It's only obvious to me because I only know a handful of real numbers so I assume you're talking about sqrt(2). But it's not a matter of laziness, no finite amount of terms would have allowed me to eliminate any possibility. From this view (when there is no algorithm) it seems like the only important number in a Cauchy sequence is the last one...and there is no last one! Anyway, sorry for putting you in a position having to defend a position you don't support! — Ryan O'Connor
Yes, something magical happens at infinity... — Ryan O'Connor
I think this question is very important. In my view, the topological graphs that I drew actually exist. The geometric graphs that we imagine imagining don't exist, but they are incredibly convenient approximations of what we could do in reality to topological graphs. — Ryan O'Connor
True, but what if we reinterpret real numbers as real processes which describe continua, not points? Wouldn't we be able to keep the same math? Can't we just say that our algorithms for calculating the 'number' pi can never output the number completely and that pi actually corresponds to those (potentially infinite) algorithms? Why do we need the number pi anyway? We have never precisely used it as a number anyway. — Ryan O'Connor
Sometimes numbers/equations are needed to describe a system, sometimes graphs are. We can't avoid the pictures. — Ryan O'Connor
Why do believe in a single pi in the first place? — norm
I haven't been able to understand Ryan's approach. As in Apocalypse Now, when Kurtz asks, "Are my methods unsound?" And Willard responds: "I don't see any method at all, Sir." — fishfry
We can do so much with potentially infinite processes. Not only can we interrupt them to produce rational numbers, but we can work with the underlying algorithms themselves. For example, the following program to outputs the entire list of natural numbers. This program can never be run to completion, but it still is a valid program...I'm talking about it after all and it makes sense even though I've never run it. The same can be said about irrational processes. We need to embrace potential infinity for what it is, not reject it. — Ryan O'Connor
In my continuum-based constructions there are still points, it's just that there are only ever finitely many of them and they are not fundamental. Can you expand on a situation where points need to be fundamental? — Ryan O'Connor
As for your continuum ideas, almost twenty years ago Peter Lynds wrote a paper appearing in Foundations of Physics Letters that postulated time having no instants and instead being composed of intervals. — jgill
I like your quote and I see where you're coming from, especially given that I'm talking so informally. — Ryan O'Connor
It was wrong of me to poke fun at you without responding to your last two lengthy posts to me. Fact is I wouldn't know where to start so it's better for me to leave it alone. If you can't graph a simple polynomial then there's no conversation to be had. Note that the poly I gave you has a real root at 2 which none of your pictures show. And your claim that there's no such thing as instantaneous velocity is falsified by your car's speedometer. — fishfry
What makes me uneasy is using a concept like topological equivalence and then discussing slopes and derivatives, which, as you know, do not carry over in that way. — jgill
We are OK with proving or assuming the existence of a number logically without having to specify that number. — norm
For a long time, the spirit was 'calculate! faith will come.' — norm
So even your attachment to algorithms is threatened by problems with infinity. Before long we're back to lots of hand waving, no strict definitions. That's fine for practical purposes perhaps, but it's basically an abandonment of math as a distinct discipline. — norm
The danger here is replacing strict symbolic reasoning with pictures. In some analysis books, there's not a single picture, for epistemological reasons you might say. Pictures often mislead, while obviously being of great use pedagogically for applied math's well-behaved functions. — norm
Why do believe in a single pi in the first place? — norm
In other words, problems with the real numbers in pure math don't go away when we switch to computability theory --which is itself pure math, awash in idealities. If you want necessary non-empirical truths, I think you are stuck with infinity, unless there's a largest integer. — norm
Since then, infinitesimals have been rehabilitated via the hyperreals (they were made rigourous, without any metaphysical commitment), and a small minority of calc students learn calculus this way. — norm
