[in reference me me asking why everything must be sharp] That's just how math is — Gregory

It's true that in my video I showed a blurry curve in between the measurements, but prior to that I showed the curve being 'topological' [ I use 'topological' only in the sense of the graph having properties which are preserved through continuous deformations]. If instead of saying that I'm 'blurring' the graph, what if we say that I'm reinterpreting it from being a geometric object to a topological object? With this view, your argument can't simply be 'that's just how math is' because topology is certainly acceptable math.

I don't think you solved Zeno's paradox because you're putting the infinite quantity into philosophically blurry box and focusing just on finite results. — Gregory

Isn't that what we do with quantum mechanics? We have finite results corresponding to our actual measurements and everything between the measurements is a 'blurry' superposition? Why must everything be 'sharp'?

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. — Metaphysician Undercover

Never in the history of mathematics or physics has discovering truth set us backwards. The fact that your philosophy would result in a weaker mathematics is a red flag that you're on the wrong track. Don't get me wrong, I agree that there's a problem, I just don't agree with your resolution.

I made this video on my proposed resolution to Zeno's Paradox. What do you think?

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

I think you're taking my words too literally. Clearly, the process of taking my dog for a walk is not an object with mass and momentum. When I say that processes are valid objects of mathematics, I simply mean that they can be studied in themselves, just as one might write a book entitled 'The Art of Dog Walking'.

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. — fishfry

I believe that's a false equality. The correct statement is "the potentially infinite process defined by 0.333... converges to the number 1/3" not "the number 0.333... equals the number 1/3". Decimal notation is flawed in that it cannot be used to precisely represent some rational numbers, like 1/3. If we want a number system which can give a precise notation for any rational number, we should use Stern-Brocot strings, where 1/3 = LL.

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. — fishfry

If photographs can't capture motion but videos can, why not conclude that motion happens in the videos? The reason why we are reluctant to come to this conclusion is because we reject the notion of videos being fundamental.

We want points (photographs) to be fundamental and continua (videos) to be composite and as long as we hold this view we will not find a satisfactory resolution to Zeno's paradoxes. If you flip things upside down and see continua as fundamental and points as emergent, then everything makes perfect sense. There's no problem with pausing a video to produce a static image.

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. — fishfry

It is clear that you appreciate the profoundness of Zeno's Paradox. Zeno presented these paradoxes in response to the criticisms from the 'one from many' camp calling his views ridiculous. Why not consider the 'many from one' view that he supported? He was wayyy ahead of his time so his view did seem to have problems of their own...but in light of modern advancements in physics his view no longer seems crazy.

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. — fishfry

Don't stop here, you may just be on your way to becoming a crank! With this admission you have placed yourself on a slippery slope. Instantaneous velocity is no different from the tangent of a function at a point. Do you accept that the derivative corresponds to a limiting process of secants rather than the output of a completed infinite process (i.e. tangent at a point)?

An object moves with constant velocity. Does it have a velocity at a given instant? — fishfry

Only if you consider 0/0 a valid velocity.

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.

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. 

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!  

@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.

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.

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.

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!

That would be the empty set, Ryan. We were all math majors and the course was taught in the math department. — jgill

Mathies keep the fun stuff to themselves. :P

That's a well made video. FWIW, I do like the continua-based approach. Have you looked into smooth infinitesimal analysis ? It seems similar. One issue worth noting is your description of a quasi-Riemann integral as an endless process. In an actual Riemann integral, for f which is continuous on [a,b], there exists a definite sum. In other words, we know that it's a particular real number, even if we only ever approximate it (like the areas under the standard normal curve.)
In SIA, certain issues are circumvented, because every function is smooth (infinitely differentiable). Some strange logic is involved. — norm

Thanks for checking it out! I'm glad you like the approach. I have not looked into SIA. However, just a few points on the wiki page seem concerning to me, like I have no problems with discontinuous functions but I do have a problem with infinitesimals. Nevertheless, I will check it out. For Riemann integrals, how do we know that it corresponds to a real number if we are only ever able to approximate it?

Interesting! So, you think the square root of 2 could be something other than a number. Well, the square root operation is closed over real numbers i.e. a square root of a real number has to be a real number. Does that answer your question? — TheMadFool

Yes, I think it's an algorithm for calculating a number (but the algorithm cannot be executed to completion). I'll need to look up a proof of your statement and hopefully I can understand it! Perhaps it will convince me.

Consider though that ellipsis are just shorthand that lazy mathematicians use for one another. In this case, it should be obvious how the sequence proceeds. Lots of different algorithms can give the exact same sequence, and that's why equivalence classes are necessary. — norm

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!

Don't forget the jump between finite and infinite sets! — norm

Yes, something magical happens at infinity...

I'm all for bold ideas. I don't know of any paradoxes. I think 'discomforts' is better. AFAIK, mainstream math works, is correct (even if we can't prove it.) The only problem is that it offends lots of peoples intuition here and there. — norm

There are a lot of infinity-related paradoxes which offend students' intuitions. Given that it drew you in, perhaps the strangeness is a strength (not a weakness) of infinity. I have no doubt that math works, it's 'why' that has puzzled me for many years.

This idea would require a radical change in the foundations, sounds like even set theory is jettisoned. If you could rewrite a calculus textbook so that calculations come out the same (so as not to clash with mainstream math in applications), it could be presented as a pedagogical alternative. — norm

I haven't worked it out, but IF this is a valid perspective I don't think the math would change much. I think that (in a way) this is consistent with how we've been thinking all along (as I described to jgill below). We'd write the same equations and draw the same pictures, we'd just think about it differently. I think it's a matter of philosophy, not math. But you're totally correct, this is only an intuition...an idea...a formal theory is a whole different thing. And you're right...I'd have my hands full fleshing it out. And to be honest, I don't have the necessary skills to flesh it out.

What would you do with limits? Infinite sums? — norm

I believe that my view is in agreement with limits and infinite sums as long as you think of them as descriptions of unending processes. For example, 1+1/2+1/4+1/8+... corresponds to the unending process of computing larger and larger partial sums which approach but never arrive at 2.

The fundamental question is something like: what are we approximating? — norm

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.

A limit is a real number, a point, and not the process (in the mainstream view). — norm

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.

We can draw the symbol root(2) confidently because we can prove that it exists from the axioms, (IVT) entirely without pictures. The desire to free math from pictures should perhaps be addressed here. Can your system free itself from pictures? A theory of continua would presumably have to be symbolically established. Would classical logic work? Would you still find the system charming if the pictures were secondary and only props for the intuition? (Just trying to ask productive questions. Hope they inspire you!) — norm

These are good questions indeed, thanks!! I don't know the answers to them, but here are my thoughts. There's a convenience to the completeness of the real numbers. We get a one-to-one correspondence between our numbers/equations and our graphs. If in our graphs, lines cross at a point, then we can always find the coordinates of that point (and most often they'll be irrational numbers). If we calculate the derivative of a function at a point, then we can always draw a tangent at that point. With my view, we cannot always do these as they would amount to completing an infinite process. We need to be clever to avoid actual infinity and I think the way to do it is to use both numbers/equations and graphs. Sometimes numbers/equations are needed to describe a system, sometimes graphs are. We can't avoid the pictures.

