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

This is one of the weird thing about pure math (including computability theory.) We are OK with proving or assuming the existence of a number logically without having to specify that number. The terms of a Cauchy sequence get arbitrarily close, but we don't when that will happen with an arbitrary Cauchy sequence. I'm used to this stuff. Analysis came pretty easy to me, once I entered a logical frame of mind and let go of metaphysics. Intuition is still in play, but it works in tandem with a logical instinct. There's nothing else like it. I also love to program, but programming is different. One builds logical spiderwebs, trapping numbers with inequalities. I found it seductive.

Yes, something magical happens at infinity... — Ryan O'Connor

Indeed! But calculus was always haunted by infinity. For a long time, the spirit was 'calculate! faith will come.' The results were good. Reliable technology and predictions emerged from the fast and loose application of a calculus that bothered the philosophers. It's very hard to avoid intuitions of infinity. Ilike the spirit of finitism, but it gets cramped and awkward quickly. Call the largest integer Z. Then Z+1 is larger. Physical arguments against this (like Wildberger's, which I browse) don't convince me, because there's an ideality to math that seems close to its essence. A Turing machine either halts or not on a certain input. I may not know which, but intuitively that's clear to me. Note that a Turing machine is a completely imaginary entity in the first place, at the heart of computability theory. 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.

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

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.

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

Again, the question is what are we approximating? Why do believe in a single pi in the first place? The circle is already an ideal object. If we decide to think of Turing machines that give decimal expansions as real numbers, then we still need equivalence classes. For each computable real number there are an infinite number of Turing machines that approximate it. Which one will we call pi? Do we put some bound on the number of steps needed to give us digits? Turing machines could be made arbitrarily slow (programmed with wasted motion). (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. )

Sometimes numbers/equations are needed to describe a system, sometimes graphs are. We can't avoid the pictures. — Ryan O'Connor

But pure math has successfully avoided the pictures. It finally triumphed over an uncertain prop. The issue here is systematic reasoning in a strict language which seemingly must be discrete. Perfectly formal proofs of theorems are strings of symbols. The whole 'right or wrong' charm of math is caught up in this. As you may know, some engineers continued using infinitesimals when they were ejected by pure math, simply because they found them convenient and gave good results. 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. I have a thin Dover book that presents them.

*Read lots of math books and you may end up an open-minded skeptic who sees the pros and cons of different approaches. You'll like system A for this charming thing and system B for another. They'll all get something right (for your intuition) and something wrong. Meanwhile all of them are correct in the sense of working logically, not appearing to lead to contradictions, and giving the expected applied results that help us build bridges that don't fall down. I'm no expert, but I think Newton's and Euler's calculus (lacking clear foundations) would be enough for most human practical purposes.

Why do believe in a single pi in the first place? — norm

Pi only encodes a finite amount of information. $\displaystyle \pi = 4 \sum_{k = 0}^\infty \frac{(-1)^k}{2k + 1}$. That's 16 characters if I counted right.

I don't see why this discussion is hung up on such an obvious point. Pi is no more mysterious than 1/3 = ...3333, another number that happens to have an infinite decimal representation. Decimal representation is handy for some applications and not for others. Decimal representation is broken. Some real numbers have infinite representations and others have two distinct representations. You should never confuse a number with any of its representations, so all of the other expressions for pi are just as valid.

https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

I agree completely. I'm trying to show Ryan the difficulties with his approach.

Of course I realize that. I'm trying to show Ryan the difficulties with his approach. — 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."

I love that movie.

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.

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

I like your quote and I see where you're coming from, especially given that I'm talking so informally. While I do not have the ability to formally present the idea, I did make a video in which I attempt to describe the intuition behind my method.

Would you please consider watching this 2 minute clip (watch from 1:56-3:48)?

Thanks!

