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# Of what does Space consist?

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Grames,

You said motionless is "0 distance-traveled over any time interval". The problem is, the term "time interval" invokes motion. How did you measure this time interval? You watched a clock hand, or a photon went from an atom to your eye, or whatever. Therefore, the object in question moved relative to *something* (the clock hand, the photon, etc.) even though it may not have moved relative to your ruler.

In order for an object to have 0 velocity relative to anything (like a ruler) it must have a finite velocity relative to something (like a clock hand). The object it physically at TWO locations and may have 0 velocity relative to the ruler but not relative to everything! Ironically in order to have 0 velocity relative to anything an object must have a finite velocity relative to something. Which of course means it's not motionless, by definition.

No I am not falling into the Fletcher's paradox. That's moronic and I don't even care to go into the fallacies in it.

I reiterate. Motionless is defined as ONE location of an object. That's it. As soon as you say anything about a "time interval" you are invoking motion (2 locations) and the object is no longer motionless by definition. Saying anything about "time interval" or "0 velocity" in the definition of "motionless" is a nonsensical contradiction. Both of them imply that the object in question is moving relative to something.

How do you measure "one location"? What coordinate system, from which rest frame? Because for any frame in which you can say an object is motionless because it has only one location, I can find a frame in which it is in motion because it is changing locations. This is exactly the same issue as above where you said "Ironically in order to have 0 velocity relative to anything an object must have a finite velocity relative to something."

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In other words, there is no such thing as "motionless." There is no such thing as "true position, in the absolute sense" - or "true velocity, in the absolute sense." One can observe locations and velocities relative to his own, but he cannot ascertain the "true" positions or velocities of either other objects or himself, because there is no such thing.

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Nothingness, non-existence, void.

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How do you measure "one location"? What coordinate system, from which rest frame? Because for any frame in which you can say an object is motionless because it has only one location, I can find a frame in which it is in motion because it is changing locations. This is exactly the same issue as above where you said "Ironically in order to have 0 velocity relative to anything an object must have a finite velocity relative to something."

The definition of "motion" or "motionless" has nothing to do with measurement. Do we have to perform a measurement for an object to move? Of course not! If all humans died object still move. The definition of motion is simply "2 locations of an object". Nobody has to do any observing or measuring. Nature doesn't care about measurements, rest frames, preferred frames, etc. If the object was at 2 or more locations, it moved by definition, whether anyone was around to see it do so or not.

Velocity and time invoke measurements and reference frames. Velocity demands a consciousness to record/remember the locations, and possibly look at their watch or a clock. A human needs to define an inertial frame, a coordinate system, etc. Nature doesn't. An object has location all on its own. Is it possible to "measure one location"? Since location is "the set of distances from an object to every other object" it's doubtful that a human will ever measure location at all. What we do measure is "distance-traveled". The distance-traveled by the leading edge of my tape, my ruler, a runner, etc relative to the table, the road, etc. We use distance-traveled to infer velocity. The man ran 4 miles while the long hand moved 2*pi radians, we have two distances-traveled of two objects respectively. We have *TWO* sets of distances traveled: (0,0) ; (4,2*pi)

The point is that instantaneous invokes a single location whereas motion invokes at least 2. The words are mutually irreconcilable.

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Nature doesn't care about measurements, rest frames, preferred frames, etc.

Demonstrably false - at least in the way you meant the sentence.

It is not possible for anyone to attempt to discover his own location or his own velocity, let alone the position or velocity of anything else. The reason for this is: there is no such thing as "absolute position" or "absolute velocity." They do not exist. As things which do not exist, they are not available to observation. This statement is an integration from all of the facts we know about the universe - not a single one of which implies the possibility of discovering a body's "absolute position" or "absolute velocity" (and many of which imply their nonexistence).

All that is open to a person is to calculate his own position and velocity with respect to a second "reference" body (such as the Earth) and then to calculate the position and velocity of another object with respect to his own position and velocity or with respect to those of the "reference" body.

Thus, "rest frames" and "coordinate systems."

Edited by y_feldblum
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The definition of "motion" or "motionless" has nothing to do with measurement.

I'll be damned, but you are a troll. Well done sir, well done. But you've exposed yourself at last so we're done here.

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I'll be damned, but you are a troll. Well done sir, well done. But you've exposed yourself at last so we're done here.

You can't move unless you take a measurement?

