# 1=0.999... infinities and box of chocolates..Phliosophy of Math...

Continuity is assumed in Euclidean geometry (Tarski's axioms for Euclidean geometry explicitly cover continuity), it is proven in analytic geometry from the properties of the real numbers. In point-set topology the line is the locus of all points that share the property of co-linearity and so continuity of the line is a function of continuity of the space.

Is the continuity of space just assumed or there is also a proof for it?

The continuity of mathematical spaces like $$\mathbb{R}^3$$ are built-in to the definitions. The continuity of physical space was assumed starting with at least the ancient Greeks.

The 'assumption of ancient Greeks' is not a valid proof for 'continuity of points' in the physical space.

True, but there is no evidence of discrete space to date. Nor is there a mathematical model of discrete space that explains rotational symmetry. So physical space is mathematically modeled as a continuous space until empirical evidence is discovered to replace that hypothesis. That's how science deals with all physical theories.

When talk is made about points and especially the concept of adjacent points, I don't see the faintest connection to physicality in evidence in the discussion. So lets stick to mathematical spaces in which I'm still waiting for you guys to demonstrate the concept of adjacent points makes any sense at all in geometry. It certainly isn't part of any standard take on Euclidean geometry, analysis or point-set topology. In analog to $$\mathbb{R}^3$$ there is a cubic lattice of integers $$\mathbb{Z}^3$$ which clearly does have adjacent points at distance one, but no circles or continuous rotational symmetry.

True, but there is no evidence of discrete space to date. Nor is there a mathematical model of discrete space that explains rotational symmetry. So physical space is mathematically modeled as a continuous space until empirical evidence is discovered to replace that hypothesis. That's how science deals with all physical theories.

I am not saying space is discrete. What i am saying is that points in the space may be discrete.

I fail to see a difference between those claims. If you are saying two distinct points have no points between them then you are talking about a failure of the continuity of the space.

I fail to see a difference between those claims. If you are saying two distinct points have no points between them then you are talking about a failure of the continuity of the space.

What i am saying is that 'a point' may not be the constituent of space. Constituent of space may be something else other than a point.

Because, by definition 'a point' is dimension-less ie radius zero. Whereas space or a line has some dimension which is non-zero.

Adding infinite zero's will result in zero only. It will not produce something non-zero.

Adding infinite non-zero's will result something non-zero.

So, constituent of space or a line has to be something which has some non-zero dimension.

Hi handsa, rpenner, everyone. As already advised earlier, I don't have time to spare for prolonged discussion at this time because I am busy finalizing my stuff ready for publication. I only came in to give you a heads up regarding the respective stances in this particular side-discussion between handsa and rpenner, mainly because my upcoming publication will, among many other things, provide that which rpenner has been requesting regarding re-definitions/improvements etc; as well as explaining those subtle things which QQ and handsa have been trying to convey.
I fail to see a difference between those claims. If you are saying two distinct points have no points between them then you are talking about a failure of the continuity of the space.

What i am saying is that 'a point' may not be the constituent of space. Constituent of space may be something else other than a point.

Because, by definition 'a point' is dimension-less ie radius zero. Whereas space or a line has some dimension which is non-zero.

Be advised that I long ago identified a logical mathematical argument/example which proves handsa's take on this aspect is correct and consistent with the physical reality as well as the mathematical treatment of that reality 'space' IF the axiomatic definition of 'point' is changed to reflect that reality. I even (in my various past posts here and elsewhere) identified the actual nature and effect (in mathematical and physical terms) of what the point actually represents in both. I already changed the axioms/postulates accordingly, and even coined a phrase/term for this mathematical-physical 'entity'. This is in addition to the various changes I have made to the mathematical/physical axioms/postulates involving 'infinity' and 'infinitesimal' and 'zero' etc, and explained the consistent treatments in a contextual-reality 'space' in physics/maths systems accordingly, such that the treatments/equations no longer 'explode' into undefined/infinite/meaningless results!

Anyhow, just to let you all know that the conversation has been (thanks to you all!) interesting to me in more ways than one, being as I have been observing the discussion from a totally new contextual-reality maths-physics understanding which bridges the current maths-versus-physics views of 'spaces' and 'points' etc.

