Monday, March 13, 2006

I strongly disagree, namely for the reasons given in my response to his post.

Additionally, here are some other reasons that velocity at an instant must be defined to have some numerical value in some manner:

1. The derivative of the velocity is very well defined (namely, it is the accelleration.) If it were truly the case that the velocity was not defined at each slice of time, but instead had only a numerical value over a broad set of times, then the velocity curve could not be said to be continous. If the velocity curve were not continuous, then one could not find a value for its derivative at all times. This is easier to see if you imagine plotting the velocity versus the time. For any point along one axis, there is a corresponding point along the other. This is the definition of a function. In other words, since velocity can be expressed as a function of time, the cardinality of the set of velocities is identical to the cardinality of the set of times (they are both continuum); furthermore, for every member of the set of velocities, there is a corresponding member of the set of times. If this were not the case, velocity would not be a function; if velocity were not a function, then you could not take its derivative. Thus, if velocity were not defined at every time t then accelleration would not exist.

2. There is one object whose velocity is the same irrespective of time. That object is light, whose velocity, c, is independent of time (this arises out of Maxwell's equations and is used as a postulate in Einstein's Special Relativity.) If velocity could only be defined by using a large number of space-time slices, then it would be the case that the velocity of light wasn't defined at any particular point in time (but rather had to be taken over a series of times.) However, this is not the case - the velocity of light is defined at a time precisely because the equations for it are independent of time.

3. Could it not be that a limiting operation gives the behavior of a function at a point? Why are you resistent to think that it might be the case that the limit of a function at a point is identical to the function's behavior at that point? Although it is true that not all functions behave at a given point as their limits do, there do exist functions which do behave exactly as their limits do. For these functions, I believe it is safe to assume that the behavior of the point is identical to the behavior of the limits precisely because we already know, a priori, that this is so. For instance, the limit of the function y=5x at x=3 is 15, but the value of y at that point is also 15. Therefore, for that particular function, the value of the limit is identical to the value of the function - something that we all have to agree upon precisely because 15=15. There is no reason to suppose, as far as I can see, that the limit involved in the definition of the derivative would be any different for many functions.

By the way, are you supposing that if every individual member of a given set does not have some property or feature F then the whole set might? Please note that the argument against motion given in class depends entirely upon the opposite (i.e. that if every individual member of a given set does not have F then the whole set will not either.) If some property of a set (such as the set of times) can only be defined over several members of the set (such as your idea of motion) then this must be the case. However, I believe that there exist several difficulties in supposing that this is, in fact, the case. For instance, if for all times t, some statement A is not true, then it cannot be the case that collectively, over the times t, A is true. Under your ideas of motion, you would be forced to say something like, "Motion is not defined for all times t, but it is defined for the set collectively", which would seem to contradict the statement I just made. That is, if a logical statement is false at one time, then it must be the case that it is false for all times, simply because such statements are independent of time.

Here's some sites you might find useful:

Velocity versus time graphs (high school physics and kinematics): http://www.glenbrook.k12.il.us/gbssci/phys/Class/1DKin/U1L4a.html

Definition of a function (including how it relates to set theory): http://mathworld.wolfram.com/Function.html

Definition of the derivative: http://mathworld.wolfram.com/Derivative.html

Maxwell's Equations (including the definition of the speed of light arising therefrom): http://en.wikipedia.org/wiki/Maxwell%27s_Equations

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