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

I'm in complete agreement that an infinite process is a logically valid process. That's what I said, but it's counterintuitive to think that any process could 'live' forever. This is the same issue which the ancients had with the immortal soul. It's valid logic, but there's something wrong with the premises, which makes the conclusion unsound, despite the fact that the logic is valid. Furthermore, the "immortal soul" served as a very useful moral principle, just like the "infinite process" serves as a very useful mathematical principle, but usefulness does not necessitate truth, if truth is what we are ultimately after..

To say that a process is infinite is to say that it will run forever. That is what you are claiming with "infinite processes". Notice, that to say "X process is infinite", is to say "it will run forever", and that's a statement of necessity, just like to say "the soul is immortal" is to say it will, necessarily live forever. But of course you respect the possibility that the process may for some reason, at some time, cease. Therefore you say that it is "potentially" infinite. It could run forever, if it's allowed to. There is a clear difference between "it will run forever" and "it could run forever".

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. 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.

You say: "I have no doubt that math works, it's 'why' that has puzzled me for many years." The simple answer is that it works because it conforms to the constraints of our universe. We can dream up a seemingly infinite number of logically possible axioms which will be completely useless in our universe. It's correspondence with the physical reality which makes them useful. Some people will make a distinction between coherence and correspondence, claiming that coherence is all that is necessary within a system of logic, but coherence itself is fundamentally a correspondence. Fundamental laws of coherence, like non-contradiction, work because they correspond with our universe. So a coherent logical structure has a basic correspondence simply by being coherent.

You might think that this completely contradicts what I said above, "usefulness does not necessitate truth". However, we must maintain the distinction between sufficient and necessary. Proof requires necessity. So despite the fact that correspondence (truth) works, we need to also be aware that there are other reasons why principles, like mathematical axioms, work. And this is dependent on the end, the goal which we have in mind, by which "works" is judged. If the goal is not itself truth, then the axioms are formulated toward that alternative goal, and "work" for that alternative purpose. 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.

So I think you need to adjust your enquiry from 'why does math work?', to 'what does math do?' The point being that so long as people are applying it, it will work, otherwise they wouldn't be using it. The reason it works is because it's designed, shaped, conformed to the purposes which it is put to, like a finely honed tool. But if that purpose is something other than giving us truth, then sure it "works", but is what it's doing really good?

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

The issue here is with the way that I conceive of the relation between space and time. It is not conventional. With evidence derived from modern observations of phenomena like spatial expansion, 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. This means that we cannot model spatial existence with a static 3d representation, adding time as a fourth dimension, because we need to allow that the principles for geometrical figures which model 3d space must actually differ being time dependent.

There is a need to produce two dimensions of time, one consistent with our present modeling of space as a static continuity of 3d existence, and the other to allow for the changes which occur to space, they require time as well, but this cannot be the same dimension of time. The finite points you refer to are the points where the two dimensions of time relate, or intersect with each other. So there is a continuum of spatial existence, extended in time, that is the classical 3d modelling. The points within that continuum need to be more fundamental because they represent where the other dimension of time intersects, thus constituting the real possibility for spatial existence. If we propose an infinite continuum of space, and we want to map onto this continuum real finite points of possible existence, then the limited location of those points must be derived from something real. That "real thing" must be more fundamental, because it represents real existence whereas the infinite continuum is the artificial map produced by us. The real points are points of spatial expansion (in this proposed model), which dictate the contortions to classical 3d spatial representations required to account for spatial expansion.

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

The problem with this "intervals" of time is that some sort of instants are still require to separate one interval from another. Anything posited to break the apparent continuity of time would require a distinct aspect of time, calling for a second dimension. And if the intervals in any way overlap then there is also a need for multidimensional time.

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. I'm sure you have some interesting ideas but I probably won't engage much going forward, and I'll refrain from indirect remarks even if they are from great movies.

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

No problem at all. I appreciate the message! Although I don't expect a response, 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). Had I chosen to also plot y=0 then you would have seen the points corresponding to the roots. Your speedometer is measuring the average velocity but one measured over quite a short time interval. And I enjoy the quips, even that's all I hear from you.

