With my view many paradoxes (Zeno, Dartboard, Liar's, etc) are easily resolved — keystone

I take it you must mean dis-solved?

@keystone

You might be interested in Norman Wildberger on YouTube. He seems to hold positions on infinity similar to yours.

With circular reasoning. Perhaps a label for endless but not quite infinite in a physical sense ? — magritte

I wasn't endorsing the infinite universe argument, just pointing out that it has been seriously considered by qualified scientists.

You say there exists a number called pi with infinite digits and you use a truncated approximation of it when you calculate the approximate area of a circle.

I say that what exists is a (finitely defined) algorithm called pi that doesn't halt but you can prematurely terminate it to produce a rational number to calculate the approximate area of a circle. — keystone

I still don't get it. I don't see any advantage in your way of seeing things. For me, pi is clearly a number. I guess numbers and mathematics began with counting. Even that simple step was an simplified abstraction. Since then, the mathematical universe has been expanded to include non-counting elements - 0, rational numbers, negative numbers, real numbers, imaginary numbers. All of those are also simplified abstractions and are also numbers, even if it's hard to find a real world analogue.

Seems like you're asking for an abstract, human invention to match up with your understanding of reality. It doesn't work that way. As they say, the map is not the terrain.

The difference is that you are asserting the existence of an infinite object, something beyond our comprehension. My approach seems more in line with what us engineers actually do, so why bother asserting the existence of something impossible to imagine if you don't even need to? — keystone

A number is not an object. It doesn't have a physical existence. Also, it's not beyond my comprehension. That way of seeing things has always made sense to me.

Do you believe that 0+0+0+0+... can equal anything other than 0? If not, then how can you claim that 0-length points could be combined to form a line having length? — keystone

Abstract entities, i.e. all human concepts, are always simplified reflections of the world. I can't think of any that aren't. That's why math is so wonderful. It's a game of pretend that just happens to work really, really well.

Sounds like double-think from 1984. There are no contradictions in wave-particle duality. — keystone

Of course there are. Particles and waves are different kinds of physical entities. One is extended, spread out, in space and the other is found in a specific location. That's contradictory, and, just like numbers, both are simplified, abstract ideas. The fact that they seem contradictory, at least to most people, is a failure of human imagination.

I'm talking about the philosophy of mathematics, not the application of it. — keystone

Agreed. I think, like many mathematicians, you are expecting math to have a precise correspondence with reality. That never works.

Yes, all reality is void of actual infinities. So why do we need to believe that reality is just an approximation of some ideal infinity-laden object that we can't comprehend or observe? Why can't we stop at reality?...

...They think it's possible only because modern math welcomes actual infinity. If mathematicians rejected actual infinity then I'm sure physicists would be less inclined to accept it. — keystone

That's kind of a circular argument:

You - Mathematics shouldn't include elements with infinite properties because that doesn't match reality. Nothing infinite actually exists.
Me - There are qualified people who believe that infinite phenomena exist.
You - They've been fooled by their reliance on mathematics which include infinite elements.

Once again, calculus is about LIMITS, — jgill

True, but in many a calculus problem and theorem the limit IS infinity. — god must be atheist

Normally $x\to \infty$ arises in the following context:

$\underset{x\to \infty }{\mathop{\lim }}\,f(x)=L\text{ }\Leftrightarrow \text{ }For\text{ }\varepsilon \gt0\text{ }\exists M\gt0\text{ }such\text{ }that\text{ }x\gt M \Rightarrow \left| f(x)-L \right|\lt\varepsilon$

or $\underset{x\to \infty }{\mathop{\lim }}\,\,f(x)=\infty \text{ }\Leftrightarrow \text{ }For\text{ }N\gt0\text{ }\exists M\gt0\text{ }such\text{ }that\text{ }x\gt M\Rightarrow \left| f(x) \right|\gt N$

Infinity as a mathematical object is not used in calculus with one exception that I can think of, in complex analysis (calculus of complex variables) where the "point at $\infty$" corresponds in a projective sense with the north pole of the Riemann sphere.

See? You started an entertaining discussion that drew in some pretty good thinkers. Probably better than paying a PhD student. :cool:

Yep. So construction gets replaced by constraint. And then my point is you go the next step of seeing construction and constraint as the two halves of the one system. — apokrisis

The need to complicate things by forming your triadic metaphysics is unclear to me. You say that the idea of an absolute continuity as the alternative is offensive to the ontic intuition. Please explain why.

