But the rationals fail to be Cauchy-complete. For example the sequence 1, 1.4, 1.41, ... etc. that converges to sqrt(2), fails to converge in the rationals because sqrt(2) is not rational. There's a hole in the rational number line. — fishfry

That's right, the existence of irrationals really throws a wrench into the rational number line. Where do those irrationals exist in relation to that line?

That's exactly why the reals are regarded as a mathematical continuum, and the rationals aren't. — fishfry

The real numbers cannot fulfill the conditions of a proper definition of "continuity". Real numbers produce a sequence of contiguous units. Contiguity implies a boundary of separation between one and another. This boundary must produce an actual separation between one number and the next, to allow that each has a separate value. This is contrary to "continuity" which is the consistency of the same thing.

So mathematicians have created a term, "continuum", which applies to a succession of separate units, allowing that each is different, so there is something missing in between them, and that "something", which is the difference in value, is unaccounted for. Therefore "continuum" means something completely different from "continuity".

The rational numbers are an attempt to account for this "something", the difference in value, which exists between the reals. This an attempt to create a true continuity. However, the irrationals appear, and foil this attempt. So mathematics still does not have a continuity.

We are discussing a point as being dimensionless. The stake, blob, pixel, is used to give the point some visibility. The point has to be associated with a physical object in order to be useful.

We are discussing a point as being dimensionless. The stake, blob, pixel, is used to give the point some visibility. The point has to be associated with a physical object in order to be useful. — sandman

Right, but the point is only a location when there is a physical object to mark the location. The point, as dimensionless, cannot be a location because there needs to be something physical to mark a location. Therefore, what I've been explaining, is that the point cannot be both dimensionless and a location at the same time.

1) Things are composed of infinite parts

2) For rigor, we can use Cantor's arguments for taking infinities out of infinities

3) So Banach-Tarski comes right from Cantor

4) Therefore you can make a universe out of a pebble

The conclusion is that the limits of calculus go out the window!

I've merely had the courage to take what Metaphysician Undercover is saying to it's logical conclusion

Therefore you can make a universe out of a pebble — Gregory

Assuming God has laid forth unto us the Axiom of Choice.

The conclusion is that the limits of calculus go out the window!
I've merely had the courage to take what Metaphysician Undercover is saying to it's logical conclusion — Gregory

This is quite entertaining. :smile:

The point though is that #1 is wrong, things are really composed of finite parts. Things are finite. However, we can dream up an imaginary thing, a set of numbers or something like that, and stipulate that this thing is composed of infinite parts. But what good is that? It's just an imaginary thing, and the description of that thing, 'composed of infinite parts', is not consistent with any real thing which are all composed of finite parts, so it's all just a fiction.

If things are purely finite, there is a smallest unit of space. That is impossible though because you just divide it further. If you can't it's zero and has nothing to do with the object.

"The principle of the excluded middle has only a scholastic and heuristic value, so that theorems that in their proof cannot avoid the use of this principle lack of any mathematical content." Brouwer

If things are purely finite, there is a smallest unit of space. That is impossible though because you just divide it further. If you can't it's zero and has nothing to do with the object. — Gregory

I think modern science has demonstrated that there is a smallest unit of space. That's what quantum physics is all about. It's definitely not zero though, because these quanta of space contain our entire physical reality.

The division goes on forever. Physics assigns a lowest limit we can fathom. Gee are you guys computers, or just think with Hobbean calculation? Computers cannot fully deal with the finite and the infinity in relation to space. It doesn't have reason so it can't know what those things mean. They "think" along the lines of Hibert's geometry. Zeno's paradox otoh leads directly the paradox of the large and the small. How can you ever get to the small when where ever you go you simply enter more of the large?

If the Plank length has any length at all, it has a front and back. Divide further. The descent goes on forever in a finite space. Hence our contradiction.

"In reality, space is therefore amorphous, a flaccid form, without rigidity, which is adaptable to everything, it has NO properties of its own. To geometrize is to study the properties of our instruments, that is, of solid bodies." Poincare

I think modern science has demonstrated that there is a smallest unit of space. — Metaphysician Undercover

This is a common misconception. What modern science has demonstrated is that there is a smallest observable unit of space (and time), which does not entail that space (or time) is discrete in itself.