I'm trying to meet him half way, because I do find these issues fascinating. I don't think Ryan has the experience to see math as mathematicians see it. In physics and engineering classes, one can go very far without resolving these issues or even seeing a proof. So a kind of 'outsider's mathematics ' (like outsider art) is a natural result when a person gets mathematically creative. I agree that one has to study some actual proof-driven math to genuinely enter the game, but I also understand the impatience to talk about exciting ideas now. I'm sure that Ryan is learning, and I get to dust off some math training. — norm
What I am requesting is that you recognize the constraints of the physical world, to acknowledge that an infinite process is not possible within our universe. Therefore both "it could run forever", and "it will run forever" are excluded as impossible, therefore false premises. — Metaphysician Undercover
From this perspective we see that any problem which requires an infinite process to resolve, is actually not solvable. If a question is not solvable, it is not properly posed, it is not a valid question. — Metaphysician Undercover
The simple answer is that it works because it conforms to the constraints of our universe. — Metaphysician Undercover
I find that in the modern sciences, which are the principal users of mathematics, the goal has shifted from truth to prediction, and these two are not the same. — Metaphysician Undercover
The reason it works is because it's designed, shaped, conformed to the purposes which it is put to, like a finely honed tool. — Metaphysician Undercover
it becomes clear to me that we need to give time priority over space in our modeling, to allow that space itself changes with the passing of time. — Metaphysician Undercover
And your claim that there's no such thing as instantaneous velocity is falsified by your car's speedometer. — fishfry
Velocity is always an average over a duration of time. So-called "instantaneous velocity" is just a derivative from an average. Since velocity is a measure of change, and change without a duration of time is impossible, then also true "instantaneous velocity" is also impossible. — Metaphysician Undercover
It's just a term of convenience, to be able to say that at x point in time, the velocity was such and such. — Metaphysician Undercover
What is really taken is an average over a duration, — Metaphysician Undercover
and from that we can say that the velocity at any particular point in time within that duration was such and such. — Metaphysician Undercover
But you can see from the applicable formula, that "instantaneous velocity" is really just another average. — Metaphysician Undercover
And it's quite obvious that the idea that something has velocity at a point in time, when there is no duration, is nonsensical. — Metaphysician Undercover
No problem at all. I appreciate the message! Although I don't expect a response, — Ryan O'Connor
I do want to say a couple of things. I obviously know how to plot a polynomial in the traditional sense (and I also know how to use plotting programs). If you don't see a polynomial in my graphs it's because you don't understand my view (I'm not blaming you, this may be entirely my fault). — Ryan O'Connor
Had I chosen to also plot y=0 then you would have seen the points corresponding to the roots. — Ryan O'Connor
Your speedometer is measuring the average velocity but one measured over quite a short time interval. — Ryan O'Connor
And I enjoy the quips, even that's all I hear from you. — Ryan O'Connor
I don't think it's a trivial assumption. — Ryan O'Connor
You have a good point so please allow me to soften my position. Perhaps pictures are only a handy prop in my view but the lack of symbolic reasoning may only reflect that my view is not mature. — Ryan O'Connor
I've read the Dover book on infinitesimal calculus by Keisler. It must be different from yours because mine isn't so thin. I'm not convinced that there are irrational numbers between the rationals, I'm even less convinced that there are infinitesimals in between the reals. But you're the professional and you've seen the proofs to conclude that the reasoning is rigorous so I don't want to debate about this issue. — Ryan O'Connor
A potentially infinite process is one which will not end (unless prematurely terminated). Does this work for you? — Ryan O'Connor
Well, can't the answer to the question simply be the infinite process? — Ryan O'Connor
For instance, consider the question 'what is the area of a unit circle?' Is this a valid question? In one sense, I think you're right since no rational number will do. But in another sense, I think you're too strict in only accepting rational numbers. I think it's valid to say that the answer is pi, which I believe corresponds to a potentially infinite process. (Well my beliefs are changing a bit as I talk here with norm but I think you get what I'm saying). — Ryan O'Connor
No, actually. Not even a computer program doing the same. Rather, there is a little induction motor attached to the driveshaft. The faster you travel, the faster the drive shaft spins, the faster the induction motor turns, the more current it outputs. And that current directly drives the needle of your speedometer.