We start with the highest dimensional continuum of interest.
— keystone

Which would be the "infinite dimensional" continuum — apokrisis

What is the dimension of purely empty abstract space? One might say that it is infinite dimensional, another might say that it is 0-dimensional. What matters is what you do in that space. If you're in elementary school and learning to draw X-Y graphs, then the highest dimensional continuum of interest is 2D. I see no need for one to assert the existence of continua of which they're not interested in.

So I am saying I wouldn't deal with the metaphysics of the number line in isolation. It is illustrative of the far bigger conversation we need to have about how holism in mathematical conception plays out. The same principles have to cover mathematical structure in general — apokrisis

One step at a time :)

This is a rather basic level of discussion. Again, how could it even be a continua unless it could be cut? How could it even be a 0D point except as the positive absence of any dimensioned extension? — apokrisis

The continua that I'm describing can be cut. Consider cutting a string (continua). When doing so, end-points emerge. I see no need to say that infinite points reside within the string.

So if you want to apply the strength metaphysics to questions about mathematical structure, you have to count to three in terms of "fundamental things". — apokrisis

In reality I think that the three fundamental things are space, strings, and observers.
In math I think that the three analogous fundamental things are continua, cuts, and computers/minds.

Or in other words, no matter how many times I cut up a piece of paper, never will it vanish to nothingness.
— keystone

But each piece also gets more pointlike. — apokrisis

No, no matter how many times you cut a continua it never becomes more point-like. In a similar way, no matter how many times you cut a string it never becomes 'nothing-like'...since it always remains 'something-like'.

The cut has to be sandwiched between the two ends of two lines. Each end of the line is a point. At what point does the point marking the cut – that is, the absence of a point at that point – get marked off from the other two points marking the starts of a pair of now separated continua? — apokrisis

I would draw it like this: ----o o---- (note that the o is like an open interval)
Of this diagram, the cut is this: o o (note there's nothing actually there, the point is not an actual object)

So it is easy to picture just forever cutting a line. Or instead, just forever gluing points. — apokrisis

I can picture cutting a line and gluing lines back together. I can't picture gluing points together...that's just gluing nothing. I don't see the need for points being actual objects.

The thing is that we can't go the limit.
— keystone

But the fact that we can approach the limit – both limits – with arbitrary closeness is how we know they are there. The limit is precisely that which isn't reachable in the end. But it certainly defines the direction we need to keep going from the start. — apokrisis

0+0+0+0+0+0+.... approaches 0. Nothing comes from nothing, no matter how much of it you have. Continua are not the limiting case of points.

Now consider the following summations:
5
2.5 + 2.5
1.25 + 1.25 + 1.25 + 1.25
...

I can keep going down this route whereby each term gets smaller and smaller, but the overall sum of each line remains 5. Something evolves to something, no matter how many times you cut it up. Points are not the limiting case of continua.

There is not a duality between points and continua where they define each other. Continua are fundamental whereas points are not.

With this parts-from-whole construction, objects are finite and processes are potentially infinite...and there are no paradoxes.
— keystone

Again, this suffers all the usual problems of an object-oriented ontology. Reality is better understood in terms of relations – processes and structures. — apokrisis

I don't understand your objection. What is the usual problem of an object-oriented ontology that I'm facing?

For me, pi is clearly a number. — T Clark

Pi is a ratio. Diameter~circumference. So it is actually an algorithm. And it can vary between 1 and infinity as it is measured in a background space that ranges from a sphere to a hyperbolic metric.

How all that actual physics translates to claims one might want to make about numberlines and irrational values is another issue.

There are so many ways to undermine the metaphysics implicit in the continuum that perhaps this ought to be taken as confirmation that it is a useful concept that doesn’t demand further justification in terms of realist explanations?

Maths just defines it and gets on with it. And that is fine. It is what maths does.

I like to highlight the many unreal aspects of the conception from a realist metaphysics point of view. Another big one is the assumption the ground of counting ain’t divergent as you zero in on some arbitrary point.

Chaos theory and fractals illustrate how this might be the case. Scale matters. As you zoom in, you can no longer be sure that any point belongs to the line or a gap in the line if you are dealing with something fractal but space filling like a Cantor dust or Peano curve. If everything diverges on the finest scale, how do you plonk down your finger on some defined spot with any true certainty?