As for logic, this puts me even further out of my comfort zone but I'm guessing some hybrid would be required: classical logic for the discrete points and fuzzy logic for the continua. This may seem like a patchwork solution to connect the discrete with the continuous but it has parallels to what we do with quantum mechanics to describe our universe (measured objects have definite states and unobserved objects remain in a superposition of potential existence).

I know what you mean here, but reading it makes me uneasy. — jgill

I appreciate why it makes you feel uneasy, but consider this claim: Every** graph that you have ever seen was topologically precise and geometrically approximate. For example, if you and I were to independently sketch out a graph of y=x^2-2, our graphs would share few geometric properties. They would be of different sizes and neither would perfectly capture the curvature of the parabola. But when we compare our graphs we still see them being the same because we continuously deform the graphs in our mind to see that they correspond to the same system. And this goes for computer generated plots as well. A computer plotting y=x^2-2 does not place infinite points on your screen but instead calculates the rational coordinates of a finite set of points and imprecisely connects the dots. But once again, we have no problem with it because it is topologically precise. The only difference is that in my plots I continuously deformed the graphs to a state where it was clear that they were not geometrical.

With 'parts-from-whole' constructions, the systems are topological and described perfectly (without approximation). There's no need to imagine a cloud of infinite points, what you see is what you get. It's just that these plots have infinite potential to be cut.

**When I said 'every graph' in the first sentence of this response I was exaggerating a little because some systems cannot be drawn with perfect topological precision. For example, y=sin(1/x) and y=1 cannot be plotted in the same graph because we'd have to draw infinite points where they interest and that's impossible. In such cases I'd just conclude that that pair of functions cannot be plotted simultaneously. I know it's an odd claim, but it's not too far from the complementarity principle in quantum mechanics which holds that objects have certain pairs of complementary properties which cannot all be observed simultaneously.

I see that you're a retired math professor so I'm especially keen to hear your feedback, especially if you see a flaw in my argument. Thanks!

My belief is that we need to go one step further, and apprehend an infinite process, or "irrational process" as actually impossible. But since this process is a potential process, as you describe, this means that it is a possible process which is actually impossible. Therefore the infinite process must be rejected as logically invalid, because it's contradictory. — Metaphysician Undercover

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.

while ( i > 0):
    print (i)
    i = i + 1

I believe that this distinction between the continuum perspective and the points perspective is a very good start, but I don't think it's an either/or question. We need to allow for both. It is the application of both, the two being fundamentally incompatible, which leads to infinity, and the appearance of paradoxes. However, we cannot simply exclude one or the other as unreal, and unnecessary, because there is a very real need for both non-dimensional points, and dimensional lines. You cannot remove the points because this would invalidate all individual units, therefore all number applications would be arbitrary. — Metaphysician Undercover

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?

What I propose is a fundamental division between numerical arithmetic and geometry, which recognizes the incompatibility between these two. — Metaphysician Undercover

I think we might be on the same page regarding this issue. I mentioned to norm above that I think numbers/equations and graphs are complementary, not equivalent. The problems occur when we think there is a one-to-one correspondence between the two.

It's unfortunate the word "foundations" is used in mathematics. Foundational set theory (that intersects with philosophy significantly) and all that revolves around it is a marvelous intellectual area, but a great deal of mathematics flows unimpeded by its pronouncements. Do most seasoned mathematicians worry about the existence of irrational or non-computable numbers? Or non-measurable sets? Not likely. — jgill

I liken it to physics. Many engineers get by with classical physics. They don't worry that it predicts singularities because it works great for them in their applications. But the singularities are the loose thread which suggest that classical physics is not fundamental. It doesn't lie at the foundation. We need to go quantum. If set theory properly lies at the foundation of mathematics then (I believe) it should have no loose threads (e.g. paradoxes). With that said, obviously set theory works to a significant extent so even if further refinements needed to be made to the foundation, I'm sure that the essence of set theory will play a significant role.

Afterwards the prof - a young energetic fellow - made the statement, "Unless you are really fascinated with the material in this course, I recommend you never take another foundations of mathematics class." — jgill

LoL. I'm sure the philosophy students in the room were aghast.

I don't understand your idea at all. Suppose the position of a particle at time tt is given by f(t)=t3−5t2+9t−6. Find the acceleration of the particle at t = 47.

How do you do that problem after you've thrown out 350 years of calculus and our understanding of the real numbers? What happens to the whole of physics and physical science? Statistics and economics? Are you prepared to reformulate all of it according to your new principles? And what principles are those, exactly? That there aren't real numbers on the real number line? — fishfry

Because I'm lazy, I'm going find the velocity of the particle instead of acceleration. First, I want to show you three valid graphs of y=t3-5t2+9t-6 (valid in my construction, that is).



You have to remember that these plots are topological so even though it looks linear in the first image I could have just as well drawn it with squiggles. The key thing is that I've only made a finite number of 'cuts' (exactly like cutting the string in my last post to you, but this time I'm cutting a 2D continuum). All of the information contained in these three graphs is correct. It's just that the graphs have the potential to be cut infinitely more times. The more cuts I make, the more points will emerge and I (a 'computer') will make sure that the points have coordinate values consistent with the functions.

First, I need to point out that (in my view) velocity at an instant is meaningless. It's equivalent to 0/0. Velocity applies, not to points, but to curves. In the third graph I've highlighted a section of the function between x=47 and x=48. As normal, I can calculate the average velocity across this interval using the coordinates as follows (99498-93195)/(48-47)=6303.

But obviously we're not satisfied with that. We want to shrink this interval as much as possible. And we can do so by making cuts closer and closer to x=47 and finding the average velocity across those shrinking intervals. This is what the limit describes (in my construction), it is a potentially infinite process.

What calculus does is describe the potential of that process. And I believe that when calculus was made rigorous by going from numbers (infinitesimals) to processes (limits) some 'infinite-like' numbers (irrational numbers) were left behind. I believe that to complete the job, we need to reinterpret irrational numbers as irrational processes. Calculus is the study of potentially infinite processes. In my view, the math is the same, dy/dt=3t2-10t+9. It's just that the philosophy is different.

This may seem like a trivial difference, but I believe that with this continuum-based view (as opposed to the standard points-based view) many paradoxes are no longer paradoxical. In fact, I can't even think of a paradox with this view (especially given our refined intuitions developed through quantum-mechanics).

I hope I was clear in my explanation. I'd love to hear your feedback, especially if you have criticisms. Thanks!

For instance: 1, 1.4, 1.41, 1.414, 1.4142, .... just 'is' the square root of 2 — norm

I find this example unsatisfying given that everything important is contained in the ellipses. You are no better just writing "For instance: ... just 'is' the square root of 2". And so that equivalence class could just as well correspond to 42. The only way to give it meaning is to state the algorithm used for generating the sequence, which is why I think non-computable numbers are questionable since there is no algorithm behind them.

Have you looked into Dedekind cuts? — norm

Here's a dumb question for you: how can the rational numbers (of which there are only aleph-0) can be cut in c unique ways? For example, if there are 2 numbers, then there's only 1 unique cut. If there are 3 numbers, then there are only 2 unique cuts. If we approach the limit, how do we end up with more cuts than numbers?

