dt ramblings

[Dave]

Besides the fact that we do use the reals to model time (and most of the above post doesn't make any sense), I really can't see where this is going. What you've said in the past few posts has nothing to do with what the actual argument is.

 

I won't reply to anything else that uses time or any other physical quantity when you should be using nothing but pure mathematics to prove your points. We're not talking about modelling things, we're talking about defintions of fundamental concepts in mathematics.

[Johnny5]

Besides the fact that we do use the reals to model time (and most of the above post doesn't make any sense)

 

 

 

What part didn't make sense?

[Dave]

First of all, there is no reason to define "moment in time" mathematically, because you can define it operationally. There is nothing forbidding it I suppose, but I don't suggest it. But I know the answer you want is point[/i']. But certainly not 'point' in any spatial sense.

 

This makes little sense.

 

I would like you to define your idea of dt, without using any concept of time or physical quantity and using only mathematics.

 

Saying that dt = t₂ - t₁ is quite obviously non-sensical for the reasons that I've already stated.

[Johnny5]

Well first off, I said that you can operationally define before. When you do so, you will encounter something which occurred to umm, what the heck was his name... the river guy... you cannot step twice into the same river... oh yeah Heraclitus i think was his name.

 

What Heraclitus realized, is that the universe will never be in the same state twice.

 

Here... use this operational definition... then you will see...

 

Go to the moon, and drop a feather from a height h above the surface of the moon. You are well aware of two distinct moments in time, the one where you release the feather, and the one at which the feather collides with the surface of the moon. So you define the binary relation BEFORE, using that experiment... the free-fall experiment, made famous by Galileo.

 

So the domain of discourse, is the set of "moments in time"

 

Your binary relation takes this form:

 

X before Y

 

The X,Y are elements of the domain of discourse.

 

So that the meaning of "moment in time" comes from that experiment which operationally defines "before."

 

That's what I mean.

 

So there is no need to define "moment in time" mathematically, because you can know what X,Y denotes from the Galileo experiment, which defines the binary relation before, on the set of moments in time.

 

As for where Heraclitus' comment fits into all this...

 

Is the following statement true or false:

 

$X: X before X

 

Read, "There is at least one X, such that X before X" or can translate into English as follows: "There is at least one moment in time X, such that X before X."

 

 

I would like you to define your idea of dt, without using any concept of time or physical quantity and using only mathematics.

 

I will try.

 

Let S denote the set of states of the universe.

 

Let X, Y be elements of S.

Let it be true that X before Y.

 

Additionally, let it be the case that X,Y are adjacent moments in time, as previously described.

 

Let there be a clock, which maps natural numbers onto each state, such that there is a 1-1 mapping.

 

The following notation could be used to refer to elements of the domain of discourse:

 

S = {S₁, S₂, S₃, ...}

 

Each S_n, refers to a different configuration of matter in this universe.

 

You asked for a mathematical definition of dt. I am certaint that means that you want to incorporate numbers into the definition of dt.

 

I have chosen the natural number system, because the clock is digital.

 

The clock ticks out elements of the following set:

 

N={1,2,3,4,5,...}

 

But they are just readings on this clock.

 

Let S_m, S_n, denote arbitrary elements of S.

 

An "amount of time as measured by this clock " would be defined as follows:

 

Dt = S_n - S_m

 

Where necessarily, S_m before S_n.

 

So to have a time differential, one merely needs m,n to be consecutive natural numbers. To finish the definition, you can just use first order logic which I will do.

 

I have NOT claimed that this clock ticks at a constant rate, only that it ticks once and only once, for each moment in time.

[Dave]

So much for not using any concepts of time or physical quantity

[Johnny5]

So much for not using any concepts of time or physical quantity

 

I am not introducing anything to do with a pre-conceived notion of time.

 

I was just relying upon that which follows from the operational definition of 'before'.

 

An "amount of time" becomes a difference in clock readings. This is in fact what we do in practice, so that is why it is there.

 

Here is the definition of dt which you asked for...

 

Let S _n denote an arbitrary moment in time.

Let S denote a state of the universe.

 

dS = S _n+1 - S _n

 

Let there be a clock in some arbitrary frame, which can map natural numbers onto S, in a 1-1 fashion, regardless of whether or not the clock is in an inertial or non-inertial reference frame.

 

dt = (n+1) - n = 1

[Dave]

I want you to define it using mathematics only, and using no concept of time, the universe, what is observed by people looking at clocks or any other physical notions. Purely by considering a continuous function f, mapping elements from R -> R (reals), define what you mean by "dt" and how it ties in with the definition of the derivative of f.

[ydoaPs]

perhaps johnny's reliance on logic leads not only to pseudoscience but also pseudomathematics.