But the fact that the real world undermines the simplicities of the metaphysics that maths finds useful is part of the epistemic game here. The more holes there are in the story, the more we can take it as all just a story about reality - that works with “unreasonable effectiveness.”

We can take comfort in the transparency of it being a model. We can get on with using it within the limits in which it looks to be useful.

∞∞ isn't and object like for example an elephant or the number 10100 or the word "elephant", it's simply a shorthand for the procedure 1. n = 0; 2. print n; 3. n = n + 1; 4. go to 2]. :chin: — Agent Smith

I agree. However, I would go one step further and say that infinity-laden objects such as real numbers are also not objects like the number 10100. Instead, they are simply a shorthand for a procedure.

I have a feeling that some ideas like ∞∞ and nothing cause brain damage - Cantor lost his mind (theia mania) and spent his later years in a lunatic asylum for instance. These concepts & paradoxes of which there are many seem to have a deletorious effect on the brain/mind - constantly mulling over them may lead to a nervous breakdown. Such ideas are more than our brains can handle at present. And yet ... there have been no reports of an epidemic of mental problems among mathematicians. Why I wonder. — Agent Smith

Or maybe a certain type of unstable mind searches to understand infinity as it grapples for an absolute to hold on to.

:zip: I have nothing more to contribute.

I see you point. Calculus uses points to approach; infinity is not a point. You can't approach infinity, as you will never stop. There is no point in a continuum into infinity that you can call "this is the point of infinity".

The following is philosophy, not mathematics. Please treat it as such. I only have an undergraduate degree in math and at the university where I got the degree calculus was just a wee bit tougher than high school calculus. On the whole, the whole thing had been 40 years ago, I haven't used any of it since then, and my memory is not perfect; I don't remember much math.

What about approaching something infinitely small? dividing a given integer by a larger and larger integer in succession. Until the ratio almost becomes zero.

f(x)=sin(x)/x. It can't be evaluated at zero. Yet calculus succeeds in doing so. It does by using smaller and smaller numbers for x, and not actually using zero, but yet it gets an evaluation at zero, which means that it accepts that there is such a thing as an infinitely small number. It does have an end point, which can be only achieved by having an end point in the denominator of the fraction in the previous example. The end point in the denominator approaches infinity for x to approach 0; yet it's not a point. Yet, the actual value of sin(x)/x given by calculus is a real number, and that can only happen if Calculus does use infinity as a mathematical point. If infinity were not a mathematical point in calculus, sin(x)/x could not be evaluated at zero. yet it is evaluated at zero. So infinity, despite itself not being a point, does act as a mathematical point in calculus.

Never could a continuum be decomposed into points
— keystone

For physics, isn't that the driving force behind quanta, to put a stopper into space leaking out ? — magritte

Somewhat related, my understanding is that the planck length applies to measurement, not space itself. My understanding is that the continuity of space is still to be understood. Until then, it is possible that there is no limit to how small one can divide space.

With my view many paradoxes (Zeno, Dartboard, Liar's, etc) are easily resolved
— keystone

I take it you must mean dis-solved? — magritte

Yes.

An object is at rest. It is not moving.

Now the object is moving at a velocity V.

How many different velocities did the object move at, to get from zero velocity to V velocity?

If your answer is not "infinite" then you don't deserve the name "mathematician". — god must be atheist

In a quantum reality we can only talk about it's velocity when measurements were made. Since we can only ever make a finite number of measurements in any given time interval, I would answer 'a finite number'. I believe that the answer in pure mathematics should be the same.

I'm basically warning against logicism — sime

I see. I'm not really heading down the logicism route at the moment.

the algorithm for approximating sqrt(2) to any desired degree of approximation can itself be used to denote sqrt(2) without being executed. — sime

Agreed. That is why I what to give existence to the algorithm and not the completed output of the algorithm (which would be for example the complete decimal expansion of pi).

You might be interested in Norman Wildberger on YouTube. He seems to hold positions on infinity similar to yours. — emancipate

I really like Norman Wildberger. I think his issues with the foundations of mathematics are valid. However, I do not like his resolution to the issues. He still believes that points are fundamental. At the bottom he explains natural numbers using tic marks, which is no different than spaced points on a number line.

Sin(x)/x as x approaches zero is an entity itself, a ratio that converges to one. Look at the simpler ratio (x^2)/x as x approaches zero. It's an indeterminate form that reduces to x, so goes to zero.

