The paradox of Gabriel's horn.

Ryan O'Connor

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.

fishfry

2) The limit of the journey corresponds to the final destination, which if anything would be (∞,0). — Ryan O'Connor

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

Ryan O'Connor

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.

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. — Ryan O'Connor

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

Yes that's what I'm saying, there is no final destination, so to even produce any representation (such as ∞,0), as if it is a final destination, is a misrepresentation amounting to contradiction.

There is a particular line we are talking about, and #1 ought to state that this line is extended "without bound", which means "there is no limit". So #3 ought to read "the point (∞,0) does not exist because there is no limit". Now we could add #4: "∞" is a description of the entirety of the line (not a point on the line), and 0 is completely unrelated to the line, therefore irrelevant.

Here's another way to look at the position of zero. It is an ideal, like infinite is an ideal. The two are opposing ideals, like hot and cold are opposing ideals. The line takes the characteristic of the one ideal, the infinite, therefore the opposing ideal, zero, is excluded. It's just like if we were talking about the absolute, ideal hot, cold would be completely excluded.

Ryan O'Connor

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.

Metaphysician Undercover

↪Ryan O'Connor

Did you read the rest of my post? What I'm saying is that 0 is not even relevant. That's what I've been trying to explain, that to describe the value of y as approaching 0 is a false representation. Y is always infinitely far away from zero, because zero is impossible on that line. The value for y never "approaches 0". 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. 0 is not at all relevant to this line.

To say that y approaches zero is an inaccurate simplification, nothing but a rounding off in your description. The real description is that the value of y gets lower and lower without ever approaching zero. Of course the true description is "not workable", that's the nature of any infinity. The appearance of infinity is the result of something being not workable. To make an infinity into something workable is to provide a false representation.

In short, I believe our disagreement is simply the result of us having a different definition of limit. — Ryan O'Connor

I think where we disagree is in the role that zero can play in this measurement. You think 0 can play a role, as the value which y approaches. I think that since the line has no start nor end, 0 is not an applicable number.

Ryan O'Connor

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!

jgill

(and perhaps you would even agree that the greatest value which y never reaches is 0) — Ryan O'Connor

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. Your attempts at the philosophy of mathematics may never bear fruit if you consider this a cogent statement.

Ryan O'Connor

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!

Metaphysician Undercover

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! — Ryan O'Connor

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.

What is misleading in the example of Gabriel's horn, is that the x and y axes are set to converge at 0, at a right angle in relation to each other. This proposed point of convergence creates the illusion that 0 is a valid value, where x and y are 'the same". However, as I described earlier, the spatial representation of two dimensions at right angles to each other is actually a false representation, making the two dimensions incommensurable, as demonstrated by the irrationality of the square root of two. This incommensurability indicates that the two proposed lines, x and y, cannot actually be modeled as intersecting, and sharing a common point at 0.

So this false idea that x and y actually meet each other at that point, 0, is what misleads you into thinking that 0 is a valid measurement.

Metaphysician Undercover

To put it simply, non-dimensional existence, which is represented by the point, 0, is incompatible with our representations of dimensional existence. So 0 cannot enter into our scales for measuring dimensional existence, as a valid measurement point until we establish commensurability between non-dimensional and dimensional existence. This problem with zero becomes very relevant when we start to consider motions, and acceleration from rest. An infinite acceleration is required to go from rest to moving. The problem is somewhat avoided with relativity theory which denies the reality of rest, making acceleration simply a change in direction. But what that does is make the mathematics extremely complex, still working with points and vectors, rather than resolving the problem of how the non-dimensional truly relates to the dimensional.

fishfry

An infinite acceleration is required to go from rest to moving. — Metaphysician Undercover

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? According to special relativity, you should be pressed against the back of your seat with infinite force. You'd be crushed before you drove a foot. What do you say?

ps -- Let's do the math. Say I'm at rest and start moving at 1 unit/second or whatever. In physics we need to give the units but in math we'll just say the velocity is 1. So at 1 second we've gone 1 unit, at 2 seconds we've gone two units, etc.

So our position function is p(t) = t; and our velocity is always 1, which is consistent with the first derivative of position being the derivative of t with respect to t, or 1.

