If dX=F(X) and X=F(dX) then it means at all magnitudes of a thing (X), the thing (X) is an element of itself (X). That is, irrespective of number, a set of potatoes is 'a' potatoe. In this case, in oeder to quantify X we have to use a fixed quantum of X and if integral dX=X then X is the number of quanta. If we do not quantify X in terms of number of quanta then integral of dX is 1. The problem arises when dX=F(X) but X is not the function of dX. In this case integral of dX must occupy either space (like a gravitational field or a magnetic field) or it must occupy time (like natural radio active decay) and any number of successive differentiation of the integral of X [when dX=F(X)but X is not the function of dX] will not produce a constant. Here integral of dX is not a number. Here X decreases or increases exponentially. Exponential increase is like chain reaction - increase increases, and, exponential decreases means decreases decreases. Continuous increase in decreases cannot make the value negative because the activity is like an implosion - either it must stop by itself without any external cause, or it should become a black-hole - dark areas of our knowledge. I am unable to imagine any real number which a mathematician can integrate as well as differentiate at his will. Can anybody help me?
> If dX=F(X) and X=F(dX) then it means at all magnitudes of a thing (X), > the thing (X) is an element of itself (X). That is, irrespective of > number, a set of potatoes is 'a' potatoe.
X is a variable, not a set. So, saying "X is an element of X" just doesn't make sense.
BTW, what do you mean by dX? Differential? With respect to what?
> In this case, in oeder to > quantify X we have to use a fixed quantum of X and if integral dX=X > then X is the number of quanta. If we do not quantify X in terms of > number of quanta then integral of dX is 1. > The problem arises when dX=F(X) but X is not the function of dX.
What is the function, and what is the variable? I was under the impression that F() is a function, X is a variable, and dX is just a integration symbol. Please explain.
<snip>
> I am unable to imagine any real number which a mathematician can > integrate as well as differentiate at his will. Can anybody help me?
Numbers can't be differentiated or integrated; functions can.
Please check your calculus knowledge.
-- --------------------------------------- Duran Castore (duran_cast...@yahoo.com)
> > If dX=F(X) and X=F(dX) then it means at all magnitudes of a thing (X), > > the thing (X) is an element of itself (X). That is, irrespective of > > number, a set of potatoes is 'a' potatoe.
> X is a variable, not a set. So, saying "X is an element of X" just > doesn't make sense.
> BTW, what do you mean by dX? Differential? With respect to what?
> > In this case, in oeder to > > quantify X we have to use a fixed quantum of X and if integral dX=X > > then X is the number of quanta. If we do not quantify X in terms of > > number of quanta then integral of dX is 1. > > The problem arises when dX=F(X) but X is not the function of dX.
> What is the function, and what is the variable? I was under the > impression that F() is a function, X is a variable, and dX is just a > integration symbol. Please explain.
> <snip>
> > I am unable to imagine any real number which a mathematician can > > integrate as well as differentiate at his will. Can anybody help me?
> Numbers can't be differentiated or integrated; functions can.
> Please check your calculus knowledge.
I wish to know whether the 'process' of integration has contiguity in space or in time or at least in our mind. Whether the process of integration must be in one direction in time (and space) or integration can proceed in a chaotic manner in all directions in space and by changing directions in time. If integration has to have contiguity and direction (like acceleration)then the process of integratin is irreversible and we cannot put the process of 'integration' in the reverse gear and find the differential from its integral. If we are NOT integrating or differentiating 'number' then we must be integrating 'functions'. I cannot understand what purpose mathematical symbols (+, -, / and x) can serve in the process of integration of functions. Integration is useful in geometry in the calculation of area and volume but the calculation is irreversible. I can not imagine a reversible integration of functions, other than mechanical functions - motion of interconnected parts.
<snip> > I wish to know whether the 'process' of integration has contiguity in > space or in time or at least in our mind. Whether the process of > integration must be in one direction in time (and space) or > integration can proceed in a chaotic manner in all directions in space > and by changing directions in time.
What do you mean by "contiguity"?
As far as I know, the only direction which has something to do with integration is the one of the values of the independent variable of the function being integrated. Example:
integral [0 to 1] x dx = 1/2
The integration goes from x varying from 0 to 1. If it's done from 1 to 0 instead, the result is -1/2.
> If integration has to have > contiguity and direction (like acceleration)then the process of > integratin is irreversible and we cannot put the process of > 'integration' in the reverse gear and find the differential from its > integral.
Do you know the difference between definite integral and indefinite integral? If not, please check a Calculus book.
Please check the Fundamental Theorem of Calculus, at
> If we are NOT integrating or differentiating 'number' then > we must be > integrating 'functions'. I cannot understand what purpose mathematical > symbols (+, -, / and x) can serve in the process of integration of > functions.
They are used, as operators which they are, to write the functions' formulas.
> Integration is useful in geometry in the calculation of > area and volume but the calculation is irreversible. I can not imagine > a reversible integration of functions, other than mechanical functions > - motion of interconnected parts.
"Irreversible" in what sense? One can calculate the value of a definite integral, if such value exists, and it's clearly impossible to return to the original funcion based only in one numeric result. Now, indefinite integrals yield a function as a result, and such function can be differentiated, and the result is the original function.
Hope this is clearer now,
-- --------------------------------------- Duran Castore (duran_cast...@yahoo.com)
> <snip> > > I wish to know whether the 'process' of integration has contiguity in > > space or in time or at least in our mind. Whether the process of > > integration must be in one direction in time (and space) or > > integration can proceed in a chaotic manner in all directions in space > > and by changing directions in time.
> What do you mean by "contiguity"?
> As far as I know, the only direction which has something to do with > integration is the one of the values of the independent variable of the > function being integrated. Example:
> integral [0 to 1] x dx = 1/2
> The integration goes from x varying from 0 to 1. If it's done from 1 to 0 > instead, the result is -1/2.
> > If integration has to have > > contiguity and direction (like acceleration)then the process of > > integratin is irreversible and we cannot put the process of > > 'integration' in the reverse gear and find the differential from its > > integral.
> Do you know the difference between definite integral and indefinite > integral? If not, please check a Calculus book.
> Please check the Fundamental Theorem of Calculus, at
> > If we are NOT integrating or differentiating 'number' then > > we must be > > integrating 'functions'. I cannot understand what purpose mathematical > > symbols (+, -, / and x) can serve in the process of integration of > > functions.
> They are used, as operators which they are, to write the functions' > formulas.
> > Integration is useful in geometry in the calculation of > > area and volume but the calculation is irreversible. I can not imagine > > a reversible integration of functions, other than mechanical functions > > - motion of interconnected parts.