I still don't get it. I don't see any advantage in your way of seeing things. For me, pi is clearly a number. — T Clark

What I'm saying is that pi isn't the string of digits that begin with 3.14. Pi is the algorithm(s) such as the infinite series beginning with 4/1 - 4/3 + 4/5 - 4/7 + .... Since we can't actually complete the computation of an infinite series, we never produce a number. So let's just say that pi is the algorithm. The beauty of the algorithm is that it's definition is entirely finite (I just wrote it in finite characters) and it's execution is potentially infinite (i.e. it would compute to no end). There's no need to say that pi has any association with actual infinity. If we say that it's a decimal number then we must say that it has actually infinite digits.

Seems like you're asking for an abstract, human invention to match up with your understanding of reality. It doesn't work that way. As they say, the map is not the terrain. — T Clark

I don't need mathematics to align with reality. I just think reality has a clever way of avoiding actual infinity and making sense. Reality is a good sign post. If a computer can't do the math (in principle), maybe there's something wrong with the math.

A number is not an object. It doesn't have a physical existence. Also, it's not beyond my comprehension. That way of seeing things has always made sense to me. — T Clark

A number is an object of computation. Computers do stuff with numbers. I don't think you can fill your head with all the digits of the decimal expansion of pi. The best you could do is fill your head with an algorithm for calculating pi. That's what I'm saying exists - the algorithm.

Abstract entities, i.e. all human concepts, are always simplified reflections of the world. I can't think of any that aren't. That's why math is so wonderful. — T Clark

Interesting view. I doubt that many hold this view. I think the traditional view is the inverse, that abstract objects are the ideals and reality is just an approximation of the ideal. Nevertheless, I don't hold either view. The concept of a unicorn is not a simplified reflection of any real world object.

Particles and waves are different kinds of physical entities. One is extended, spread out, in space and the other is found in a specific location. That's contradictory, and, just like numbers, both are simplified, abstract ideas. The fact that they seem contradictory, at least to most people, is a failure of human imagination. — T Clark

Imagine me flipping a coin. While it's in the air is it heads or tails? I'd say it's neither. Instead it has the potential to be heads or tails. Only when it lands does it hold an actual value. In between quantum measurements objects are waves of potential. When they are measured they hold an actual state. I see no reason why the potential should behave the same as the actual so I see no contradiction. In fact, I think if they behaved the same then change would be impossible.

I think, like many mathematicians, you are expecting math to have a precise correspondence with reality. That never works. — T Clark

I'm not expecting that, I just believe that truth rhymes.

That's kind of a circular argument:

You - Mathematics shouldn't include elements with infinite properties because that doesn't match reality. Nothing infinite actually exists.
Me - There are qualified people who believe that infinite phenomena exist.
You - They've been fooled by their reliance on mathematics which include infinite elements. — T Clark

The universe has a wonderful way of avoiding actual infinities. Maybe we could to the same in math. If we do, maybe people would be less open to the unsupported idea that the universe is infinite.

See? You started an entertaining discussion that drew in some pretty good thinkers. Probably better than paying a PhD student. :cool: — jgill

This is true, and I sincerely appreciate everyone's responses.

You say that the idea of an absolute continuity as the alternative is offensive to the ontic intuition. — keystone

Think of how the speed of light is an absolute limit on the velocity of a mass. The mass can be accelerated to some arbitrary speed approaching light speed, but it can’t actually arrive at light speed. The limit bounds the velocity of mass as an absolute. But the velocity of the mass is always some shade within that limit.

The reciprocal argument makes that explicit. The approach to a limit is asymptotic as it is always yoked to its divergence from its other pole. For a mass, it likewise can never be absolutely at rest, although it can approach that minimum velocity with arbitrary closeness.

So as I argued, continuity is measurable as the absence of discreteness. The fact you can choose to truncate your decimal expansion in search of some specific numberline value only shows you didn’t exhaust its capacity for discreteness and thus also failed to demonstrate it is as securely continuous as you might want to assume.

What is the dimension of purely empty abstract space? One might say that it is infinite dimensional, another might say that it is 0-dimensional. — keystone

I think the maths of manifolds and topology would want to give a more sophisticated answer than that.

And physics likewise would give you something more complex as all actual spaces come with time and energy too. Vacuums are quantum.

As I mentioned, if we are talking spaces with algebraic dimension, then there is a whole structuralist story about that as well.

