Where's the issue? — Shamshir

:smile:

A line is NOT infinitely divisible. Numbers are.

Let's suppose it is.
And you passed this infinitely divisible line, just as you pass from one infinitely divisible digit to another.

Is there an issue?

Zeno is the issue!

A line is NOT infinitely divisible. Numbers are. — TheMadFool

At which point we should try to figure out what the ontological facts about time are supposed to have to do with the concept of numbers.

At which point we should try to figure out what the ontological facts about time are supposed to have to do with the concept of numbers. — Terrapin Station

Quantification (numbers) is the problem and also the solution.

It's the solution because once we have the numbers we can understand.

Mathematics is the language of the universe — Galileo

It's the problem because what can be done with math can't be done with reality. Zeno's paradoxes.

I think it can be best explained as:

All things in the universe are things that are mathematical

BUT

Some mathematical objects are not things in the universe

Quantification (numbers) is the problem and also the solution.

It's the solution because once we have the numbers we can understand. — TheMadFool

It's just the problem, because there's no reason to believe that time (or space for that matter) works just like our concepts re numbers.

It's just the problem, because there's no reason to believe that time (or space for that matter) works just like our concepts re numbers. — Terrapin Station

:up:

One of Zeno's most famous paradox has to do with Achilles never being able to catch a tortoise that's been given a head start in a race because of the impossibility of having to traverse an infinite number of points between the two.

I am guessing that when moving, we are not traversing infinite points because of how impossible that is. I am guessing that, just like movement that we see in a computer screen, we disappear and appear in a different point.

Because how can something be finite and infinite at the same time

It's not only the mathematics, Zeno's tortoise paradox and Arrow paradox just show how time is related to movement and change.

Assume everything, to the smallest particles would be still without anything moving anything else. The fundamental forces wouldn't effect anything. What would it matter if a microsecond or a milennium would pass by? If after two milennia things started to move again, we wouldn't notice the two milennia that past just by.

Zeno's paradox shows that two or three dimensions are illogical. That they naturally imply other dimensions to make them real. Or something spiritual, if something simple (in the Scholastic sense) could make sense of it (something being finite and infinite)

The solution to Zeno's paradox is that time is an interval, thus cannot go to zero, meaning that since velocity is distance/time the distancing will still happen.

Zeno paradox asks how a segment can be finite when it has an infinity of parts

The soul is immortal, your body is not. Ignore the Zeno crap. Focus on the eternal truths. What is love? What is justice? What is beauty etc... This is true philosophy. I am not castigating Zeno and his thought, I am simply saying that if you can get your head around the eternal truths, you realise that a whole load of Philosophy is a waste of time

I don't believe wisdom comes with age. Wisdom doesn't exist except after death. Only knowledge of facts increase, and character

The mistake in the OP, going all the way back to Zeno, is thinking that discrete dimensionless positions in space and discrete durationless instants in time are real. Instead, it is continuous motion through continuous spacetime that is real, while positions and instants are useful fictions that we create for the sake of description and measurement.

The problem is that mathematics is a way that we think about relations. The world isn't required to match that. — Terrapin Station

Indeed, mathematics is the science of drawing necessary conclusions about hypothetical states of things, which may or may not match up with any real states of things.

If we take time to be on a number line how many points of time are there between 1976 and 2019? — TheMadFool

Again, a continuous line or interval of time does not consist of discrete points or instants at all, but we can mark any multitude of points or instants along it to suit our purposes. In other words, contrary to Cantor, there is a fundamental difference between a continuum and a collection.

there are infinite numbers between 0 and 1, but it is intrinsical that there are more numbers between 0 and 2 — Filipe

What Cantor got right is that there is likewise a fundamental difference between an infinite collection and a finite collection, such that we cannot reason about them in the same way. The multitude of real numbers between any two arbitrary values is the same, because they can be put into one-to-one correspondence with each other.

You can put a one to one correspondence between any infinite set, because infinite sets have units. Likewise, unless you are speaking of process philosophy, an object must have parts. These can be divided endlessly, so it is neither discrete nor continuous. There are simply other dimensions, like a stick man on paper wondering about the 3rd dimension

You can put a one to one correspondence between any infinite set, because infinite sets have units. — Gregory

No, the collection of all combinations of the subjects of a collection--even an infinite collection--is always of greater multitude than that collection itself. The integers and the rational numbers can be put into one-to-one correspondence with each other, but not with the real numbers, because those are of the next greater multitude. There is another multitude greater than that, and another greater than that, and so on endlessly--which is why an infinite collection of any multitude can never be "large" enough to qualify as a continuum.

Likewise, unless you are speaking of process philosophy, an object must have parts. These can be divided endlessly, so it is neither discrete nor continuous. — Gregory

That depends on what you mean by "parts." The portions of a continuum are indefinite, unless and until they are deliberately marked off by limits of lower dimensionality to create actual parts. For a one-dimensional continuum like a line or time, those limits are discrete and indivisible points or instants that serve as immediate connections between portions, but the portions themselves remain continuous--which is why they can always be divided further by inserting additional limits of any multitude, or even exceeding all multitude.