Now this is tricky and this is where you got yourself confused. What was our instantaneous acceleration at 0? After all we weren't moving and then a tiny moment later we were. Well, the graph of our position look like this:

             /
           /
----------o

This function is not differentiable at zero. There is no instantaneous velocity at zero and no definite acceleration either. I agree that this is counterintuitive, and your intuition is not uncommon. But it's wrong. Clearly it's wrong. If you experienced infinite acceleration even for a moment, every atom in your body would be flattened like so many pancakes.

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. Well I guess I do know. If we're a steel ball in Newton's cradle, or we're a ball on a pool table, we start moving when we get smacked by another ball that transfers its momentum to us. But how does our velocity go instantaneously from zero to nonzero? The Newtonian physics works out, but not the intuition.

resolving the problem of how the non-dimensional truly relates to the dimensional. — Metaphysician Undercover

A little woo-woo-y there @MU. 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, despite the fact that we have a mathematical formalism that says $\int_0^1 dx = 1$ . The math works but we have no metaphysical explanation that I know of.

Janus

We cannot 'paint' the horn in that sense because the volume required would be the area (infinite) multiplied by the thickness (nonzero), which means an infinite volume. — andrewk

Is the area infinitely large or merely infinite in the sense of 'unbounded'?

Metaphysician Undercover

This function is not differentiable at zero. There is no instantaneous velocity at zero and no definite acceleration either. I agree that this is counterintuitive, and your intuition is not uncommon. But it's wrong. Clearly it's wrong. If you experienced infinite acceleration even for a moment, every atom in your body would be flattened like so many pancakes.

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. Well I guess I do know. If we're a steel ball in Newton's cradle, or we're a ball on a pool table, we start moving when we get smacked by another ball that transfers its momentum to us. But how does our velocity go instantaneously from zero to nonzero? The Newtonian physics works out, but not the intuition. — fishfry

I addressed the issue in my post. There is only a need to conclude infinite acceleration if we assume absolute rest, zero velocity, but relativity denies absolute rest. If something had an absolute zero velocity, and changed from that zero velocity to having a positive velocity, this would imply a point in time (not a short duration) when the thing goes from zero velocity to a positive velocity. At that point in time, since there is no duration, but there is acceleration, the acceleration would be infinite. We avoid this problem with relativity theory which denies the reality of rest, and makes any supposed zero point in time into an extended duration.

tim wood

relativity theory which denies the reality of rest, — Metaphysician Undercover

Are you quite sure of this? Sure enough so that if you're wrong you will once and for all stop with nonsense?

There's at rest in a given inertial frame. Which is to say, really, that any acceleration, on your theory, should involve instantaneous infinite acceleration. My hand is at rest on the table. I raise it to type. Space-time not locally crushed in the process.

Ryan O'Connor

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.

fishfry

I'm not convinced that Diogenes appreciated that profundity of Zeno's paradoxes. — Ryan O'Connor

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 addressed the issue in my post. There is only a need to conclude infinite acceleration if we assume absolute rest, — Metaphysician Undercover

I addressed this in my post. The position and velocity functions are not differentiable at time zero. So there's no well-defined acceleration. Nor as others pointed out does relativity bail us out. Relative to your own frame of reference, you are at zero velocity at time zero and nonzero velocity a short time afterward. You have to come to terms with that.

SophistiCat

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. Well I guess I do know. If we're a steel ball in Newton's cradle, or we're a ball on a pool table, we start moving when we get smacked by another ball that transfers its momentum to us. But how does our velocity go instantaneously from zero to nonzero? The Newtonian physics works out, but not the intuition. — fishfry

Collision is a notoriously messy scenario - both physically and mathematically. Better to think of a ball in Newton's cradle at its highest point: at that point it is instantaneously at rest, then it starts moving again. Voila, motion from rest. Or easier still, just pick up that ball, gently release your grip and let it fall to the ground. Same deal, and we even know pretty exactly what its acceleration is when it starts moving. This doesn't seem so unintuitive to me.

I remember struggling with the concept of acceleration when it was first introduced - in middle school, I guess. It started making sense after a while. But some people just can't come to grips with such abstract concepts. Most of them have the good sense to leave it alone and apply themselves to something they are better at. Those who can't leave it alone become lifetime cranks, like MU. Or philosophers :)

norm

Yes that's what I'm saying, there is no final destination, so to even produce any representation (such as ∞,0), as if it is a final destination, is a misrepresentation amounting to contradiction. — Metaphysician Undercover

FWIW, and because no one has mentioned it yet, 'infinite limits' are taught in calculus as usefully specific ways to indicate divergence.