If you want to use algorithms (an idea I like), it seems you need to either use mainstream computability theory or rebuild that too. But the computable numbers have measure 0, so you'll have to rebuild measure theory or stick with early analysis. — norm

I think our problem is that we're using numbers to model a continuum. As I'm discussing with fishfry in this thread, I think we should do the opposite and instead use a continuum to model numbers. I think flipping this on its head avoids the paradoxes, allows objects to have non-zero measure, and does not require us to decide between the discrete and continuous because they actually do play well together.

Why call this "potential infinite" then? If you are certain that the process goes without end, then you are certain that it is actually infinite. — Metaphysician Undercover

This is standard terminology. Check out this wikipedia page

This is why I didn't like your use of "infinity". You used it as if it signified something with actual existence, which one could be approaching. — Metaphysician Undercover

Who says only actual things be approached? I can certainly approach a mirage.

the real question is why brilliant people have pokéballs tattooed on their arms..... — TaySan

One day they may reveal the pokémon stored in that ball and the answer will be clear.

@norm: Thanks for the book recommendations, I plan to read both "Analysis By Its History" and "Groundless Grounds".

The mainstream real line is a vast darkness speckled by bright computable numbers, numbers we can actually talk about, numbers with names, while most of them are lost in the darkness and inferred to exist only indirectly. — norm

Very poetic, I like it!

But when I do math, I don't think of R in terms of that glorious set-theory mess at all...In my POV, foundations is its own fascinating kind of math. It doesn't really hold up the edifice of applied calculus, IMO. It's a decorative foundation. Humans trust tools that work most of the time. Full stop. — norm

To me it is concerning that the foundations are so disconnected from the applications. Could this be an indication that further foundational work is required? I'm not sure if you're following this thread closely but I pointed GrandMinnow to a few links on my YouTube channel. Here's one that's somewhat related to our discussion on integrals (Dartboard Paradox). You might be interested in this perspective as it offers a different perspective (granted, probably wrong and certainly half-baked). Nevertheless, I'd love to hear your thoughts on this view, especially if you can find flaws in it...but no pressure at all!

@GrandMinnow: Imagine having a discussion with a child. If they ask a question, one way of addressing it is to add layers of complexity to the issue such that it is beyond their grasp, pile on a dozen textbooks and say 'ask me when you know what you're talking about'. Another way is to simplify the issue to the bare essentials and provide an informal (perhaps imperfect) answer to inspire them to continue their quest for knowledge. And you may even learn something in the exercise of simplifying the issue to the bare essentials. I certainly find the repeated 'why...why...' questions from kids quite revealing of my own lack of understanding.

I'm not here to pick fights or spread misinformation. I'm here to learn. I've been quite open about my educational background and (I think) I've asked you many more questions than claimed that I have the answers. We don't need gatekeepers to 'scenic trails', we need people to help the litterers learn how not to litter. In joining this forum you did not sign up to teach others so feel free to ignore my messages, but if you're inclined to help then I welcome it. I certainly could benefit from someone with your knowledge.

FWIW, at the end of all 3 videos that I linked you to I include a message, like this one from my derivative paradox video:

"Now it’s worth repeating that the complete point-based construction is how we do math. The incomplete construction simply offers a different perspective on the paradox. We should not discard established ideas just because a different view might offer a more appealing resolution to a single paradox...Can we even do math with incomplete constructions? Or are there insurmountable problems with that approach? Let’s talk about it."

There are two issues being discussed here: (1) potential problems with the current philosophical foundations for math (2) potential problems with my proposed half-baked alternative to the philosophical foundation for math. I don't think you will enjoy us talking informally about (1) so I recommend that we set that aside.

Here are some replies to a subset of your comments which I think are most relevant:

Zeno's paradox (at least as it is usually presented) is not a formal mathematical problem — GrandMinnow

I think the beauty of paradoxes, such as Zeno's, is that they capture the essentials of a profound problem in a way that anybody can discuss. If you don't want to talk informally, or if you want to disregard Zeno's paradox due to its informal presentation that is fine, we can go our separate ways.

I watched the one about Zeno's paradox up to the point you said, "mathematicians begin with an assumption [that] [an] infinite process can be completed". What exact particular quote by a mathematician are you referring to? Please cite a quote and its context so that one may evaluate your representation of it in context, let alone your generalization about what "mathematicians [in general] begin with as an assumption" — GrandMinnow

You're right that I shouldn't have made that generalization. My apologies. Here's one quote by James Grime on Numberphiles video on Zeno's Paradox:

"I want to give you the mathematician's point of view for this, because, well, some say that the mathematicians have sorted this out........So something like this-- an infinite sum-- behaves well when, if you take the sum and then you keep adding one term at a time, so you've got lots of different sums getting closer and closer to your answer. If that's the case, if your partial sums--that's what they're called-- are getting closer and closer to a value, then we say that's a well-behaved sum, and at infinity, it is equal to it exactly. And it's not just getting closer and closer but not quite reaching. It is actually the whole thing properly."

To be fair, he follows that by saying that that's the paradox. Do you believe that infinite processes cannot be completed? If so, how can I move from A to B to C. I'll never get to C because I'll never complete the infinite steps required to get to B.

Anyway, I'm not sure where you want our conversation to go, but I'd be glad to hear your feedback on the rest of the videos that I linked if you care to give me a chance.

Again, set theory rises to the challenge of providing a formal system by which there is an algorithm for determining whether a sequence of formulas is indeed a proof in the system. So whatever you think its flaws are, that would have to be in context of comparison with the flaws of another system that itself rises to that challenge. — GrandMinnow

Agreed.

It's worth noting that the challenges in the first post of this thread have been met. But hell if I know whether the poster understands that by now. — GrandMinnow

Should we move this discussion to a new thread?

if someone worked out pi to hundreds of decimal points, and was still not convinced that it would go forever. That person would say that it appears to approach infinity, and it is potentially infinite, but I think it might still reach an end at some point, so I won't admit that it's actually infinite. — Metaphysician Undercover

When I say potential infinite, I don't mean 'something that might be infinite'. I mean a process that certainly goes on to no end. We are certain that if you begin to write out the digits of pi that that process would never end.

But what is the meaning of that point which you label as (∞,0)? How can ∞ represent a point? You say it's a "pseudo-point". I assume that this means that it's not a valid point. What's the point in having a non-valid point? I can see how it's useful in practice, but this is an exercise in theory. — Metaphysician Undercover

I call it a pseudo-point because it's not an actual point on the graph but instead a potential point. It's a point that you can approach but never arrive at. By 'never arrive at' I mean you can't plug one of its coordinates into a calculator to compute the other.

But if one does reject non-computable numbers, then R has measure 0, which completely breaks modern analysis. — norm

Can you give me an example of what would break down without non-computable numbers?

but no foundation has ever seemed 'just right ' to me. There's always some ugly weeds. In the end I'm a relatively carefree antifoundationalist who enjoys math as an excellent if imperfect language. — norm

I like that you admit that there are ugly weeds. So you're just satisfied ignoring the weeds? But you must enjoy the philosophy to some extent, you're here after all? Actually, I'd love to hear what you think these weeds are...

IMO, there's a 'know how' at the bottom of things that perhaps can never be formalized or made explicit. — norm

Can you explain what lies at the bottom that you don't think can be explained?

