L2k=1gk(z)=g2∘g1(z)=g2(g1(z)) — jgill

Iterated composition. Hadn't seen that notation before. Thanks.

This notation was suggested by a Japanese mathematician. I was starting to use something else, but switched to his.

This notation was suggested by a Japanese mathematician. I was starting to use something else, but switched to his. — jgill

A cursory search shows that I can't find it used anywhere. "Today I learned!"

Oh -- L and R. Nice. I thought iterated functions were getting a lot of attention these days. Fractals and such.

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.

When teaching almost thirty years I would publish a paper each year just to avoid collapsing intellectually, and all were readily accepted and published, but after I retired in 2000 I decided to do the research for me and anyone else who might be interested, avoiding the formalities of publishing. Hence a collection of informal notes on researchgate. It's amazing the audience one reaches there. Virtually every civilized nation has cropped up with reads of my stuff. But fewer than half read the entire note. Still, that can be 20-30 a week that do. And it's free, unlike most journals - and that irritates me since few journals pay the referees, at least when I was active. (in all fairness, institutions take up the slack with fewer teaching hours, etc.)

I consider the major journals corrupt in their financial practices. But that's just me. Pay no attention.

@fishfry

I'm not denying measurement at all. If an event can be completely described by a photograph then it is instantaneous. If it needs a video to be accurately described then it is a transient event. The only way to report a transient event at an instant is to compress the transient data, e.g. by time averaging it. I have no problem with (at some given instant) reporting a velocity, as long as we recognize that that quantity is not the velocity at that instant, but instead some average velocity over a short interval.

And in the case of a speedometer, I wasn't trying to say that everything is relative to everything else so measurement is meaningless. I was only saying that the problem is not solved by pushing it downstream (from a moving car to moving electrons). Measuring current is a giveaway that you're measuring a transient phenomenon. If you were measuring capacitance on the other hand (which is analogous to a spring) then you could be measuring an instantaneous event.

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.

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

What do you think speedometers measure?

I take your point about instantaneous motion, it's related to one of Zeno's paradoxes. If the arrow is not moving at a given instant, how does it know it's moving, or something like that. I don't think I have to resolve that ancient mystery to read my speedometer and know that it's telling my my instantaneous velocity. Even if it's only actually reading a current from an induction motor.

Bonus question. What is the speed of light at any given instant?

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 do like this idea and I need to find out why it's disturbing.

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

That's a reasonable statement.

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

It reminds me of the Chinese room argument in AI. Someone locked in a room could be blindly following a set of instructions to take chinese character inputs and generate outputs to convince someone that they speak chinese. It sounds like your view is that math is like this chinese room, a mere set of rules and symbols. I believe that it's more than that. I believe that the messages have content (like the Chinese messages), and if we better understand what it's saying perhaps math will be easier.

How many rational numbers are there? — norm

My response is 'how many rational numbers are where? Present me with a graph and label all points explicitly and I can tell you what the largest number on your graph is.

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

"Without mathematics we cannot penetrate deeply into philosophy. Without philosophy we cannot penetrate deeply into mathematics. Without both we cannot penetrate deeply into anything." Leibniz.

I don't think we have to decide between the two...but you're right, at this point I'm only philosophizing. But hey, I'm on a philosophy forum!

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

Yes, I like the halting problem and anything else that deals with incompleteness. I don't know if such an algorithm exists. But as it is we define pi as an equivalence class of particular cauchy sequences. All I'm proposing is that instead of pi being the equivalence class itself that it is the description of that equivalence class. Surely our description is finite, right?

@jgill
Incredible!  

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. 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.

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.

You seem to be missing the point fishfry. — Metaphysician Undercover

Not for the first time I'm sure.

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

Yeah yeah. One of Zeno's complaints. If you look at the arrow at a particular instant it's not moving. How does it know what to do next in terms of direction and speed? Not a bad question actually, one that I won't be able to answer here.

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

Yes you already said that. I take the point. If you show me a photo (taken over a sufficiently short time interval, since even a photograph takes time) the arrow appears stationary and you can't determine its velocity.

Yet, it still HAS a velocity, wouldn't you agree? And what does a speedometer measure? A current. And as @Ryan pointed out, even that's a flow of electrons. Since current is a flow, does it exist at an instant? Well I'm sure that a modern physicist would point out that the electromagnetic field exists at every moment. But if you have a magnet in a coil, you have to move the magnet to create a current. I'm really not enough of a physicist or a philosopher to know these things. Good questions though.

Still, would you at least grant me that velocity over a short but nonzero distance exists? And likewise a current flow? Then a moving car has a velocity that can be determined without recourse to formal symbolic manipulations, which was my original point.

Moderator note: When I hit the @ button to search for @Ryan O'Conner his handle doesn't come up, any ideas why?

Yet, it still HAS a velocity, wouldn't you agree? — fishfry

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.

