This notation was suggested by a Japanese mathematician. I was starting to use something else, but switched to his. — jgill
But it doesn't actually addressing the different point that I'm making: That moving objects have a velocity, which we can approximately measure directly, without needing formal symbolic methods of calculus. — fishfry
Your original claim was that my rejection of instantaneous velocity is falsified, which I think is false. If you now claim that the speedometer must necessarily be reporting some average or approximate velocity then I have no problem with that. — Ryan O'Connor
Consider some great mathematicians were attracted to Brouwer's ideas, but they found that it was not worth all that had to be sacrificed for it...There's something like true, false, and undetermined. — norm
I see your view as gestating. It's born for a mathematician when there are axioms and a logic. I hope you continue with it as long as you keep enjoying it. — norm
Consider that the reasoning is dry and formal with no 'metaphysickal' commitment. A person could not even 'believe' in integers and still be great at pure math. — norm
How many rational numbers are there? — norm
Until a formal system is erected for examination, we're not doing math but only philosophy (but then I love philosophy, so I'm not complaining.) If you remember my first response, I suggested that the issue of fundamentally social. Who are your ideas ultimately for? Mathematicians or metaphysicians? — norm
Do you know about the halting problem? This is some of my favorite math. How do you know that there is a finite algorithm that always halts that can determine if other algorithms generate pi? — norm
In short, your objection is valid, but overly general. We can't measure any physical quantity at all by your logic. What if I want to measure the wavelength of a beam of light? Well I use a spectrometer, but all that really measures is the prism or the glass or however spectrometers work. — fishfry
You seem to be missing the point fishfry. — Metaphysician Undercover
Velocity is a measurement of motion, and motion only occurs when time is passing. At an instant zero time passes. Therefore there is no motion at an instant, and no velocity at an instant. — Metaphysician Undercover
A measurement of velocity requires a determined distance over a determined duration of time. It requires two instants, to determine a duration of time, one to mark the beginning of the period of time, the other to mark the end of the period of time, just like it requires two points to determine a distance. One instant (point in time) is insufficient for a determination of velocity, just like one point is insufficient for a determination of distance. — Metaphysician Undercover
Yet, it still HAS a velocity, wouldn't you agree? — fishfry
Still, would you at least grant me that velocity over a short but nonzero distance exists? — fishfry
What do you think speedometers measure? — fishfry
And as you drive your car continually accelerating and decelerating, the spring behind the needle is continually playing 'catch-up' and thus reporting some sort of average. In fact, it is most meaningful to say that it is always reporting an average. — Ryan O'Connor
Sure, the object is described as moving, it must have a velocity. But it cannot have a velocity at an instant, if no time passes at an instant, just like a point has no spatial extension. That's why points and lines are incompatible, and a line is not composed of points, but points mark off line segments.
So the solution to the issue with velocity, is not to say that it has no velocity, it is to say that there is no such thing as the instant. Time is not composed of instants. So the arrow, or car always has velocity, all the time that it is moving, but that time has no instants. The instant is just an arbitrary point which we insert for the purpose of making a measurement. — Metaphysician Undercover
Sure, but the whole point I am arguing in this the thread is that the inclination to reduce the nonzero distance to zero, or even define it as somehow related to zero, produces theoretical absurdities. And this is well demonstrated by these Zeno type paradoxes which speak of time as consisting of instants. — Metaphysician Undercover
Ok. Maybe. Let me put to you a hypothetical. An object moves with constant velocity. Does it have a velocity at a given instant? — fishfry
I'm kind of done with this topic, the point I'm making isn't worth all this ink. You don't need calculus to do analog measurements. And yes physical measurements depend on time, even if those intervals are tiny. There aren't any actually physical instants as far as we know. Or as far as we don't know. The matter is not answered by current science. — fishfry
The arrow may be momentarily stationary, but it has momentum. — jgill
Does it have, say, "metadata," a data structure attached to it that says, "Go due east at 5mph?" You can see that this is problematic. — fishfry
Can you explain this to Metaphysician Undercover and @Ryan whose handle doesn't show up when you use the @ button? — fishfry
But actually it's a good question. Suppose there were such a thing as an instant of time, modeled by a real number on the number line. Dimensionless and with zero length. So the arrow is there at a particular instant, frozen in time, motionless. Where does its momentum live? How does it know where to go next? — fishfry
However, just because we apply delineation through our act of retrograde analysis, and create a mathematical notion of temporality, that doesn't mean that the thing did not have a momentum at a particular instant. It's only a question of accuracy. — emancipate
The point being, that you cannot take the arrow at a particular moment in time. — Metaphysician Undercover
Look up Shota Kojima and infinite compositions. A lot of stuff out there on fractals and simple iteration, but composing different functions endlessly not very much, especially if one considers complex functions that are not holomorphic, which I have done. They are far more interesting IMO. — jgill
I understand this, but my point is that due to the nature of our universe, any such "potentially infinite process" will be prematurely terminated. So it doesn't make any sense to say that such and such a process could potentially continue forever, because we know that it will be prematurely terminated. Therefore, if we come across a mathematical problem which requires an infinite process to resolve, we need to admit that this problem cannot be resolved, because the necessary infinite process will be terminated prematurely, and the problem will remain unresolved. — Metaphysician Undercover
We end up believing that the real figures which we are applying the artificial (perfect circles) to are actually the same as the artificial, because all the discrepancies are covered up by the patches. — Metaphysician Undercover
I agree. By treating rationals and irrationals both as the same type of object (i.e. numbers) we blur the line between the output of a finite algorithm and the output of a potentially infinite algorithm. — Ryan O'Connor
Kinda off topic, but have you ever seen a generalisation of the iterated composition operator to non-natural indexes? — fdrake
If you reject potentially infinite processes as valid mathematical objects then you must reject calculus, and nobody will buy into your philosophy. — Ryan O'Connor
The mathematical object is the process itself. — Ryan O'Connor
I'm not looking for people to buy in, I'm looking for truth — Metaphysician Undercover
So we ought to conclude that "objects" and "processes" are distinct categories. — Metaphysician Undercover
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