The personal context you present is based in gross ignorance of the subject. I don't opine as to all discussions, but this discussion engenders an unfortunate trail of waste product - gross misinformation, misunderstanding, and confusion. Sometimes people who have an appreciation of the subject wish not to see such otherwise beautiful scenic trails left without being de-littered. — GrandMinnow

Perhaps we are both making this 'scenic trail' unpleasant for the other in different ways. Do you want to live in a country where the 'scenic trails' are exclusive to the 'privileged rich'? You are trying to find a way to reject my ideas without understanding them. Don't waste your time and simply disregard my posts. 

What does [that a line is not composed of points, but instead points emerge from lines] mean? — fishfry

Imagine that lines are fundamental, not composite objects. Take a string and mentally label the two endpoints -∞ and ∞. In my world, this string only has two points - the endpoints (I'd actually call them pseudo-points but that's not important here). Cut the string somewhere and mentally label the cut (i.e. the gap) 42. This string now has an additional 'point'. It doesn't matter where this cut was along the string as long as you follow this rule: if you cut the string somewhere between 42 and ∞ then that cut will be mentally labelled with a number greater than 42 (and if you cut the string somewhere between 42 and -∞ then that cut will be mentally labelled with a number less than 42).

The only object in my example was a string, which I'm using to describe a line. There are only a finite number of points on the string and they correspond to the cuts...the absence of string. The points only emerge when you make cuts and the numbers and their proper ordering are contingent on you (a 'computer') actively thinking about them. If you forget about the mental labels, then the numbers no longer exist. You are left with a string...a line...a continuum....and nothing more.

This way of building the parts from the whole is not new. It was (loosely) how Aristotle resolved Zeno's Paradox. I believe that this construction is paradox-free, but I'd love to hear your criticisms.

The real number line is composed of real numbers. How can you disagree with that? — fishfry

What if instead of 'the real number line', I defined the object that's constructed from all of the real numbers 'the real number point'? Does my definition make it so? It may seem like I'm giving a silly example but I truly think you'd have a tough time explaining why your definition is better than mine, and I think this because I believe that the real numbers are singular. But since you are not convinced by Wildberger's arguments, what are the chances that an untrained engineer could convince you with informal reasoning. Perhaps your time is better spent telling me why my 'parts-from-whole' view is wrong rather than hearing me informally complain about why I think your 'whole-from-parts' view is wrong...

we also can draw the unit square and its diagonal and try to measure it 'perfectly' or 'ideally' and discover irrational numbers. — norm

We can perhaps use Newton's method or some other algorithm to produce better and better approximations of sqrt(2) but trying to measure a 'perfect' value doesn't imply that you've discovered it. Perhaps all that you've discovered is an algorithm...and not an irrational number.

A mathematical line is composed of points. But there is no "next" point after any given point. — fishfry

How can we travel from one point to the another without traversing through the intermediate points in sequence? This is essentially Zeno's paradox and if you cannot offer a resolution to it then you are not justified to claim that a line is composed of points.

By the way, how do the zero-dimensional, zero-length points in the unit interval make up the one-dimensional, length 1 unit interval? That's actually another mystery... — fishfry

And you even admit that your view is shrouded in mystery. Why not consider the alternative...that a line is not composed of points, but instead points emerge from lines? Why won't you consider my...line...of thought?

I can't tell whether 0.999999...[?] is different than 1 until I finally find a non-9 in the expansion somewhere, so there's no bound on the check for equality. — norm

You can't tell by inspecting the digits, but at least 0.999... is computable so you can make some assessments by comparing the algorithms used to generate 0.999... and 1. The same cannot be said about non-computable numbers, which is what I was getting at.

have you looked into Zeilberger? He's a maverick too, a bit of a finitist. — norm

I've read about him and listened to him being interviewed on a podcast but I have only very briefly skimmed his website. I can't recall why but I left the podcast not being interested in pursuing his ideas further.

If I point out to Metaphysician Undercover that he can get in his car and drive to the store without being crushed before he drives the first inch; am I failing to appreciate the profundity of his beliefs? I think not! — fishfry

I think he's touching on something important. If time can be broken down into a collection of instants and if at one instant we're stationary and the next instant we're not then in one sense it does appear that we have undergone infinite acceleration. Pointing to our physical reality and suggesting that it proves he's wrong is besides the point. The real issue is with our assumption: that time can be broken down into a collection of instants. Or more generally, that a line is composed of infinite points. Anyway, you've already mentioned that you are puzzled by this so the analogy to Diogenes does not exactly fit so don't put much weight on my 'throwaway' comment.

If you can apprehend this, then why can't you turn it around, and see that when infinity is at the low end, there is no such thing as "the lowest number", and it doesn't make any sense to say that someone counting lower and lower is "approaching" the lowest number? — Metaphysician Undercover

I don't have a problem with saying 'x approaches infinity' in the context of a potentially infinite process. I interpret it as 'the value of x is continuously growing'. 'x approaches infinity provides some information about the journey, even if we never can arrive at some final destination.

Consider the graph linked here: https://imgur.com/FZANGZ8

This is not a typical graph in that it spans all possible values of x and y. Think of it topologically in that it is a single system which maintains its topological properties when undergoing continuous deformations. In this plot, there is a point at (1,1) and a pseudo-point at (∞,0). In this context, it makes sense to say that we're starting at (1,1), travelling along y=1/x and heading towards the pseudo-point at (∞,0). By plotting Gabriel's Horn like this, (∞,0) is no longer out of sight, it's right there in front of us. And because of that we have the ability to use it in some contexts without requiring infinite measuring capacity.

In practice the world is continuous (time passes continuously), but in theory the world is discrete (represented by distinct units, numbers). — Metaphysician Undercover

I would argue that objects are continuous but measurements are discrete. This allows us to use the richness of mathematics that calculus offers while avoiding the paradoxes of actual infinity.

Where do they live? And what else lives there? The baby Jesus? The Flying Spaghetti Monster? Pegasus the flying horse? Platonism is untenable. There is no magical nonphysical realm of stuff. And if there is, I'd like to see someone make a coherent case for such a thing. — fishfry

The eternal truths that I am referring to are different from your eternal truths because mine are finite. I don't need to assert the existence of an actually infinite entity beyond our comprehension. Nevertheless I believe that all truths are contingent on a 'computer', it's just that there exists a 'computer' that lives outside of time. For example, I believe that the laws of nature exist outside of time and that these 'eternal' laws reflect the 'personality' of the grand computer. If on the other hand the laws of nature somehow did exist within time, what laws allowed them to pop into existence? If such deeper laws exist, then it is those laws which I'm referring to as external truths. I believe that the only object which can live outside of time is the unmeasured wave function of the universe...or more generally, a continuum filled with infinite potential.

Well I don't understand. Contingent on what? If there is a Platonic realm after all, surely mathematical truths live there if nothing else. — fishfry

I believe that everything in actual existence is finite, even 'the grand computer' in which eternal truths live. And so 'the grand computer' can not actualize the set of all natural numbers any better than us. The actual existence of 5 is contingent on a computer 'thinking' about it. When no computer is thinking about it, it does not actually exist and it is meaningless to say that it has definite properties (akin to Schrödinger's cat). With that being said, I am comfortable saying that when a computer is 'thinking' about 5 that it will certainly be prime.