Once again, an object doesn't care about reference frames. Do you two honestly think a block of wood looks around and decides which inertial frame to be in?

If one object in the universe moves, every other object moves by definition, because they were at at least 2 locations. How hard is this!?

Additionally, despite all the disagreeing with me, not a single person has reconciled "instantaneous" with "motion" or "velocity". Quite the opposite. I've presented copious arguments and not a single valid counterargument. Not a single person has defined "instantaneous" and "velocity" or "motion" such that the two words are not mutually irreconcilable.

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y_feldblum:

IIf an object was at more than one location it moved by definition, and so did every other object in the universe, by definition. We don't need to invoke reference frames or observers to define motion this way. Velocity and time are relative to a specific "frame". Time/velocity = motion + observer. Motion = 2 or more locations of an object. Motionless = 1 location of an object.

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y_feldblum:

IIf an object was at more than one location it moved by definition, and so did every other object in the universe, by definition. We don't need to invoke reference frames or observers to define motion this way. Velocity and time are relative to a specific "frame". Time/velocity = motion + observer. Motion = 2 or more locations of an object. Motionless = 1 location of an object.

Every object is at all locations, all the time. Because location is not absolute. You need to invoke reference frames in order to measure an object's position with respect to your own or with respect to another object's position. The same applies to velocity.

You don't have to take my word for it. Feel free to attempt to discover your own absolute location or velocity (not your location or velocity with respect to the planet Earth, but with respect to the entirety of the universe). You may make use of anything you wish, including scientific instruments and scientific books. But you will never succeed, because success is outside of the realm of possibility.

As Newton and Einstein observed in different contexts, it is impossible to set up a scientific experiment which will permit one to discover his own absolute location or his own absolute motion.

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It seems to me that location has no meaning apart from distance, and distance is a relationship between two entities. This means that absolute location (i.e. the location of an entity without reference to other entities) is meaningless.

A question for the more experienced in epistemology: Does the above mean that "absolute location" is a stolen concept?

I suppose one could define absolute location as the location of an entity relative to some primary entity, but I defy anyone to justify the selection of some entity as the primary entity.

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The primary entity should be Jesus.

...

Or maybe not. Maybe if only he were a raptor. Then it'd be awesome. We could get paleontology and physics and religion into the same sphere of thought.

Yes, what I said in this post is just as meaningful as all this banter about "instantaneous velocity being contradictory".

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"Every object is at all locations, all the time." -- y_feldblum

This is nonsense. An object cannot be both here and there. This is simple locality.

"You need to invoke reference frames in order to measure an object's position with respect to your own or with respect to another object's position. The same applies to velocity." -y_feldblum

But I am not measuring, I am defining the word "motion". Certainly objects move even when nobody's taking a measurement. On its own (when we're not looking and thinking) an object just has shape and location. These are the two most essential qualities of existent entities and have nothing to do with measurement. At any given instant an object is at a specific location, i.e. the set of distances from it to every other object. An object does not have to engage in a measurement, it has shape and location all on its own.

"It seems to me that location has no meaning apart from distance, and distance is a relationship between two entities. This means that absolute location (i.e. the location of an entity without reference to other entities) is meaningless." -- Jake

Of course, location is the set of distances from an object to every other object. Of course it's a relationship between entities. Where location and distance differ is that distance is a relationship between TWO entities and location is a relationship between an entity and every other entity.

'

Of course the object has an "absolute location". There is a specific distance from it to every other entity in the universe.

"I suppose one could define absolute location as the location of an entity relative to some primary entity, but I defy anyone to justify the selection of some entity as the primary entity." -- Jake

Nope, we just need the definition of location.

Distance is a static concept. At an instant there is a specific distance between two entities and, by extension, a specific location. This doesn't involve a measurement.

Often the equivocation is made, because of loose language in relativity, between distance and distance-traveled. The latter is a dynamic concept. In relativity the latter is what changes in different measurements. Relativity claims that, because I measure a different distance-traveled by a photon from A to B, that the physical distance between A and B actually changed. This is the conceptual error. Relativity only shows that *distance-traveled* is based on perspective. This dynamic concept has nothing to do with the incident issue we are discussing, which involves the static concepts of location and distance.

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"Every object is at all locations, all the time." -- y_feldblum

This is nonsense. An object cannot be both here and there. This is simple locality.