[Johnny5]

I want you to define it using mathematics only, and using no concept of time, the universe, what is observed by people looking at clocks or any other physical notions. Purely by considering a continuous function f, mapping elements from R -> R (reals), define what you mean by "dt" and how it ties in with the definition of the derivative of f.

 

You know what, this is extraordinarily complicated, becase I am not explaining what I do, but rather trying to answer your question.

 

Real clocks can fall in and out of sync, and I am fundamentally aware of this. But dt in the formulas of physics, is a theoretical aid.

 

You are complicating things, furthermore, you should not try to map elements from R to R.

 

The universe changes state discretely.

 

Let S1 denote the first moment in time.

Let S2 denote the second moment in time.

Let S3 denote the third moment in time.

And let S4 denote the fourth moment in time.

 

Now, consider changes of state of the universe.

Define them as follows:

 

DT1 = S2 - S1

DT2 = S3 - S2

DT3 = S4 - S3

 

Answer me this, "what is the amount of time which elapsed by the fourth moment in time?"

 

You are NOT to assume that each state change represents an equivalent amount of time.

 

I can explain.

 

Suppose that:

 

DT1 = DT2 = DT3 = DT

 

Then you can say the answer is 3 DT.

 

But that is a supposition, one which I do not know the truth value of. Actually, I think it's false, because of something to do with acceleration. But you tell me what you think.

[Dave]

You know what, this is extraordinarily complicated, becase I am not explaining what I do, but rather trying to answer your question.

 

Well, isn't this the point? I'm trying to show you that you can't define dt by itself.

 

You are complicating things, furthermore, you should not try to map elements from R to R.

 

Why not? I can do this if I choose to do so. In fact, if I really wanted, I could map things from higher dimensional spaces and it wouldn't make a fat lot of difference.

 

The universe changes state discretely.

 

<snip>

 

Good for the universe. Now, back to the problem...

[Johnny5]

Well' date=' isn't this the point? I'm trying to show you that you [i']can't[/i] define dt by itself.

 

What do you mean that dt cannot be defined by itself. Explain that right there, that.

[Dave]

I've already stated my reasoning for this. How do you define something to be infinitessimally small?

[Johnny5]

Is this the reasoning you're talking about here:

 

[math]\lim_{x\to c} f(x) = l \Leftrightarrow \forall \epsilon > 0 \exists \delta > 0 \text{ such that } |x-c| < \delta \Rightarrow | f(x) - l | < \epsilon[/math].

 

?

[Dave]

No, that's the definition of a limit.

[Johnny5]

Yes, its the famous definition of the limit, i know that.

 

Well run through your own reasoning again for me.

 

As far as how do I define something which is infinitessimally small, the answer is almost trivial.

 

Take the difference between the quantity over two consecutive moments in time.

[Dave]

Yes, its the famous definition of the limit, i know that.

 

Then why do you ask?

 

Well run through your own reasoning again for me.

 

Look at it this way: it's useful for you to think of dy/dx as a fraction. However, it's clearly not. d/dx is an operator, and you can't just play around with dx like it's a triviality.

 

As far as how do I define something which is infinitessimally small, the answer is almost trivial.

 

Take the difference between the quantity over two consecutive moments in time.

 

This is just getting silly. This is the last time I will say this: I do not care about time, or any other physical quantity. If you refuse to acknoweldge this, then I'll just stop replying. It doesn't really look like you're listening to me anyway.

[Johnny5]

Well I don't know what you want from me.

 

The question you asked is, "how do you define something to be infinitessimally small?"

 

I don't see that you can. If you know better, then tell me. I thought you were going to use the limit concept to do exactly that.

[Dave]

No, I'm saying you can't do that. Which is kindof my point.

[Johnny5]

Well then we might not have any kind of disagreement here.

 

Answer some questions please.

 

Is space continuous?

Is time continuous?

Is motion continuous?

Is angle measure continuous?

 

Thank you.

[Dave]

I don't really care to think about this from a physical viewpoint, as I've already mentioned.

[Johnny5]

Ok, we are getting nowhere very fast.

 

Is there any error in the definition of the derivative?

[Dave]

No.

[Johnny5]

I've seen any number of people who would disagree with you (about the definition of derivative). Do you know what they are talking about?

[Dave]

Not really. The only people who tend to disagree are those who don't like the definition (or maybe even concept) of the limit. Personally, I think it's a great definition.

[Ducky Havok]

Okay, early Dave you said you couldn't think of dy/dx as fraction, but don't you have to do it to do implicit integration (actually I don't think that's what it's called, I think I just made that up)? example:

 

[math]\frac{dy}{dx}=\frac{x}{y}[/math]

[math]ydy=xdx[/math]

[math]\int_a^by\,dy=\int_a^bx\,dx [/math]

[math]\frac{y^2}{2}=\frac{x^2}{2} [/math]

 

I understand that dy/dx isn't really a fraction, but how would you do something like that without treating it as one?