Wildberger is a nut, his math doctorate notwithstanding. He does have some very nice historical videos and some interesting ideas. But his views on the real numbers are pure crankery. You should not use him in support of your ideas, since that can only weaken your argument. — fishfry

I believe that, like Zeno, Wildberger is able to take our 'whole-from-parts' view to its limit to suggest that there are fundamental problems with it. We should value people who identify paradoxes as paradox leads to progress. However, I don't agree with Wildberger's resolutions. They do not offer the richness that infinite math does. If there indeed are paradoxes, the resolution should not be to weaken mathematics.

The axiom of infinity lets us take all of the numbers given by the Peano axioms and put them in a set. That's the essential content of the axiom.

The Peano axioms gives us 0, 1, 2, 3, ...

The axiom of infinity gives us {0, 1, 2, 3, ...}

The former will not suffice as a substitute for the latter. For example we can form the powerset of {0, 1, 2, 3, ...} to get the theory of the real numbers off the ground. But we can't form the powerset of 0, 1, 2, 3, ... because there's no set. — fishfry

If there is no way to reinterpret the Axiom of Infinity to apply it to potential infinity, then I'm inclined to reject it on some level. However, that doesn't necessarily mean that I entirely reject ZF. When working with ZF, we are always dealing with finite statements. Is it possible that these statements are the mathematical objects, not the sets which they are talking about? By 'actually exist', I'm trying to identify the objects that computers are actually working with. A finite computer can never work with an actually infinite set, it can only work with finite objects and potentially infinite algorithms.

You're talking about how you'd like mathematics to be, but you do it entirely in a castles-in-the-air manner without regard to even a minimal understanding of the mathematical context. Philosophizing about mathematics is fine. But when the philosophizing concerns actual mathematical concepts, then, unless there is an understanding of the actual mathematics and the demands of deductive mathematics, that philosophizing is bound to end as heap of half-baked gibberish. — GrandMinnow

I agree that I can only half-bake a mathematical idea. But even when mathematicians 'bake' an original idea at some point it's half-baked. And why must I bake it all by myself? I understand that you may not want to invest your time in evaluating half-baked ideas, but isn't a forum like this a good place to discuss them?

Ordinary calculus does use infinite sets. — GrandMinnow

Do limits require the existence of infinite sets?

The set theoretic axiomatization of mathematics is very straightforward, easy to understand, and eventually yields precise formulations for the notions of the mathematics of the sciences. — GrandMinnow

Do you believe that there are any paradoxes related to the set theoretic axiomatization of mathematics, and if so, is it fair to conclude that it's straightforward? ZF Axioms are rarely if ever mentioned in applied math (science, engineering, etc.).

If you are sincerely interested in the subject, even from a philosophical point of view, you should learn the set theoretic foundations and then also you could learn about alternative foundations that bloom in the mathematical landscape. — GrandMinnow

I have learned a lot of math independently, but I certainly have much to learn and realistically not enough time to learn what is needed to do everything by myself. May I ask, how much education should a person have before initiating a discussion on a philosophy forum like this?

And neither is there a Zeno's paradox with set theoretic infinity. — GrandMinnow

What resolution of Zeno's paradox are you satisfied with? Limits can be used to describe a process of approaching a destination but they cannot describe arriving there. So how does one arrive at some new destination?

I wouldn't begrudge philosophical objections to the notion of infinity. My point though is that one does not have to be platonist to work with theorems that are "read off" in natural language as "there exists an infinite set". The axiom that is (nick)named 'the axiom of infinity' does not mention 'infinity' and, for formal purposes, use of the adjective 'is infinite' can just as well be dispensed in favor of a purely formally defined 1-place predicate symbol. — GrandMinnow

Are your and fishfry's posts in agreement? As I said to fishfry earlier in this post, I believe that 'there exists an infinite set' could be a valid mathematical object. But I think one does have to be a platonist if they think that such an infinite set exists.

as you mention what you consider to be flaws in classical mathematics, as I said before, you have not offered a specific alternative that we could examine for its own flaws. — GrandMinnow

I totally understand where you're coming from. I'm sure you've dealt with many infinity-cranks in the past and this probably feels like deja-vu. I get that. But I don't like that you are labelling my view incoherent. That is entirely my fault since I haven't communicated it well enough in my post. I've actually produced a collection of videos roughly explaining my views, would you consider reviewing a couple of them:

Derivative Paradox: https://youtu.be/PSzqoQ9J5yg
Dartboard Paradox: https://youtu.be/LQmwZZNUMNA
Zeno's Paradox: https://youtu.be/_96tczP_eaY

Please note that I'm not forwarding you these links in an attempt to generate views. I gave up on my channel last year and have no plans to do anything with it. I'm only linking you to it since it may better communicate my view. If you don't want to watch the videos I could try explaining my views further on this chat...

Also, even though you're providing much resistance, you (and others like fishfry) nevertheless have generously given me some of your time by reviewing my posts and writing responses so please note that I'm very appreciative of that.

Then the real number 42 is the equivalence class that contains f and all g such that |f(n) - g(n)| --> 0 as n --> inf. — norm

I understand your limit-based 'algorithm' but would there ever be an instant in time when you would be sure that it's not 42?

we often think we are talking about numbers when we are really talking about talk about numbers. — norm

Yes, and I think we do the same about actual infinity. We don't conceive of actual infinity, we conceive of conceiving of actual infinity (using potentially infinite algorithms).

This looks good! Thanks!!

As I explained, by way of example, to assume such a "greatest value", or "lowest value" is contradiction. When we say that the natural numbers are infinite, and therefore have no highest value, it's contradiction to say that 20 is closer to the highest value than 10. Likewise, when there is no lowest value, it's contradiction to say that .01 is closer to the lowest value than .02. — Metaphysician Undercover

You are equating 'approaching' with 'arriving at'. If I could only tell you one piece of information about my trip I would tell you my destination. But if my trip never ends there are some situations where I could still give you some useful information since in some situations I could still tell you which direction I'm pointing (e.g. what I'm approaching). But I think we are both firm with our incompatible definitions and we've said what could be said. I think our words would be much better spent on the new topics of this thread.

No it's not. When you get in your car and start driving to the store, do you experience infinite acceleration? What's that feel like, exactly? — fishfry

This reminds me of Diogenes the Cynic's rebuttal to Zeno's paradoxes. According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions. I'm not convinced that Diogenes appreciated that profundity of Zeno's paradoxes.

I actually agree with you about the intuition. If we're not moving, how do we start moving? It's a bit of a mystery actually, I'm not sure what physicists say about this. — fishfry

Again, Zeno's Paradox. The issue is that we're looking at things with a 'whole-from-parts' view. We want to advance forward one point at a time and can't seem to get anywhere...our intuition is saying that it doesn't make sense how a line can be formed from points. But with a 'parts-from-whole' view it's easy. We start with the whole (the unobserved wave function of the universe spanning all of time) and then we make (quantum) measurements. At one measurement we're here and then at the next measurement we're there. Change doesn't happen at points (or instants in time). Instantaneous velocity makes no sense. Change it happens in between the points. And if we draw a graph using the 'parts-from-whole' view as I mentioned in the Have we really proved the existence of irrational numbers? thread, the change of a function happens in between the points...across the unmeasured curves.