You missed the meaning of my statement, entirely. The meaning was: "location" cannot in an absolute sense be defined. The word has no meaning. The word certainly has meaning when describing the displacement of one entity with respect to another entity.

"You need to invoke reference frames in order to measure an object's position with respect to your own or with respect to another object's position. The same applies to velocity." -y_feldblum

But I am not measuring, I am defining the word "motion". Certainly objects move even when nobody's taking a measurement.

They move with respect to other objects, whether or not anyone is looking. Likewise, they do not move with respect to absolute stillness, whether or not anyone is looking, because there is no such state as absolute stillness.

On its own (when we're not looking and thinking) an object just has shape and location. These are the two most essential qualities of existent entities and have nothing to do with measurement. At any given instant an object is at a specific location, i.e. the set of distances from it to every other object. An object does not have to engage in a measurement, it has shape and location all on its own.

Shape is a primary of sense-perception. Location with respect to oneself is also a primary of sense-perception. But location in an absolute sense is not. You cannot observe location in an absolute sense. You have no way to define it. The entirety of the field of physics has no way to define it. Newtonian physics rests on the demolition of the idea of absolute location.

"It seems to me that location has no meaning apart from distance, and distance is a relationship between two entities. This means that absolute location (i.e. the location of an entity without reference to other entities) is meaningless." -- Jake

Of course, location is the set of distances from an object to every other object. Of course it's a relationship between entities. Where location and distance differ is that distance is a relationship between TWO entities and location is a relationship between an entity and every other entity.

'

Of course the object has an "absolute location". There is a specific distance from it to every other entity in the universe.

That is not the meaning of "absolute location."

"I suppose one could define absolute location as the location of an entity relative to some primary entity, but I defy anyone to justify the selection of some entity as the primary entity." -- Jake

Nope, we just need the definition of location.

The concept of "absolute location" rests upon the concept of some primary entity which is by its nature absolutely still.

Distance is a static concept. At an instant there is a specific distance between two entities and, by extension, a specific location. This doesn't involve a measurement.

Often the equivocation is made, because of loose language in relativity, between distance and distance-traveled. The latter is a dynamic concept. In relativity the latter is what changes in different measurements. Relativity claims that, because I measure a different distance-traveled by a photon from A to B, that the physical distance between A and B actually changed. This is the conceptual error. Relativity only shows that *distance-traveled* is based on perspective. This dynamic concept has nothing to do with the incident issue we are discussing, which involves the static concepts of location and distance.

Distance, and distance-traveled are identical in meaning, when the journey is along a geodesic (which, in a non-curved space, is a straight line). A photon travels along a geodesic. In fact, relativity argues that all objects not under the influence of a force (where gravity is not a force) travel along geodesics and cannot do otherwise, light included.

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"You missed the meaning of my statement, entirely. The meaning was: "location" cannot in an absolute sense be defined. The word has no meaning. The word certainly has meaning when describing the displacement of one entity with respect to another entity." -y_feld

You mean it can't, in an absolute sense, be measured/quantified. It absolutely has meaning, simply "the set of distances from an entity to every other entity". The entity doesn't care if you took a measurement or if a consciousness is alive. The fact that it is a distance from every other entity is not dependent upon human intervention.

"Displacement" is an action, a verb. It deals with motion. Distance-traveled has meaning when describing the displacement of one entity with respect to another. Distance is a static concept.

"Shape is a primary of sense-perception. Location with respect to oneself is also a primary of sense-perception. But location in an absolute sense is not. You cannot observe location in an absolute sense. You have no way to define it. The entirety of the field of physics has no way to define it. Newtonian physics rests on the demolition of the idea of absolute location."

Shape means finite, i.e. has a border. Shape is also a static concept. It does not depend on human intervention either or on motion. Humans may describe shapes in terms of their sense perceptions, but entities have shape regardless of this sense perception.

Location is the set of distances from an entity to every other entity. Again it is not dependent upon human intervention. Hopefully an entity was finite and at some location whether we ever did or are capable of measuring or sensing it.

I never said I could "observe location in an absolute sense". I am *defining* the words entity, shape, location, and motion so that I can use these terms consistently and unambiguously. You and I may measure different distances-traveled and come to different conclusions about location or distance, but by definition the distance/location of an entity was specific. Measurement and observation may be dependent upon the observer and his/her perspective, but the actual primary qualities of an entity are not. We can resolve our different conclusions through reason and logic, i.e. by discovering that differences in perspective are a result of different assumptions (am I stationary or are you stationary, and is the speed of light constant or not?). But hopefully the entity itself didn't care what our assumptions, premises, measurements, or calculations were. It was just at a specific location.