So arguing against the infinite numberline in terms of a Euclidean geometry conception might already be heading off in the wrong direction - even if it is a hardy perennial of philosophical debate.

No, no matter how many times you cut a continua it never becomes more point-like. In a similar way, no matter how many times you cut a string it never becomes 'nothing-like'...since it always remains 'something-like'. — keystone

But a string has a width. And so you can eventually chop it so much that the width exceeds the length. At which point, your analogy is in trouble.

The width of your string would have to shrink every time the length of your continua is cut. That would preserve your claimed geometric relation.

But now we are into the log/log realm of the fractal. We are into the reciprocal relation of two processes yoke together that my triadic metaphysics describes, and not the monistic notion of a single process - a continuity of cutting - that you want to claim.

I would draw it like this: ----o o---- (note that the o is like an open interval)
Of this diagram, the cut is this: o o (note there's nothing actually there, the point is not an actual object) — keystone

There is a huge literature on how to handle terminal points of continua. I’m not arguing against the maths that maths finds useful.

But I am pointing to the deeper metaphysical issue that this kind of discussion reveals. We wind up with a threeness because you are demanding that two continua separated by a cut is still also the one continuum.

So rather than finding a point on a line, we create two lines with a cut that leaves them with a point sealing their bleeding ends, and some kind of gap inbetween … that is not a point, just the absence of even points now? An anti-point perhaps? Or what?

It all falls apart to the degree we try to apply everyday folk metaphysics - a Euclidean form of realism.

A more sophisticated metaphysics would let you analyse the situation in terms of a unit of opposites. A dialectical process. The numberline doesn’t need to get cut, but neither is it ever whole. The numberline instead always exhibits its twin reciprocal properties of being both limitlessly integrated and limitlessly differentiable.

To have this kind of character - to have as emergent properties the opposite conceptions of being cleanly cut and being smoothly unbroken - then requires the third thing of a logic of vagueness.

The line that can be both cut and connected is describing a state that is maximally binary in its ontology. That claim of absolute bivalent crispness in turn must find its grounding contrast in its own opposite of being the “minimally vague”. The numberline could be other. It could be just a swamp of vagueness. It could be a fractal Cantor dust for instance, where you could never know whether you land on a cut or a line.

So again, maths can take a simple view and dispose of its metaphysical issues as cheaply as it wants. The whole numberline debate is then of metaphysical interest perhaps largely as it reveals how quickly we indeed do stop short in our metaphysics generally.

We find some kind of monistic formula that identifies “a fundamental thing”. And that’s it. Job done.

I’m just arguing that the numberline debate is another example revealing that any holist ontology has to be triadic.

I can keep going down this route whereby each term gets smaller and smaller, but the overall sum of each line remains 5. Something evolves to something, no matter how many times you cut it up. Points are not the limiting case of continua. — keystone

But with a Dedekind cut approach, aren’t you just stabbing your finger down on the point marking one or other side of the cut continua … and never knowing which of the two terminating points you have touched. There is always the third ground thing of a vagueness?

Each cut leaves a left and right point. But you don’t know which side of the divide you are pointing to. And so “left vs right” is a radically indeterminate claim. The PNC fails to apply.

What is the usual problem of an object-oriented ontology that I'm facing? — keystone

It binds you to a monistic and reductionist conception of nature.

As I say, thinking that way is fine if all you want to do in life is construct machines. That can be your reality model.

But for natural philosophy and metaphysics, not so much.

Pi is a ratio. Diameter~circumference. So it is actually an algorithm. And it can vary between 1 and infinity as it is measured in a background space that ranges from a sphere to a hyperbolic metric.

How all that actual physics translates to claims one might want to make about numberlines and irrational values is another issue. — apokrisis

I have no trouble with this way of seeing it.

Maths just defines it and gets on with it. And that is fine. It is what maths does. — apokrisis

Agreed, but I think many people don't see it that way. Some think that math somehow produces reality. That if math doesn't track common sense, everyday reality exactly, there needs to be an explanation. That something is wrong.

But the fact that the real world undermines the simplicities of the metaphysics that maths finds useful is part of the epistemic game here. The more holes there are in the story, the more we can take it as all just a story about reality - that works with “unreasonable effectiveness.” — apokrisis

Agreed. That's consistent with my understanding of metaphysics in general - it is not true or false, it works or it doesn't.

Somewhat related, my understanding is that the planck length applies to measurement, not space itself. — keystone

The Planck length emerges out of the triad of dimensional constants, c,G and h. Which happen to be reciprocally yoked.

So zoom in on the Planck scale and you find the same metaphysics I have described. The smallest length is also the hottest temperature. Spacetime becomes so buckled that it dissolves into the vagueness of a quantum foam. It has neither length nor points, flatness nor curvature, in any proper contextual fashion.

Event and context become the same size. And so neither can be distinguished from its other. The Planck scale speaks to a fundamental cut-off for all such metrical relations.

Some think that math somehow produces reality. That if math doesn't track common sense, everyday reality exactly, there needs to be an explanation. That something is wrong. — T Clark

Sure. But reality scales. It “runs its couplings” in physics-speak. The maths that best describes reality has to do the same.

So the everyday folk conception or reality - and the maths that might describe that - is based on the current experienced state of the Cosmos, when it is vast, cold, and a couple of degrees from the limit of its heat death.

That is a world in which an object#oriented ontology of “medium sized dry goods” seems to make “fundamental sense”.

But physics tells us that this is not fundamental, just a passing stage. The Big Bang had quite a different kind of ontology. And physics has worked up a decent account of the maths required to track how each stage evolved into its next.

And again, it is the kind of triadic/holistic/dialectic systems view I’m talking about. Peircean semiosis.

Since we can't actually complete the computation of an infinite series, we never produce a number. So let's just say that pi is the algorithm. The beauty of the algorithm is that it's definition is entirely finite (I just wrote it in finite characters) and it's execution is potentially infinite (i.e. it would compute to no end). — keystone

I don't see any advantage to the fact that your way of conceptualizing pi is "entirely finite."

If a computer can't do the math (in principle), maybe there's something wrong with the math. — keystone

There's always something wrong with the math. That's why people keep having to add on new concepts to keep up with our understanding of reality.

A number is an object of computation. Computers do stuff with numbers. I don't think you can fill your head with all the digits of the decimal expansion of pi. The best you could do is fill your head with an algorithm for calculating pi. That's what I'm saying exists - the algorithm. — keystone

It is my understanding that computers do not generally store the algorithm for generating pi, they store the actual number rounded to a specified number of decimal laces. If computers think pi is a number, why shouldn't I?

abstract objects are the ideals and reality is just an approximation of the ideal. — keystone

Perhaps Plato and some mathematicians and logicians think this way. Not most people.

Imagine me flipping a coin. While it's in the air is it heads or tails? I'd say it's neither. Instead it has the potential to be heads or tails. Only when it lands does it hold an actual value. In between quantum measurements objects are waves of potential. When they are measured they hold an actual state. I see no reason why the potential should behave the same as the actual so I see no contradiction. In fact, I think if they behaved the same then change would be impossible. — keystone

When I measure light one way, it's always a wave. When I measure it another way, it's always a particle. It's not a wave that becomes a particle. It's always both at the same time.

The universe has a wonderful way of avoiding actual infinities. — keystone

Again, sez you.

I think you and I have taken this as far as we're going to get. I don't see the need for or value of the way of seeing things you propose. You obviously disagree. Neither of us is going to convince the other.

runs its couplings — apokrisis

I think I understand what you are saying about scaling, but I am not familiar with this phrase. What does it mean?

But physics tells us that this is not fundamental, just a passing stage. The Big Bang had quite a different kind of ontology. And physics has worked up a decent account of the maths required to track how each stage evolved into its next. — apokrisis

Seems like physics is always trying to compress all this multitude of stories about reality at many scales and stages into a single narrative that covers everything at once.

I don't think you and I are disagreeing much.

The fundamental forces run their couplings. They are all fractured and very different in the cold/large universe of today. But all their strengths and properties (probably) converge at the Planck scale in one simple Grand Unified Theory – a vanilla form of quantum action that is the contents of a general relativity spacetime container of smallest scale.

The fundamental forces run their couplings. They are all fractured and very different in the cold/large universe of today. But all their strengths and properties (probably) converge at the Planck scale in one simple Grand Unified Theory – a vanilla form of quantum action that is the contents of a general relativity spacetime container of smallest scale. — apokrisis

Thanks.