> "Irreversible" in what sense? One can calculate the value of a definite > integral, if such value exists, and it's clearly impossible to return to > the original funcion based only in one numeric result. Now, indefinite > integrals yield a function as a result, and such function can be > differentiated, and the result is the original function.
> Hope this is clearer now,
You say: Integration goes from x varying continuously from 1 to 0. This integration is irreversible if there is continuity from 1 to 0, because then we cannot have a definite integral of 1/x when x varies from 1 to 0. When X varies from 1 to 0, 1/x varies continuously from 1 to infinity. In this case can we prove that x*1/x=1? That is, can we prove that number of numbers between 1 and 0 is same as that between 1 and infinity? I believe that we can only integrate functions of fields. Irreversibility implies that there is a continuous change is reference, continuous forgetting of the past and some times past is continuously falsified by the present during the process of integration. Any idea that has contiguity and direction (like 0 to 1) is irreversible.
>> Please check the Fundamental Theorem of Calculus, at <snip> >> "Irreversible" in what sense? One can calculate the value of a
<snip>
Please don't ignore my questions.
> You say: Integration goes from x varying continuously from 1 to 0.
Or 0 to 1, or a to b, where a and b are real numbers. Definite integral, that is.
> This integration is irreversible if there is continuity from 1 to 0, > because then we cannot have a definite integral of 1/x when x varies > from 1 to 0.
Non-sequitur; f(x) = 1/x is not continuous at x=0. But one can integrate f(x) = 1/x in the interval [e,1], e > 0, e arbitrarily small.
> When X varies from 1 to 0, 1/x varies continuously from 1 > to infinity. In this case can we prove that x*1/x=1?
x*1/x = 1 due to properties of real numbers (1 is the multiplicative identity in R). Nothing to do with the function f(x) = 1/x.
> That is, can we > prove that number of numbers between 1 and 0 is same as that between 1 > and infinity?
Yes. One can find easily a bijection between the intervals A=]0,1] and B= [1,oo[. It is, not surprisingly, f:A -> B, f(x) = 1/x (x in A). One of the intervals is just taken in reversed order.
> I believe that we can only integrate functions of fields.
Please clarify: what you mean by "field"?
> Irreversibility implies that there is a continuous change is > reference, continuous forgetting of the past and some times past is > continuously falsified by the present during the process of > integration. Any idea that has contiguity and direction (like 0 to 1) > is irreversible.
You aren't making sense. Integration of functions has nothing to do with time at all; even if one integration variable is time, the _process_ of integration does not depend on time.
As an aside: I saw, in my mail inbox, a huge Word document, whose style is suspiciouly similar to yours. Please refrain from sending such long e- mails to me without my permission.
Bye for now,
-- --------------------------------------- Duran Castore (duran_cast...@yahoo.com)
In <38af3945.0209110724.c2a9...@posting.google.com>, on 09/11/2002 at 08:24 AM, vgopa...@rediffmail.com (V.Gopal) said:
>I wish to know whether the 'process' of integration has contiguity in >space or in time or at least in our mind.
Is that supposed to mean something?
>Whether the process of >integration must be in one direction in time (and space) or >integration can proceed in a chaotic manner in all directions in >space and by changing directions in time.
Integration is a mathematical operation; it is not a physical process in space-time. The term "integration" covers several concepts, some of which refer to 1-1 mappings and some of which don't.
>I cannot understand what purpose mathematical >symbols (+, -, / and x) can serve in the process of integration of >functions.
1. Defining what we mean by integration
2. Defining a function that we would like to integrate
>I can not imagine >a reversible integration of functions, other than mechanical >functions - motion of interconnected parts.
Fourier.
-- Shmuel (Seymour J.) Metz, SysProg and JOAT Atid/2, Team OS/2, Team PL/I
Any unsolicited commercial junk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail.
I mangled my E-mail address to foil automated spammers; reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamt...@library.lspace.org
> In <38af3945.0209110724.c2a9...@posting.google.com>, on 09/11/2002 > at 08:24 AM, vgopa...@rediffmail.com (V.Gopal) said:
> >I wish to know whether the 'process' of integration has contiguity in > >space or in time or at least in our mind.
> Is that supposed to mean something?
> >Whether the process of > >integration must be in one direction in time (and space) or > >integration can proceed in a chaotic manner in all directions in > >space and by changing directions in time.
> Integration is a mathematical operation; it is not a physical process > in space-time. The term "integration" covers several concepts, some of > which refer to 1-1 mappings and some of which don't.
> >I cannot understand what purpose mathematical > >symbols (+, -, / and x) can serve in the process of integration of > >functions.
> 1. Defining what we mean by integration
> 2. Defining a function that we would like to integrate
> >I can not imagine > >a reversible integration of functions, other than mechanical > >functions - motion of interconnected parts.
> Fourier.
> -- > Shmuel (Seymour J.) Metz, SysProg and JOAT > Atid/2, Team OS/2, Team PL/I
> Any unsolicited commercial junk E-mail will be subject to legal > action. I reserve the right to publicly post or ridicule any > abusive E-mail.
> I mangled my E-mail address to foil automated spammers; reply to > domain Patriot dot net user shmuel+news to contact me. Do not > reply to spamt...@library.lspace.org
Contiguity of functions: When we conceive an organisation with a goal in mind we 'integrate functions' in our mind and then SUBSTITUTE ENTITIES FOR FUNCTIONS. In this case we are able to explain how the organizational goal is achived by 'an integration of functions of its parts'. In mathematics an integral must be a continuum in space or in space-time. The sum of series of numbers in arithmetic progression, or G.P or in H.P is not an integral. We cannot expres or communicate (the structure) of a continuum. Unform velocity is insensile and incommunicable because it is like 'time without change'. In L/T we can never prove that there is no change within T or within unit of time. L/T is average velocity. Newton took the average of the average velocities and called it 'acceleration' and assumed it to be a constant! If time average of time averages is constant then it is an endless 'time without change'. We can never express or convey continuous change like acceleration or a state of change. The whole structure of 'mathematics' is erected on a self-contradictory or paradoxical assumption: Number of cycles per cycle is constant all the time in all environments. Number of units of time per unit time is constant all the time----------. Number of units of length per unit length is constant all the time------. Number of space-time intervals, of same duration, within any two consecutive integers (0 and 1, 1 and 2, 2 and 3, 10^10 and (10^10)+1 is same and constant. 0, 1, 2, 3 ------------N, (N+1)------ are equally spaced. Number of numbers between 0 and 1, 1 and 2, 2 and 3 or between N and N +1 is same all the time. On a number line if integers are equally spaced or if 'scale' is conventional then there is a pardox - X*1/X cannot be equal to 1, because we have to accommodate the reciprocals of all the conceivable numbers between 1 and infinity within all the conceivable numbers between 1 and 0. We can understand the problem with continuity from the following example: Suppose X and Y are two sides of a rectangle and the area of the rectangle, given by XY is constant. When XY=constant, if X decreases continuously Y should increase continuously. If X decreases continuously at a constant speed 'S' (so that at the end of time T the side is X-ST, then what is the rate at which Y increases? YOU WOULD SAY that when X becomes 1/2, Y is double its original size. But, can we correlate the RATE OF DECREASE IN X and RATE OF INCREASE IN Y? Here we have failed to (we cannot) demonstrate that X is continuously decreasing within unit of time used specify speed S=L/unit time. As X decreases, the increase in Y increases. X cannot reach zero within a finite time (=X/S) because then we will know the 'exact time' when Y would HAVE REACHED infinity. As X approaches 0 the INCREASE IN GAP between the corresponding values of Y, INCREASES (Here increase increases). Y increases as a function of itself - exponentially and its rate of increase is inexpressible. If you say, 'when X becomes half, Y is doubled' (or X/2*2Y or X/10*10Y or X/N*NY=constant) then your statement does not correlate the rate of decrease in X with rate of increase in Y. A continuous chnage in a number cannot be expressed as function time or any other variable , it seems to take place without any cause as a function of it-self on its own like acceleration in gravity. If we want to describe the gravitational field using particles called 'gravitons' then we have show how functions of the 'gravitons' are integrated so that increase in velocity is continuous in one direction in space-time. I am ready to clear any more doubts about contiguity and continuity.