So, physics does indeed have a way to define location, it's in mathematics/measurement which we cannot. In physics we first define our terms objectively, in a way that does not depend on testimonials from individuals. Then we can proceed to test ideas empirically. If we find empirically that we can only measure relative motion, so be it. Nature does not bend to our wishes or our limitations. Nature laughs at our insistence that She conform only to our ability to measure while discarding our ability to think, imagine, and reason.

"That is not the meaning of "absolute location."

So you've spent all this time arguing against your own definition of absolute location instead of the one I presented? What's wrong with the one I have been repeating over and over? Besides the unfortunate aspect that a human cannot quantify it?

"The concept of "absolute location" rests upon the concept of some primary entity which is by its nature absolutely still."

You mean your definition of absolute location. All this time I thought you were disagreeing with me, when you were really just disagreeing with your own definition of absolute location.

"Distance, and distance-traveled are identical in meaning, when the journey is along a geodesic (which, in a non-curved space, is a straight line).

This is incorrect, and the source of a lot of misintegration.

Distance:

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0

0

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Distance-traveled:

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0

0

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0

0

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Distance is merely the separation between two objects. Distance-traveled is the distance between an object and a now imaginary version of that object. The latter is dynamic, it requires at least two locations of an entity. The former is static, each entity is at a location. An observer moving to the side will infer a diagonal distance-traveled, which will necessarily be a larger quantity. Does this mean the actual separation between A and B expanded? No, it means the observer is moving away (or the objects are moving away from the observer) which means there is an observed horizontal component of the velocity. The distance-traveled relative to the observer is greater because s/he is in relative motion, of course. But the static separation between A and B does not spontaneously fluctuate depending on who decides to take a measurement. This is the difference between distance-traveled and distance.

"A photon travels along a geodesic. In fact, relativity argues that all objects not under the influence of a force (where gravity is not a force) travel along geodesics and cannot do otherwise, light included."

Indeed they do, relativity's empirical success indicates objects traverse curvilinear paths and that the degree of curvature is dependent upon the empirical parameter "mass" and their relative velocity. This has nothing to do with distance. It has solely to do with motion.

Another equivocation that is often made is between length and distance. Length is a quality of ONE object, its extent in a direction. Distance is a quality of TWO objects, their separation. "Distance lies between lengths" is a general rule. Length is to 1 as distance is to 2.

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Since nobody has taken up the challenge of actually defining "instantaneous" and "velocity" such that they are not mutually exclusive, I can only conclude that the resistance to recognizing this simple observation is the inertia that comes from multi-generational habitual usage of the phrase.

The objection that is routinely raised seems to be the issue of measurement. That, because I cannot measure an entity's location (location = set of distances from an entity to every other entity) this concept is rendered invalid (some seem to like to call it "absolute location"). It is quite easy to show why this is both arbitrary and illogical. Concepts are often defined in terms of what one could imagine doing, although it may be practically, physically, or theoretically impossible.

In the case of absolute location we imagine freezing the universe. A person lays down bricks from an entity to every other entity. S/he now knows its location. Of course this is impossible, but nevertheless we can imagine it. Additionally, one argues that, just because this location is not measurable in practice, does not in fact mean the entity does not have this location. Why should Nature care what is practically measurable to Man? Does it make any sense to discard concepts because they are, for whatever reason, considered physically unrealizable? Theoretical physics constantly pushes the envelope in terms of defining concepts of dubious physicality. A primary difference is that, at least in the case I am describing, we can imagine it; while many concepts of theoretical physics (4D space-time, 10/11D entities, etc.) can't even be visualized.

Imagine we all consider that it is impossible to jump over the Eiffel Tower. We come to this conclusion by simply looking, we conclude that it's practically impossible. A physicist, for whatever reason, decides to define 1 glerg as the force exerted by a 70 kg human on contact with the earth after jumping over the Eiffel Tower with minimum effort. Is a glerg of force physically unrealizable? Of course not. Suppose a team of biologists and physicists somehow "prove" that it is not only practically impossible, but beyond the bounds of any living entity to jump over the Eiffel Tower. Is a glerg now physically unrealizable?