The complete and consistent maths-physics TOE will be published soon, and all (including the necessary new axioms/postulates set) will be revealed. Until then, I challenge you all to find the obvious flaw in the CURRENT maths axioms/arguments regarding 'point' and 'continuity' etc, which inevitably leads to the same maths/physics contradicting the initial axioms/assumptions/notions regarding 'point' and 'continuity' (especially contradicted by certain geometry and topology 'exercises'; see if you can spot them before I publish!).

I can't say too much more now in case I let the cats out of the bag prematurely; so I must leave you to identify these contradictory results for yourselves until I publish everything!).

In the meantime I would just observe in passing that Quantum Quack seems to be the only other person here that is anywhere near understanding the subtle realities attending 'infinitesimal' and 'infinity' and 'zero' and 'point' per se. While handsa seems to be the only other person here to see the logics/realities involved which necessarily make both mathematical and physical 'spaces' NON-continuous in effective/logical reality (ie, any arbitrary system 'number system' for defining/treating 'numbers' which leads to the INcorrect extrapolation/extension into 'physics spaces' and then also INcorrectly claim "there is always a mid-point between two distinct points" can be easily proven to be a non-sequitur in both logics and reality).

Gotta go! I look forward to some reader(s) here coming to the necessary insights before reading my own publication of same in my book. Good luck!....and enjoy your polite and interesting science/humanity discussions!

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What i am saying is that 'a point' may not be the constituent of space. Constituent of space may be something else other than a point.

Because, by definition 'a point' is dimension-less ie radius zero. Whereas space or a line has some dimension which is non-zero.

Adding infinite zero's will result in zero only. It will not produce something non-zero.

Adding infinite non-zero's will result something non-zero.

So, constituent of space or a line has to be something which has some non-zero dimension.

I think now it is mathematically proven that there can not be continuity between two points in a line. So, all the distinct points in a line or space are discrete. There is a reality for consecutive points. This distance between two consecutive points can be considered as the constituent of a line and the space. Thus a line or space is continuous but the points are discrete.

In the meantime I would just observe in passing that Quantum Quack seems to be the only other person here that is anywhere near understanding the subtle realities attending 'infinitesimal' and 'infinity' and 'zero' and 'point' per se. While handsa seems to be the only other person here to see the logics/realities involved which necessarily make both mathematical and physical 'spaces' NON-continuous in effective/logical reality (ie, any arbitrary system 'number system' for defining/treating 'numbers' which leads to the INcorrect extrapolation/extension into 'physics spaces' and then also INcorrectly claim "there is always a mid-point between two distinct points" can be easily proven to be a non-sequitur in both logics and reality).

Thanks for accepting my explanation that, points in the space are discrete.

hansda said:
I think now it is mathematically proven that there can not be continuity between two points in a line.
No, the opposite is true if the points are not the same point. Are you implying that the surface of an object is discontinous from its interior? You say the line is discontinous somewhere between two points on it? How do you go about showing this?
So, all the distinct points in a line or space are discrete.
Distinct generally means discrete. (...)
There is a reality for consecutive points.
In a space of discrete points or objects, that makes sense. How does it make sense in a continuous space?
This distance between two consecutive points can be considered as the constituent of a line and the space.
What space are you talking about?
Thus a line or space is continuous but the points are discrete.
Points cannot be other than discrete objects, adjacent points are by definition separated by a distance.
Really all you've said is that vertices in a graph are discrete, but any edges are continuous. That's a basic axiom.

No, the opposite is true if the points are not the same point.

Can you prove that, two points are continuous in a line(ie two points are located in the line touching each other.)?

Are you implying that the surface of an object is discontinous from its interior?

Points on the surface and the points in the interior are discontinuous.

You say the line is discontinous somewhere between two points on it? How do you go about showing this?

What i am saying is that, a line is continuous but the points in it are discontinuous.

Distinct generally means discrete. (...)

Two points are distinct means, these two points are at two different locations. Discrete means discontinuous.

In a space of discrete points or objects, that makes sense. How does it make sense in a continuous space?

It makes sense because points are not the constituent of space. In the space all the points are discontinuous. "Distance between two consecutive points" is the constituent of a line. "Space of the smallest cube" is the constituent of the space.

What space are you talking about?

It is the Physical Space, which is reality.

Points cannot be other than discrete objects, adjacent points are by definition separated by a distance.

Points are by definition dimension-less. That means their radius is zero. There is no continuity of points in the space(ie its non-zero distance between two distinct points).

Really all you've said is that vertices in a graph are discrete, but any edges are continuous. That's a basic axiom.