The context of modern logic and mathematics involves formal axiomatization. Anyone can come up with all kinds of philosophical perspectives on mathematics and even come up with all kinds of ersatz informal mathematics itself. But that does not in itself satisfy the challenge of providing a formal system in which there is objective algorithmic verification of the correctness of proof (given whatever formal definition of 'proof' is in play). I mentioned this before, but you ignored my point. This conversation is doomed to go nowhere in circles if you don't recognize that what you're talking about is purely philosophical and does not (at least yet) provide a formalization that would earn respect of actual mathematicians in the field of foundations. — GrandMinnow

My ideas are so many steps away from a formal mathematical theory and in any one of those steps my ideas can be revealed to be inconsistent or useless. With that said, it's unreasonable to expect a formal theory to be perfected in isolation. And while my ideas are far from publishable standards, I do think they're at a level where they can be discussed on a forum. So please allow me to pick your brain, and please demolish my ideas. All I can guarantee to you is that you are dealing with someone incredibly receptive to criticism so the probability of going in circles is likely far less than what you expect.

please show your system of axioms and rules by which one may make an evaluation of such circumstances. — GrandMinnow

I don't have axioms for you and to be honest, I don't have the technical skills to do much of the mathematical heavy lifting. I'm an engineer not a mathematician nor a philosopher. But I still think I can contribute. When Descartes developed analytic geometry, I suspect that he didn't present axioms but instead presented an intuitive way of thinking that proved incredibly useful (I'm particularly referring to Cartesian coordinates graphs). I think I could do something similar.

I believe it was his work which catapulted us to our actual infinity point-based world-view. When we draw a graph, we think that it is completely filled with points. It is a 'whole-from-parts' view where a continuum is constructed from infinite points. But consider this alternate 'parts-from-whole' view. We start with a continuum, perhaps a 2D square whose dashed edges correspond to x=∞, x=-∞, y=∞, and y=-∞. This continuum has no points, only the 'pseudo-points' (∞,∞), (∞,-∞), (-∞,∞), and (-∞,-∞). Draw curve from (-∞,∞), passing through the interior, and ending at (∞,∞). Label this curve y=x^2-2. Next, draw a somewhat horizontal curve through the interior and crossing y=x^2-2 at 2 points. Label these points (?,0) and (?,0). Finally label the starting pseudo-point (-∞,0) and the ending pseudo-point (∞,0). This graph does not have the infinite points to give it a fixed geometry. Instead, it can be thought of a topological system, whereby the relationship between what was actually drawn is maintained through continuous deformations. This covers just an overview of where I'm coming from, but let me just say that so many paradoxes (especially Zeno's paradox, derivative paradox, dartboard paradox) are not paradoxical with this view because this view is void of actual infinity. Instead, this graph has limitless potential to be further refined by adding additional lines.

If it's an object, then it exists. At the very least, in formal terms, there is an existence theorem. — GrandMinnow

I argue that we cannot talk about existence without including a qualifier: actual or potential. An object can potentially exist. In my graph example, I can think of potentially infinite points that could be introduced should I decide to add more lines, but until I do so, those points only potentially exist. And so in the context of that graph the only number that actually exists is 0 (and perhaps 2 if you're including the function's definition).

The justification is from axioms from which we prove that there exists an infinite set. — GrandMinnow

Axioms are not my strength, but could we perhaps reinterpret the Axiom of Infinity to assert the existence of an algorithm for generating an infinite set, without requiring that the infinite set actually exists? For example, with a few lines of code I can create a program to list all natural numbers even though it is impossible for the program to be executed to completion. I hope but cannot demonstrate that we can replace the actual infinities in ZF with potential infinities. Cantor was a 'Whole-from-Parts' guy so he built up set theory from points. What if we instead took a 'Parts-from-Whole' view?

the mathematics itself stands whether the mathematician regards it platonistically or not. — GrandMinnow

I believe that if we abandon Platonism by replacing the actual infinities with potential infinities that the mathematics will stand. This is what I mean when I say it's a philosophy issue, not a mathematical issue.

That is patently false. Of course, usually physical measurements, as with a ruler, involve manipulation of physical objects themselves and as humans we can't witness infinite accuracy in such circumstances. But that doesn't meant that mathematical calculations are limited in that way. — GrandMinnow

I think your comment is a result of me not communicating myself properly. All that I'm saying is that when I use pi precisely I do not evaluate it, I keep it as...pi. I consider this to be a real algorithm. But when I actually evaluate it to produce an actual number, the number that I produce is always a rational number.

I gave that as a supposition in the context of separating the two questions I mentioned, But, meanwhile "there is a real number x such that x^2 =2" is proven from the ordinary axioms. — GrandMinnow

Let's set this aside for now since I'm not in a good position to debate with you about what exactly is proven by the axioms.

Fine. Then your remark to the other poster about undecidability is not pertinent. By the way, in classical mathematics a statement that is undecided is not undecided simpliciter but rather it is undecided relative to a given system. — GrandMinnow

With the 'Whole-from-Parts' view we can't put brackets around everything and call it math. There is no set of all sets. We cannot talk about division by zero. We can't avoid Gödel statements. With a 'Parts-from-Whole' view it's easy to talk about this stuff...they're just unmeasured potential objects. In a way, by embracing incompleteness nothing gets left out.

No it's not [in total agreement with the foundations of calculus]. Clearly. — GrandMinnow

Limits have a precise meaning in a 'Parts-from-Whole' view of calculus - they describe potentially infinite processes. If you want the area under a curve, approximate it with a set of smaller and smaller rectangles, to no end. There's no need to rationalize how lines can be assembled to create an object of area. In this example, if we work with potential infinity, the objects of study already have area, because we are not studying points...we are studying continua.

This reminds me of the construction of the real numbers from Cauchy sequence of rational numbers. — norm

Have you ever seen the cauchy sequence of a non-computable real number? If I claim that that Cauchy sequence is for the number 42, how could you challenge that claim?

Have you looked into Brouwer? Or Chaitin's Metamath: The Quest For Omega? In short, I think some experts have problems with the real numbers, but these problems are philosophical/intuitive rather than technical. — norm

Thanks for the recommendations. Yes, there are a few experts (e.g. Norman Wildberger) who have problems with real numbers, but they're few and far between. His criticism is quite technical but I agree with you that it's a philosophical issue.

In general, we don't know exactly what we are talking about, but math tempts us to forget that. — norm

You may be right, but I'm of the view that we don't know exactly what we're talking about because there's more work to be done.

Why do you think that mathematicians, simply by believing in something have the capacity to grant that something "existence", while for other groups of people, simply believing in something is insufficient for the existence of what is believed in? — Metaphysician Undercover

My view is that we can only think objects into existence internally. For example, a computer program can simulate a reality where internal to that reality the flying spaghetti monster is real. But a computer cannot do any amount of computing to make the flying spaghetti monster real external to the simulation.

Well, either a number can be expressed as a ratio of two integers or not. If it can be the number is rational and if can't be it's irrational. When there are only two choices, the fallacy of the false dichotomy can't be committed. — TheMadFool

At what point did we prove that it was a number?