In <38af3945.0209162031.4dc8c...@posting.google.com>, on 09/16/2002 at 09:31 PM, vgopa...@rediffmail.com (V.Gopal) said:
>Contiguity of functions: When we conceive an organisation with a goal >in mind we >'integrate functions' in our mind and then SUBSTITUTE ENTITIES FOR >FUNCTIONS.
That sounds like you're not asking about Mathematics, but about psychology. In Mathematics "integration" has a very different meaning.
>In >mathematics an integral must be a continuum in space or in >space-time.
No.
>The sum of series of numbers in arithmetic progression, or G.P or in >H.P is not an integral.
I don't know what you mean by GP and HO, but the sum of a series of number is most definitely an integral.
>We cannot expres or communicate (the >structure) of a continuum.
Of course we can.
>Unform velocity is insensile and >incommunicable because it is like 'time without change'.
"Uniform velocity" is a concept in Physics, not in Mathematics. But the physical concept can be expressed in straightforward mathematical language.
>In L/T we can >never prove that there is no change within T or within unit of time.
I'm not sure what you are trying to say; it reads like a paraphrase of Xeno, whose paradoxes did not stand up to close examination.
>Newton took the average of the average velocities
No.
>and assumed it to be a constant!
Likewise no. Otherwise F = m*A would not be meaningful.
>The whole structure of 'mathematics' is erected on a >self-contradictory or paradoxical assumption:
Why don't you learn some Mathematics before guessing as to what its structure is?
>Number of cycles per cycle is constant all the time in all >environments.
What do cycles and time have to do with Mathematics?
>Suppose X and Y are two sides of a rectangle and the area of the >rectangle, given by XY is constant. When XY=constant, if X decreases >continuously Y should increase continuously. If X decreases >continuously at a constant speed 'S' (so that at the end of time T >the side is X-ST, then what is the rate at which Y increases? YOU >WOULD SAY that when X becomes 1/2, Y is double its original size. >But, can we correlate the RATE OF DECREASE IN X and RATE OF INCREASE >IN Y?
Yes. And, again, your subsequent text confuses Mathematics with Physics and is Mathematically incorrect. Read up on conic sections, hyperbolas and quadratic equations
-- Shmuel (Seymour J.) Metz, SysProg and JOAT Atid/2, Team OS/2, Team PL/I
Any unsolicited commercial junk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail.
I mangled my E-mail address to foil automated spammers; reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamt...@library.lspace.org
> "Shmuel (Seymour J.) Metz" <spamt...@library.lspace.org.invalid> wrote > in message <news:3d83e009$10$fuzhry+tra$mr2ice@news.patriot.net>... >> In <38af3945.0209110724.c2a9...@posting.google.com>, on 09/11/2002 >> at 08:24 AM, vgopa...@rediffmail.com (V.Gopal) said:
>> >I wish to know whether the 'process' of integration has contiguity >> >in space or in time or at least in our mind.
>> Is that supposed to mean something?
<snip>
> Contiguity of functions: When we conceive an organisation with a goal > in mind we > 'integrate functions' in our mind and then SUBSTITUTE ENTITIES FOR > FUNCTIONS. In this case we are able to explain how the organizational > goal is achived by 'an integration of functions of its parts'.
What type of "organization" you have in mind? Note that "integration", in the sense of "gathering resources together", is entirely different of "integration" of functions, in the mathematical sense.
> In > mathematics an integral must be a continuum in space or in space-time.
Nope. "An integral must be a continuum" does not make sense at all. And most integrals don't have any physical meaning, nor are done in space or space-time.
> The sum of series of numbers in arithmetic progression, or G.P or in > H.P is not an integral. We cannot expres or communicate (the > structure) of a continuum.
> Unform velocity is insensile and > incommunicable because it is like 'time without change'. In L/T we can > never prove that there is no change within T or within unit of time. > L/T is average velocity.
You are making almost no sense here. One can assume, in some model of a physical phenomenon, that the speed of an certain object is constant (or "uniform", if you like it). What is "incommunicable" about it?
> Newton took the average of the average > velocities and called it 'acceleration' and assumed it to be a > constant! If time average of time averages is constant then it is an > endless 'time without change'. We can never express or convey > continuous change like acceleration or a state of change.
Two errors: 1) The acceleration, as a function of time, is the second derivative of the space function, not an "average of averages". 2) "time average" does not make sense, since no one is deriving time as function of itself when calculating velocity or acceleration.
> The whole structure of 'mathematics' is erected on a > self-contradictory or paradoxical assumption: > Number of cycles per cycle is constant all the time in all > environments. > Number of units of time per unit time is constant all the > time----------. > Number of units of length per unit length is constant all the > time------. > Number of space-time intervals, of same duration, within any two > consecutive integers (0 and 1, 1 and 2, 2 and 3, 10^10 and (10^10)+1 > is same and constant. > 0, 1, 2, 3 ------------N, (N+1)------ are equally spaced. > Number of numbers between 0 and 1, 1 and 2, 2 and 3 or between N and N > +1 is same all the time.
"Number of <fitb> per <fitb> is constant" appears to be a truism, when it makes sense.