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

If I didn't use the word topology would you have any other problems with my view? I glimpsed the article and paper. I certainly agree with his postulate: 'there is not a precise static instant in time underlying a dynamical physical process.' It seems obvious, really. But there are statements like 'there is no physical progression or flow of time' and 'a body in relative motion does not have a precisely determined relative position at any time' which I'm not convinced by. Overall, his paper is more like an essay than the type of paper I'd expect to see in a journal (but to be clear, my ideas are no closer to journal standards).

We are OK with proving or assuming the existence of a number logically without having to specify that number. — norm

I don't think it's a trivial assumption.

For a long time, the spirit was 'calculate! faith will come.' — norm

Or as some quantum physicists say 'shut up and calculate'. I'm an engineer, I get that. But the armchair philosopher in me is not satisfied.

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

Maybe, but maybe my lack of strict definitions is simply because my ideas are not mature yet.

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

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.

Why do believe in a single pi in the first place? — norm

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.

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

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?

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'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.

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

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!

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

A potentially infinite process is one which will not end (unless prematurely terminated). Does this work for you?

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

Well, can't the answer to the question simply be the infinite process? 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).

The simple answer is that it works because it conforms to the constraints of our universe. — Metaphysician Undercover

I don't think math is subordinate to physics. Both offer a path to truth.

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

Only when we understand (truth) can we make a prediction. I think they're connected, but predictions make $$$.

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

An engineer may see math as a tool but I imagine a mathematician sees math as something to be understood for the sake of understanding. Like a beautiful painting, it doesn't need any other purpose.

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

What does time mean in the absence of a ticking clock? In other words, if all objects and space are static, has time actually passed? I don't think you have a good reason to believe that time has priority over space. I also don't see why 3D space needs 2 temporal dimensions to change. All of our experiences point to there being only 1 temporal dimension. What evidence do you have to support this claim? I find it a bit hard to follow your later statements, but anyways, until I understand the motivations behind your view, there's no point talking about fundamental points.

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. It's just a term of convenience, to be able to say that at x point in time, the velocity was such and such. What is really taken is an average over a duration, and from that we can say that the velocity at any particular point in time within that duration was such and such. But you can see from the applicable formula, that "instantaneous velocity" is really just another average. And it's quite obvious that the idea that something has velocity at a point in time, when there is no duration, is nonsensical.

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

Funny you should mention that, since it's so easily disproven.

Consider the speedometer in your car. How do you suppose it works? Is there a tiny little freshman calculus student in there, frantically calculating the limit of the difference quotient moment by moment?

No, actually not. 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.

You are confusing the velocity of an object, with the procedure we teach calculus sufferers students to find the velocity of points moving in the plane or in space.

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

No, as I have just explained, velocity is a directly measurable physical quantity, like the mass or volume of an object, or the wavelength or luminous intensity of a beam of light.

What is really taken is an average over a duration, — Metaphysician Undercover

No, as I've just explained.

and from that we can say that the velocity at any particular point in time within that duration was such and such. — Metaphysician Undercover

You're confusing freshman calculus with the actual, directly measurable instantaneous velocity of a moving body.

But you can see from the applicable formula, that "instantaneous velocity" is really just another average. — Metaphysician Undercover

No, because the calculus formalism is not the velocity, it's merely the way we determine velocity given the position function. But we don't need to do that if we have a direct way of measuring the velocity.

But even your remark about the formalism is wrong, because although the value of the difference quotient at any point is not the true velocity, but rather the slope of the secant line; the limit of the difference quotient is exactly the velocity. It is not an approximation.

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

Yet another instance of the phenomenon whereby something false appears "quite obvious" to you solely by virtue of your lack of knowledge. You wield your ignorance like a weapon. A comic book character, Ignorance Man, whose slogan is "Believe the science!" while knowing none of it. Come to think of it there's rather a lot of that about these days, wouldn't you agree?

No problem at all. I appreciate the message! Although I don't expect a response, — Ryan O'Connor

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.

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

I agree that I don't understand your viewpoint. I have no trouble with adjoining points at plus/minus infinity to the real line, that's just the two point compactification of the real line. But the rest of it I couldn't follow.

