answering and without offering any reason, as to why you or him reject the notion of Hopkins, of my example and the spaceship, of Theodores example of evolution and the zygote. You dismissed them and you offered no reason as to why you dismissed them, and you still continue quoting a guy, who thinks he discovered the Sun. With his definition of the moment, we become unable to measure, and thus what is the bloody use of it? What is its purpose?

One can redefine the terms as he wishes, after all everything is relative indeed, but if i argue for something that serves absolutely no purpose, what is the use of it?

Also he claims that this theory is the Buddhist theory when in fact this is the Jewish theory of the torah:

See here the philosophical proof agsint the non assignment of a cyclo, of a point of reference, of a dot. The dot becomes the point of reference and it gives us a known X, once we got the known X, we can find all the unknows, it is up to us, with no known X we are able to do nothing, and use no Logic, and hence we become animals, not noemon Humans.

http://www.politicsforum.org/forum/view ... 9&start=50

The stream is a stream of ever changing nows, indeed, noone denies this(in fact this man is the one who does, and you who quote him contradict yourself for you have already accepted the notion of the stream), it is linear, however assigning a dot, a point zero, we become able to measure the stream, by staying into the torah, we stay ignorant, we become unable to measure, and this serves no purpose at all. And hence philosophically and practically it becomes invalid, for philosophy presupposes a purpose. What is this purpose? Oh and something else, this man calls his theory the theory of TIME, which means that he accepts a priori that time actually exists.


And the ultimate LOL:


In the beginning of his discourse he states that this philosophy is tottaly incompatibe with Western Philosophy and that it cancels Western Philosophy and their perception of time, at the end if his discourse he states:

According to the philosophy of abstract Buddhist atomism, the basic building blocks—which are atoms of energy, irreducible momentary pieces of existence—emerge in a lawless manner from the void. This also appears to possibly be in accord with the description of the quantum foam, as it is occasionally refereed to, that physicists have theorized makes up all of reality. In writing about the very tiniest levels of reality that physicists discuss, the physicist Brian Greene shows that the nature of reality described by physicists may appear similar to the reality of abstract Buddhist atomism:


As gravitational fields are reflected by curvature, these quantum fluctuations manifest themselves as increasingly violent distortions of the surrounding space… By probing to even smaller scales,… we see that the random quantum mechanical undulations in the gravitational field correspond to such severe warpings of space that it no longer resembles a gently geometrical objects… Rather, it takes on… frothing, turbulent, twisted form. John Wheeler coined the term quantum foam to describe the frenzy revealed by such an ultramicroscopic examination of space (and time)—it describes an unfamiliar arena of the universe in which the conventional notions of left and right, back and forth, up and down (and even before and after) lose their meaning… [The ultramicroscopic level is a] roiling frenzy of quantum foam… [A]s we recede to more ordinary distances…, the random, violent small-scale undulations cancel each other out… [T]he fabric of space-time appears to be smooth except when examined with ultramicroscopic precision (Greene, 1999 : 127-29).

Which the above, the Void is the basis of all Western philosophy the Chaotic apeiron, the Void, the Holon, and the space time assignment conclusion has been reached through our current Western philosophical Hypothesis of the Void, which were eventually proven correct through experiment and the scientific method. Quantum mechanics do not cancel Einstein's space-time nor vice versa, they simply deal with different stories, we shall assign more dimensions that is 11 and we shall make these 2 theories to co-exist once we assign the domain that they are able to co-exist. If we cancel the dimension of time, as you say Scoops then we are moving backwards, and not forward in assigning more dimensions and this assignment of a domain that contains more of the Void, of the Holon.

Quantum mechanics theory are in accord with the Western philosophical concepts that he fights against, and not only in accord, this is their Hypothesis, and thus proving himself, his own invalidity. The concept of the Void(Chaotic apeiron) created the notion of the manifestations of the Void that we term as spacetime and that we term yet and now, as structured for we have not assigned all the dimensions. It is this Hypothesis of the Void we used to reach to these conclusion adn it is this Hypothesis that our scince uses, it is this Hypothesis that einstein used and defined one more characteristic of the Void, and this man, uses our conclusions that derive from this Hypothesis and aim to explain the Void by building the block case by case to fight our conclusion and then goes on to claim his accord just because of the abstractness with our own notion of the Void. LOL's. We figured out everything comes from the Void, and we try to find the dimensions of the void, in order to do so, we assign a perfect circle, a structured entity, a moment that functions as a point of reference in order to create a known X, and thus become able to explain the unknowns through the linear mathematics, and by finding the unknowns the assignement of dimensions(time included) comes by default, that will eventually explain the Void. He thinks that he will explain the Void, without the assignment of a structured point of reference, but abstractly. He tries to go the other way around....May God be with him.