There are not "space-time" intervals between integers, or any other numbers. There are just intervals, abstract, with no relation to physics.
Please check your facts. I strongly suggest to you to study abstract algebra and real analysis before making such claims about the structure of mathematics. Below are some references to begin studying:
> On a number line if integers are equally > spaced or if 'scale' is conventional then there is a pardox - X*1/X > cannot be equal to 1, because we have to accommodate the reciprocals > of all the conceivable numbers between 1 and infinity within all the > conceivable numbers between 1 and 0.
As I said in another message, there is no paradox - both intervals [1, oo[ and ]0,1] are infinite and of the same cardinality.
> We can understand the problem with continuity from the following > example: > Suppose X and Y are two sides of a rectangle and the area of the > rectangle, given by XY is constant. When XY=constant, if X decreases > continuously Y should increase continuously. If X decreases > continuously at a constant speed 'S' (so that at the end of time T the > side is X-ST, then what is the rate at which Y increases? YOU WOULD > SAY that when X becomes 1/2, Y is double its original size. > But, can we correlate the RATE OF DECREASE IN X and RATE OF INCREASE > IN Y?
The rate of change of y varies with x, s, and t, due to the constraints of the problem: xy = A (constant). One can't prove any discontinuity in y or x this way.
Rephrasing your problem: Let x(t) = X - st be a linear function in t. X and s are constants. Suppose A is a real constant > 0, and let be Y = A/X. Let y(t) = A / x(t) be a function in t.
The answer to your questions above is: calculate the function dy/dt, and evaluate it at value 1/2. dx/dt and dy/dt are related, just use the chain rule in derivation. (dx/dt and dy/dt denote, at each point t, what is the increasing/decreasing rate). I suppose you know Calculus fairly well.
> Here we have failed to (we cannot) demonstrate that X is continuously > decreasing within unit of time used specify speed S=L/unit time. As X > decreases, the increase in Y increases. X cannot reach zero within a > finite time (=X/S) because then we will know the 'exact time' when Y > would HAVE REACHED infinity. As X approaches 0 the INCREASE IN GAP > between the corresponding values of Y, INCREASES (Here increase > increases). Y increases as a function of itself - exponentially and > its rate of increase is inexpressible.
x(0) = X, x(X/s) = 0. So, lim [t->(X/s)] y(t) = +oo. One can calculate dy/dt(t) at any t < X/s.
What is "inexpressible" about this?
BTW, evaluations of a function do not take time to happen; time, when needed, is just one more variable.
<snip irrelevant argument>
In case I didn't make myself clear before: mathematical entities do not have physical meanings (like space or time) in themselves; they are abstract. One can use math to create a model of the physical world, but the math behind the model does not need to match the properties of the physical world.
You failed in understanding all of above.
Bye for now,
-- --------------------------------------- Duran Castore (duran_cast...@yahoo.com)
> > "Shmuel (Seymour J.) Metz" <spamt...@library.lspace.org.invalid> wrote > > in message <news:3d83e009$10$fuzhry+tra$mr2ice@news.patriot.net>... > >> In <38af3945.0209110724.c2a9...@posting.google.com>, on 09/11/2002 > >> at 08:24 AM, vgopa...@rediffmail.com (V.Gopal) said:
> >> >I wish to know whether the 'process' of integration has contiguity > >> >in space or in time or at least in our mind.
> >> Is that supposed to mean something? > <snip>
> > Contiguity of functions: When we conceive an organisation with a goal > > in mind we > > 'integrate functions' in our mind and then SUBSTITUTE ENTITIES FOR > > FUNCTIONS. In this case we are able to explain how the organizational > > goal is achived by 'an integration of functions of its parts'.
> What type of "organization" you have in mind? Note that "integration", in > the sense of "gathering resources together", is entirely different of > "integration" of functions, in the mathematical sense.
> > In > > mathematics an integral must be a continuum in space or in space-time.
> Nope. "An integral must be a continuum" does not make sense at all. And > most integrals don't have any physical meaning, nor are done in space or > space-time.
> > The sum of series of numbers in arithmetic progression, or G.P or in > > H.P is not an integral. We cannot expres or communicate (the > > structure) of a continuum.
> The only continuum I know is the cardinality of the real set. Please see > http://mathworld.wolfram.com/Continuum.html > for more. So, the continuum is perfectly explainable.
> > Unform velocity is insensile and > > incommunicable because it is like 'time without change'. In L/T we can > > never prove that there is no change within T or within unit of time. > > L/T is average velocity. > You are making almost no sense here. One can assume, in some model of a > physical phenomenon, that the speed of an certain object is constant (or > "uniform", if you like it). What is "incommunicable" about it?
> > Newton took the average of the average > > velocities and called it 'acceleration' and assumed it to be a > > constant! If time average of time averages is constant then it is an > > endless 'time without change'. We can never express or convey > > continuous change like acceleration or a state of change.
> Two errors: 1) The acceleration, as a function of time, is the second > derivative of the space function, not an "average of averages". 2) "time > average" does not make sense, since no one is deriving time as function of > itself when calculating velocity or acceleration.
> > The whole structure of 'mathematics' is erected on a > > self-contradictory or paradoxical assumption: > > Number of cycles per cycle is constant all the time in all > > environments. > > Number of units of time per unit time is constant all the > > time----------. > > Number of units of length per unit length is constant all the > > time------. > > Number of space-time intervals, of same duration, within any two > > consecutive integers (0 and 1, 1 and 2, 2 and 3, 10^10 and (10^10)+1 > > is same and constant. > > 0, 1, 2, 3 ------------N, (N+1)------ are equally spaced. > > Number of numbers between 0 and 1, 1 and 2, 2 and 3 or between N and N > > +1 is same all the time.
> "Number of <fitb> per <fitb> is constant" appears to be a truism, when it > makes sense.
> There are not "space-time" intervals between integers, or any other > numbers. There are just intervals, abstract, with no relation to physics.
> Please check your facts. I strongly suggest to you to study abstract > algebra and real analysis before making such claims about the structure of > mathematics. Below are some references to begin studying:
> > On a number line if integers are equally > > spaced or if 'scale' is conventional then there is a pardox - X*1/X > > cannot be equal to 1, because we have to accommodate the reciprocals > > of all the conceivable numbers between 1 and infinity within all the > > conceivable numbers between 1 and 0.
> As I said in another message, there is no paradox - both intervals [1, oo[ > and ]0,1] are infinite and of the same cardinality.