Had I chosen to also plot y=0 then you would have seen the points corresponding to the roots. — Ryan O'Connor

I'll take a more detailed look at what you wrote and try to frame some specific questions.

Your speedometer is measuring the average velocity but one measured over quite a short time interval. — Ryan O'Connor

Not so, please see my response to @Metaphysician Undercover here. Briefly, 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.

And I enjoy the quips, even that's all I hear from you. — Ryan O'Connor

Often that's all I can manage. And I find many subjects in philosophy are best responded to with an old pop tune or a line from a film.

I don't think it's a trivial assumption. — Ryan O'Connor

That's the gist of constructivism. It don't exist unless I can grab it! I was/am strongly attracted to constructivism, but it comes at a cost. 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. If you check out constructivist logic, you might like it but also find it disturbing in its own way. Intuitively, some things are true or false even if we can't say which. A Turing machine does or does not halt in some logical sense. There is or there is not a string '7777777' somewhere in the expansion of pi. If I understand correctly, a constructivist would disagree, since a constructivist acknowledges time. There's something like true, false, and undetermined. (Or that's how I remember it. Perhaps someone who knows better will chime in.)

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 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.

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

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. (This is what I've seen people not realize, that math is agnostic on pretty much all ambiguous matters.) Do I believe in chess kings? Doesn't matter what I think if I publicly play by the rules. Nothing is hidden, or whatever counts epistemologically is not hidden. The primary obstacle I've seen in the transition to pure math from the calculus sequence is an implicit metaphysics that gets in the way. (I know a graph theory guy who thinks the continuum is a convenient fiction, and so on. The fictions as such are fun to play with. )

A potentially infinite process is one which will not end (unless prematurely terminated). Does this work for you? — Ryan O'Connor

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.

Well, can't the answer to the question simply be the infinite process? — Ryan O'Connor

I don't think so, because the process is the means by which the answer is produced. If the answer requires an infinite process, and the infinite process will be prematurely terminated, then the answer will not be produced.

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

The reason I am "too strict", is that I don't believe in coincidence, when it comes to mathematics. Call me superstitious, but I believe that in mathematics, there is a reason for everything being the way that it is. So when it turns out, that a circle cannot have a definite area, then I believe that there is a reason for this. The most likely reason, is that the circle is not a valid object. By "valid" here, I mean true, sound, corresponding with reality.

Here's a sort of anecdote. Aristotle, in his metaphysics posited eternal circular motions for each of the orbits of the planets. Motion in a perfect circle could continue forever because there could be no beginning or ending point on the circumference of the circle, as each point is the same distance from the centre. Of course we've since found out that the orbits are not perfect circles. What we can learn from this, is that despite the fact that the circle is an extremely useful piece of geometry, there is something fundamentally wrong with it, as a mode for representation. It is not real. And, with the irrational nature of pi, the circle actually indicates directly to us, that it is not real. So if we ignore this fact, insisting that we want the circle to be real, or that it must be real because it's so useful, and then we work around the irrational nature, creating patches, and fancy numbering systems to deal with all these seemingly insignificant problems which crop up from employing perfect circles, we are simply deceiving ourselves. 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.

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

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?

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

In the broader context of my general philosophical views, words don't have exact meanings and context is a dominant factor in what meaning they do have. So I can only grope at what 'actual infinity' means. This is the charm of pure math. There are strict definitions, cold and dry. But I think I've already answered your question. If you have the concept of a rational number, you're already there. Do you believe in a largest rational number? How many rational numbers are there? In general concepts shimmer with infinity.

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

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?