Still, would you at least grant me that velocity over a short but nonzero distance exists? — fishfry

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.

What do you think speedometers measure? — fishfry

Imagine water flowing uniformly out of the faucet onto a flat plate. Below the flat plate is a spring which compresses. We can determine a relationship between the spring compression and the water flow rate. This is essentially how an analogue speedometer works, but with electrons instead of water.

This description may give the impression that the spring can measure instantaneous velocity but it cannot.

Consider this: the first instant the water hits the plate, the spring is not compressed. It then takes some time for the spring to find the equilibrium position, and only at that time will it report the correct flow rate. During this transient period the spring is not reporting the correct flow rate, but instead some value between zero and the actual flow rate. It's reporting some sort of average.

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. 

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

I'll accept this point. It still has nothing to do with what I originally said, which is that you don't need calculus to determine the instantaneous velocity of a moving object. And I'll concede that by instantaneous I only mean "occurring over a really short time interval."

I have to say I'm not nearly as invested in this point as the number of words written so far, I should probably stop.

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

Ok. I can live with that. Whether it's a moving arrow or a current driving the speedometer, it's a change occurring over a short interval of time. But my original point was that we don't need calculus to determine the velocity. Actual velocities are not subject to the ancient philosophical mysteries of calculus.

Still, would you at least grant me that velocity over a short but nonzero distance exists?
— fishfry

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?

Likewise does the speed of light have velocity 'c' at a given instant?

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.

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

No, because "a given instant" is not anything real which can be adequately identified. We can attempt to arbitrarily assign an instant to time, to mark a point for the purpose of measurement, but that assignment becomes much more difficult than it appears to be, at first glance. To mark a temporal point in one process or activity, requires a comparison with another process or activity, thus requiring a judgement of simultaneity. According to special relativity such judgements are dependent on the reference frame. Therefore any "given instant" may not be the same instant from one frame to the next, and the question of what a thing's velocity is at a given instant is rather meaningless because it depends on what frame of reference you measure it in relation to.

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

I don't think you've adequately considered what is required to produce accuracy in a time related measurement.

If you look at the arrow at a particular instant it's not moving. How does it know what to do next in terms of direction and speed? — fishfry

The arrow may be momentarily stationary, but it has momentum.

The arrow may be momentarily stationary, but it has momentum. — jgill

Can you explain this to @Metaphysician Undercover and @Ryan whose handle doesn't show up when you use the @ button?

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?

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.

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

Isn't this just confusing a snapshot of a thing with the thing itself? The movement is not divided into discrete moments until we try to model it using mathematics. Yet mathematics is distinct from the event, as it is merely a way to model the event (albeit, an extremely useful and accurate enough model). We necessarily measure momentum in retrospect, so when we analyse a things momentum at a particular instant it is already a passed instant. 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.

Or pehaps one day scientists will discover some kind of 'instance particles'. Now that would be interesting.

Edit - I guess I lean towards the (Bergsonian) idea of an essential indivisibilty of time/movement.

Can you explain this to Metaphysician Undercover and @Ryan whose handle doesn't show up when you use the @ button? — fishfry

The point being, that you cannot take the arrow at a particular moment in time. This is an impossibility because time is always passing, and this would require stopping time at that moment. So, despite the fact that using mathematics to figure hypothetical conditions at particular moments is a very useful thing to do, what it provides us with is a representation which is actually a falsity. Then if people start talking about this situation, with the underlying implication that this mathematics provides us with some sort of truths about these situations, this talk is really a deception or misinformation.

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

This is the key point to understanding temporal continuity, inertia, Newton's first law, and the overall validity of inductive reasoning in general. We observe that things continue to be as they were, as time passes. Intuition tells us therefore, that they will continue to be as they were, unless something causes them to change, and this intuition is what validates inductive reasoning. However, there is a very real, and very big problem, and this is the reality of change. We see that human beings have the capacity to interfere with, and change the continuity of inert things. Because of the reality of change, we are forced to accept the fact that this continuity is not a necessity. This appears to be the hardest thing for some people, especially those with the determinist mentality, to accept, that the continuity of existence, which we observe, is not necessary. This means that the supposed brute fact, which underlies all inductive principles as supportive to those principles, that things will continue to be as they have been, is itself contingent, not necessary.

When we take a law, like Newton's first law, we view it as something taken for granted. The law tells us the way things are, and it's assumed that it's impossible for things to be otherwise, that's why it's "the law". However, when we apprehend that this is not necessary, then we can grasp the fact that there is a need for a reason why the law upholds. Through principles like the law of sufficient reason, we see that if there is a temporal continuity of existence, described as momentum or inertia, and this feature of existence is not necessary, then there must be a reason for it, a cause of it.