> > We can understand the problem with continuity from the following > > example: > > Suppose X and Y are two sides of a rectangle and the area of the > > rectangle, given by XY is constant. When XY=constant, if X decreases > > continuously Y should increase continuously. If X decreases > > continuously at a constant speed 'S' (so that at the end of time T the > > side is X-ST, then what is the rate at which Y increases? YOU WOULD > > SAY that when X becomes 1/2, Y is double its original size. > > But, can we correlate the RATE OF DECREASE IN X and RATE OF INCREASE > > IN Y?
> The rate of change of y varies with x, s, and t, due to the constraints of > the problem: xy = A (constant). One can't prove any discontinuity in y or x > this way.
> Rephrasing your problem: > Let x(t) = X - st be a linear function in t. X and s are constants. > Suppose A is a real constant > 0, and let be Y = A/X. > Let y(t) = A / x(t) be a function in t.
> The answer to your questions above is: calculate the function dy/dt, and > evaluate it at value 1/2. dx/dt and dy/dt are related, just use the chain > rule in derivation. (dx/dt and dy/dt denote, at each point t, what is the > increasing/decreasing rate). I suppose you know Calculus fairly well.
> > Here we have failed to (we cannot) demonstrate that X is continuously > > decreasing within unit of time used specify speed S=L/unit time. As X > > decreases, the increase in Y increases. X cannot reach zero within a > > finite time (=X/S) because then we will know the 'exact time' when Y > > would HAVE REACHED infinity. As X approaches 0 the INCREASE IN GAP > > between the corresponding values of Y, INCREASES (Here increase > > increases). Y increases as a function of itself - exponentially and > > its rate of increase is inexpressible.
> x(0) = X, x(X/s) = 0. So, lim [t->(X/s)] y(t) = +oo. One can calculate > dy/dt(t) at any t < X/s.
> What is "inexpressible" about this?
> BTW, evaluations of a function do not take time to happen; time, when > needed, is just one more variable.
> <snip irrelevant argument>
> In case I didn't make myself clear before: mathematical entities do not > have physical meanings (like space or time) in themselves; they are > abstract. One can use math to create a model of the physical world, but the > math behind the model does not need to match the properties of the physical > world.
> You failed in understanding all of above.
> Bye for now,
It is wrong to say that no one is deriving time as a function if itself. When we want to express or convey angular acceleration (rate of change of frequency) we have to express time as a function of itself. In general, frequency means 'density'. In general, 'RATE' means number of one thing within each unit of another thing. Rate of change of frequency means rate of change of density or even rate of change of rate. In general 'acceleration' means rate of change frequency or density or rate. A unit of time is always equivalent a fixed displacement - angular or linear. In order to avoid any confusion about the nature of time we must replace 'unit' of time (the constant that we place in the dinominator, to 'quantify' velocity) by a fixes angle or a fixed length. Then velocity becomes L/l - displacement of the object in question divided by the displacement in the clock showing unit time. (Acceleration is rate of change of rate, both in mechanics and in economics) A formula for 'Prediction' always demands that we express the object (actually its state) as a function of itself (its original state) and not include any knowledge without the object. In general 'acceleration' means rate of increase in the number of numbers within unit or one. The graph showing the relation between linear displacement (or number of units of length)or angular displacement (or number of cycles) and time, during acceleration, cannot give a continuous and smooth open curve. L=UT+1/2aT^2 cannot be a smooth open curve. These graphs can be smooth only without the units of time and length - the 'displacemet' to be measured and correlated with time, elongates - the information we are seeking is changing! If dL/dT is the 'instantaneous' velocity then what is its reciprocal dT/dL? Note that this expression is equivalent to infinite time divided infinite distance.
<snipping thru the context of the point you chose to debate>
>> > Newton took the average of the average >> > velocities and called it 'acceleration' and assumed it to be a >> > constant! If time average of time averages is constant then it is >> > an endless 'time without change'. We can never express or convey >> > continuous change like acceleration or a state of change.
>> Two errors: 1) The acceleration, as a function of time, is the second >> derivative of the space function, not an "average of averages". 2) >> "time average" does not make sense, since no one is deriving time as >> function of itself when calculating velocity or acceleration.
<big snip>
> It is wrong to say that no one is deriving time as a function if > itself. > When we want to express or convey angular acceleration (rate of change > of frequency) we have to express time as a function of itself. In
You are attacking the wrong argument; I was meaning linear acceleration. "Straw man" fallacy.
> general, frequency means 'density'.
Nope. Density has unit kg/m^3; frequency has unit s^-1.
> In general, 'RATE' means number of one thing within > each unit of another thing. Rate of change of frequency means rate of > change of density or even rate of change of rate. In general > 'acceleration' means rate of change frequency or density or rate.
Too vague.
> A unit of time is always equivalent [to] a fixed displacement - angular > or linear.
If the (linear or angular) velocity is constant, that is.
> In order to avoid any confusion about the nature of time we > must replace 'unit' of time (the constant that we place in the > dinominator, to 'quantify' velocity) by a fixes angle or a fixed > length. Then velocity becomes L/l - displacement of the object in > question divided by the displacement in the clock showing unit time.
No one is questioning the nature of time. Relating time to a form of measuring time (a clock) just complicate things; time is a SI base unit already. Velocity will always have unit m/s (linear) or rad/s (angular).
Furthermore, taking a "displacement in the clock showing unit time", such displacement must take time, so substituting time by a "reference" length or angle just begs the question of where the time variable must be used.
> (Acceleration is rate of change of rate, both in mechanics and in > economics) A formula for 'Prediction' always demands that we express > the object (actually its state) as a function of itself (its original > state) and not include any knowledge without the object. In general
Economics has nothing to do with the argument; the point appears to be Physics.
> 'acceleration' means rate of increase in the number of numbers within > unit or one.
Please review the standard definition for acceleration. Any good book of Mechanics will do.
> The graph showing the relation between linear > displacement (or number of units of length)or angular displacement (or > number of cycles) and time, during acceleration, cannot give a > continuous and smooth open curve.
(variables: E, E0 length; V velocity; A acceleration; t time) E(t) = E0 + V*t + (A/2)*t^2
is a polynomial function in t, so continuous, differentiable, etc, in t. What you mean by "open"? "Smooth", in my book, means having at least a continuous second derivative. Appears to me that you are using a different definition of "smooth".
> L=UT+1/2aT^2 cannot be a smooth open > curve. These graphs can be smooth only without the units of time and > length - the 'displacemet' to be measured and correlated with time, > elongates - the information we are seeking is changing!
This does not make sense.
> If dL/dT is the 'instantaneous' velocity then what is its reciprocal > dT/dL?
This is the inverse of the velocity. I don't know if such thing has physical significance.
> Note that this expression is equivalent to infinite time divided > infinite distance.
Nope; it's more like "zero time divided by zero distance". Please check the definition of derivative; I already gave you references.