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

I think you'd probably enjoy looking into computability theory. Do you know about the halting problem? This is some of my favorite math. How do you know that there is a finite algorithim that always halts that can determine if other algorithms generate pi? But then you said that the algo that determines whether another algo generates pi is itself pi? This doesn't make sense. What a person might do is declare a particular Turing machine given a particular input to be a representative of pi and then include all equivalent-in-some-sense Turing machines as different representatives. This would have its own issues, but perhaps you see the charm of equivalence classes? You can start with something relatively concrete and define a notion of equivalence to scoop up the other entities that should be included in the concept. I don't need to know how many such machines there are. I can do some further proofs that show that addition and multiplication are independent of the representatives used. (Actually this technique blew my mind at first. It was when I first started feeling like a mathematician. AFAIK, there's nothing comparable in engineering.)

Pi only encodes a finite amount of information — fishfry

Bet you haven't seen this:


$\begin{align}& ArcTan(z)=\underset{k=1}{\overset{\infty }{\mathop L}}\,\frac{2z}{1+\sqrt{1+\tfrac{1}{{{4}^{k}}}{{z}^{2}}}},\text{ }\underset{k=1}{\overset{n}{\mathop L}}\,{{g}_{k}}(z)={{g}_{n}}\circ {{g}_{n-1}}\circ \cdots \circ {{g}_{1}}(z),\text{ } \\ & \pi =4ArcTan\left( 1 \right) \\ \end{align}$

That's pretty rad!

↪jgill

That's pretty rad! — norm

One of my inventions (probably! You can't tell in mathematics.) :smile:

I'm going to guess that the proof is nontrivial. (Well, I'd be shocked if it was easy!)

I hadn't seen that symbol for composition before. I can actually use that in something that I need to get around to writing up.

Do you think you could render that in plain English in not too many short sentences? Leaving unnecessary details aside?

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

Jeez man it's an analog computer. It gives a direct measurement of a physical quantity. You're saying there's no such thing as velocity. Your scientific nihilism is spreading from math to physics.

ArcTan(z)=L∞k=12z1+1+14kz2‾‾‾‾‾‾‾‾√, Lnk=1gk(z)=gn∘gn−1∘⋯∘g1(z), π=4ArcTan(1) — jgill

Arctan(1) is the proof of the Leibniz formula. What's the meaning of $L$?

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

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.

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

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.

You: Your instantaneous velocity is 10 km/h.
Me: How do you know?
You: Because I'm running right next to you and my instantaneous velocity is 10 km/h.

This begs the question, how do you know your instantaneous velocity? For instantaneous velocity to make sense, it needs to be based only on static quantities that exist at that instant. But just as you can't look at a photograph and determine how fast I'm running, you cannot come up with a meaningful definition of instantaneous velocity.

Now, if you were referring to my GPS-based speedometer then yes, inside my phone is a little freshman calculus student that does the math, not calculus, just a simple delta_s/delta_t.

Edit: If the needle position in your speedometer is indeed only based on instantaneous information it must be doing so like a spring, which deforms as a function of the force applied. One could come up with a correlation between spring deflection and velocity, but this is only approximate. It is not a true measure of instantaneous velocity.

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

On the contrary, I have too much time on my hands. I've been over active on this forum lately and I'm feeling the need for a break. 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.

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

Yes but this is true of any physical quantity. How do I measure the mass of a bowling ball? Well I can put it on a scale, but that only measures weight and not mass. The weight would be different on the moon.

So to measure mass, we must observe the bowling ball's acceleration response to force, as described in this fascinating thread.

https://physics.stackexchange.com/questions/179269/how-do-we-measure-mass

How can we do that? We can suspend it from a spring with a known spring constant using the formula for a harmonic oscillator. But then you (and @Metaphysician Undercover) will object that all I'm doing is measuring the springiness of the spring.

Or I can measure the centripetal force on a centrifuge, or use a small angle pendulum. But in each case aren't we just measuring something about the apparatus and not the mass itself?

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.

What if I want to measure the temperature of air? I use a thermometer, but that's only measuring the response of mercury or the coil of a metal spring or however thermometers work these days.

