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As usual, character assassination but no counter arguments.

This user has been deleted and all their posts removed.

Try to focus on the current discussion - my OP - and give me some counter arguments.

This is a philosophy forum, not a mud slinging contest.

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As with the teaching of infinity, something which is just an assumption is taught to us as absolute knowledge. I feel our maths teachers are letting us down — Devans99

The Cantor-Dedekind axiom is not an axiom in the usual sense. You can construct a complete ordered field over the Euclidean plane from the axioms of synthetic geometry. Since every complete ordered field is essentially the real numbers, this justifies the methods of analytic geometry. This justification required the development of axiomatic methods for geometry and algebra. A classic and highly readable reference is Hartshorne's Geometry: Euclid and Beyond (Chapters 2-3).

Now the prevailing wisdom is that [1] holds - a line segment is composed of an infinite number of zero length points. I cannot make any sense out of this. How can anything zero length (dimensionless) be said to composed something with non-zero length? This is the view Aristotle held. — Devans99

The only "explanation" I can offer is:

Consider a line of length 1 unit extending from 0 to 1.It can be repeatedly halved by multiplying with 1/2
Each multiplication will yield a point and a length corresponding to that point. For example the first halving will give us 1/2 which is a point and dimensionless but don't forget the distance from 0 to 1/2 which is a 1 dimensional length. You're failing to consider the length that corresponds to each point in a line. So, although points are dimensionless, the distance between points have a dimension viz. length.

Considered another way there are an infinite number of points in any given line but the line is constituted of the distances between these points and not the points themselves.

Nevertheless, the point I am making is that I can find no workable mathematical description of continua. This might lend credence to the idea that, like matter, time and space are discrete? — Devans99

If by workable you mean conformity to your private intuition of the continuum, then actual mathematicians have famously wrestled with this. https://plato.stanford.edu/entries/weyl/

Or let's say that workable means conformity to your metaphysics. Fine, you haven't found technical and objective treatments that agree with your non-technical home-brewed metaphysics. But have you really shopped around?

https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
https://en.wikipedia.org/wiki/Constructive_analysis
https://en.wikipedia.org/wiki/Computable_analysis

I'm not sure how such options can be intelligible in the first place to someone who hasn't learned at least the basics of mainstream real analysis. Or at the very least a semi-rigorous calculus. As far as I can tell from your posts, you think that math is some strange form of metaphysics that uses symbols as abbreviations for fuzzy concepts. And then proofs are just fuzzy arguments to be interpreted like mystical literature on the profundities of time, space, matter. Not so. The black queen on a chess board rules no kingdom. Legal moves are strictly defined and computer checkable. While the queen piece may be called a 'queen' and not a 'pawn' because of her exquisite mobility, what we call her and the intuition associated with that name is not important to the computer that checks whether a move is legal. It's only her place within the structure that matters. Math has to be this 'dry' or machine-like for the vivid objectivity it has in contrast to vague qualms about 'actual infinity' or the 'continuum.'

*I'll extend this point to metaphysics in general by quoting Wittgenstein. What I mean by it in this context is that none of us even have access to what is 'meant' by the 'continuum' or 'actual infinity.' For that matter even your private intuition corresponding to 'finite' cannot play a role. And so on. Only signs are public. In ordinary language, we have comparative freedom. In math no.

If I say of myself that it is only from my own case that I know what the word "pain" means - must I not say the same of other people too? And how can I generalize the one case so irresponsibly?

Now someone tells me that he knows what pain is only from his own case! --Suppose everyone had a box with something in it: we call it a "beetle". No one can look into anyone else's box, and everyone says he knows what a beetle is only by looking at his beetle. --Here it would be quite possible for everyone to have something different in his box. One might even imagine such a thing constantly changing. --But suppose the word "beetle" had a use in these people's language? --If so it would not be used as the name of a thing. The thing in the box has no place in the language-game at all; not even as a something: for the box might even be empty. --No, one can 'divide through' by the thing in the box; it cancels out, whatever it is.

That is to say: if we construe the grammar of the expression of sensation on the model of 'object and designation' the object drops out of consideration as irrelevant. — Wittgenstein

'The object [the continuum in our 'minds' or 'reality'] drops out of consideration as irrelevant.'