A note: Thank you to making me review some mathematics and physics from far ago, which I needed to counter your almost nonsensical arguments and confused mathematical/physical concepts. This is the only reason why I did not *PLONK*ed you before.
Bye.
-- --------------------------------------- Duran Castore (duran_cast...@yahoo.com)
> <snipping thru the context of the point you chose to debate>
> >> > Newton took the average of the average > >> > velocities and called it 'acceleration' and assumed it to be a > >> > constant! If time average of time averages is constant then it is > >> > an endless 'time without change'. We can never express or convey > >> > continuous change like acceleration or a state of change.
> >> Two errors: 1) The acceleration, as a function of time, is the second > >> derivative of the space function, not an "average of averages". 2) > >> "time average" does not make sense, since no one is deriving time as > >> function of itself when calculating velocity or acceleration.
> <big snip>
> > It is wrong to say that no one is deriving time as a function if > > itself. > > When we want to express or convey angular acceleration (rate of change > > of frequency) we have to express time as a function of itself. In
> You are attacking the wrong argument; I was meaning linear acceleration. > "Straw man" fallacy.
> > general, frequency means 'density'.
> Nope. Density has unit kg/m^3; frequency has unit s^-1.
> > In general, 'RATE' means number of one thing within > > each unit of another thing. Rate of change of frequency means rate of > > change of density or even rate of change of rate. In general > > 'acceleration' means rate of change frequency or density or rate.
> Too vague.
> > A unit of time is always equivalent [to] a fixed displacement - angular > > or linear.
> If the (linear or angular) velocity is constant, that is.
> > In order to avoid any confusion about the nature of time we > > must replace 'unit' of time (the constant that we place in the > > dinominator, to 'quantify' velocity) by a fixes angle or a fixed > > length. Then velocity becomes L/l - displacement of the object in > > question divided by the displacement in the clock showing unit time.
> No one is questioning the nature of time. Relating time to a form of > measuring time (a clock) just complicate things; time is a SI base unit > already. Velocity will always have unit m/s (linear) or rad/s (angular).
> Furthermore, taking a "displacement in the clock showing unit time", such > displacement must take time, so substituting time by a "reference" length > or angle just begs the question of where the time variable must be used.
> > (Acceleration is rate of change of rate, both in mechanics and in > > economics) A formula for 'Prediction' always demands that we express > > the object (actually its state) as a function of itself (its original > > state) and not include any knowledge without the object. In general
> Economics has nothing to do with the argument; the point appears to be > Physics.
> > 'acceleration' means rate of increase in the number of numbers within > > unit or one.
> Please review the standard definition for acceleration. Any good book of > Mechanics will do.
> > The graph showing the relation between linear > > displacement (or number of units of length)or angular displacement (or > > number of cycles) and time, during acceleration, cannot give a > > continuous and smooth open curve.
> (variables: E, E0 length; V velocity; A acceleration; t time) > E(t) = E0 + V*t + (A/2)*t^2
> is a polynomial function in t, so continuous, differentiable, etc, in t. > What you mean by "open"? "Smooth", in my book, means having at least a > continuous second derivative. Appears to me that you are using a different > definition of "smooth".
> > L=UT+1/2aT^2 cannot be a smooth open > > curve. These graphs can be smooth only without the units of time and > > length - the 'displacemet' to be measured and correlated with time, > > elongates - the information we are seeking is changing!
> This does not make sense.
> > If dL/dT is the 'instantaneous' velocity then what is its reciprocal > > dT/dL?
> This is the inverse of the velocity. I don't know if such thing has > physical significance.
> > Note that this expression is equivalent to infinite time divided > > infinite distance.
> Nope; it's more like "zero time divided by zero distance". Please check the > definition of derivative; I already gave you references.
> A note: > Thank you to making me review some mathematics and physics from far ago, > which I needed to counter your almost nonsensical arguments and confused > mathematical/physical concepts. This is the only reason why I did not > *PLONK*ed you before.
> Bye.
It seems that all mathematics is aimed at developing Artificial intelligence. Artificial intelligence is digital and it cannot feel the meaning of 'continuous change'. What I am saying has to 'felt' to realize the truth. How do we decrease finite number X or 'delta X' and arrive at a 'differential' say, dX? There are two possibilities: dividing 1 (one or a finite number) by a number approaching infinity OR by deducting a finite number or its integral multiples, from 1? By dividing 1 by continuously increasing number we can not reach infinitesimal or dX or what you call as zero. We can reach dX or infinitesimal or zero only when the divisor suddenly jumps from finite (known) to infinite - the unknowable. The idea of 'delta X' approaching zero gives an impression as if the decrease is continuous. A 'continuous decrease' from finite to infinitesimal or zero is not possible, just as continuous increase from the 'finite' cannot take us to the infinite or the unknown. Your mind only works like a 'digital' machine and I am trying to make you understand a continuous change. Obviously I cannot succeed.
In <38af3945.0209191943.4d57d...@posting.google.com>, on 09/19/2002 at 08:43 PM, vgopa...@rediffmail.com (V.Gopal) said:
>It seems that all mathematics is aimed at developing Artificial >intelligence.
No, any more than all Mathematics is aimed at developing a piano concerto.
>Artificial intelligence is digital
Speculation, unless you have working AI.
>What I am saying has to 'felt' to realize the truth.
Mathematics deals with what you can proove via logic. For feelings, see a shrink.
>How do we decrease finite number X or 'delta X' and >arrive at a 'differential' say, dX?
You don't; your question has no meaning.
>The idea of 'delta X' approaching zero gives an impression as if the >decrease is continuous.
No, the idea of taking half a phrase out of context and expecting it to have a meaning is fundamentally wrong. The complete phrase involves a variable (Delta X) and some function of that variable, e.g.,
Lim DeltaX -> 0 (f(X + DeltaX)-f(X))/DeltaX
>A 'continuous decrease' from finite to >infinitesimal or zero is not possible,
Xeno was wrong. Get over it.
>Your mind only works like a 'digital' machine
Neurons are analog.
-- Shmuel (Seymour J.) Metz, SysProg and JOAT Atid/2, Team OS/2, Team PL/I
Any unsolicited commercial junk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail.
I mangled my E-mail address to foil automated spammers; reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamt...@library.lspace.org
> In <38af3945.0209191943.4d57d...@posting.google.com>, on 09/19/2002 > at 08:43 PM, vgopa...@rediffmail.com (V.Gopal) said:
> >It seems that all mathematics is aimed at developing Artificial > >intelligence.
> No, any more than all Mathematics is aimed at developing a piano > concerto.
> >Artificial intelligence is digital
> Speculation, unless you have working AI.
> >What I am saying has to 'felt' to realize the truth.