So what you and @Meta are saying is that we can't measure ANY physical attributes at all. This is hardly an objection to my point about velocity being directly measurable without recourse to formal calculus. It's a philosophical objection to the idea that we can do any measurements whatsoever, or to the idea that objects even have physical attributes before we measure them by proxy. But that doesn't actually address the different point that I'm making: That moving objects have a velocity, which we can measure directly (by proxy with an induction motor coupled to the driveshaft), without needing formal symbolic methods of calculus.

After all, bowling balls fall to earth with an acceleration of -32 feet/sec^2, and this was true even before Galileo discovered it and Newton modeled it with his law of gravity. You and @Meta can not deny that bowling balls fall down and that they do so with a measurable velocity at any instant of time; without denying the whole of physical science. You don't need calculus to know that falling bowling balls have a velocity. You're both confusing the mathematical model with nature itself. And the fact that all measurements require some intervening apparatus is a red herring.

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

Hey no probs, really nothing personal. I thought it might have been relevant, that's all. I find your knowledge of philosophy of maths really interesting.

I was one of the many students who failed terribly at maths. I would feel my grasp of the work slipping away in class, and could never catch up. But later in life, I've come to appreciate maths aesthetically, even though I can't understand it very well. I also think the question of the nature of the reality of number really is important. I was just perusing that article on Hossenfelder's website again, and came across this remark in the combox:

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.

I think that comment is dead wrong. There is a matter of fact about this issue. I think fictionalism really is a white flag in philosophical terms. It might not matter for applying mathematics, but it makes a major difference to your conception of the nature of reality. 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'?

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

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"

My former and late father-in-law's good friend Eugene Wigner raised this question years ago in a famous paper. I agree - I don't perceive it as a useful fiction.

:up: That essay was one of the first I encountered when I started posting on Forums.

What's the meaning of L? — fishfry

Here's what's going on, with a simple example:

$\begin{align}& {{g}_{2}}\circ {{g}_{1}}(z)={{g}_{2}}\left( {{g}_{1}}(z) \right) \\ & {{g}_{1}}(z)=2z+1,\text{ }{{g}_{2}}(z)={{z}^{2}}-3\text{ }\Rightarrow \text{ }{{g}_{2}}\circ {{g}_{1}}(z)={{\left( 2z+1 \right)}^{2}}-3=4{{z}^{2}}+4z-2 \\ & \underset{k=1}{\overset{2}{\mathop{L}}}\,{{g}_{k}}(z)={{g}_{2}}\circ {{g}_{1}}(z)={{g}_{2}}\left( {{g}_{1}}(z) \right) \\ \end{align}$

$\underset{k=1}{\overset{\infty }{\mathop{L}}}\,{{g}_{k}}(z)=\underset{n\to \infty }{\mathop{\lim }}\,\underset{k=1}{\overset{n}{\mathop{L}}}\,{{g}_{k}}(z)$

The process arises using techniques from functional equations. For example,

$\begin{align}& f(z)={{e}^{z}}-1\text{ }\Rightarrow \text{ }f(2z)=f(z)\left( f(z)+2 \right) \\ & f(z)=z(z+2)\circ f(z/2)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ 2z\circ f(z/2)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ 2f(z/2) \\ & \text{ =}\left( \frac{{{z}^{2}}}{4}+z \right)\circ \left( 2{{z}^{2}}+4z \right)\circ f(z/4)=\left( \frac{{{z}^{2}}}{4}+z \right)\circ \left( \frac{{{z}^{2}}}{8}+z \right)\circ 4z\circ f(z/4)=\cdots \\ & \text{ =}\underset{k=1}{\overset{n}{\mathop R}}\,\left( \frac{{{z}^{2}}}{{{2}^{k+1}}}+z \right)\circ {{z}_{n}},\text{ }{{z}_{n}}\to z \\ \end{align}$

But here
$\underset{k=1}{\overset{3}{\mathop R}}\,{{f}_{k}}(z)={{f}_{1}}\circ {{f}_{2}}\circ {{f}_{3}}(z)$
etc. Going the other way involves rules for inverses of compositions.

This sort of thing appears in a general study of infinite compositions, a topic of practically no interest in the larger mathematics community. Abstractions and generalizations are more attractive.