The exceeding of all magnitude seems to be what describes segments and objects. The only way something finite can have infinite parts is to posit something which they are a part of (dimensions). The problem with Cantor is that there is no proof odd numbers can be put in a one to one correspondence to the rational numbers. Any attempt at lining them up applies to any infinity. Try it. Line them up and send them off to infinity. The more likely solution is that all infinities are the same

Again, all the different combinations of subjects of any collection--including any infinite collection--is of greater multitude than the collection itself; i.e., there are not "enough" subjects to be put into one-to-one correspondence with their combinations. The real numbers correspond to all the different combinations of rational numbers, so the real numbers are of greater multitude than the rational numbers; i.e., there are not "enough" rational numbers to be put into one-to-one correspondence with the real numbers. Put another way, the real numbers are a "larger" infinity than the rational numbers.

Cantor's diagonal problem is that, although he finds an infinity of real numbers not within the rational numbers, there are even numbers that are not within the odd, yet he wants to put them in correspondence. Most people don't notice this

Mathematicians are well aware of it, and it is not a problem at all. The real numbers are of greater multitude than the rational numbers, but the even and odd numbers are of the same multitude. We would never "run out" of even numbers to pair with the odd numbers in a one-to-one correspondence. We would never even "run out" of even numbers to pair with the integers, despite the fact that there are only half as many of them on any finite interval. Again, we cannot reason about an infinite collection in the same way as a finite collection, and we also cannot reason about a true continuum in the same way as an infinite collection.

I think mathematicians take Cantor as dogma without considering other problems. If I can find all the even numbers but line all the odd numbers with all the whole numbers, why can't I do this with all the real numbers? Nothing has been settled to be countable or uncountable at that point yet

If I can find all the even numbers but line all the odd numbers with all the whole numbers, why can't I do this with all the real numbers? — Gregory

Because the real numbers correspond to all the possible combinations of rational numbers, and therefore are necessarily of greater multitude than the rational numbers themselves--which are of the same multitude as the natural numbers, along with the even numbers, the odd numbers, the whole numbers, the integers, etc.

Nothing has been settled to be countable or uncountable at that point yet — Gregory

"Countable" is defined as being of the same multitude as the natural numbers, and thus applies to the rational numbers, the even numbers, the odd numbers, the whole numbers, the integers, etc. "Uncountable" is defined as being of a multitude greater than that of the natural numbers, and thus applies to the real numbers.

By the way, I am using Peirce's terminology by referring to the "multitude" of a "collection," rather than the standard terminology that refers to the "cardinality" of a "set."

A combination of other numbers is no guarantee that its a greater set since the whole numbers are more dense than the odd numbers

Density is irrelevant to multitude, and in any case the whole numbers are of the same multitude as the odd numbers.

For any collection A that has n subjects, the collection B of all the possible combinations of A's subjects has 2^n subjects, and 2^n > n for any value of n (whether finite or infinite). Therefore, B is always of greater multitude than A; it is commonly called the "power set" of A. The real numbers correspond to all the possible combinations of the rational numbers, so the collection of real numbers is of greater multitude than the collection of rational numbers.

So far l've seen is opinion about infinite sets here. Nobody has proven anything about this, most especially Cantor. He got in to many paradoxes that drove him insane. Banach-Tarski came latter. The conclusion does not follow that the real numbers outnumber the rational numbers, because I can as easily find an even number that is not on the odd numbers. The one-to-one correspondence thing is badly used by mathematicians

Suit yourself, but I will go with the mathematicians on this. Cheers.

The infinity of the continuum would suggest that all objects have the same infinity, Thus thought Cantor. But Banach and Tarski essentially pointed out that this would mean you could take a mountain out of a pea. There are so much mystery about the infinite that I don't think mathematicians really know anything about it. We just know there is more out there than we can grasp. Cheers!

The infinity of the continuum would suggest that all objects have the same infinity, Thus thought Cantor. — Gregory

Cantor wrongly thought that the real numbers constitute a continuum, but as I noted previously, they can only constitute an infinite collection--one whose multitude is greater than that of the rational numbers. His own theorem proves that there is another collection of even greater multitude, and another greater than that, and so on endlessly. Consequently, a true continuum cannot consist of discrete subjects (like numbers or points) at all.

But Banach and Tarski essentially pointed out that this would mean you could take a mountain out of a pea. — Gregory

Right, but that paradox stems from the same mistake of treating discrete points as if they were somehow continuous. It reflects a limitation of such standard models of continuity, which are adequate for most mathematical and practical purposes. Banach-Tarski does not arise in a better model of true continuity, such as synthetic differential geometry (also called smooth infinitesimal analysis).