> Mathematics deals with what you can proove via logic. For feelings, > see a shrink.
> >How do we decrease finite number X or 'delta X' and > >arrive at a 'differential' say, dX?
> You don't; your question has no meaning.
> >The idea of 'delta X' approaching zero gives an impression as if the > >decrease is continuous.
> No, the idea of taking half a phrase out of context and expecting it > to have a meaning is fundamentally wrong. The complete phrase involves > a variable (Delta X) and some function of that variable, e.g.,
> Lim DeltaX -> 0 (f(X + DeltaX)-f(X))/DeltaX
> >A 'continuous decrease' from finite to > >infinitesimal or zero is not possible,
> Xeno was wrong. Get over it.
> >Your mind only works like a 'digital' machine
> Neurons are analog.
> -- > Shmuel (Seymour J.) Metz, SysProg and JOAT > Atid/2, Team OS/2, Team PL/I
> Any unsolicited commercial junk E-mail will be subject to legal > action. I reserve the right to publicly post or ridicule any > abusive E-mail.
> I mangled my E-mail address to foil automated spammers; reply to > domain Patriot dot net user shmuel+news to contact me. Do not > reply to spamt...@library.lspace.org
I do not deny that 'Unit' of frequency is s^-1. What I insist is that angular acceleration or 'state of change' has no unit. Frequency and angular velocity indicate the same activity. Angular velocity is radians/sec. or radians/year. We cannot quantify angular acceleration either with one radian or within 1 second or 1 year. We place acceleration without angle and without time. If X-axis shows time and Y-axis shows frequency, is constant frequency a straight line or a point? If frequency is changing (there is angular acceleration) can we draw a SMOOTH curve showing that shows the relation time (number of seconds) and frequency 0, 1, 2 ,3 etc. You have to feel the 'twist' that acceleration produces.
In <38af3945.0209210507.3f0d3...@posting.google.com>, on 09/21/2002 at 06:07 AM, vgopa...@rediffmail.com (V.Gopal) said:
>I do not deny that 'Unit' of frequency is s^-1. What I insist is that >angular acceleration or 'state of change' has no unit.
The unit of angular accelleration is s^-2.
>Frequency and angular velocity indicate the same activity.
No.
>We place acceleration without angle and without time.
?
>If X-axis shows time and Y-axis shows frequency, is constant >frequency a straight line or a point?
A line, of course.
>If frequency is changing (there is angular >acceleration) can we draw a SMOOTH curve showing that shows the >relation time (number of seconds) and frequency 0, 1, 2 ,3 etc.
If you the graph to integral time than of course it's not smooth. Both if you're talking about acceleration than you have to admit continuous time, and the graph is smooth in the sense you mean it.
>You have to feel the 'twist' that acceleration produces.
If the acceleration is not zero than the graph is not a line. Is that what you mean by a twist?
-- Shmuel (Seymour J.) Metz, SysProg and JOAT Atid/2, Team OS/2, Team PL/I
Any unsolicited commercial junk E-mail will be subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail.
I mangled my E-mail address to foil automated spammers; reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamt...@library.lspace.org
> In <38af3945.0209210507.3f0d3...@posting.google.com>, on 09/21/2002 > at 06:07 AM, vgopa...@rediffmail.com (V.Gopal) said:
> >I do not deny that 'Unit' of frequency is s^-1. What I insist is that > >angular acceleration or 'state of change' has no unit.
> The unit of angular accelleration is s^-2.
> >Frequency and angular velocity indicate the same activity.
> No.
> >We place acceleration without angle and without time.
> ?
> >If X-axis shows time and Y-axis shows frequency, is constant > >frequency a straight line or a point?
> A line, of course.
> >If frequency is changing (there is angular > >acceleration) can we draw a SMOOTH curve showing that shows the > >relation time (number of seconds) and frequency 0, 1, 2 ,3 etc.
> If you the graph to integral time than of course it's not smooth. Both > if you're talking about acceleration than you have to admit continuous > time, and the graph is smooth in the sense you mean it.
> >You have to feel the 'twist' that acceleration produces.
> If the acceleration is not zero than the graph is not a line. Is that > what you mean by a twist?
> -- > Shmuel (Seymour J.) Metz, SysProg and JOAT > Atid/2, Team OS/2, Team PL/I
> Any unsolicited commercial junk E-mail will be subject to legal > action. I reserve the right to publicly post or ridicule any > abusive E-mail.
> I mangled my E-mail address to foil automated spammers; reply to > domain Patriot dot net user shmuel+news to contact me. Do not > reply to spamt...@library.lspace.org
I am only concerned with the duration when acceleration is not zero and angular velocity begins to increase from zero. What is the shape of the line (straight or curved) that shows the relation between time (passing time, as the clock shows - 0 sec, 1 second, 2 seconds etc.) and frequency during angular acceleration 'a'=1/s^2? If 'a'>0 or if 1/s^2 is not zero then what is the shape of the curve that shows the relation between 1/s (frequency or angular velocity) and duration 'T' in 'seconds' starting from T=0 and 1/s=0? Is it a straight line or a parabola? How do we calculate 1/s or angular acceleration between T=0 (clock reading) and 1/s=0 and T=1 (clock reading) and 1/s=1. How can we calculate rate of change of 1/s (or the rate of change of frequenncy or rate of change of angular velocity) between any two consecutive frequencies (or two consecutive values of angular velocitiy) during angular acceleration? We are talking of rate of change in (displacement) angle/time within the same angular displacement. If there is no change withing the same angular displacement then angular accleration is never 'placed within angle of rotation. Then how can we know acceleration?
> > > "Shmuel (Seymour J.) Metz" <spamt...@library.lspace.org.invalid> wrote > > > in message <news:3d83e009$10$fuzhry+tra$mr2ice@news.patriot.net>... > > >> In <38af3945.0209110724.c2a9...@posting.google.com>, on 09/11/2002 > > >> at 08:24 AM, vgopa...@rediffmail.com (V.Gopal) said:
> > >> >I wish to know whether the 'process' of integration has contiguity > > >> >in space or in time or at least in our mind.
> > >> Is that supposed to mean something? > > <snip>
> > > Contiguity of functions: When we conceive an organisation with a goal > > > in mind we > > > 'integrate functions' in our mind and then SUBSTITUTE ENTITIES FOR > > > FUNCTIONS. In this case we are able to explain how the organizational > > > goal is achived by 'an integration of functions of its parts'.
> > What type of "organization" you have in mind? Note that "integration", in > > the sense of "gathering resources together", is entirely different of > > "integration" of functions, in the mathematical sense.
> > > In > > > mathematics an integral must be a continuum in space or in space-time.