Thanks for the links…

If by workable you mean conformity to your private intuition of the continuum, then actual mathematicians have famously wrestled with this. https://plato.stanford.edu/entries/weyl/ — "softwhere

Wrestled with - and consistently failed to achieve - a sound mathematical description of continua - as I also failed to in the OP.

Weyl was not a believer in the ‘Cantor–Dedekind axiom’. He saw the real number as a discrete concept in contrast to the (alleged) continuous nature of time and space:

“The conceptual world of mathematics is so foreign to what the intuitive continuum presents to us that the demand for coincidence between the two must be dismissed as absurd” - Weyl

So he admits that construction of a valid mathematical model of a continuum is an impossibility - and he never achieves such in his work.

The Cantor–Dedekind axiom is highly questionable to my mind. A real number is a purely imaginary concept. It is like a label so it cannot be said to have any width. So an infinite number of real numbers on a finite line segment is acceptable - in our minds only. A line however represents something that can have objective reality. It must be constituted of something - points or sub-line segments, and the parts must have non-zero, non-infinitimsal width - else something becomes nothing.

Weyl was a supporter of the Brouwerian continuum. As I understand it, the Brouwerian continuum has strange attributes - ‘numbers’ in the Brouwerian continuum are allowed to be dynamical, constantly evolving, quantities in that such a ‘number’ does not have a complete, decimal expansion at any point in time - rather it is in a state of constant evolution as its digits grow with time. This means that in the Brouwerian continuum:

- For real numbers a, b either a < b or a = b or a > b does not hold
- The law of excluded middle: for any real numbers a, b, either a = b or a <> b does not hold

I do not class a system with the above two properties as ‘mathematical’ - in the sense that for me, valid mathematics should be built upon the principles of basic arithmetic and logic. I think once these principles are discarded, then we enter the realm of ‘pure maths’ - maths that does not reliably tells us about the world we actually live in - it may tell us interesting stuff about other realities - virtual worlds with different rules to ours - but it does not describe the universe we live in.

So again, we have mathematics failing to come up with a mathematical description of continua.

https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis — "softwhere

I am somewhat discouraged that this system denies the law of excluded middle and by the definition: ‘nilpotent infinitesimals are numbers ε where ε² = 0 is true, but ε = 0 need not be true at the same time’ - this is again contrary to basic arithmetic and logic.

It may well have applications, but a reliable description of the nature of our universe is not one of them.

https://en.wikipedia.org/wiki/Constructive_analysis
https://en.wikipedia.org/wiki/Computable_analysis — "softwhere

"The continuum as a whole was intuitively given to us by intuition; a construction of the continuum, an act which would create by means of the mathematical intuition "all its points" is inconceivable and impossible" - Brouwer

“Space and time are quanta continua… points and instances mere positions… and out of mere positions views as constituents capable of being given prior to and time neither space nor time can be constructed” - Kant

As far as I can tell from your posts, you think that math is some strange form of metaphysics that uses symbols as abbreviations for fuzzy concepts. And then proofs are just fuzzy arguments to be interpreted like mystical literature on the profundities of time, space, matter. — "softwhere

Most of the maths you have linked to falls into the category of contrary to basic logic or arithmetic. Now you might call me closed minded, but maths for me has to obey the principles of basic arithmetic. When advanced maths differs from basic arithmetic I feel it is no longer telling us about the real world - it is describing some virtual reality that is not our reality - so it is therefore not helpful in the pursuit of understanding the nature of our universe.

I have tried and failed using my basic maths skills to construct a description of continua. You claim the logic I use is 'fuzzy' but then do not point out any examples of my ‘fuzzy’ thinking - making me think that you are unable to identify any such - please advise.

My research also indicates that no mathematician has ever come up with a sound mathematical description of a continuum - so I would be interested to learn what your favoured mathematical prescription for a continua is?

You're failing to consider the length that corresponds to each point in a line. So, although points are dimensionless, the distance between points have a dimension viz. length.

Considered another way there are an infinite number of points in any given line but the line is constituted of the distances between these points and not the points themselves — TheMadFool

I suppose you can view a line segment as constituted of points or sub-segments. Whichever way though, the length of the constituents has to be non-zero.

I think the core of the issue is in in assuming that a line can be constructed from points on the line.

If we are talking about pure and abstract maths, and I think we are,, then there is no reason for this.

A line can be defined by an equation such as X=Y, where all numbers that satisfy that equation lie on the line. It can be considered to be a continuum as for any two points on the line another point can be identified that is between those two points.

There is no need to consider that the line is made up of points.

I'm more interested in the nature of our reality and the nature of real lines rather than abstract/imaginary mathematical lines. IE is it possible for there to exist in nature something truly continuous (IE of the same nature as our mental models of lines / the real number line). Obviously the question is settled for matter (discrete) but for time and space it is still an open question.

If you assume that the Cantor–Dedekind axiom is true - that the real numbers correspond to points on a line - then we can take the purely mental model we have of real numbers as being infinitely divisible - and apply it to a mathematical line segment - resulting in a mental model of the segment as containing an infinite number of zero-length points (something than does not make physical sense - but we can do it purely in our minds).

But I'm not sure that the Cantor–Dedekind axiom can be said to apply to 'real life' lines - when I wrote the OP, it was buried deep in my consciousness under the category of 'unquestionably the way the world works'. I have since changed my mind about this.

A real life line segment (as opposed to a purely imaginary, mathematical line segment) is something substantial - and it must be composed of something - and the components must have non-zero length - else it is nothing. So I do not see how a physical line segment could ever correspond to our mental model of the real number line (obviously here I assuming some sort of continuous substance in reality to construct the real line segment with - time and space are the only two candidates - no form of continuous matter is known).

Real numbers I think are more like logical/mental labels - they have zero width - they are just a way to label parts of a line rather than the constituents of a real line.

So I think there is probably an equivalence between the real number line and an imaginary/mathematical line segment - both exist in our minds only - so the impossible stuff like them being composed of an infinite number of zero-length points happens in our minds only (where the impossible is possible), but I doubt the equivalence holds to any line with real life existence.

There is obviously also the question of whether time and space qualify as 'something' rather than 'nothing' (Relationism Versus Substantivalism). I am in the 2nd camp on that question.

Well, software, being of a modern generation of mathematicians, has opened my eyes again:

Wiki:"Other mathematical systems exist which include infinitesimals, including non-standard analysis and the surreal numbers. Smooth infinitesimal analysis is like non-standard analysis in that (1) it is meant to serve as a foundation for analysis, and (2) the infinitesimal quantities do not have concrete sizes (as opposed to the surreals, in which a typical infinitesimal is 1/ω, where ω is a von Neumann ordinal). However, smooth infinitesimal analysis differs from non-standard analysis in its use of nonclassical logic, and in lacking the transfer principle. Some theorems of standard and non-standard analysis are false in smooth infinitesimal analysis, including the intermediate value theorem and the Banach–Tarski paradox. Statements in non-standard analysis can be translated into statements about limits, but the same is not always true in smooth infinitesimal analysis.

Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved. The construction of discontinuous functions fails because a function is identified with a curve, and the curve cannot be constructed pointwise. We can imagine the intermediate value theorem's failure as resulting from the ability of an infinitesimal segment to straddle a line. Similarly, the Banach–Tarski paradox fails because a volume cannot be taken apart into points."

It IS possible to learn something on this forum! Thanks.

I suppose you can view a line segment as constituted of points or sub-segments. Whichever way though, the length of the constituents has to be non-zero. — Devans99

I'd like to continue the discussion if you don't mind because I see what you mean but I feel, given that mathematicians don't make a fuss about points being zero-dimensional, you're in error.

Points are locations. Location are distances. Distances are lengths. Lengths are spaces between two points. As you can see there's a definitional circularity: points defined in terms of lengths and lengths in terms of points and I think this is the reason why a point belongs to the list of undefined terms in geometry.

Anyway...

When we mark off a point, say C, on a line AB= 5cm, it needs an identifying label and that's its distance from one of the endpoints of the line. Suppose C is 3cm from A. We label point C = 3cm. C, the point, is zero-dimensional but remember it divides AB into two lines viz. AC = 3cm and BC = 2cm. AC + BC = 3cm + 2cm = 5cm. The line AB can be infinitely divided into infinitesimally small non-zero lengths and each length will always have a point associated with it. Can you now see that, since a line can be divided into an infinite number of non-zero lengths and each length has a point associated with it, there'll be an infinite number of points in any line. In some sense, a point is just a label/name for a length and while a length is a building block/structural component of a line the point is nothing more than the name for lengths.

I'd like to continue the discussion if you don't mind because I see what you mean but I feel, given that mathematicians don't make a fuss about points being zero-dimensional, you're in error. — TheMadFool

Aristotle made a fuss about zero-dimensional points being the components of lines so I feel the question can be regarded as an open philosophical question.

The line AB can be infinitely divided into infinitesimally small non-zero lengths and each length will always have a point associated with it. — TheMadFool

That does not appear to be the case - see the argument in the OP that sums an infinity of infinitesimals to zero (length).

Now you might disagree with the maths in the OP and say an infinity of infinitesimals sums to ∞ * 1/∞ = 1. But this (questionable) maths leads to the conclusion that all continua have the same length - both a segments and its sub-segments are continua so they would both have the length 1.

Then I suppose one could further argue that the infinitesimals on the segment are somehow longer than the infinitesimals on the subsegment - but that's contrary to the definition of continua as all having identical structure (and identical cardinality - uncountably infinite).

Aristotle made a fuss about zero-dimensional points being the components of lines so I feel the question can be regarded as an open philosophical question. — Devans99

What were Aristotle's objections to points being zero-dimensional?

But this (questionable) maths leads to the conclusion that all continua have the same length - both a segments and its sub-segments are continua so they would both have the length 1. — Devans99

I don't get this part.
Are you saying that because there are an infinity of points in any given line that all lines have to be of the same length?

What were Aristotle's objections to points being zero-dimensional? — TheMadFool

- He felt that something with zero extent like a point could not constitute something with non-zero extent, like a line.
- He also regarded points on a line as a 'potential infinity' in that the points are not 'actualised' until the line is divided. In this way, he justified his believe that 'actual infinity' is impossible.

There maybe others too, I'm no expert on Aristotle.

Are you saying that because there are an infinity of points in any given line that all lines have to be of the same length? — TheMadFool

The mainstream definition of continua (Dedekind-Cantor) does seem to lead to this conclusion:

1. Points have zero dimension
2. A continua has an uncountably infinite number of points
3. All continua have the same structure and cardinality
4. Therefore it follows that all continua have the same length

So point 4 is expressing the fact that:
(size of a point) * (cardinality of the continua) = (size of the continuum in question)
0 * ∞ = 0

Equivalently, if we assume that points have 'infinitesimal' length, then, depending on how you do the math, you get:
1/∞ * ∞ = 0
or
1/∞ * ∞ = 1

And you always calculate the same length for all different lengths of continua.

1. Points have zero dimension
2. A continua has an uncountably infinite number of points
3. All continua have the same structure and cardinality
4. Therefore it follows that all continua have the same length — Devans99

I think 4 doesn't follow.

Consider two lines AB = 2 cm and CD = 4 cm

Divide these lines i.e. find points on these lines by dividing them with k

1. k = 1; for AB, 2/1 = 2; for CD, 4/1 = 4
The point on AB is 2 and the point on CD is 4

2. k = 2; for AB, 2/2 = 1; for CD 4/2 = 2
The point on AB is 1 and the point on CD is 2

3. k =3; for AB, 2/3; for CD, 4/3
The point on AB is 2/3 and the point on CD is 4/3
.
.
.
n. k = n; for AB, 2/n; for CD, 4/n
The point on AB is 2/n and the point on CD is 4/n

As you will notice for every point on AB there will always be a unique point on CD i.e. the cardinality of the set of points on AB = cardinality of the set of points on CD. They're both infinite.

However, notice that a point on AB has a different numerical value to the corresponding point on CD. They are different quantities and so add up to different, not same, lengths.

As you will notice for every point on AB there will always be a unique point on CD i.e. the cardinality of the set of points on AB = cardinality of the set of points on CD. They're both infinite. — TheMadFool

Agreed.

However, notice that a point on AB has a different numerical value to the corresponding point on CD. They are different quantities and so add up to different, not same, lengths. — TheMadFool

That numerical value does not tell us the width of a point and length is the sum of widths of the constituent points / sub-line segments.

@ Devan read the following link. and follow the link explaining limits
https://en.wikipedia.org/wiki/Integral
Hopefully that will answer your questions.

I understand what an integral is and the concept does not lead to a sound definition of a continuum - unless you can correct me on this?

In order to understand the justification for analytic geometry, you need to study synthetic geometry. In particular, Hilbert's work on the foundations of geometry. Your questions were resolved more than a century ago. From the axioms of geometry we can construct models of the real numbers that satisfy our intuitive notion of a coordinate system. There isn't much interesting to see here.

The result of that work from a century ago is the Dedekind-Cantor continuum - which is a nonsensical proposition. I'm not sure you read / understood the OP?

I'm saying that your objections are more than a century out of date. In order to understand why, you need to learn a bit of synthetic geometry and abstract algebra. I don't see any way around this.

- Yet the Dedekind-Cantor continuum is taught in school along with the fact that a point has zero width. So my objections are bang upto date, as far as I can see.
- No-one has yet pointed out any logic/math error in my OP.
- If I have something wrong, then someone should set me straight, rather than vague hand waving
- At least a link to your preferred definition of the continuum would be nice

I'm saying that your objections are more than a century out of date. — quickly

How is the date of the objection relevant? If it's a reasonable objection then it's a reasonable objection, regardless of the date.

And, as Devans99 indicates, the issues have not been "resolved more than a century ago". They've simply been ignored.

A topologically structured set that only possesses the notion of "neighbouring" elements does not in itself possess 'holes', regardless of the finiteness of the set.

The notion of a continuum is relative to the notion of a 'hole', which is describable in terms of the absence of a topology-preserving surjection between two topologically structured sets.


"Gaps" only exist within the line of countably computable reals for platonists who think that non-computable numbers exist. Even if they are granted the existence of an uncountable number of non-computable reals to occupy those "gaps", the resulting model of the reals cannot control their cardinality, suggesting to the Platonist further holes, forcing him into constructing hyperreals and so on, without ever being able to fill the gaps in the resulting continuum.

This is another good reason for rejecting the non-constructive parts of mathematical logic.

"If we look back on what consciousness formerly took upon itself, and now takes upon itself, what is previously ascribed to the thing, and now ascribed as to it, we see that consiousness alternately makes itself, as well as the thing, into both a pure atomic manyless One and an Also resolved into independent constituted elements, materials, or matters. Conssciousness thus finds through this comparison that not only it's way of taking the truth contains the diverse moments of apprehension and returns upon itself, but that the truth itself, the thing, manifests itself in this twofold manner... ;in other words, the thing contains within itself it's opposite aspects of truth, a truth whose elements are antithesis to one another.. It's being One contradicts the diversity of has." Hegel is his first published work dealing with zeno

Zeno's argument is really about Aristotle's "common sense" understanding of the world. In calculus numbers are not added one at a time. The value/limit that the addition is approaching is what is determined. There is still an infinity in the equation because infinitesimal are the quintessence of smallness. The question seems to be whether "many is always many" or whether "many" can be something else. Aristotle was wrong to say that an object is potentially divisible but not actually so, because something contains the same volume whether divided or not. What if God divided this computer infinitely? How can it be spatially endless yet finite when I type on it? I think Devans99 has a point. Maybe Minkowski's lecture on "staircase wit" will help

"If the fractional times decrease in a harmonic series, while the distances decrease geometrically, such as: 1/2 s for 1/2 m gain, 1/3 s for next 1/4 m gain, 1/4 s for next 1/8 m gain, 1/5 s for next 1/16 m gain, 1/6 s for next 1/32 m gain, the distances form a convergent series, but the times form a divergent series, the sum of which has no limit." Wikipedia