As for the thread title and OP, Cantor's analytical definition provides an adequate model for most mathematical and practical purposes. However, it is a bottom-up construction that wrongly attempts to assemble a continuum from distinct parts corresponding to the real numbers. In other words, it presupposes that any line, surface, or solid of any length, area, or volume is composed of all such parts--namely, points--which is why the Banach-Tarski theorem follows from it.

By contrast, a true continuum is a top-down conception in which the whole is ontologically prior to its parts, all of which have parts of the same kind and the same mode of immediate connection to each other. Those parts are indefinite unless and until we arbitrarily mark them off for a particular purpose, such as measurement, by inserting limits of lower dimensionality, which can hypothetically be of any multitude or even exceed all multitude. A line is composed of lines that are contiguous at points, a surface is composed of surfaces that are contiguous at lines, and a solid is composed of solids that are contiguous at surfaces.

I think there is a lot to be learned from Eric Temple Bell on this matter. He had a real change in perspective in his later years. Mathematics is like any other pattern that a biological creature recognizes in its semiosphere. To understand the nature of 'continuum' and that there is definitely continuity in all of existence, not only do I look to C .S. Peirce, but I factor in Heraclitus's view ... "what is drawn together and what is drawn asunder, the harmonious and the discordant. The one is made up of all things, and all things issue from the one." ... We need to remember this when we think we are so sure of our 'particular' points and patterns. .... I delve into this in episode 4 of my podcast, and I reference a Janus-like cosmic law when we consider music. .... Just some thoughts to share here.

Peirce's rather poetic analogy between semeiosis and music in "How to Make Our Ideas Clear" (1878) seems relevant here: "In a piece of music there are the separate notes, and there is the air ... Thought is a thread of melody running through the succession of our sensations ... [Belief] is the demi-cadence which closes a musical phrase in the symphony of our intellectual life."

Yes. What I say in my podcast about the circular character of unconscious processes and the Janus-like aspect is this.... To understand this means to understand that even in the case of a contrary, something is only recognized as a contrary in 'relation' to that which appears to be its opposite. This can be applied when considering a musical 'moment', which is typically thought of as 'standing alone' in the larger composition.This can also be applied to the importance of the silence between notes. There is no 'standing alone' or 'silence' without that which is not standing alone, or that which is actually a played note. There is always a relationship between what appears to be contraries <or distinctions>, meaning there is always 'continuity' in all things. ..... I also discuss how we only recognize 'self' in relation to that which is not self (referencing brain development). ...... Unfortunately, 600+ years of nominalism in western culture has affected our ability to understand this. Thank goodness an understanding of emergence in biology is helping us learn to recognize this. It's a shame that the materialists, dualists, and nominalists are still holding us back.

Was Spinoza holding us back? Purpose has nothing to do with science, and synthesis in natures has nothng to do with the infinite divisibilty of the finite

The spiritual is founded on the material. Even teilhard leaned in this direction

Spinoza has to be understood from his time period. People understood him as a materialist who was merely pointing out the holiness and sacrality of matter through obscure geometric arguments. Nondualists like actualized dot org don't believe in sin or true crime because they think we are the supernatural God

Spinoza is best understood in relation to C. S. Peirce.
No, Spinoza was not holding us back. Spinoza was not fully understood in the shadow of Descartes. On my YouTube channel I have a playlist that I call 'My Freedom from Nominalism Worldview'. Most people believe they have to choose an either this or that from what is conventionally taught in western culture academia and theology. There is a path to understanding that didn't make the popular cut due to the lure of materialism and the ontological individualism of 'I think therefore I am'. Do a search for this ... "Shaken by Nominalism: The Theological Origins of Modernity" to get some insight, or visit my YouTube playlist for some excellent learning videos. I recommend watching them in order to best understand them. Make sure you click on the playlist tab.

Hegel was much more explicit about the spiritual than Spinoza, but I understand Hegel as neologism, so the spiritual is nothingness, the flip side of the coin. The other side is sacred matter

Believing in natures is random. It depends on culture and psychology

I understand your perspective. I hope you will take some time to understand mine.

What modern science has demonstrated is that there is a smallest observable unit of space (and time), which does not entail that space (or time) is discrete in itself. — aletheist

Space and time are concepts derived from our observations. Therefore it makes no sense to say that either space or time extends beyond what is observable. What we call "space" and what we call "time", are abstract ideas created to account for what we observe. If some aspects of reality are beyond what is observable, then they are neither spatial nor temporal because these are empirically based concepts.