5 is prime, and that seems to be true independently of the opinions of people. Mathematical truth has a necessity that's forced on us in some way nobody can understand. — fishfry

I like to think that there exist truths independent of consciousness, whether it's certain axioms of mathematics or the laws of nature. But I think '5 is prime' is a contingent truth...

thousands of minds together can cover far more intellectual terrain and see into one another's blindspots. — norm

I like that idea that together we have no blindspots.

Metaphysicians, being trained in this field, are best able to say whether something exists or not. — Metaphysician Undercover

Perhaps metaphysicians have an important voice, but I'm more inclined to say that philosophers of mathematics and philosophers of physics are best equipped on this matter.

There is no such value. Might it be 100? Or 1000?. y=1/x, x>1 has a greatest lower bound, 0, which it never reaches — jgill

You're right. You said what I was intending to say. Thanks for the correction!

Thanks jgill! Stack Exchange is a good format for Q&A but I don't think it offers a very good format for discussion. I think TPF is a pretty sweet spot it's good to know that I've come to the right place. Cheers.

It is correct to say that the value gets lower and lower, but it is incorrect to say that it approaches 0, because no matter how low it gets it never approaches 0. — Metaphysician Undercover

The real description is that the value of y gets lower and lower without ever approaching zero. — Metaphysician Undercover

Our disagreement is also due to us having different definitions of approach. We both agree that y gets lower and lower (and perhaps you would even agree that the greatest value which y never reaches is 0) but I call that approach and you call that not approaching. Let us agree to disagree on definitions!

Constructivists deny the law of the excluded middle. — fishfry

I believe that the only measurable states of a proposition are true or false so in one sense I accept the Law of the Excluded Middle. Where my view deviates from the norm is that I believe there can also be an unmeasured 'potential' state where a proposition is neither true nor false (akin to Schrödinger's Cat). And returning back to numbers, numbers which are not a part of any computation are in this unmeasured potential state.

Different issue. Landing on side should be included in the outcome space. — fishfry

Is it a different issue though? Isn't 'not a number' a reasonable option to be included in the outcome space of the proof √2.

I see your point, but then existence becomes contingent on what everyone's thinking about and/or computing. 3 might or might not exist depending on what 7 billion people and a few hundred million computers are thinking or computing at this exact instant. What kind of standard for existence is that? Not one that would have much support from ontologist, I'd wager. — fishfry

Assume for the moment that our universe is like a computer simulation. Wouldn't your existence be contingent on the simulator 'thinking' about you? To me, it seems reasonable to think that, like us, numbers are contingent. Why must everything eternally and actually exist?

But your particular flavor of ultrafinitism is untenable, granting existence to only those things that someone is thinking about or computing at any given instant. Under such a philosophy we can never say whether a given number or mathematical object exists. — fishfry

Why do we need all mathematical objects to actually exist? And moreover, why do we need to know that all numbers actually exist. To me it seems sufficient to know that you have the potential to keep counting the natural numbers, we don't need to actually count to 'the end'...in fact we can't. Perhaps if we fully embrace potential infinity and potential existence, we will find that we don't need actual infinity or the platonic realm.

If you deny the mathematical existence of the natural numbers you not only deny ZF, but also Peano arithmetic. That doesn't let you get any nontrivial math off the ground. Not only no theory of the real numbers, but not even elementary number theory. You may be making a philosophical point but not one with much merit, since it doesn't account for how mathematics is used or for actual mathematical practice. — fishfry

Consider Euclid's Theorem. What of the following did Euclid actually prove?

Interpretation 1: Any finite list of primes is incomplete.
Interpretation 2: There exist infinitely many primes.

These two interpretations may seem equivalent but they're not because the second makes an unjustified leap to assert the existence of an infinite set. Nevertheless, the math is the same, it's the philosophy (the interpretation) that is different. I believe ZF and Peano arithmetic just need to be reinterpreted.

Well then you're an ultrafinitist. You not only deny the existence of infinite sets; you deny the existence of sufficiently large finite sets. — fishfry

While I don't believe in the existence of infinite sets, I wouldn't attach myself to the finitist label. Finitists usually have an uphill battle trying to establish mathematics as rich is infinite mathematics. I simply think we shouldn't interpret ZF in certain platonic ways.

Why doesn't pi exist? It has a representation as a finite-length algorithm. — fishfry

Pi the infinite digit number cannot exist. Pi the finite-length algorithm can certainly exist.

Well in any sufficiently interesting mathematical system we are always missing some truths. — fishfry

With your view these truths are missing. With my view the 'missing' statements are not missing at all, they're fully accounted for in the unmeasured 'potential' state. It's just that some statements, like 'this statement is false' must permanently remain in the unmeasured 'potential' state.

Because otherwise the real number line has holes in it. The intermediate value theorem is false. There's a continuous function that's less than zero at one point and greater than zero at another, but that is never zero. Of course the constructivists patch this up by limiting their attention to only computable functions. The constructivists have answers for everything, which I never find satisfactory. — fishfry

If we pluck the irrational numbers off the number line, we are not left with holes in between the rational numbers...we are left with lines in between the rational numbers. With this view, we don't have to believe that infinite 0-D points can somehow be assembled to form a 1-D object. Instead, lines (or more generally, continua) are fundamental, not points. Think of how you draw a graph: you start with a piece of paper (a continua) and you draw a grid on it. At each intersection (point) you label it with coordinates (numbers). Everything about this is finite. In between the points/numbers lies a continua. But somehow we think about it all backwards. We think that we start with infinite points and numbers and then they someone assemble to form a continua. It's because of this thinking why calculus seems so paradoxical. I don't believe that the intermediate value theorem is entirely false, it just needs to be reinterpreted to apply to continua instead of points.

The volume of a pizza of radius z and height a is pi z z a. — fishfry

Ha! Love it!!

You can say whatever you want and call it philosophy. And if no one agrees and you are truly OK with that, then you win. I mention this because of experience with infinity-deniers, anti-Cantorians, etc. — norm

I'm not okay with that, which is why I truly appreciate this discussion. I also don't consider myself an infinity-denier, after all I'm a huge proponent of potential infinity (I only reject actual infinity). And I also believe Cantor's work was extremely important, I just think it needs to be reinterpreted.

Approximation and vagueness is the rule, and integers are glowing exceptions to just about everything else that's human. — norm

Or could 'Approximation and vagueness' just be the norm because we haven't fully figured it out? Math has changed so much in the past ~100 years since Cantor, what reason do we have to think that all of the foundations have been set? There are still way too many paradoxes to think that.

The real numbers as a complete ordered field provide a foundation for the mathematics of the physical sciences. — GrandMinnow

Real numbers are never used in applied mathematics. Every number that we have ever used in a calculation is rational. Real 'algorithms' on the other hand are very useful. When I use pi in an equation I use it to refer to a potentially infinite series which I dare not try to calculate...so I keep it in algorithmic form.

if you reject certain set theoretic axioms, then we may ask: What are your alternative axioms? — GrandMinnow

I hope that my views are largely in agreement with ZF and that ZF just needs to be reinterpreted. This is largely an issue of philosophy, not mathematics.

Perhaps (I don't opine for the purposes of this particular post) proof supplied to answer (2) may be called into question constructively by rejecting the dichotomy that a real number is either rational or it is not rational. — GrandMinnow

Aren't you beginning your proof with an assumption, that irrationals are numbers?