With this definition of location we define motion as simply 2 or more locations of an entity. Now it is actually impossible for an observer O to conclude that an entity A is motionless.

"Instantaneous" means at an instant, a single moment, a single "time", i.e. a single location. Motion and velocity necessarily invoke two locations. These are mutually exclusive.

Calculus is the mathematics of infinitesimals and uncountables, not of 0 and infinity. This is a crucial distinction. If there were true "instantaneous velocity" then the concept of limits would never have been necessary, we would simply calculate/measure an entity's velocity at a single location and be done with it. The elegance of the math of calculus allows one to calculate the velocity over an arbitrarily small interval, but never "no interval".

Edited by altonhare
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Since nobody has taken up the challenge of actually defining "instantaneous" and "velocity" such that they are not mutually exclusive, I can only conclude that the resistance to recognizing this simple observation is the inertia that comes from multi-generational habitual usage of the phrase.

Isaac Newton has already done this, with the invention of the differential calculus, with his application of the differential calculus to problems in physics, and with his systematization of the physics of his era under just a few abstractions. I refer you to his definition.

The objection that is routinely raised seems to be the issue of measurement. That, because I cannot measure an entity's location (location = set of distances from an entity to every other entity) this concept is rendered invalid (some seem to like to call it "absolute location"). It is quite easy to show why this is both arbitrary and illogical. Concepts are often defined in terms of what one could imagine doing, although it may be practically, physically, or theoretically impossible.

Location is not a set of distances. Location has a very precise mathematical meaning, and from high school geometry through theoretical astrophysics, location precedes distance. Moreover, displacement (relative location) precedes distance as well. Nevertheless, "absolute location" - that is, the notion that the universe itself is unmoving and has a center, and that the locations and velocities of other objects are to be defined in terms of their locations relative to the position and velocity of the universe itself - is a faulty concept, even though the general way we think about locations and velocities are as relative to the center (or surface) and motion of the Earth.

In the case of absolute location we imagine freezing the universe. A person lays down bricks from an entity to every other entity. S/he now knows its location.

This illustration is wrong-headed and, to a physicist, impossible to imagine. Not because this is practically impossible, but because the concept "freezing the universe" doesn't make snese, and "displacement from an entity to every other entity" doesn't make sense when one takes into account Special Relativity.

With this definition of location we define motion as simply 2 or more locations of an entity. Now it is actually impossible for an observer O to conclude that an entity A is motionless.

That's not what motion is. It is merely how we observe it. Objects move in and of themselves, whether or not there is anyone to pick out what positions they happen to occupy at any particular pair of times.

"Instantaneous" means at an instant, a single moment, a single "time", i.e. a single location. Motion and velocity necessarily invoke two locations. These are mutually exclusive.

And since your understanding of motion is wrong, your objection to "instantaneous motion" falls apart.

Calculus is the mathematics of infinitesimals and uncountables, not of 0 and infinity.

And a wrong one, betraying a fundamentally misguided understanding of the calculus.

If there were true "instantaneous velocity" then the concept of limits would never have been necessary, we would simply calculate/measure an entity's velocity at a single location and be done with it. The elegance of the math of calculus allows one to calculate the velocity over an arbitrarily small interval, but never "no interval".

The method of calculating instantaneous velocity does not determine what instantaneous velocity is. We calculate instantaneous an object's velocity based on observing what an object's location is at various points in time, because that is the method of calculation that our brains are equipped to handle.

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hey AMAI. Space consists of nothing. It is emptyness. If you find something *in* space, what you are looking at is *something*, but not space. Unless there IS something which we now call space, but in that case we're talking about *something*, making it not space.

Rob

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Location is not a set of distances. Location has a very precise mathematical meaning, and from high school geometry through theoretical astrophysics, location precedes distance. Moreover, displacement (relative location) precedes distance as well.
I don't understand this idea of location preceding distance. What is your precise mathematical meaning of location?

Anytime you specify an object's location, you must specify its location in relation to something else (e.g. a chosen origin or another object). In any orthogonal basis, specifying a location requires as many coordinates as dimensions, and at least one of those coordinates must be a distance. So, distance is conceptually prior to location, because the definition of location involves distance(s).

I'm assuming that when you defined displacement as "relative location", you meant time-relative as opposed to space-relative. Since, as you stated later in your post: there is no "absolute" location.