What Aristotle did was posit "matter" as the cause of the temporal continuity of existence. So contrary to the common notion that matter is some sort of physical substance, "matter" by Aristotle's conception is really just a logical principle, adopted to account for the observed temporal continuity of physical existence. It is, in a sense, a placeholder. He didn't know the cause, but logic told him there must be a cause, so he identified it as "matter". In your example of the arrow, we do not know "how does it know where to go next", but we do know that it does. Aristotle attributes this to its "matter", or more precisely he posits "matter" as what causes it to go, where it does go, next. Therefore the theoretical points in time are in reality connected to one another by what is called "matter".

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

I'm confused by your post. You make the correct point that the math is just a model of reality, not reality itself. Then you say that the thing DOES have a momentum at a particular instant. Which is what the mathematical model says.

The point being, that you cannot take the arrow at a particular moment in time. — Metaphysician Undercover

I'm mostly in agreement with you on this point, as reading the rest of my post would have indicated.

Yes, I read the rest of your post, as reading the rest of my post should have indicated to you.

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

Kinda off topic, but have you ever seen a generalisation of the iterated composition operator to non-natural indexes? Like... does the following notion make sense in general:

Let's say we have $g=f \circ f$, does it make sense to think of $f$ as like half an application of $g$?

Are there analogous constructs for an $f$ which is $\frac{1}{\pi}$ applications of $g$?

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

The beauty of calculus is that in performing a finite number operations (e.g. in manipulating the exponents and coefficients of a polynomial to determine the derivative) we can talk sensibly about a potentially infinite process. If you reject potentially infinite processes as valid mathematical objects then you must reject calculus, and nobody will buy into your philosophy. You've got to stop thinking of the output of a potentially infinite process as the mathematical object. The mathematical object is the process itself. If you're challenging the orthodox view on real numbers then your point is valid, but if you're challenging my view then your point is misdirected.

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.

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

What do you make of 1/3 = .333...? Can't you distinguish between a number and one of its representations that you don't happen to like? After all 1/3 is just a shorthand for the grade school division algorithm for 3 divided into 1.

Kinda off topic, but have you ever seen a generalisation of the iterated composition operator to non-natural indexes? — fdrake

Yes. Goes back a hundred years if my memory serves. Sometimes it's very easy. For example, here is a linear fractional transformation written in terms of fixed points and multiplier.(I'm working on a theorem right now involving this). I think a more general case was dealt with in the discipline of functional equations. Can't recall the work offhand.


$\begin{align}& f(z)=\frac{(\alpha -K \beta )z+\alpha \beta (K -1)}{(1-K )z+(\alpha K -\beta )},\text{ }{{F}_{1}}(z)=f(z),\text{ }{{F}_{n}}(z)=f\left( {{F}_{n-1}}(z) \right) \\ & \Rightarrow {{F}_{n}}(z)=\frac{(\alpha -{{K }^{n}}\beta )z+\alpha \beta ({{K }^{n}}-1)}{(1-{{K }^{n}})z+(\alpha {{K }^{n}}-\beta )} \\ & Define\text{ }{{F}_{t}}(z):=\frac{(\alpha- {{K }^{t}}\beta )z+\alpha \beta ({{K }^{t}}-1)}{(1-{{K }^{t}})z+(\alpha {{K}^{t}}-\beta )},\text{ }t\in R \\ \end{align}$

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

I'm not looking for people to buy in, I'm looking for truth. If others are looking for the same thing, they might like to join me. Otherwise I don't really care if people might deceive themselves into thinking that they are engaged in infinite processes. Many think that the soul is eternal, and this doesn't both me either. I consider those two beliefs to be very similar.

The mathematical object is the process itself. — Ryan O'Connor

There is a fundamental incompatibility between an object and a process, which was demonstrated by Aristotle. If an object changes, it is no longer what it was. We assume a change (process), to account for the object becoming other than it was. So we have object A, then a process, then object B, whereby object A becomes object B. If we represent the intermediary between A and B as another object, C, then object A becomes object C which becomes object B. Now we need to assume a change (process) to account for object A becoming object C, and a process to account for object C becoming object B. We might represent the intermediaries between A and C, and C and B, as objects again, but you can see that we're heading for an infinite regress. So we ought to conclude that "objects" and "processes" are distinct categories.

I'm not looking for people to buy in, I'm looking for truth — Metaphysician Undercover

Time for you to develop a new axiomatic system, then, that leads to "Truth".

So we ought to conclude that "objects" and "processes" are distinct categories. — Metaphysician Undercover

Agreed.

Pi is a finite number because it's inbetween 3 and 4. But if the length of a circumference is multiplied by pi than you have a length with space corresponding to each number, so the circle has infinite space within a definite finite limit (like being inbetween 3 and 4). Aristotle never understood this stuff

Aristotle never understood this stuff — Gregory

I'm thankful for that. :roll:

If you have a two foot segment and make it into a circle, suddenly it's pi/r/squared instead of two feet. Hmm. It seems that we must "round to the finite" in everything we do in geometry