I am not talking about graphs but the real space, real line and the real points.

hansda said:
Can you prove that, two points are continuous in a line(ie two points are located in the line touching each other.)?
What does "touching each other" mean?
Points on the surface and the points in the interior are discontinuous.
On the surface and interior of "what"?
What i am saying is that, a line is continuous but the points in it are discontinuous.
Ok, but you also seem to be saying something else. For instance
There is no continuity of points in the space(ie its non-zero distance between two distinct points).
appears to contradict "a line is continuous".
I am not talking about graphs but the real space, real line and the real points.
Actually you are talking about graphs, but that's not important right now.

What does "touching each other" mean?

When two objects( 0D, 1D, 2D or 3D) have atleast one common point; it can be said that the two objects are touching each other.

On the surface and interior of "what"?

Any 3D object has outer surface and inner or interior space/volume.

Ok, but you also seem to be saying something else. For instance appears to contradict "a line is continuous".

A line is continuous. I am not denying this fact. I am only saying that the points in the line are not continuous. Because a line is not constituted by points. A line is constituted by "infinitesimal segment of line". This "infinitesimal segment of line" is consisting of two consecutive points in the line.

Actually you are talking about graphs, but that's not important right now.

Well, you can consider my 'model of space' as the "3D graph"; where all the points are discontinuous and at an "non-zero infinitesimal distance" to each other.

When two objects( 0D, 1D, 2D or 3D) have atleast one common point; it can be said that the two objects are touching each other.

A non-useful definition because you are talking about two distinct 0D objects that consists of one point apiece and by your definition can never satisfy your definition of "touching each other." Because you are lazy about formal definitions and formal logic, you don't stand much chance of figuring out what you yourself mean by touching points and thus have little hope of communicating what you think you have.

A 1-D lattice of allowable points makes sense to me, but that is identical to the model of the "integers" for which there is always a "next number" in either direction. That does not uniquely generalize to higher dimensions, but a cubic lattice works in any number of dimensions. The problem with the cubic lattice however is that the "nearest point" in some directions is grossly different from the "nearest point" in other directions and in some directions doesn't exist at all. You can see this in a 2-D square lattice, is that the line with slope $$\sqrt{2}$$ that passes through the point (0,0) does not hit any other point. This phenomena is general to 2-D and higher dimension lattices of all types. Thus, I don't think Euclidean geometry is recoverable without denseness and continuity, two properties that Tarski explicitly called out.

Anyway a correct use of both point-set and analytic geometry is seen in [post=3151091]Post #2 of the thread Pentagon-Hexagon-Decagon[/post] where the straight lines and circles are labeled a-h and are defined as all points that satisfy the relevant equation and the points are labeled A-F and are the intersection of 2 or more of the sets. Then we can easily computed the distance between all pairs of points -- the answer in every case is a real number which is where the "analysis" goes into "analytic geometry."

A non-useful definition because you are talking about two distinct 0D objects that consists of one point apiece and by your definition can never satisfy your definition of "touching each other."

That's precisely the point.

Two distinct points can not touch each other without losing their distinct identity. For example, consider A and B as two distinct points in the space. So, there will be some non-zero distance between A and B. If this distance between A and B is gradually reduced, A and B will come closer to each other. When the distance between A and B becomes zero, these two points will touch each other and become one point. Thus the two distinct points loose their distinct identity when they touch each other.

Hence all the distinct points in the space do not touch one another. So, all the distinct points are discontinuous.

Continuity and denseness are not (logically, they cannot be) properties of any finite number of points. They can be properties of a collection of points with a cardinality larger than any finite number.

It's logically impossible for any finite collection of two or more points to have the property of denseness because that property says there is at least one point between any distinct pair of points and if there are n points, then there always at least n-1 empty gaps.
It's logically impossible for any finite collection of two or more points to have the property of continuity because that property says we can always find a point at a location corresponding to any real number fraction of the distance between two points. (Or that it is impossible to name a position between two points that is not the location of a point.) Thus continuous is actually denser than mere denseness.

Continuity and denseness are not (logically, they cannot be) properties of any finite number of points. They can be properties of a collection of points with a cardinality larger than any finite number.

It's logically impossible for any finite collection of two or more points to have the property of denseness because that property says there is at least one point between any distinct pair of points and if there are n points, then there always at least n-1 empty gaps.
It's logically impossible for any finite collection of two or more points to have the property of continuity because that property says we can always find a point at a location corresponding to any real number fraction of the distance between two points. (Or that it is impossible to name a position between two points that is not the location of a point.) Thus continuous is actually denser than mere denseness.

