Disproving Parmenides and Zeno: A Logical and Mathematical Analysis

[wolf921]

Hello

I don't know a whole lot about mathematic philosophy, but I am truly interested.

I am learning about Parmenides and then of course Zeno and I was asked if I could disprove Zeno logically or mathematically.
Specifical to disprove the theory of motion and change is an illusion.
And even more Specifically I want to disprove Zeno's Achilles racing the turtle paradox.
Can anyone help me out I have pondered for quite a while and can't figure out a way (do i seem dumb?).

At first I thought of a possible unit of measurement that is indivisible.
Theoretically is there no such thing?
When trying to logically disprove Parmenides no movement thing I thought about a possible mistake in his reasoning of the language of the word "nothing".

This is new to me but apparently this is ground breaking due to the fact that the logic is good but it is still wrong.

Also I have herd of del x "a name I cannot remember"'s uncertainty theory
well that just confused me a whole bunch.

So can anyone help me out. Could you say, if it is not to much trouble and if you know, both ways to disprove both Parmenides and Zeno but any help what so ever is greatly appreciated.

 

[Roberth]

Zeno's problem: That old kettle of fish

Hello

I don't know a whole lot about mathematic philosophy, but I am truly interested.

I am learning about Parmenides and then of course Zeno and I was asked if I could disprove Zeno logically or mathematically.
Specifical to disprove the theory of motion and change is an illusion.
And even more Specifically I want to disprove Zeno's Achilles racing the turtle paradox.
Can anyone help me out I have pondered for quite a while and can't figure out a way (do i seem dumb?).

At first I thought of a possible unit of measurement that is indivisible.
Theoretically is there no such thing?
When trying to logically disprove Parmenides no movement thing I thought about a possible mistake in his reasoning of the language of the word "nothing".

This is new to me but apparently this is ground breaking due to the fact that the logic is good but it is still wrong.

Also I have herd of del x "a name I cannot remember"'s uncertainty theory
well that just confused me a whole bunch.

So can anyone help me out. Could you say, if it is not to much trouble and if you know, both ways to disprove both Parmenides and Zeno but any help what so ever is greatly appreciated.

Hi there,

From a mathematical point of view the problem is related to infinitesimal calculations, ie. calculus.

The non-rigorous ("rigorous" is a somewhat pompous term used by mathematicians and physicists =me) answer is:

The [sum of the distances per infinitesimal time unit] covered by Achilles is larger than the [sum of the distances per infinitesimal time unit] covered by this super fast turtle.

∫s1 dt > ∫s2 dt … where s1 and s2 are the distances covered by Achilles and the turtle respectively.

You are certainly not dumb. The question was not satisfactorily answered until calculus was invented or discovered (whichever you look at it). Some people still find it unsatisfactory.
------------------

But there is some stuff that is a lot more exciting:

In theoretical physics there is good reason to believe that there exists a minimum length unit in the universe. It is called Planck's length and is around 1.6 x 10^-35m. That is 0.000000000000000000000000000000000016m.

Now, since Achilles is faster than the turtle, he can cover that distance in shorter time than the turtle and therefore will eventually overtake the turtle.

There is also a theoretical shortest time in the universe, which is (not surprisingly) called Planck's time. It is 5.4x10^-44 seconds.

It is the time that light, the fastest 'thing' in 4 dimensional space-time, takes to transfer the Planck's distance.

Now the argument goes like that: since Achilles is faster than the turtle, he can cover more distance per time unit. Since there is a minimum time unit that both Achilles and the turtle are bound by, Achilles will eventually reach the point of the turtle and overtake the turtle.

Exciting stuff, hey?
----------------------

Finally, there is an funny way to look at infinitesimal from a psychological point of view. It is a 'question game' that brings out human's problems in "thinking the infinite", so to speak.

Ask somebody:

Q: "Can you imagine a never-ending, flat road?
Most will answer: "No, there must be an end to it"
Q: "Can you imagine that anything is stretching forever?
Most will answer: "No, there must be an end to it"
Q: "Ok, can you imagine an absolute end, a final wall somewhere?"
Most will answer: "Hmmm, no there must be something behind the wall"

Well, if there cannot be a final wall then the road can always go on behind it.

So, one cannot imagine a final end but also not the logical opposite that whatever this non-existing final end would bound must therefore be boundless, ie. go on forever.

What does this have to do with Zeno's paradox? It is a similar cognitive imagination problem:

Q: "Can you imagine that you infinitely cut a piece into smaller and smaller pieces forever?"
Most will answer: "nonsense, there must be an end to it"

Therefore there must be a smallest piece, but why can you then not cut the smallest piece in 2?

Roberth

 

[selfAdjoint]

My own private notion about Zeno is that he took the matrix of four alterantives generated by the two dichotomies (time is continuous or discrete), (space is continuous or discrete) and devised a paradox for each cell. So the arrow considers (time continuous) (space discrete), and the stadium considers (time discrete)(space discrete).

Now on this view, Achilles and the Tortoise considers (time discrete)(space continuous). The space between the racers can be infinitely subdivided, and to cross each of an infinite sequence of subintervals costs a minimum of one unit of time. But that adds up to an infinite number of time units, or in other words an eternity.

The modern way to explain A&T is to use both space and time continuous and take limits, and then speak smugly about Zeno's ignorance of the limit concept. But in fact the limit concept was old in Greek thought, the presocratics almost certainly knew of it in the relevant case of $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = 1$$. So the modern way must be missing some essential point. Hence my theory.

 

[arildno]

This was very interesting, SelfAdjoint!.
What would then Zeno's paradox for (t-continuous,s-continuous) be?

