General Relativity is a theory of a continuous space-time, and Quantum Field Theory os a theory of continuous fields in a continuous space-time. — tom

Loop quantum gravity is a theory of discrete space-time, and causal sets is a theory of discrete space-time.

These are viable alternatives to quantum field theory and general relativity (themselves known to be incompatible and incomplete); they're not pseudo-science.

We don't a have a theory of everything. Nothing explains everything; every theory has shortcomings.

There is precisely zero evidence for a discrete space-time. — tom

That's not sufficient to dismiss any theory that suggests that it is discrete, hence why such theories are studied. And I would argue that the logic of Zeno's paradox and supertasks would suggest that such theories must be correct.

And, despite your claim, String Theory also takes place in a continuous space-time. — tom

My mistake.

My guess...

The real numbers are represented in terms of length or distance with a number line. The number line is a model for real numbers. But the reverse is not true. Physical length/distance can't be mapped onto the real numbers.

So, Zeno's paradox arises from confusing the model (length/number line) with the actual stuff, the real numbers.

I said in that very post you qouted it converges to a finite number.

Real numbers are all whole numbers, rational numbers and irrational numbers and they are absolutely on the number line. Distance is definitely mapped with real numbers.

Numbers not on the number line are imagery numbers and infinity, as infinity is not a number. If you mean something else I would suggest you don't use the phrase real numbers.

I said in that very post you qouted it converges to a finite number. — Jeremiah

But THAT is not the paradox.

The series obviously converges, but how can it ever reach the limit, if the entity performing the infinite sum is a finite creature in finite time? That is the (apparent) paradox.

Distance is definitely mapped with real numbers. — Jeremiah

You will never encounter a distance expressed in anything more than the set of computable numbers.

That's the flawed assumption of Zeno's Paradox, that is what sequence and series resolved, a finite being can converge an infinite partial sums.

Watch, I a finite creature will converge an infinite partial sums to a finite number.

.3 +.03 + .003 + .0003 + .00003 . . .

On forever. That is an infinite series, do you recgonize it?

I could rewire it this way. .33333....

Which is 1/3 a finite sum of the infinite parts.

Watch, I a finite creature will converge an infinite partial sums to a whole.

.3 +.03 + .003 + .0003 + .00003 . . . — Jeremiah

Why stop there?

Because in math ". . ." means the pattern repeats forever and that was easier to type. You know shorthand.

That's not sufficient to dismiss any theory that suggests that it is discrete, hence why such theories are studied. And I would argue that the logic of Zeno's paradox and supertasks would suggest that such theories must be correct. — Michael

I am sorry, but this is where I lose respect for philosophy. Zeno's paradox certainly doesn't prove such theories "must be correct". That is a very bold claim off the back of a few conceptual paradoxes whose relation is a subjective classification called supertask. As far as I can tell, there is not even consensus that all supertask are impossible, or even relevant.

In mathematics Zeno's paradox does not prove partial sums, it lends to the notion yes, but those are backed by formal proofs that have been through the works. Deciding something must be correct just off the back of this paradox is sloppy and lazy. These paradoxes are guides, not proof.

I am sorry, but this is where I lose respect for philosophy. Zeno's paradox certainly don't prove such theories "must be correct". That is a very bold claim of the back of a few conceptual paradoxes whose relation is a subjective classification called supertask. — Jeremiah

Though experiments in physics are very real things.

I know they are a real thing, but then they are backed by real evidence.

Because in math ". . ." means the pattern repeats forever and that was easier to type. You know shorthand. — Jeremiah

You mean literally FOREVER? Literally infinite time?

Though experiments in physics are very real things. — Michael

Yes they are, and the whole point of Zeno's paradox is that it purports to be one, but it is not.

Partial sums only explains how theoretically in math you can have infinite points in finite space. It doesn't solve the paradox, rather it lends itself to justifying the paradox.

Yes. Unless a number is given at which to stop.

2, 4, 6, 8, ...
Implies all even numbers to infinity.

2, 4, 6, 8, ... , 200
Implies all even numbers until 200.

Yes. Unless a number is given at which to stop. — NKBJ

Sheesh, I must have missed that when I did my maths degree.

Not sure what you're getting at? Maybe your math degree needed to be supplemented by some English classes so you could learn how to express yourself clearly.

It's neat what I can do that with three little dots, right? Math is very cool.

It explains how we can have a finite net change from a to b with infinite partial sums, which is what the paradox was.

How does Achilles achieve a net change with infinte small intermediate distances to cross. Zeno's paradox only existed because no one knew how to sum infinite parts to a finite amount. Now we do, so paradox resloved.

You have math degrees, but you never saw any of this in class?

Even if the net sum is finite, if it were infinitely divisible, you'd always have one more halfway point to reach. In fact, the paradox would damn us all to complete inertia, because there's halfway points between us and the halfway points, and halfway points to those, etc.

Even if the net sum is finite, if it were infinitely divisible, you'd always have one more halfway point to reach. In fact, the paradox would damn us all to complete inertia, because there's halfway points between us and the halfway points, and halfway points to those, etc. — NKBJ

Quite. The paradox has nothing whatsoever to do with the finiteness of the sum, it has to do with a finite entity being unable to perform the infinite sum in finite time.

If my memory serves me correctly, there is a similar paradox referring to the impossibility of firing an arrow, or taking a single step.

Yes, there is.
Anyway, that's why I suggest that Planck units solve the paradox--space is not infinitely divisible.
Kant explains in the Critique of Pure Reason why it's hard for us to accept finite divisibility--it's outside of anything humans ever experience, so we can't wrap out heads around it.

You are describing infinite partial sums and if your series converges then you have net change, aka movement.

Correct me if I'm wrong, but to converge, in mathematics, means a<1. To be able to complete the movement it would have to be a=1.

A lot of you are missing the point when they start applying Zeno's paradox to real world circumstances. It was an allegory for a mathematical argument he was having with other Greek philosophers. It isn't intended as a theory of motion but as argument against the then prevailing ideas that a mathematical line is build up of points (atoms) and a finite number could not be divided infinitely (again, atoms!). It's an argument against there existing an indivisible mathematical quanta that they thought existed at the time.

As a mathematical argument it's quite good and easily imagined but the allegory is just an aid for understanding the mathematical argument not intended as to say anything sensible about the real world. So once you realise it isn't about physical reality, the paradox disappears.

Anyway, that's why I suggest that Planck units solve the paradox--space is not infinitely divisible. — NKBJ

Not a smart move. All physical theories that work require the continuum.

A series of infinite numbers can have a finite solution if the numbers are all greater than zero and less than one, therefore we are able to cross the room.

I am not sure what you are referring to , perhaps you are thinking about one of the test for convergence. I would have to review them to be sure.

I don't know how to input math notation into here so you'll have to bear with my lack of proper notation but by definition the sum of an infinite series is the limit S= lim_n→∞ S_n, if the limit exist then the infinite series converges to the sum S. If the limit does not exist the infinite series diverges.