Do you mean to say that 'finite number of points can not be continuous' whereas 'infinite number of points can be continuous'?

What do you understand by the term "continuity"( if it it is not "touching each other" of different objects)?

A continuous object is a object that you cannot slice in two without the knife hitting part of the object.

When applied to objects in point-set topology, the "parts" are points.

The rational numbers along the number line are dense, but not continuous as a knife cutting at $$\sqrt{2}$$ will not hit any of the points.
The algebraic numbers along the number line are dense, but not continuous as a knife cutting at $$\pi$$ will not hit any of the points.
The real numbers are by construction, continuous, as any knife cut does in fact define a real number. This property of continuity is why every bounded continuous function that changes sign in an interval must have a zero in that interval.

The real numbers are by construction, continuous,...

This is just an assumption. May be the ancient Greek assumption, as you mentioned earlier.

... as any knife cut does in fact define a real number.

A point on a line corresponds to the mathematical number 0(zero). Every non-zero number corresponds to a segment of line whose length is non-zero.

Adding a number is same as adding its corresponding segment of line.

A point can be considered as a segment of line, whose length is zero.

So, adding a point is same as adding the mathematical number 0(zero).

Adding infinite number of points is same as adding infinite numbers of 0's(zeroes). And the result will be a point or the number 0(zero). We will not get any non-zero finite segment of line or a non-zero number.

So, it is wrong to assume a point as the constituent of a line.

hansda said:
A point can be considered as a segment of line, whose length is zero.
So, it is wrong to assume a point as the constituent of a line.
So, a length 0 segment of a line isn't a "constituent" of that line?
Perhaps the problem is this:
A point on a line corresponds to the mathematical number 0(zero). Every non-zero number corresponds to a segment of line whose length is non-zero.
What you describe is a string of 0s with no distance defined between any of them.
This doesn't seem all that useful, somehow.

I think it's always a good idea to refer to an everyday example when you want to understand the abstract, mathematical version.

If you have a finite number of coins in your pocket, what does the 'distance' between each coin correspond to physically? This distance, in the abstract, is an arbitrary thing.
If you throw the coins on a table or some other surface, this 'operation' doesn't change the number of coins no matter how many times you repeat it. Each throw will 'assign' a distance between coins to the set of coins.

On the other hand, coins have a discrete value apart from their discreteness as (continuous) physical objects, and they aren't 0-dimensional. So the value of each discrete coin is also a kind of distance.
Anyway, since a real distance is arbitrary, there is an indefinite number of ways to assign distances between points on a real line, or to cut a piece of string into sections. Once you have a set of discrete sections with different lengths you can order them by 'size'. If you just count them all that makes them 'look like' 0-dimensional objects or vertices in a graph, but with graphs you can say a vertex has any dimension you like, it's just a representation. Hence the natural numbers are like a graph which has every vertex degree 2 except the first and last .

The real numbers are by construction, continuous, as any knife cut does in fact define a real number.
This is a fundamental fact about analysis and the real numbers as defined and used for over 100 years.

This is just an assumption.
No, it is not an assumption about physical reality, but a true statement (true by definition) for the real numbers and the Euclidean line. You can't have two lines cross in Euclidean space and fail to have them intersect in a point.

Example from analytic geometry.

$$A_1 x + B_1 y = C_1$$ is the equation of a straight line in the x-y plane. $$A_2 x + B_2 y = C_2$$ is the equation of a different straight line. If $$A_1 B_2 - A_2 B_1 \neq 0$$ then it is guaranteed that these two lines are not parallel and thus cross. Thus they intersect in exactly one point. What is that point?
$$\begin{pmatrix} x_0 \\ y_0 \end{pmatrix} = \begin{pmatrix} A_1 & B_1 \\ A_2 & B_2 \end{pmatrix} ^{-1} \begin{pmatrix} C_1 \\ C_2 \end{pmatrix} = \frac{1}{A_1 B_2 - A_2 B_1} \begin{pmatrix} B_2 & -B_1 \\ -A_2 & A_1 \end{pmatrix} \begin{pmatrix} C_1 \\ C_2 \end{pmatrix} = \begin{pmatrix} \frac{B_2 C_1 - B_1 C_2}{A_1 B_2 - A_2 B_1} \\ \frac{- A_2 C_1 + A_1 C_2}{A_1 B_2 - A_2 B_1}\end{pmatrix}$$