I fully agree that there are aspects of reality which are not observable, but these are non-spatial, and non-temporal aspects of reality, as "space and "time" apply to observations.. That there are such non-observable aspects of reality, is what validates the concept of the "immaterial", and "non-physical".

By contrast, a true continuum is a top-down conception in which the whole is ontologically prior to its parts, all of which have parts of the same kind and the same mode of immediate connection to each other. — aletheist

The problem here is the difference between a continuum, and a continuity, which I explained already. "Continuum" as implied by common usage means a collection of contiguous, but separate individual units. "Continuity" implies one uninterrupted whole. So if we assume "a whole", it could be composed of separate parts, like a continuum, or it could be a continuous whole, a continuity, but it cannot be both, because of contradiction. So to speak of a whole, as a continuity, is to speak of one thing, and to speak of a collection of parts, as a continuum, is to speak of a completely different thing. "Continuum" and "continuity" have very different meanings.

What we call "space" and what we call "time", are abstract ideas created to account for what we observe. — Metaphysician Undercover

This confuses an abstract idea with its object--i.e., what it represents. The fact that the concepts of space and time account for what we observe does not entail that real space and time are entirely observable in themselves.

"Continuum" as implied by common usage means a collection of contiguous, but separate individual units. — Metaphysician Undercover

That would be Cantor's analytical definition, which again is incorrect but adequate for many purposes. As George Box famously put it, "All models are wrong, but some are useful."

"Continuum" and "continuity" have very different meanings. — Metaphysician Undercover

Only in the sense that continuity is a property, while a continuum is anything possessing that property.

Matter has the property of the Continuum and Continuity. That is why there is a contradiction in matter, which we should adore ("I worship my own eyes" says the guru in Zen Pivots). This in no way leads to a spiritual not based on the material

See Ehrlich, Philip (2006), "The rise of non-Archimedean mathematics and the roots of a misconception." Calculus tried to overthrow Archimedes in at the end of the 18 hundreds and beyond.

This confuses an abstract idea with its object--i.e., what it represents. The fact that the concepts of space and time account for what we observe does not entail that real space and time are entirely observable in themselves. — aletheist

There's no such thing as real space and times. We see things, we create things, distinct objects, and we see how things move around and change, and we make these concepts of space and time to account for these perceptions. What would real space, or real time even be like? We have no such concept, of real space or time.

What the concepts of space and time represent, is our understanding of the existence and movement of objects. There are not any things represented by "space" and "time". Kant covered this thoroughly, these are fundamental "intuitions" he called them, which provide for us the possibility of understanding the physical existence of things.

That would be Cantor's analytical definition, which again is incorrect but adequate for many purposes. — aletheist

This is the commonly accepted definition of "continuum" in mathematics. You can't say that the mathematical definition is wrong, because it's a mathematical term. It's not used anywhere else, so when mathematicians use the term, this is what they are talking about. What we have to respect is that "continuum" is not the same thing as a "continuity". If you think that "continuum" ought to be used to refer to a continuity, then it's you who is wrong, because this is not how the word is used in mathematics, which is its normal place of usage.

Only in the sense that continuity is a property, while a continuum is anything possessing that property. — aletheist

This is not true, and that's the point I've been trying to make. A "continuum" consists of a series of distinct elements, whereas "continuity", or "continuous" refers to one uninterrupted whole. Therefore a "continuum" as the word is commonly used in mathematics cannot be continuous, nor can it have the property of continuity. If it had the property of continuity it would be one continuous thing, and not a series of distinct things, as "continuum" implies.

There's no such thing as real space and times. — Metaphysician Undercover

If you adamantly deny the reality of space and time, then there is nothing more for us to discuss on that front.

You can't say that the mathematical definition is wrong, because it's a mathematical term. — Metaphysician Undercover

The issue is not so much the mathematical definition itself, which I have acknowledged is adequate for most practical purposes. It is the widespread misconception that what most mathematicians call a continuum--anything isomorphic with the real numbers--is indeed continuous, and thus has the property of continuity. We seem to agree that it is not and does not.

I for one accept the reality of the world as we perceive it. It's a round triangle of a world and I think thats cool

Perhaps Peter Lynds' essay on the impossibility of "points" in time has appeared in this forum. If not, some might be interested in his discussion of time's continuity.

https://arxiv.org/ftp/physics/papers/0310/0310055.pdf

This paper inspired considerable controversy. You be the judge. :chin: