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. — Gregory

There is a distinction between:

1. The process of division - we cannot go on dividing an object forever because we would never finish the division process - so we can not take an actual object and make an actual infinity of pieces out of it
2. Something that exists and was never created could exist in a form that is already divided infinitely

So it seems it is impossible to create/manufacture the actually infinite, but actually infinite things could exist - uncreated things - as some hold spacetime to be uncreated (not my view).

If God created the world, he created a dualism or hybrid of finitude and infinity.

Is God the nexus, or other dimensions (as in the other world's interpretation of QM)?? Zeno's stadium paradox, when applied to the smallest unit of space (which seems to be zero), needs to be connected to modern physics. For the latter, each object has its own time perspective (like a "noise when nobody listens"). The relative motion of the columns seems to imply a smaller unit of time than the smallest unit of time

My dad, i, and my twin brother all independently thought of Zenos dichotomy before we read of him. Thoughts like that caused my mind strabismus, which is how I am now ale to speed read. TMI

I directed the OP towards a (highly online) reference that explains how mathematicians disarmed his or her objections over a century ago. The date is relevant only because the OP ignored this reference and continued to insist that the methods of analytic geometry are unfounded. The relevant foundations were provided by mathematicians operating around the turn of the 20th century.

I found the book you are referring to here:

https://download.tuxfamily.org/openmathdep/euclid/Euclid_and_Beyond-Hartshorne.pdf

It is 500 pages long and the words continua and continuum are never mentioned. It talks about 'points' a lot without even defining the term as far as I can see.

?

Slight correction, they do reference Euclid's non-sensical definition of a point: "a point is that which has no part" on page 27.

I directed the OP towards a (highly online) reference that explains how mathematicians disarmed his or her objections over a century ago. The date is relevant only because the OP ignored this reference and continued to insist that the methods of analytic geometry are unfounded. The relevant foundations were provided by mathematicians operating around the turn of the 20th century. — quickly

If you think that the objections have been resolved, then you're simply wrong. And pointing to some "highly online" reference (whatever that means) does not make you right. If you would take the time to produce the supposed resolution we could show you how it simply covers up the problem rather than solving it.

Continuum is a set of points where for every two points in the set there exists a point in the set that is in between the two points.

The fact that points have no size (which, by the way, does NOT mean that points have zero-size), whereas line segments (whether continuous or discrete) do, does not mean that line segments are not made out of points. It merely reflects the fact that distance is something that exists between points.

You don't measure the length of a line segment by counting how many points it has, you measure it by counting how many pairs of points-at-certain-distance it has. "This line is 10cm long" means "This line is made out of 10 pairs of points-at-1cm-from-each-other".

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

But it is made out of points. It's just that the length of a line is not measured the way @Devans99 thinks it is measured. You don't measure the length of a line by summing the lengths of its smallest parts (which are points.) Points have no length. They do not have such a property. Length is something that exists between two points. In order to measure the length of a line you must count the number of pairs of points-at-a-certain-distance that constitute it.

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). — Devans99

Points have no zero-length. They have no length at all. I think this is part of the problem. A lot of people do not understand that the statement "Points have no length" does not mean "Points have zero-length".

To have no length is by definition to not exist in the realm of geometry, and also geometry obviously applies to the physical world. Russell said at the turn of the century points were banished from calculus. Now non-standard analysis now says "infinitesimals are inevitable". Google that last quotation

“motion can be understood as the position occupied by an object in a
continuous series of points in a continuous series of instants.”; Bertrand
Russell, I Principi della Matematica, (Milano: Einaudi, 1963), 637.

Yet he also said points were non-sensical.

“It is just as impossible for anything to break forth from it [the One] as to break into it; with Parmenides as with Spinoza, there is no progress from being or absolute substance to the negative, to the finite.”; Hegel, Science of Logic, 94-95.

Math cannot solve this. Maybe philosophy can

"It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever" says Bertrand Russell on Zeno's arrow, this is simply "a plain statement of an elementary fact" he says. I see motion has have an impetus, so I would disagree. Motion IS a supertask though, which makes it a huge problem. A haunting one

According to Russell, it was Weierstrass who banished points from calculus. I don't see how it worked without them though. Weierstrass said a continuous function is a static completed unity

To have no length is by definition to not exist in the realm of geometry — Gregory

It depends on how you define the word "existence". You can define it any way you like. You can define it in a way that implies length. That which exists has length. If we accept that definition then it follows that points (among many other things, such as sounds, colors and other sensations) do not exist. And since definitions have no truth-value, you can't argue against such a definition by saying "That's not true definition of existence!" Unless, of course, you're arguing against it on the basis of use-value. And that would be my response. Such a definition of existence has a limited (even questionable) use-value. It's certainly not how most people define existence. Most people define existence (not necessarily verbally but certainly intuitively) in such a way that even things that have no length (such as points, colors, sounds, etc) can be said to exist.