> > Nope. "An integral must be a continuum" does not make sense at all. And > > most integrals don't have any physical meaning, nor are done in space or > > space-time.
> > > The sum of series of numbers in arithmetic progression, or G.P or in > > > H.P is not an integral. We cannot expres or communicate (the > > > structure) of a continuum.
> > The only continuum I know is the cardinality of the real set. Please see > > http://mathworld.wolfram.com/Continuum.html > > for more. So, the continuum is perfectly explainable.
> > > Unform velocity is insensile and > > > incommunicable because it is like 'time without change'. In L/T we can > > > never prove that there is no change within T or within unit of time. > > > L/T is average velocity. > > You are making almost no sense here. One can assume, in some model of a > > physical phenomenon, that the speed of an certain object is constant (or > > "uniform", if you like it). What is "incommunicable" about it?
> > > Newton took the average of the average > > > velocities and called it 'acceleration' and assumed it to be a > > > constant! If time average of time averages is constant then it is an > > > endless 'time without change'. We can never express or convey > > > continuous change like acceleration or a state of change.
> > Two errors: 1) The acceleration, as a function of time, is the second > > derivative of the space function, not an "average of averages". 2) "time > > average" does not make sense, since no one is deriving time as function of > > itself when calculating velocity or acceleration.
> > > The whole structure of 'mathematics' is erected on a > > > self-contradictory or paradoxical assumption: > > > Number of cycles per cycle is constant all the time in all > > > environments. > > > Number of units of time per unit time is constant all the > > > time----------. > > > Number of units of length per unit length is constant all the > > > time------. > > > Number of space-time intervals, of same duration, within any two > > > consecutive integers (0 and 1, 1 and 2, 2 and 3, 10^10 and (10^10)+1 > > > is same and constant. > > > 0, 1, 2, 3 ------------N, (N+1)------ are equally spaced. > > > Number of numbers between 0 and 1, 1 and 2, 2 and 3 or between N and N > > > +1 is same all the time.
> > "Number of <fitb> per <fitb> is constant" appears to be a truism, when it > > makes sense.
> > There are not "space-time" intervals between integers, or any other > > numbers. There are just intervals, abstract, with no relation to physics.
> > Please check your facts. I strongly suggest to you to study abstract > > algebra and real analysis before making such claims about the structure of > > mathematics. Below are some references to begin studying:
> > > On a number line if integers are equally > > > spaced or if 'scale' is conventional then there is a pardox - X*1/X > > > cannot be equal to 1, because we have to accommodate the reciprocals > > > of all the conceivable numbers between 1 and infinity within all the > > > conceivable numbers between 1 and 0.
> > As I said in another message, there is no paradox - both intervals [1, oo[ > > and ]0,1] are infinite and of the same cardinality.
> > > We can understand the problem with continuity from the following > > > example: > > > Suppose X and Y are two sides of a rectangle and the area of the > > > rectangle, given by XY is constant. When XY=constant, if X decreases > > > continuously Y should increase continuously. If X decreases > > > continuously at a constant speed 'S' (so that at the end of time T the > > > side is X-ST, then what is the rate at which Y increases? YOU WOULD > > > SAY that when X becomes 1/2, Y is double its original size. > > > But, can we correlate the RATE OF DECREASE IN X and RATE OF INCREASE > > > IN Y?
> > The rate of change of y varies with x, s, and t, due to the constraints of > > the problem: xy = A (constant). One can't prove any discontinuity in y or x > > this way.
> > Rephrasing your problem: > > Let x(t) = X - st be a linear function in t. X and s are constants. > > Suppose A is a real constant > 0, and let be Y = A/X. > > Let y(t) = A / x(t) be a function in t.
> > The answer to your questions above is: calculate the function dy/dt, and > > evaluate it at value 1/2. dx/dt and dy/dt are related, just use the chain > > rule in derivation. (dx/dt and dy/dt denote, at each point t, what is the > > increasing/decreasing rate). I suppose you know Calculus fairly well.
> > > Here we have failed to (we cannot) demonstrate that X is continuously > > > decreasing within unit of time used specify speed S=L/unit time. As X > > > decreases, the increase in Y increases. X cannot reach zero within a > > > finite time (=X/S) because then we will know the 'exact time' when Y > > > would HAVE REACHED infinity. As X approaches 0 the INCREASE IN GAP > > > between the corresponding values of Y, INCREASES (Here increase > > > increases). Y increases as a function of itself - exponentially and > > > its rate of increase is inexpressible.
> > x(0) = X, x(X/s) = 0. So, lim [t->(X/s)] y(t) = +oo. One can calculate > > dy/dt(t) at any t < X/s.
> > What is "inexpressible" about this?
> > BTW, evaluations of a function do not take time to happen; time, when > > needed, is just one more variable.
> > <snip irrelevant argument>
> > In case I didn't make myself clear before: mathematical entities do not > > have physical meanings (like space or time) in themselves; they are > > abstract. One can use math to create a model of the physical world, but the > > math behind the model does not need to match the properties of the physical > > world.
> > You failed in understanding all of above.
> > Bye for now,
> It is wrong to say that no one is deriving time as a function if > itself. > When we want to express or convey angular acceleration (rate of change > of frequency) we have to express time as a function of itself. In > general, frequency means 'density'. In general, 'RATE' means number of > one thing within > each unit of another thing. Rate of change of frequency means rate of > change of density or even rate of change of rate. In general > 'acceleration' means rate of change frequency or density or rate. A > unit of time is always equivalent a fixed displacement - angular or > linear. In order to avoid any confusion about the nature of time we > must replace 'unit' of time (the constant that we place in the > dinominator, to 'quantify' velocity) by a fixes angle or a fixed > length. Then velocity becomes L/l - displacement of the object in > question divided by the displacement in the clock showing unit time. > (Acceleration is rate of change of rate, both in mechanics and in > economics) A formula for 'Prediction' always demands that we express > the object (actually its state) as a function of itself (its original > state) and not include any knowledge without the object. In general > 'acceleration' means rate of increase in the number of numbers within > unit or one. The graph showing the relation between linear > displacement (or number of units of length)or angular displacement (or > number of cycles) and time, during acceleration, cannot give a > continuous and smooth open curve. L=UT+1/2aT^2 cannot be a smooth open > curve. These graphs can be smooth only without the units of time and > length - the 'displacemet' to be measured and correlated with time, > elongates - the information we are seeking is changing! > If dL/dT is the 'instantaneous' velocity then what is its reciprocal > dT/dL? > Note that this expression is equivalent to infinite time divided > infinite distance.
What I wish to point out is if XY=A (a constant area) and X/Y is a variable (SHAPE of the rectangle is the variable) then we can only
...
|