@apokrisis

I'm just rereading through your older messages and I have a few additional comments:

• I see neither the point as infinitesimal nor the line as infinite. In terms of length, the point is exactly 0 and the line is some positive number. If you're talking about something of infinitesimal length, you are talking about some tiny line segment. If that's the case then you are only talking about continua.
• infinite = 1/infinitesimal is very problematic for hopefully obvious reasons. Why not write something that makes sense, like 1,000,000 = 1/0.000001. If you do this then it's clear that you're working solely with continua.
• The inverse relation between points and continua is that the point is nothing and the continuum is something.
• Points and continua are not the measure of each other. Nothing cannot be used to measure something (0+0+0+0... always equals 0). Whereas something can be use to measure nothing (e.g. 5-5=0). There is an imbalance here in this relationship suggesting that continua are more fundamental.
• The structure of a continuum is not defined by points, it is defined by an equation(s). Aside from being impossible, you would never try to provide an infinite list of points to completely describe a line (Cantor). You would just provide an equation - a finite string of characters to perfectly describe how points would emerge if cuts are made.
• I don't understand how continua + equations are vague. If I say I'm thinking of a plot containing the curves x=0, y=0, and y=x^2 you know exactly what I'm thinking of.
• We cut and glue continua. We can't do either with points.

So as I argued, continuity is measurable as the absence of discreteness. The fact you can choose to truncate your decimal expansion in search of some specific numberline value only shows you didn’t exhaust its capacity for discreteness and thus also failed to demonstrate it is as securely continuous as you might want to assume. — apokrisis

Let's imagine a line where cuts have been made to mark all rational points (I don't believe this is possible, but let's go with it for now). I believe you cannot mark any more points on this line. If you throw a dart in between the rational points then you will hit an indivisible line segment. That is as discrete as it gets, and even then the line is securely continuous.

What is the dimension of purely empty abstract space? One might say that it is infinite dimensional, another might say that it is 0-dimensional.
— keystone

I think the maths of manifolds and topology would want to give a more sophisticated answer than that. — apokrisis

Agreed. I don't think my statement there was critical to my argument though.

But a string has a width. And so you can eventually chop it so much that the width exceeds the length. At which point, your analogy is in trouble. — apokrisis

True, the element proportions change with further divisions but still you can't cut a string out of existence. I think your argument doesn't attack the essence of my argument.

So rather than finding a point on a line, we create two lines with a cut that leaves them with a point sealing their bleeding ends, and some kind of gap inbetween … that is not a point, just the absence of even points now? An anti-point perhaps? Or what? — apokrisis

If we cut y=0 with the 'knife' x=0, then there is a void between the two newly produced line. We then have line x<0, void x=0, and line x>0. For all practical purposes this void is a point, the only difference is philosophical: that the void is not an object. It is the absence of an object (continua).

The numberline instead always exhibits its twin reciprocal properties of being both limitlessly integrated and limitlessly differentiable. — apokrisis

Both of which are performed on continua, not points.

The numberline could be other. It could be just a swamp of vagueness. It could be a fractal Cantor dust for instance, where you could never know whether you land on a cut or a line. — apokrisis

The cuts are 0-dimensional so they are illusions of convenience. If you throw a dart at the line you will always hit the line, never the cut. The cuts have measure 0 after all. Instead of a number line, let's consider the curves x=0, y=0, and y=x^2 in 2D Euclidean space. These curves cut each other at (0,0) so we have one 'point' in this system. Is there vagueness? Yes. Without constraining the curves at all points the system is more topological than geometrical. But what's wrong with that? When anyone or any computer draws this system it's always imperfectly drawn, but by labelling the curves with their equations, we know precisely what will happen if we make additional cuts.

I’m just arguing that the numberline debate is another example revealing that any holist ontology has to be triadic. — apokrisis

In part you're preaching to the choir since my view is triadic (computers/cut/continua). What is the triad in your view? Is it continuous/discrete/vagueness?

What is the usual problem of an object-oriented ontology that I'm facing?
— keystone

It binds you to a monistic and reductionist conception of nature. — apokrisis

I suppose the triad that I'm proposing isn't solely object-oriented. It has a subject (computer), a verb (cut), and an object (continua).

So zoom in on the Planck scale and you find the same metaphysics I have described. — apokrisis

I don't want to dive into what happens at the Planck scale, in part because I'm not informed enough on the topic and in part because my understanding is that we can't observe the universe at this scale.

In a quantum reality we can only talk about it's velocity when measurements were made — keystone

we were talking in terms of Calculus, and that is a very integral and important circumstance to my question. Perhaps I should have pointed that out.

Perhaps I should have pointed that out. — god must be atheist

Damn.