The proof that there is a real number x such that x^2 = 2 comes later in the history of mathematics. It is found in many a textbook in introductory real analysis. This is desirable, for example, so that we can easily refer to a particular real number that is the length of the diagonal of the unit square. — GrandMinnow

I understand why we want it to be a number, but that doesn't mean it is. After all, when we write it out explicitly we always write it as some algorithm. Why can't it simply be an algorithm? Our classical intuitions have us wanting to be able to precisely measure any coordinate, for example, the coordinates of the points where y=0 and y=x^2-2 intersect. But with quantum mechanics, our intuitions have changed. With the uncertainty principle we know that there is a fundamental limit to the accuracy with which the values for certain pairs of quantities can be predicted. Is it possible that there is a similar limit to which certain pairs of coordinates can be measured in mathematics?

Do you think your version of non-axiomatic mathematics "decides" every mathematical statement? There are undecidable statements in the mathematics of the plain counting numbers themselves. Undecidability is endemic to the most basic mathematics even aside from the real numbers as a complete ordered field. So how does your version of non-axiomatic mathematics "decide" all mathematical questions? — GrandMinnow

My view is that we don't need to 'decide' every mathematical statement because 'undecided' is a valid state. The undecided state is not a defect of my view, it is a feature. Returning to quantum analogies, it is like how the unmeasured superposition state of a particle is a valid state for the particle.

. I would add that a complete ordered field is the ordinary foundation for calculus, which is used for the ordinary mathematics for the physical sciences. — GrandMinnow

My view is in total agreement with the foundations of calculus. In fact, I believe that when calculus was reformulated based on limits (and potential infinity) to banish infinitesimals that we didn't go far enough because we failed to banish real numbers (which are inseparably tied to actual infinity). We should have replaced real numbers with real algorithms and interpreted calculus to be not a mathematics that outputs numbers, but a mathematics that outputs processes.

fishfry, norm and GrandMinnows, thanks for your detailed and educated feedback!

Yes that's what I'm saying, there is no final destination — Metaphysician Undercover

This is where the misunderstanding is. Nobody is saying that there is a final destination or that (∞,0) exists. The standard definition of a limit does not require a final destination, it only requires one to approach a number as they advance along the journey. A limit is the process of approaching not the act of arriving. And you must admit that any workable definition of 'approach' will have one approach y=0 as they travel along y=1/x. [As mentioned before, your definition of approach involving looking at the number of intermediate numbers is not workable for number systems which are dense in the reals, e.g. the rational numbers].

In short, I believe our disagreement is simply the result of us having a different definition of limit.

Yes, Pythagoras gets credit for (but probably didn't personally have anything to do with) the proof that sqrt(2) is irrational. — fishfry

He only proved that √2 is not a rational number. He did not prove that √2 is an irrational number. Yes, I'm concerned with proof by contradiction. If I flip a coin and tell you that it is not heads that is not proof that it is tails because it may have landed on its side.

But you want to strengthen that standard to say that a number only exists when we can not only give an algorithm for it, but that we can execute the algorithm to completion given the physical constraints of the universe. — fishfry

Actually, I want to strengthen it even more so to say that a number only exists when it is being computed. If there is no computer currently thinking about 3 right now then the number 3 does not exist as an actualized number. It only has the potential to exist. I think this sort of view is required if we are to avoid actual infinity. Otherwise, how would a constructivist answer the question: how many numbers are there? My response to such a question is 'how many numbers are where? In what computer?'

That's a terribly restrictive standard, for example 1/3 = .3333... doesn't exist according to you. — fishfry

I believe that the decimal representation for 1/3 cannot exist but nevertheless the number certainly can. For example, it is LL on the Stern-Brocot tree. And we can do exact arithmetic using any rational number using the Stern-Brocot tree.

For my own part, I take mathematical existence to mean anything that can be proven logically using the axioms of any given mathematical system. — fishfry

But won't there always be undecidable statements? It seems like your definition is too restrictive as it would be missing some truths.

That includes all the computable numbers and all the noncomputable ones as well, which after all are necessary to ensure the completeness of the real numbers. — fishfry

Why is it necessary to have a number system which is complete?

Just checked. Here is a link to a discussion in Wikipedia about the proof that pi is irrational — T Clark

I have no doubt that pi is irrational (i.e. not a rational number). But a sandwich is also not a rational number. My argument is that just as a sandwich is not a number, perhaps neither is pi. Maybe it's an algorithm.

What does it mean (to you) to prove that a number exists? — fishfry

Hmm, you've revealed that my original post was quite vague. I believe that an object exists if it is being computed. I exist as an artifact of the laws of nature being evaluated and the number 3 exists within my mind because I'm thinking about it right now. With this view, the number √2 (in its totality) cannot exist because it cannot exist in any finite computer. To be clear, I believe that a finite algorithm for computing √2 can exist, but the output of that algorithm (which I'm calling the number √2) cannot exist. The mainstream approach to giving the number √2 existence requires us to assert the existence the Platonic Realm (which I equate with an infinite computer), but extraordinary claims should be backed by strong evidence, for which I have come across none. Thoughts?

But perhaps it is simplest if I take existence out of my original question: Have we really proved that √2 is an irrational number?

I surely disagree. There is no "final destination." That's MU's error, why are you amplifying it? — fishfry

I'm trying to condense his argument into a few points in hopes that it brings to focus where the misunderstanding lies.

You don't seem to quite grasp why I reject "closer". — Metaphysician Undercover

Perhaps I don't grasp it, or perhaps I just don't agree with it. Let's assume it's the former. Please tell me whether the following points aligns with your view:

1) One can travel along y=1/x in the positive-x direction, without bound.
2) The limit of the journey corresponds to the final destination, which if anything would be (∞,0).
3) The point (∞,0) does not exist (since ∞ is not a number) therefore there is no limit.

So it is false to say that it gets any closer, at any point, because it's always the same, "arbitrarily close". This arbitrariness indicates that zero is completely irrelevant to any valid measurement. — Metaphysician Undercover

Your notion of 'close' that is based on the number of points between A and B can only have value in a number system which is not dense in the reals, such as the integers. For example, since there are 3 integers between 0 and 4 but 6 integers between 0 and 7, we can conclude that so 4 is closer to 0 than 7. If you want to restrict your mathematics to the integers then your notion of 'close' is suitable. However, such a mathematics is far less powerful than orthodox math so I suspect that you'd have a very tough time convincing anyone to adopt your view.

But as I said before, I sympathize with your position. In one breath we say that we use limits to avoid actual infinity but then in the next we say that the limit can be a number inseparably tied to actual infinity (e.g. pi). But I would argue that the resolution to such contradictory thinking is simple: just don't say that the limit is a number with actually infinite digits. Instead, keep the limit in algorithmic form - say that the limit is a potentially infinite process which is described by [pick your favorite] algorithm for pi, for example, 4(1-1/3+1/5-1/7+1/9-...)

Almost; I'm talking about arbitrarily large but finite amounts of paint. And the only point here is that it's not really surprising this "outside" can be indefinitely long with a layer while the inside is limited. — InPitzotl

It seems like you're implying that finite math is equally paradoxical by comparing the infinitely long needle with an infinitely long horn. If you want to say something about finite math you need to talk about a horn of finite length.