You can even write $f(x) = x$ and then $f(\infty) = \infty$ , but only as a cute abbreviation for something more technical. Later there is the extended real line, represented as $[-\infty, \infty]$ , but there's nothing magical about this, no more than there's anything magical about $0! = 1$ . It's all philosophically agnostic. Indeed, I know one mathematician who thinks the world is discrete and that continuity is a fiction, and then I know another who believes the reverse. Another dislikes philosophy altogether, and still another more has read Kant's CPR in German.

Metaphysician Undercover

There's at rest in a given inertial frame. Which is to say, really, that any acceleration, on your theory, should involve instantaneous infinite acceleration. My hand is at rest on the table. I raise it to type. Space-time not locally crushed in the process. — tim wood

It's "absolute rest" which I said is a problem, because this makes a point in time into a real situation rather than a perspective (reference frame) dependent designation. That's a point when no time passes relative to the thing at absolute rest.

There's at rest in a given inertial frame. Which is to say, really, that any acceleration, on your theory, should involve instantaneous infinite acceleration. My hand is at rest on the table. I raise it to type. Space-time not locally crushed in the process. — tim wood

This "at rest" which you refer to isn't real, because the earth is moving. Your hand is never at rest. So the moving of your hand is just a change in the existing motion of your hand, it is not an act of acceleration from rest. Physicists might represent it as an acceleration from rest, but the point I am arguing is that this is really an incorrect representation, which serves the purpose, just like representing Gabriel's horn as approaching 0 is an incorrect representation, which serves a purpose.

You are equating 'approaching' with 'arriving at'. — Ryan O'Connor

No I'm not equating these two. If there is no such thing as the lowest point, then it is impossible to be "approaching" the lowest point. In the case of the natural numbers, do you see that there is no such thing as "approaching" the highest number? We recognize that there is no such thing as "the highest number", so it doesn't make sense to say that if a person is counting higher and higher, they are "approaching" the highest number. You can never approach the highest number. 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?

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). — Ryan O'Connor

OK, this is a good point. The question here is what grounds or substantiates "direction". You imply that direction must be grounded by going toward something, but you forget that it might equally be substantiated by going away from something. In Gabriel's horn we have both, moving away from one axis, and moving toward the other. The axes are artificial confines, imposed as standards of measurement, and through the descriptive term of "infinite", the line of the form is stipulated as going beyond the capacity of the measuring scale. Therefore to understand that line we can no longer employ those measurement axes.

This is the problem we have here. Generally we assign infinite capacity to the measuring tool, and this allows us the capability to measure anything with that tool. The natural numbers are infinite for example, and this allows that we might count absolutely any multitude of objects. In Gabriel's horn, we have turned the table. We propose an infinite shape to be measured. Of course we cannot measure it, because it is defined as infinite, meaning that we cannot measure it, the thing is stipulated as going beyond the capacity of the measuring tool. So there is a trick hidden in the proposal, it's asking us to measure what cannot be measured by the tool. Then when we look at the shape, we see it getting further and further from the one axis, and we conclude, 'that's impossible to measure'. But we also see it getting closer and closer to the other axis, and intuition tells us, 'that's a finite distance which can be measured'. However, we must adhere to the principles of the construction, which state that the shape will appear to approach the axis, to a point beyond our capacity to measure the distance between them. Therefore we must resist our intuition and inclination to say that this distance is measurable.

So we must remove the axes as incapable of giving us the scale required for the measurement. The axes are what produced the form, which is by that construction, infinite and therefore immeasurable. Therefore the axes cannot be used to measure that form, because it has been constructed by them, as immeasurable. Now we have no basis for the terms of "farther from" or "closer to", because these values have been stipulated as going beyond our capacity to measure. What we are left with now, is just theoretical values, to be assumes as spatial distances, values which we acknowledge cannot actually be measured as spatial distances. Now we are really not talking about "farther from" or "closer to" any more, even though the numbering system employed was originally derived from that. We have explicitly gone beyond our capacity to determine farther from or closer to, and all we are talking about now is a higher value and a lower value. If we do not divorce the value from the spatial distance, we are just left with a spatial distance which is impossible to measure.