"Complex analysis" (where complex means "having both real and imaginary parts") is hard to justify because we get our math from the world, where many is always many and not one.

Hegel had interesting things to say on limits. Google it.


"The linear series that in its movement marks the retrogressive steps in it by knots, but thence goes forward again in one linear stretch, is now, as it were, broken at these knots, these universal moments, and fall asunder into many lines, which, being bound together into a single bundle, combine at the same time symmetrically, so that the similar distinctions, in which each separately took shape within a sphere [Parmenides's One], meet again." Phenomenology of Mind

According to Nietzsche, Hegel "systematized the riddle" of being and nothingness thru teaching that all is "obscure, evolving, crepuscular, damp, and shrouded".

Marx wrote:

"First making the differentiation and then removing it therefore leads literally to nothing. The whole difficulty in understanding the differential operation (as in the negation of the negation generally) lies precisely in seeing how it differs from such a simple procedure and therefore leads to real results."

Negations of nothing!

"Marx recognized the differential equation as an ‘operative formula’ — ‘a strategy of action’ which, when it arises, constitutes a reversal of the differential process, since the ‘real’ algebraic processes then arise out of the symbolic operational equation, which originally itself arose out of a ‘real’ algebraic process... The German mathematician Gumbel led a team to decipher them [Marx's 1000 pages of mathematical manuscripts] and published a report in 1927 listing the wide range of subjects dealt with." marxist dot com

This might be relevant, since against we understand math through the world:

http://www.hawking.org.uk/godel-and-the-end-of-physics.html?fbclid=IwAR297vm3qpeViCnrXcGXBuRo-PXCEXIOcUiQxlFQWk1e20Xvu-e90P_OhrA

Russell talked about Zeno's paradoxes in lectures and in books. He said that they had extreme subtlety. Ironically, you have to have an asymmetry between the hemispheres in order to see this

The fact that points have no size (which, by the way, does NOT mean that points have zero-size), whereas line segments (whether continuous or discrete) do, does not mean that line segments are not made out of points. — Magnus Anderson

If you check the OP, I did consider the possibility that points have no size. That leads to the size of a point being UNDEFINED and all line segments having an UNDEFINED length.

You don't measure the length of a line segment by counting how many points it has, you measure it by counting how many pairs of points-at-certain-distance it has. "This line is 10cm long" means "This line is made out of 10 pairs of points-at-1cm-from-each-other". — Magnus Anderson

You establish above that your line segment is made out of 10, 1cm sub-segments. IE if we switch to discreetly/finitely sized sub-segments (/points), suddenly we can come up with a meaningful definition of length. That should tell you something - with discrete/finite sized points/sub-segments, lo and behold, the maths suddenly starts making sense. It is when we use non-sensical definitions like ‘points have no size’ that we find the maths always leads to contradictions.

But it is made out of points. It's just that the length of a line is not measured the way Devans99 thinks it is measured. You don't measure the length of a line by summing the lengths of its smallest parts (which are points.) Points have no length. They do not have such a property. Length is something that exists between two points. In order to measure the length of a line you must count the number of pairs of points-at-a-certain-distance that constitute it. — Magnus Anderson

You cannot simultaneously hold that line segments (which have length) are made out of points and points have no length - that’s a plain contradiction.

Most people define existence (not necessarily verbally but certainly intuitively) in such a way that even things that have no length (such as points, colors, sounds, etc) can be said to exist. — Magnus Anderson

Colours have a wavelength, so do sounds so they can be said to have existence. Points however, defined as having no extent, clearly cannot not exist. A line segment has extent so can be said to have existence. Maths, claiming that non-existent things can be the constituents of existing things, is tying itself in a logical knot.

Any such mathematical object always has zero Volume in higher dimensions (here I'm using generalized Volume):
A point has zero length, a line has zero area, a plane has zero volume, a unit sphere has ...
What's so odd about that? (How many of them do you want to banish anyway?)

Volume of an n-ball (Wikipedia)

Would the Archimedean properties be worth mentioning here (since they seem to be ignored all over the place)?
Infinites and infinitesimals aren't real numbers (nor rationals etc):

$\infty \notin \mathbb{R}$
$1/\infty \notin \mathbb{R}$ (archaic Wallis notation)

Don't use them as if they were (outside of convenience in specific contexts).