Here are two Physics definitions of "displacement" that I found (bold mine):

3. Physics.

c. the linear or angular distance in a given direction between a body or point and a reference position.

3. Physics

a. A vector or the magnitude of a vector from the initial position to a subsequent position assumed by a body.

Both of these are definitions of displacement (involving distance) which I am familiar using through undergraduate Physics and Math. I remember no other uses of the term.

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I don't understand this idea of location preceding distance. What is your precise mathematical meaning of location?

Anytime you specify an object's location, you must specify its location in relation to something else (e.g. a chosen origin or another object).

Correct.

In any orthogonal basis, specifying a location requires as many coordinates as dimensions, and at least one of those coordinates must be a distance. So, distance is conceptually prior to location, because the definition of location involves distance(s).

None of the coordinates need be distance. The concept of a vector space precedes the concept of a metric over that vector space (a metric is a particular function Vector X Vector --> Scalar which defines distance for that vector space). Vectors do not have distance in and of themselves; a vector v only has a distance in the context of a metric m, and that distance is defined to be m(v, v). Displacement is a vector.

I'm assuming that when you defined displacement as "relative location", you meant time-relative as opposed to space-relative. Since, as you stated later in your post: there is no "absolute" location.

Displacement is, in Newtonian physics, space-relative. In Special Relativity, it is spacetime-relative.

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None of the coordinates need be distance. The concept of a vector space precedes the concept of a metric over that vector space (a metric is a particular function Vector X Vector --> Scalar which defines distance for that vector space). Vectors do not have distance in and of themselves; a vector v only has a distance in the context of a metric m, and that distance is defined to be m(v, v). Displacement is a vector.

Are not scalars conceptually prior to vectors? Vectors are after all comprised of pair-wise (or higher dimensioned) scalars.

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Isaac Newton has already done this, with the invention of the differential calculus, with his application of the differential calculus to problems in physics, and with his systematization of the physics of his era under just a few abstractions. I refer you to his definition.

Impossible, Newton did not even define "time, space, place, or motion".

Location is not a set of distances. Location has a very precise mathematical meaning, and from high school geometry through theoretical astrophysics, location precedes distance. Moreover, displacement (relative location) precedes distance as well.

This statement simply asserts that what I say is not the case and further that feldblum learned something different than what I said in school.

Nevertheless, "absolute location" - that is, the notion that the universe itself is unmoving and has a center, and that the locations and velocities of other objects are to be defined in terms of their locations relative to the position and velocity of the universe itself - is a faulty concept, even though the general way we think about locations and velocities are as relative to the center (or surface) and motion of the Earth.

This is completely made up in the sense that it has nothing to do with anything I said.

This illustration is wrong-headed and, to a physicist, impossible to imagine. Not because this is practically impossible, but because the concept "freezing the universe" doesn't make snese,

Another assertion that it is simply wrong, that the person can't imagine it, and it doesn't make sense to him. If it doesn't even make sense to you, then you can't even evaluate it as right or wrong.

Imagining freezing the universe is easy. Imagine you are standing away from everything in the universe with a camera and are taking photographs. The camera is special in that it detects everything in the universe simultaneously. Each photograph is an example of what one sees if they "freeze the universe". The issue here is not whether this can be actualized by a human, but whether we can imagine it.

and "displacement from an entity to every other entity" doesn't make sense when one takes into account Special Relativity.

Another assertion that it simply doesn't make sense to the poster. Still worse, feld misquotes me by using the word "displacement", which I have never actually used in this thread. He introduces this word into my definition of location for no known reason, then uses his own interpretation of this word to state that the definition doesn't make sense.

This argument is a straw man because it simply asserts I'm wrong and becuase displacement != distance as far as I'm concerned, which renders his "quotation" of me irrelevant.

SR is a theory dealing with what a person will actually measure. Here we are talking about imagining what is, irrespective of our ability to measure. At any instant every entity is at some distance from every other entity.

That's not what motion is. It is merely how we observe it. Objects move in and of themselves, whether or not there is anyone to pick out what positions they happen to occupy at any particular pair of times.

I appreciate skepticism, but not for skepticism's sake. If you don't think motion is two or more locations of an object the onus is on you to pose your own definition of motion. The definition I gave does not depend on observers. Whether you're looking or not an entity has location. If it was at more than one location it moved, again whether you look at it or not.

If you think "two locations of an object" is "merely how we observe it" then what is motion actually? Change? A rate? A rate of change? How are those better?