And so the point $$(x_0, \, y_0 )$$ is the only simultaneous solution to both equations whenever the two lines are not parallel.
$$A_1 x_0 + B_1 y_0 = A_1 \frac{B_2 C_1 - B_1 C_2}{A_1 B_2 - A_2 B_1} + B_1 \frac{- A_2 C_1 + A_1 C_2}{A_1 B_2 - A_2 B_1} = \frac{A_1 B_2 C_1 - \underline{A_1 B_1 C_2} - A_2 B_1 C_1 + \underline{A_1 B_1 C_2}}{A_1 B_2 - A_2 B_1} = \frac{ A_1 B_2 - A_2 B_1 }{A_1 B_2 - A_2 B_1} C_1 = C_1 \\ A_2 x_0 + B_2 y_0 = A_2 \frac{B_2 C_1 - B_1 C_2}{A_1 B_2 - A_2 B_1} + B_2 \frac{- A_2 C_1 + A_1 C_2}{A_1 B_2 - A_2 B_1} = \frac{\underline{A_2 B_2 C_1} - A_2 B_1 C_2 - \underline{A_2 B_2 C_1} + A_1 B_2 C_2}{A_1 B_2 - A_2 B_1} = \frac{- A_2 B_1 + A_1 B_2 } {A_1 B_2 - A_2 B_1} C_2 = C_2$$

If $$A_1, A_2, B_1, B_2, C_1, C_3$$ are all integers, it does not follow that $$x_0 \; \textrm{and} \; y_0$$ are integers. But if $$A_1, A_2, B_1, B_2, C_1, C_3$$ are all real numbers, then when the lines cross it follows that $$x_0 \; \textrm{and} \; y_0$$ are real numbers. Thus the strength of algebra of real numbers is added to the strengths of geometry without the need to ponder if the intersection point exists.

May be the ancient Greek assumption, as you mentioned earlier.
The Greeks, Romans, English, etc. all assumed the geometry of Euclidean space was the geometry of reality. This does not have to be true, and indeed, in the early twentieth century the presumption of mathematical flatness was found to be flawed. This does not provide evidence that the assumption of continuity was flawed or that the mathematical Euclidean space is not a self-consistent system.

But most important, it is impossible to say that reality is better modeled by a non-continuous mathematical model of space until you have developed a rigorous model of how such mathematics actually works and how (via the principle of correspondence) that no one has yet noticed the lack of continuity in the physical world. 1-D is simple, because the lattice spacing might be too small to notice, but higher dimensional lattices break the property of isotropy that makes circles round.

You claims about mathematics are wrong and your claims about physical reality are not argued from the evidence.

A point on a line corresponds to the mathematical number 0(zero).
You mean the distance between a point and itself is zero. Why limit it to lines?
Every non-zero number corresponds to a segment of line whose length is non-zero.
This correspondence is arbitrary. You need a segment chosen to be of one specific non-zero length before you can establish a correspondence between real numbers and lengths. (Examples, 1 inch, 1 meter, 1 Ångstrom, etc.). Since Euclidean geometry is scale-invariant this choice of unit length is completely arbitrary.
Adding a number is same as adding its corresponding segment of line.
That is the basis of much of analytic geometry when adding just finite numbers of segments together.
A point can be considered as a segment of line, whose length is zero.
That follows.
So, adding a point is same as adding the mathematical number 0(zero).
x + 0 = x.
Adding infinite number of points is same as adding infinite numbers of 0's(zeroes). And the result will be a point or the number 0(zero). We will not get any non-zero finite segment of line or a non-zero number.
There is no such operation as adding an infinite number of points. You add finite numbers of segments. The cardinality of points in a segment of non-zero length is larger than the cardinality of the set of counting numbers, so because you have two different orders of infinity at work here, your argument does not hold water. It has a big hole in it.

There is such a thing as a Cantor dust which is a infinite collection of points with just the cardinality of the counting numbers the same cardinality as the points of a line segment. It has zero measure. So this is an example of an infinite collection of points not forming a continuum. You need more than just points to make a continuum as in the classical construction of a Cantor dust one starts with a line segment and postulate away the middle thirds of of all the segments until almost all points have been removed and the structure is no longer continuous.

So, it is wrong to assume a point as the constituent of a line.
No, what is wrong is for someone uneducated in a field to make empty pronouncements about the invalidity of a field and expect people to listen to him.

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