Continuum is a set of points where for every two points in the set there exists a point in the set that is in between the two points. — Magnus Anderson

Related to dense sets:

Dense order (Wikipedia)

If you check the OP, I did consider the possibility that points have no size. That leads to the size of a point being UNDEFINED and all line segments having an UNDEFINED length. — Devans99

But that's not what it leads to. If points have undefined length it does not follow that line segments have undefined length.

It is when we use non-sensical definitions like ‘points have no size’ that we find the maths always leads to contradictions. — Devans99

What's non-sensical about the statement that points have no size?

You cannot simultaneously hold that line segments (which have length) are made out of points and points have no length - that’s a plain contradiction. — Devans99

How's that a logical contradiction? How does "Line segments have length" contradict "Line segments are made out of points that have no defined length"?

Colours have a wavelength, so do sounds so they can be said to have existence. — Devans99

Colors do not have wavelength. Rather, it is light waves that have wavelength and light waves are not colors, they are the cause of colors.

Hegel united hericlitus with parmenides. This is related to the math because volume deals with the world. Parmenides famously said you can't think nothing and that nothing is impotent because it is.no thing. But the East came up with zero and meditation where you can think nothing and feel it's power

It is when we use non-sensical definitions like ‘points have no size’ that we find the maths always leads to contradictions. — Devans99

OK. Please describe those contradictions. I've not encountered them in my many years. :roll:

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. — Devans99

I'm focusing on your claim that mathematicians assume but have not justified the methods of analytic geometry.

No-one has yet pointed out any logic/math error in my OP. — Devans99

I'm not a mathematician, but you're using mathematical language in ways that seems to betray a misunderstanding of fundamental mathematical concepts rather than problems with mathematics. Of course, I could always be mistaken.

If I have something wrong, then someone should set me straight, rather than vague hand waving — Devans99

Would you be satisfied with a proof that one can construct a complete ordered field from the axioms of geometry such that the methods of analytic geometry work out? In other words, would you be satisfied with a construction of such a field from purely geometric premises? My hand-waving is an attempt to communicate these ideas without simultaneously writing a textbook on algebra and geometry. You, on the other hand, refuse to understand basic mathematical concepts.

At least a link to your preferred definition of the continuum would be nice — Devans99

I'm satisfied with the real numbers (up to isomorphism) for the purposes of this discussion.

The fact that points have no size (which, by the way, does NOT mean that points have zero-size), whereas line segments (whether continuous or discrete) do, does not mean that line segments are not made out of points. It merely reflects the fact that distance is something that exists between points. — Magnus Anderson

OK, let's say that points have no length whatsoever, they have no size, so it is incorrect to classify them in the category of things with lengths, such as line segments. Therefore we cannot say that they have zero length, just like you say.

You don't measure the length of a line segment by counting how many points it has, you measure it by counting how many pairs of points-at-certain-distance it has. "This line is 10cm long" means "This line is made out of 10 pairs of points-at-1cm-from-each-other". — Magnus Anderson

How can you determine where a point is on a line, such that you could use this as an instrument, a tool in the act of measurement? We've already placed the no-length point as right out of the category of things to be measured, so how can a point appear on a line to be measured? How can you find and identify any point?

But it is made out of points. — Magnus Anderson

A line can be measured. A point cannot be. There is a fundamental incommensurability between a line and a point, such that it is impossible that a line is made of points. The claim that a line is made of points is illogical and irrational.

The claim that a line is made of points is illogical and irrational. — Metaphysician Undercover

What would it be made of?

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. — Devans99

I accept that a point as defined is zero-dimensional and has neither width nor length.

Your contention is that if the above statement is taken to true then a line segment can't exist for they're a collection of points and adding nothings (points in this case) together can never yield something/line segment.

I admit that most of the information I have on the topic of lines and points can be summarized in the following claim: there are an infinite number of points on any line segment. This can be easily proven, right? Afterall if a/b and c/d are any two points on a line segment, the point (a+c)/(b+d) will lie in between a/b and c/d: a/b < (a+c)/(b+d) < c/d. That out of the way we can now focus on the original statement: there are an infinite number of points on any line segment. Now, supposes, in deference to your concerns, we change the meaning of a point from a zero-dimensional object to something of a fixed, non-zero length. What do we notice?
1. There are an infinite number of points on a line segment (proven above)
2. Giving due respect to your objection, instead of a point being zero-length we now define a point is a non-zero length
So,
3. There are now an infinite number of non-zero length points.
So
4. Any and all lines would have to be of infinite length because we proved there are is an infinity of points AND if points are a non-zero quantity then doing the math we get infinity
5. It is false that any and all lines are of infinite length e.g. a line of 6 or 8.5 or pi centimeters is a finite line.
Ergo
6. We have to reject one of our premises
7. That there are an infinite number of points on a line is true and proven so has to stay
Ergo
8. We must reject that a point has a non-zero length
So,
9. A point is zero-dimensional and has neither width nor length and there are an infinite number of them on any line.

Doesn't it make sense? A point necessarily is zero-dimensional or else we come to the preposterous conclusion that all lines are of infinite length.

Compare the above argument with yours. You've swung the other way by attempting to show a line segment is impossible since they're composed of zero-dimensional points and no matter how many nothings there are we simply can't get something.

From here let's use some prepositional logic.
S = a line segment is structurally dependent on points
Z = a point is zero-dimensional
P = it is possible for a line segment to exist. For example a 5 cm line segment
I = all line segments are infinite

My argument is as follows
1. (S & Z) > ~P...(this is your claim) premise
2. (S & ~Z) > I...(this is part of my argument above) premise
3. ~I...premise - there are finite line segments
4. P...premise
5. ~S > (Z & P)...premise
6. S...assume for reductio ad absurdum
7. ~I > ~(S & ~Z)...2 contra
8. ~(S & ~Z)...3, 7 MP
9. ~S v ~~Z...8 DeM
10. ~S v Z...9 DN
11. ~~S...6 DN
12. Z...10, 11 DS
13. ~~P > ~(S & Z)...1 contra
14. P > ~(S & Z)...13 DN
15. ~(S & Z)...4, 14 MP
16. ~S v ~Z...15 DeM
17. ~~Z...12 DN
18. ~S...16, 17 DS
19. S & ~S...6, 18 conj (CONTRADICTION)
20. ~S...6 to 19 Reductio ad absurdum
21. Z & P...5, 20 MP

QED

Basically, the assumption that points are structural components of lines is false and their dimension being zero has absolutely no relevance to the length of a line segment and so lines are possible geometric objects even if points are zero-dimensional.

1) http://philsci-archive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf

2) If points are not zero dimensional, you can make a triangle out of it and half the hypotenuse would be smaller than a point!

3) Banach-Tarski paradox assumes that every object is the same size. Does not Cantor?

Calculus tries to make "many" (understood spatially) into something else. So calculus seems to be in error. But does not my opinion on this lead to nominalism?

The debate in the late Middle Ages about how many angels can dance on the head of a pin was about this question, because angels are dimension-less. Nominalism was popular in those days too btw

What about a "point" in N-dimensional space? There is no problem if a point is in essence an N-tuple of real numbers. Of course, for those who do not believe in irrational numbers there is no peace. :sad:

"Complex analysis" (where complex means "having both real and imaginary parts") is hard to justify because we get our math from the world, where many is always many and not one. — Gregory

How about the real plane where a point is denoted by (x,y) ? Still have a problem?

Complex analysis may be hard to justify for those who know little mathematics, but ask a QM physicist about path integrals and their predictive values.

Here is a fundamental question: Is it reasonable (however defined) for philosophers who have not studied mathematics to argue basic principles of the subject? This is a far reaching question that can be applied to sciences in general. Remember, math came first, then its set theory foundations, PA and ZFC, were postulated in the 1800s and 1900s. Those mathematicians knew mathematics.

Banach-Tarski paradox assumes that every object is the same size — Gregory

The Axiom of Choice sounds so very benign (one can pick an element out of each non-empty set in a collection of sets), but look what it leads to! And you think the Axiom of Infinity is bad!!! :scream:

We've already placed the no-length point as right out of the category of things to be measured, so how can a point appear on a line to be measured? — Metaphysician Undercover

Though you cannot measure how long a point is (since it has no length, as per definition) you can identify a point. And we do so through a complex process that involves the movement of our bodies (if we're talking about identifying points in physical space, that is.)

There are many things that have no size but that nonetheless exist (and are not logically contradictory, illogical or otherwise irrational.) The word "existence" does not imply size. For example, colors and feelings exist, and yet, they have no size. A typical counter-argument is that colors are light waves and that light waves have size (their wavelength.) But light waves are not colors. Rather, light waves are things that cause colors. (This is evident in the fact that light waves can exist without conscious beings whereas colors can't.)