Your speedometer is not a mathematically derived average. It is in fact a direct analog measurement of the instantaneous velocity of your car; subject of course to slight mechanical error common to any physical instrument. The velocity is an actual, physical quantity that can be directly measured -- that IS directly measured -- without recourse to any formal mathematical procedure. — fishfry
I don't think your criticisms of finitism apply to my view. In my view, every system does have a largest number, it's just that there's no universal system containing all possible numbers. For example, in the graph below the largest number is 99498. We could certainly 'cut' the continuum to produce points with coordinates having larger values, but until we actually do that it is meaningless to assign coordinates to those potential points. Could you expand on how I'm stuck with actual infinity? — Ryan O'Connor
I am thoroughly enjoying this discussion and I appreciate your pointed questions. So far, my view is that you've clearly demonstrated how far my view is from a formal theory (thanks!) but you haven't identified any flaws yet. You're right, I don't see it as mathematicians see it. And so a mathematician might say that my probability of being right is 0. Thankfully, that means mathematicians still believe I have a chance! — Ryan O'Connor
There are infinite potential chairs. Must all potential chairs actually exist to give the word chair meaning? The 'chairness' algorithm must be finite otherwise we'd never call anything a chair. Perhaps the same can be said about pi. Perhaps on the deepest level, pi is not the number pi, nor the infinite algorithms used to calculate the number pi, but instead the finite algorithm used to identify which algorithms would generate the number pi. — Ryan O'Connor
Oh come on fishfry, you're smarter than this. The current you refer to is just measuring revolutions of the driveshaft. Then the speedometer of the car is scaled to how many revolutions are required to cover a specific distance. It is not measuring the instantaneous velocity of your car. What happens when you use the wrong size tires? — Metaphysician Undercover
I'll take a run at your graphs when I get a chance. You went to some trouble to draw them, you deserve a response. — fishfry
your speedometer is driven by an induction motor coupled to your driveshaft. It gives a direct analog measurement of instantaneous velocity without any intervening computation. — fishfry
With you being a crankologist, I'd really benefit from your criticisms and I think you'd enjoy learning my view as I believe I am coming at infinity from a unique angle. As such, I think you'd need a different strategy to take down my ideas (assuming I'm wrong). But your time is short and crankery is infinite so whether you find time or not, it's all good. — Ryan O'Connor
You're definition of the instantaneous velocity of a car rests upon a dynamic quantity: the flow of electrons through a wire (i.e. current). So you've only shifted the problem from instantaneous velocity to instantaneous current. Consider this example. — Ryan O'Connor
I think my reacting negatively to Wayfarer's helpful link to a Sabine Hossenfelder video was a clue. I'm just crabby lately for the sake of being crabby and when I find myself doing that it's time for a forum break. Sorry @Wayfarer, I apologize. — fishfry
Complex numbers don't exist. For that matter, natural numbers don't exist. They are merely useful fictions.
I say this because I am a mathematical fictionalist. However, there are also many mathematical platonists who would disagree with me.
Honestly, it doesn't make any important difference. Platonists and fictionalists do their mathematics in pretty much the same way. Their philosophical differences don't actually affect the mathematics.
And then there's the Quine - Putnam indispensibility thesis, which argues for platonism to explain why mathematics works so well in physics. However, I happen to think that fictionalism makes more sense of the role of mathematics in physics.
So it is really much ado about nothing. Go with whatever makes most sense to you.
When you consider the predictive power of maths, the fact that through it you can discover things about reality that you otherwise could never know - how is that reconcilable with the idea that it's a 'useful fiction'? — Wayfarer
What's the meaning of L? — fishfry
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