And since your understanding of motion is wrong, your objection to "instantaneous motion" falls apart.

More unjustified assertions that I'm wrong.

And a wrong one, betraying a fundamentally misguided understanding of the calculus.

Unjustified claim that I do not understand calculus.

The method of calculating instantaneous velocity does not determine what instantaneous velocity is.

Then what IS instantaneous velocity? I've ventured a definition, though nobody else yet has.

We calculate instantaneous an object's velocity based on observing what an object's location is at various points in time, because that is the method of calculation that our brains are equipped to handle.

location: 1 2 4 8 16

"time": 0 1 2 3 4 5

Where's "instantaneous velocity"?

I don't understand this idea of location preceding distance. What is your precise mathematical meaning of location?

Mine was not a mathematical definition, it's a physical one. Physics precedes mathematics.

As far as a conceptual hierarchy, we present these concepts in the way it is easiest for a person to understand. Nature doesn't have to define this in terms of that, that's what humans do. It makes no sense to talk about location "preceding" distance in Nature.

In any event, it's easiest to define distance, first, as the separation between two entities. Then location is simply the set of distances from one entity to every other entity.

Anytime you specify an object's location, you must specify its location in relation to something else (e.g. a chosen origin or another object).

Sure sure. An object's location is in relation to every other object in the universe. Its distance from every other object.

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Are not scalars conceptually prior to vectors? Vectors are after all comprised of pair-wise (or higher dimensioned) scalars.

A vector space is indeed a vector space over a field, and that field is the Scalars with respect to that vector space.

However, vectors do not have any intrinsic notion of magnitude.

Some vector spaces are metric vector spaces, in that there is an additional notion of a metric, which includes the notion of magnitude of a vector. One needs to come up with a metric over the vector space of displacements in order to define what distance means (distance, a scalar, is defined to be the magnitude of displacement, a vector).

Sample formulas:

The distance from location p to location q is |p - q|2 = g(p - q, p - q), where g is the metric.

In Newtonian mechanics, the metric g(u, v) = uxvx + uyvy + uzvz.

In Special Relativity, the metric g(u, v) = utvt - uxvx - uyvy - uzvz.

As you can see, magnitude is defined as the square root of the metric applied to the same vector twice, and distance is the square root of the metric applied to the same displacement twice. But different metrics hold in different contexts, and are not embedded within the vectors themselves. Special Relativity will give different answers for "distance", but will give identical answers for "displacement".

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A vector space is indeed a vector space over a field, and that field is the Scalars with respect to that vector space.

However, vectors do not have any intrinsic notion of magnitude.

Some vector spaces are metric vector spaces, in that there is an additional notion of a metric, which includes the notion of magnitude of a vector. One needs to come up with a metric over the vector space of displacements in order to define what distance means (distance, a scalar, is defined to be the magnitude of displacement, a vector).

Sample formulas:

The distance from location p to location q is |p - q|2 = g(p - q, p - q), where g is the metric.

In Newtonian mechanics, the metric g(u, v) = uxvx + uyvy + uzvz.

In Special Relativity, the metric g(u, v) = utvt - uxvx - uyvy - uzvz.

As you can see, magnitude is defined as the square root of the metric applied to the same vector twice, and distance is the square root of the metric applied to the same displacement twice. But different metrics hold in different contexts, and are not embedded within the vectors themselves. Special Relativity will give different answers for "distance", but will give identical answers for "displacement".

Before we can quantify some concept via some combination of theories and math, we first have to qualify this concept. i.e. define it and understand it. Otherwise all we have before us is a mathematical description.

Distance is a static concept, we use this word as a noun. It indicates the separation between two objects: 0 0

Displace is a dynamic concept, a verb, which we would like to use as a noun in casual language for simplicity. So we made the word "displacement". It's obvious that, although the dictionary calls "displacement" a noun, the meaning of the word embodies the notion of a verb:

disâ‹…placeâ‹…ment

â€‚ â€‚/dÉªsËˆpleÉªsmÉ™nt/ Show Spelled Pronunciation [dis-pleys-muhnt] Show IPA

â€“noun

1. the act of displacing.

The act of displacing. These are both verbs. Displacement indicates an action, i.e. a motion.

3. Physics.

a. the displacing in space of one mass by another.

b. the weight or the volume of fluid displaced by a floating or submerged body. Compare Archimedes' principle.