The point now, is that since we have done what the example requires, and taken the values beyond our capacity for making spatial measurements, we cannot use spatial references to ground or substantiate "direction". All we have now is a higher value and a lower value, and the stipulation that each of these may continue infinitely. The two directions (values) are actually defined in relation to each other. As the one gets a lot larger, the other gets a tiny bit tinier. And so long as we allow that the one can continue to get a lot larger, we must allow that the other can get a tiny bit tinier. But these "directions" must be thought of solely as numerical values, because we have gone beyond the relevance of spatial distances as dictated by the proposed example. So we cannot look at them as spatial directions of "farther" or "closer" or else we just fall back into the stipulated impossible to measure..

I addressed this in my post. The position and velocity functions are not differentiable at time zero. So there's no well-defined acceleration. Nor as others pointed out does relativity bail us out. Relative to your own frame of reference, you are at zero velocity at time zero and nonzero velocity a short time afterward. You have to come to terms with that. — fishfry

So the point I'm making, is that zero is completely arbitrary, and represents nothing real, just like in the case of Gabriel's horn. That's why we must decline this idea of "approaching zero". It is extremely useful in practice, yes, for sure it serves the purpose. But this is an exercise in theory, and we need to be able to go beyond what works in practice to be able to see that the principles which we employ in practice mislead us in our metaphysical efforts to understand the true nature of reality. The existence of paradoxes such as Zeno's demonstrate an incompatibility between theory and practice, and these incompatibilities expose where we misunderstand the true nature of reality.

I know one mathematician who thinks the world is discrete and that continuity is a fiction, and then I know another who believes the reverse. — norm

This is a different, but related issue, the difference between discrete and continuous. The issue is not whether the world is discrete or continuous, it is to find compatibility between the two. In practice the world is continuous (time passes continuously), but in theory the world is discrete (represented by distinct units, numbers). Simply modeling the world as discrete, or modeling the world as continuous, is fine either way, until someone approaches you with an example of the other, and makes a paradox jump out at you.

tim wood

This "at rest" which you refer to isn't real, because the earth is moving. — Metaphysician Undercover

You know this how, exactly?

fishfry

FWIW, and because no one has mentioned it yet, 'infinite limits' are taught in calculus as usefully specific ways to indicate divergence. — norm

This has been mentioned to @Metaphysician Undercover repeatedly. For years.

Metaphysician Undercover

You know this how, exactly? — tim wood

Because the wind blows.

tim wood

↪Metaphysician Undercover

That's actually, imo, an excellent answer. But it seems to me to miss the point.

The question goes to what can be known. The reason there is no absolute at rest is not because there is no such place, but rather that if there were such a place, then every place is a candidate for that place. And that is the whole lesson of relativity; there's no way to tell. You were dismissive of my question without answering it and apparently not even getting the substance of it.

The question was not what works within a model, however well, but how, exactly, you know. From the wind? Plausible, ultimately sensible and no doubt correct with respect to..., but with respect to some other criteria, maybe not. So the knowledge itself is conditioned upon that which makes it knowledge. Which is how the world works, except when it encounters claims like yours. Then it's back to basics and fundamentals to uncover the absurdity.

If I am at rest in any sense whatsoever, then on your account any acceleration I'm subject to must be in the instant infinite. And that is absurd.

Ryan O'Connor

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.

fishfry

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 — Ryan O'Connor

Second clause does not follow from the first. A mathematical line is composed of points. But there is no "next" point after any given point. You are confused on this ... point.

I think he's touching on something — Ryan O'Connor

You're thinking of Andrew Cuomo.

Ryan O'Connor

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?

fishfry

that a line is not composed of points, but instead points emerge from lines? — Ryan O'Connor

What does that mean?

ps -- I'm not making a geometric statement. The real number line is composed of real numbers. How can you disagree with that?

norm

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. — Ryan O'Connor

One theory that I've toyed with is that we have intuitions of both the continuous and the discrete that don't play nice together. Measurements are clearly discrete, as you say, but we also can draw the unit square and its diagonal and try to measure it 'perfectly' or 'ideally' and discover irrational numbers. The arithmetization of analysis was maybe driven by epistemological concerns. We want proofs in a universal language, and pictures aren't computer-checkable. (?)

The paradox of Gabriel's horn.

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