Fact is that Einsteins theory is unable to explain the movement of the atom in quantum mechanics, because the dimensions are not enough, spacetime is truly not enough, we need more dimensions, not cancel the ones that we sweated to locate and reason. Another fact is that the theory of Einstein is a result of our Hypothesis of the abstract movement of the atom in the molecule, see Atomic theory and the first Atomists, Democritus, the ones that he actually fights against.

[Theodore]

Take away the presuppositions and the implications because I don't doubt them, they just don't make time true, they simply support my assertion that Time is a construct of the mind.

In what way do they do this? "Imply" and "presuppose" are simply shorthand for "the perception in question forces a reasonable man to conclude that X exists objectively". The existence of time is revealed in our experience, but that doesn't mean we create time; on the contrary, time is a necessary condition for any form of experience.

[Andres]

Indeed, since how are you going to define a non-relativistic speed without referring to time

By measuring it's momentum, in the case of a bullet for example, measuring how deep it has gone into a piece of known material. But that is not the reason why it does not prove it a fundamental object.

If we use a metre as your length

How about I use my arm.

Can you think of any other way to define it?

Originally it was defined as 1/10,000,000 of the distance from the equator to the north pole through Paris.

And if you yourself agree that your equation in no way proves that times is fundamental, why does there not exist an equation or something that proves the existence of such a fundamental thing, if it is such?

There is no equation that proves that any physical object is fundamental, so the question is not very helpful to determining a fundamental object. The equations, being part of a theory, can only 'show' that the object is fundamental within the theory.

A better question is, what evidence do you require that distances are real?

[wonder cow]

A three page thread on "Time" and I did not start it and only have two posts in it, counting this useless commentary.

I must be losing my passion.

[Maxim Litvinov]

Someone read this stuff and tell me if anyone's answered "a magazine" yet.

[SCoopsdk]

neomon,

You would have impressed me, had I not realized that you had apparently read and comprehended a 27,000 word article, and then typed a 1700 word reply, all in the time it normally takes me to go for a leisurely dump and read the latest football results. (I must be missing something here)

I read your reply which seems mostly to be filled with disparaging remarks regarding the author, a professor of Buddhist philosophy at Purdue University with the assistance of the Chair of Physics at Western Michigan University and a Philosophy Professor from Western Michigan University

This is what he argues about, and he offers no proof, only some metaphysical mumbo jumbo.

I'm sorry, but coming from you, that makes me laugh. Most of the "mumbo jumbo" contributed to this thread has come from someone else (no offence meant)

In what way do they do this? "Imply" and "presuppose" are simply shorthand for "the perception in question forces a reasonable man to conclude that X exists objectively". The existence of time is revealed in our experience, but that doesn't mean we create time; on the contrary, time is a necessary condition for any form of experience.

Theodore, I have a feeling that we are about to dance in that circle again.

By measuring it's momentum, in the case of a bullet for example, measuring how deep it has gone into a piece of known material. But that is not the reason why it does not prove it a fundamental object.

Indeed, and kinetic energy is defined using time, not to mention "how deep"

How about I use my arm.

So if we have a one fundamental particle universe and want to establish the existence of time, your arm needs to be there too?

There is no equation that proves that any physical object is fundamental, so the question is not very helpful to determining a fundamental object. The equations, being part of a theory, can only 'show' that the object is fundamental within the theory

That's a genuinely interesting comment. I cannot conclusively prove that the theory that i put forward is true, but then neither can you. I can only try to dismantle yours (and I've had a go) and you can try to do the same.The result becomes the consensus belief of reality.

A better question is, what evidence do you require that distances are real?

That is a big problem because if, as I contend, time should be removed, then everything gets turned on its head. Distance then becomes a major problem. I didn't realise that to start with but it does start a ball rolling that demolishes most of what we normally take to be true and real.

[noemon]

noemon,

You would have impressed me, had I not realized that you had apparently read and comprehended a 27,000 word article, and then typed a 1700 word reply, all in the time it normally takes me to go for a leisurely dump and read the latest football results. (I must be missing something here)

So my answer has actually impressed you, but because you understand the concept of time, the way i term it, adn not the way you do, you dismiss it and you simply claim that your reply is great, but the time needed to perfrom the reply is not enough, and hence somehting is wrong with the reply and consequently with yourself. In your theory of time, i just need a moment to read, and the next moment i am able to respond, have you just cancelled your own theory, perhaps? In any case...read below:

I read your reply which seems mostly to be filled with disparaging remarks regarding the author, a professor of Buddhist philosophy at Purdue University with the assistance of the Chair of Physics at Western Michigan University and a Philosophy Professor from Western Michigan University

Disparaging comments are the LOL;s that follow the points raised and you have failed to address. So seriously, what are you babbling about?

For info only, i did not read the whole article, my points from what i read still stand for they have not been addressed.

I'm sorry, but coming from you, that makes me laugh. Most of the "mumbo jumbo" contributed to this thread has come from someone else (no offence meant)

No offense taken ofc, coming from the only person in here who has not addressed none of the questions raised...

So if we have a one fundamental particle universe and want to establish the existence of time, your arm needs to be there too?

The fundamental particle is the dot. The emptiness, and wholiness of the point of reference.

That's a genuinely interesting comment. I cannot conclusively prove that the theory that i put forward is true, but then neither can you. I can only try to dismantle yours (and I've had a go) and you can try to do the same.The result becomes the consensus belief of reality.

This has been posted from thread page 1 and about a million times, it is in a closed system that our fundamental laws exist, one this closed system is redefined and includes a larger (space-time-x) part of the Holon, then the system "becomes" closed again and it can be used to define the fundamental laws of the new system. This system begun with 2 dimensions, width, and stuff, then we reasoned depth, for we meausred it, and Einstein measured Time, his system does not include the Holon the whole adn hence its limitation inregards to quantum mechanics, only parts of it, hence it will be reassigned, redefined. Time the new definition of space did not cancel space or volume, it enlarged them to include larger parts of the Holon, similarly once we assign more dimensions they will not cancel the previous, including time, the new dimensions will redefine the holon, and why i have telling you that its not time in itself its more like spacetime, one existence.

What you and this man have failed to address is, what is the purpose of your redefinition? What is its method? You aim to cancel time, instead of redefine it to fit the "larger" dimensions of the universe, which ofc should and must be redfined to obey the laws of quantum mechanics. And something else you cannot cancel time without cancelling space, you cannot cancel space without cancelling the existence of your own existence. You can redefine them, not cancel them fro you will have just pronounced your non-being.

[Andres]

Indeed, and kinetic energy is defined using time, not to mention "how deep"

Do you need a watch to measure how deep a bullet went into a block of wood? No. Neither kinetic energy, nor length are necessarily defined using time.

So if we have a one fundamental particle universe and want to establish the existence of time, your arm needs to be there too?

You should pay attention to your own post, specifically to what the context of your quoted text was. You asked how will I define length without referring to time, I gave as an example the length of my arm. I can measure my table solely by it, without using a clock.

I cannot conclusively prove that the theory that i put forward is true, but then neither can you.

Actually, nothing in physics is proven. It is merely validated by agreement with observations. You are barking at the wrong tree.

That is a big problem because if, as I contend, time should be removed, then everything gets turned on its head.

If you accept that distances exist, then time intervals follow. Otherwise you will have to enter a discussion where defining what 'exist' means should probably be a priority.

[SCoopsdk]

Peace brothers

It's clear that I'm not going to persuade or re-educate you guys today.

Shall we just agree to disagree and go out for a beer?

Whose round is it anyway?

[noemon]

Peace brother, and know that i like your theory, and i have thought about it many times, the only thing i get stuck into with this notion, is that the Buddha said that each moment exists on its own, when a moment exists, a point exists , a point of reference, which can be perceive as the starting point of a line, a thread as a line, and hence he is in accord with our perception of "linearnism" and the dot, the cyclo, the point. So maybe there is no fuss, really.

In any case, i just went upstairs to eat, and the city has cut off the water supply, so i had soap in my hands, and turned the thing on, and nothing, so i was like wtf?

Anyway, i called my gf, and told her to grab a bottle of mineral water and come to pour it over my hands, she was in a hurry and she poured very quickly, and she amost finished it before the soap had cleansed my hands, so i shouted, Hey, relax, slow down.

That moment i realized this simlpe notion of time that is the speed that it takes for an object (water) to reach to a X object (my hands). The higher the speed the less the time, something very simple really that takes place everytime you jog, or drive but yet so hard to measure.

As she was pouring it, she was holding the bottle on an angle the larger the angle, the more increased speed it acquired, another simple notion of spacetime, that again is so fucking hard to measure.

Anyway, thats all, and peace mate.

[SCoopsdk]

That moment i realized this simlpe notion of time that is the speed that it takes for an object (water) to reach to a X object (my hands)

I realised the same thing whilst having my dump. Strange isn't it?

So you agree that time requires motion (speed/water out of the bottle and onto your hands) in order to exist, and therefore time is simply a measurement of motion and memory producing duration.

That's great

Does that mean that you're buying the beers?

[noemon]

Ofc am bying you beers, and ill be glad, but i never said otherwise, only that motion requires by default interelation.

Something that you and the man you quoted reject.

We both agree on the notion of motion, and that time is simply a measurement of this motion, we measure it because we see it as each motion relating to the next. You reject this notion. I do not. My measurement comes from this notion of related motion, and i ask you again since you reject it how do you measure motion if its infinite moments are not related to the next or the previous?

I measure it and hence prove my theory that motion is related, how do you measure it? and thus prove your theory?

[SCoopsdk]

but i never said otherwise, only that motion requires by default interelation.

Something that you and the man you quoted reject.

And your quite right of course. The problem being that interelation requires contact of some sort of present moment with another. This is dealt with in part here:

I will first consider i), where moments contact one another. On the account where two moments contact or abut, consider that two moments are side-by-side one another, one moment being the present, p0, and the other being the moment before the present, p-1. If p0, being a moment, is a location in time,[19] then any other moment contacting or abutting p0, such as p-1, which is aside p0, can only also be located at the present, for the following reasons. Moment p0, the present, is a location in time. If p0 is a location in time, then it is not located where, for example, p-1 is, since that would mean p0 is not identical to itself: p0 would be a present moment that is located in the past (a present moment that is not the present). In other words, if p0 and p-1 are distinct locations in time, then it must be the case that p0 is not where p-1 is, and p-1 is not where p0 is. But if p0 and p-1 have any sort of interplay (such as contacting, attaching or abutting to one another), then they would apparently have to coincide (fully or partially, depending on the nature of the moments) for reasons explained next. If p-1 contacts or abuts p0, then p-1, would have to “go where p0 is” in order to contact or abut p0. p-1, which is past, must also be present—it would have to be present if it is to contact or abut p0, if it is to have anything to do with p0. Also, p0 would be a present that is located in the past, which is a contradiction.[20]

So the concept of interrelatedness is impossible and false according to the view I support and the concept of time only occurs due to what is called contingent, conceptual, or empirical causation, which involves imaginary experience of duration and time, where one static time t1 contacts and/or is related to another static time t2. Of course this sort of contact or relation is impossible in reality, according to my view.

[Andres]

SCoopsdk, your source needs to learn simple Calculus.
Do you also believe that one cannot get anywhere due to Zeno's paradox?

[noemon]

I read this part, when i read your link, sorry but i do not see any proof in this, measure it and thus prove it.

Its as simple as that.

He just assigns units Po and P1 to the moments like we all do, with 1, 2, 3, but he considers that 1+1 is not equal to 2 but to 1 and that there is no plus, that they are not interelated, that one goes and is being replaced by another one, a very simple concept that everyone can think of, however i ask you again, what is the purpose of this definition? to cancel calculus and send us back to the dinosaurs? What is its purpose? What does one gain from this?

Its as simple as that. At least philosophically reason it, for practically it is simply impossible.

[MB.]

Ugh. Please tell me that wasn't a reference to the four fundamental forces of physics

Theodore, why can't Time/change be a force? Excuse my naivety, of course.

[noemon]

Because 4 is the magic number.

4 numbers construct the infinity.

1+2+3+4=10 and once you got the 10 you can extend the combinations to infinity.

Once you got the four you can construct the infinity. You do not need more. And hence 4 forces construct the infinty and we know all about them, through our scientific method. And more....because the axiomatic system divides the circle, the Holon to 4 pieces., the division of the Holon the DIA/HOLON.

But essentially, because Time or space is not a force, if it was it would belong to the first 4.
It is an assigned dimension, it does not exist independently.

It simply does not exist, it is a term that we use to define the relation between matter.

[Andres]

1+2+3+4=10 and once you got the 10 you can extend the combinations to infinity.

There is no need to start with 1, 2, 3, 4 to construct the Natural numbers, one can do it quite simply from set theory.

[noemon]

There is no need to start with 1, 2, 3, 4 to construct the Natural numbers, one can do it quite simply from set theory.

exactly , see my answer better:

Once you got the four you can construct the infinity. You do not need more. And hence 4 forces construct the infinty and we know all about them, through our scientific method. And more....because the axiomatic system divides the circle, the Holon to 4 pieces.

And your article:

In mathematics, there are a number of ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field.

And also:

The synthetic approach axiomatically defines the real number system as a complete ordered field. Precisely, this means the following. A model for the real number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and * on R (called addition and multiplication, resp.), a binary relation ≤ on R, satisfying the following properties.

1. (R, +, *) forms a field. In other words,

For all x, y, and z in R, x + (y + z) = (x + y) + z and x * (y * z) = (x * y) * z. (associativity of addition and multiplication)
For all x and y in R, x + y = y + x and x * y = y * x. (commutativity of addition and multiplication)
For all x, y, and z in R, x * (y + z) = (x * y) + (x * z). (distributivity of multiplication over addition)
For all x in R, x + 0 = x. (existence of additive identity)
0 is not equal to 1, and for all x in R, x * 1 = x. (existence of multiplicative identity)
For every x in R, there exists an element −x in R, such that x + (−x) = 0. (existence of additive inverses)
For every x ≠ 0 in R, there exists an element x−1 in R, such that x * x−1 = 1. (existence of multiplicative inverses)
2. (R, ≤) forms a totally ordered set. In other words,

For all x in R, x ≤ x. (reflexivity)
For all x and y in R, if x ≤ y and y ≤ x, then x = y. (antisymmetry)
For all x, y, and z in R, if x ≤ y and y ≤ z, then x ≤ z. (transitivity)
For all x and y in R, x ≤ y or y ≤ x. (totalness)
3. The field operations + and * on R are compatible with the order ≤. In other words,

For all x, y and z in R, if x ≤ y, then x + z ≤ y + z. (preservation of order under addition)
For all x and y in R, if 0 ≤ x and 0 ≤ y, then 0 ≤ x * y (preservation of order under multiplication)
4. The order ≤ is complete in the following sense: every non-empty subset of R bounded above has a least upper bound. In other words,

If A is a non-empty subset of R, and if A has an upper bound, then A has an upper bound u, such that for every upper bound v of A, u ≤ v.
When we say that any two models of the above axioms are isomorphic, we mean that for any two models (R, 0R, 1R, +R, *R, ≤R) and (S, 0S, 1S, +S, *S, ≤S), there is a bijection f : R → S preserving both the field operations and the order. Explicitly,

f is both 1-1 and onto.
f(0R) = 0S and f(1R) = 1S.
For all x and y in R, f(x +R y) = f(x) +S f(y) and f(x *R y) = f(x) *S f(y).
For all x and y in R, x ≤R y if and only if f(x) ≤S f(y).
The final axiom above is most crucial. Without this axiom, we simply have the axioms which define an ordered field, and there are many non-isomorphic models which satisfy these axioms. However, when the completeness axiom is added, it can be shown that any two models must be isomorphic, and so in this sense, there is only one complete ordered field.

And the dot, the point of reference to distinguish.