Displacing, displaced, etc.

Displacement indicates TWO separations between objects: 0 0 ; 0 0

Displacement = distance-traveled = quantification of motion

Since instantaneous embodies a single location one cannot logically talk about instantaneous displacement anymore than instantaneous motion, velocity, etc.

These are qualitative, conceptual issues. Math, as a quantitative rational exercise, cannot resolve these issues. Every mathematical deduction is based on some premises. If you have not defined those premises rigorously and unambiguously your math is, at best, a correlative description with no concrete connection.

Here, I am discussing the qualitative premises, what comes before the math. Before we can speak of vectors and scalars and quantities of velocity and fields etc. we need to unambiguously distinguish qualitatively between length, distance, distance-traveled, location, motion, etc. Length is a quality of a single object, specifically the quality of extent. Distance is a quality of two objects, specifically of separation. Location consists of the distance from an object to every other object. Motion is two or more locations. Instantaneous invokes a single location. We can write all the equations we want and match them up quantitatively with observations and it will not change this simple qualitative induction. So far everyone just wants to disagree, for reasons I can't fathom, and furthermore offers no alternative, which is even more unfathomable. Everyone just kinda disagrees, makes unjustified assertions, and finally invokes Newton's authority, as if it were impossible for him to make a mistake. In actuality it was inevitable that he would make some mistakes in tying his equations of motion to reality since he did not define motion in the first place.

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Displacement indicates TWO separations between objects: 0 0 ; 0_ 0
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The definition of displacement which I am using is (this is my own definition):

displacement: n. the relationship between two points in space; the difference between two points in space; the vector from one point in space to another (mathematics, physics).

See Wikipedia for confirmation of the use of the word in this manner. Note that the article uses the word displacement in the context of analyzing motion because the word is predominant in kinematics, the study of motion; however, the concept that the word signifies is not limited to kinematics but extends to all aspects of geometrical mathematics and physics, where it no longer carries the connotation of the motion of objects.

Displacement does not mean distance traveled. It does not mean straight-line distance, either. It does not mean distance in any sense of the term. The reason is, distance is but one part of the entire relationship between two points in space, and displacement is the entire relationship between two points in space. Moreover, distance is not intrinsically a part of displacement, and in fact many branches of geometrical mathematics and physics deal with displacement which does not carry with it any conception of distance (a vector space without a metric).

Motion is a form of action. Like all action, it is continuous, not discrete. That means, if an object is moving, then it is moving at every instant during the time period in which it is moving. Motion does not describe discrete observations of an object. It describes what the object is doing continuously, at every instant. We, as humans, only have the power to observe where an object is at discrete instants in time; we are unable to observe continuous motion. Before Isaac Newton, we did not even have the tools to understand continuous motion (thus Xeno's paradox). But Isaac Newton discovered a powerful tool to enable us to understand continuous motion based on our prior understanding of discrete observations: differentiation.

Objects move continuously. We, given our limited powers of vision but our infinite powers of differentiation, proceed as follows. I know how to calculate an object's average velocity over a range of position and a range of time, given that I wrote down the initial conditions and the final conditions of my experiment. I have (xinitial, tinitial) and (xfinal, yfinal), so I calculate the average velocity over this range of position and range of time as (xfinal - xinitial) / (tfinal - xfinal). Given that I know how to perform this calculation with this formula, how do I use the same formula to calculate an objects precise position and velocity at any point in space or time? Isaac Newton introduced the rigorous mathematical technique of taking the limit, and showed how taking the limit is the answer to our question. Start with the formula for calculating average velocity given discrete initial and final observations, take the limit of this formula, and arrive at the formula for calculating exact velocity at any given position or time.

Recall: objects move continuously, but we do not have the power directly to observe continuous motion. Instead, we observe motion discretely, and so must amplify our sensory powers with mathematical powers in order to have a more complete understanding of the world. Using the technique of taking the limit, we can abstract from the basic concept of average velocity, which is based on discrete initial and final observations, to arrive at instantaneous velocity, which is the objects actual velocity. Again, we cannot observe an object's actual motion directly, so we must use mathematical technique to amplify what powers of observation we do happen to have, and so we use mathematics to go from average velocity to instantaneous velocity in a completely rigorous way.

(Obviously, I have left out the definitions of limit and left out all use of the term differentiation, because these have appeared in previous posts in this topic.)

Edited by y_feldblum

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