 

[selfAdjoint]

That's the hole in my theory. There doesn't seem to be one; perhaps there was one and it was lost. As you know, the only reference to Zeno's parqadoxes we have is Aristotle's Physics, and Aristotle might not have put in a paradox that he couldn't solve! We should try to think up one of our own: suppose time and space are both infinitely divisible, what can we say about velocity that contradicts common experience?

 

[arildno]

Hmm...I guess I am too blinded by dogmas and techniques of calculus to think up such a paradox..

 

[arildno]

I would like to say that even if we aren't able to reconstruct Zeno's (cont.,cont.)-paradox, I think your theory is possibly the best effort in order to rehabilitate Zeno as an interesting thinker.
The only other good defense I've seen, postulates that the basic paradox should be seen as an effort to elucidate the difference between countable/uncountable infinities. Somehow, I don't find those arguments entirely convincing..

 

[Hurkyl]

Maybe Zeno was trying to prove (cont, cont) via contradiction?

 

[arildno]

Maybe Zeno was trying to prove (cont, cont) via contradiction?

That is, that our received view of Zeno saying that movement is illusory is simply a misconstruction of his philosophy?

 

[Hurkyl]

I think, following selfAdjoint's hypothesis, that is the natural conclusion.

 

[selfAdjoint]

That would be amazing if true! Zeno proving (cont,cont) by contradiction! Then how would that square with his support of Parmenides' "all change is illusion"? An obvious modern way is the four dimensional continuum, in which movement, for example, is represented by the worldline. That is, a dynamic change (to us) is "really" a static geometric figure. Hmmm, aw no, it couldn't be Zeno was on his way to relativity could it?

 

[wuliheron]

Aristotle argue that there were potential infinities but no actual infinities, and Zeno's were all just potential infinites. Later when calculus was invented and set theory explored, Aristotle's limitations on what was considered proper mathematics was overturned in favor of assuming there are actual infinities, despite the complete lack of evidence. Hey, it's just math after all.

Still, the irony is striking. Using the unproven concept of infinity to disprove Zeno's paradoxes is a bit like pot calling the kettle black. To make matters worse, using the Indeterminacy of Quantum Mechanics to disprove Zeno is also a bit like the pot calling the kettle black. One paradox denies the existence of another.

Quite humorous actually. :0)

 

[Hurkyl]

assuming there are

I was going to respond, but this phrase is sufficiently vague that I'm not sure just what you're asserting here...

 

[Canute]

I'm one of those who feels that the concept of infinitessimals does not solve Zeno's paradoxes. To me Zeno was constructing various reductio ad absurdum arguments against the idea that spacetime is quantised (or, if you like, of the idea that the number line is a series of points). I feel he succeeded. However I haven't explored all the objections to this view, so I'll post it here and see what happens.

Zeno's paradoxes can only be resolved if spacetime is taken to be continuous. Infinitessimals work as a solution because although notionally they are points, in fact the idea of infinitessimals is predicated on a spacetime that is continuous, for only in or on a continuum is a point infinitely divisible.

In a continuum a point is really a range, the diameter of the point. This range is of course infinitely divisible because in a continuum there is no lower limit at which one has to stop dividing it. In a continuum there are no points, only arbitrary ranges. The way the calculus rounds off the infinities that derive from trying to quantise a continuum works very well for calculating motion etc. However any attempt to 'reify' infinitessimals lands us back with Zeno's problem again. Because of this the calculus solves the practical problem of 'mathematicising' motion, but not the logical problem of explaining how motion is possible if the fabric of reality is quantised.

This is easily seen if you imagine Achilles and the Tortoise to be two fundamental particles A and B, each one fundamental quanta of length in diameter, racing each other along a line of dashes in which each dash is one fundamental quanta in length. It soon becomes apparent that the only way they can travel at different speeds is to allow time dilation or length contraction. So I'd agree with whoever said that Zeno was exploring relativity. (The slowest B can go is 1 quanta of length per instant, otherwise it is stationary, and the fastest A can go is 1 quanta of length per instant, for otherwise it is in two places at once). This shows that the notion of fundamental quanta of space and time becomes incoherent at the limit. (This has been argued very recently by at least one physicist).

I agree also with selfAdjoint that Zeno was specifically exploring the idea of 'space-discrete' and 'time-discrete'. But I don't agree with selfAdjoint's answer to the question asked by Arildno "What would then Zeno's paradox for (t-continuous,s-continuous) be?", which was "That's a hole in my theory. There doesn't seem to be one...'. I'd say that this is not a hole in your theory, it's just that if spacetime is taken to be a continuum then motion ceases to be paradoxical.

To sA's question "suppose time and space are both infinitely divisible, what can we say about velocity that contradicts common experience?" my answer would be that no paradoxes arise from this assumption. By assuming spacetime is infinitely divisible we are assuming that it is continuous. After all if spacetime is quantised then it cannot also be infinitely divisible.

To me Zeno's paradoxes show that spacetime is not quantised. It can be treated as being quantisable into infinitely divisible points for mathematical convenience, but this is only a different way of saying that is is not quantised.

Or have I missed something?

 

[Hurkyl]

You've missed that you can have a set of points that isn't discrete. A topology consists of two parts: its points, and its neighborhoods. (for the real line, the neighborhoods can be taken to be the "ranges" -- open intervals)

 

[Canute]

You've missed that you can have a set of points that isn't discrete. A topology consists of two parts: its points, and its neighborhoods. (for the real line, the neighborhoods can be taken to be the "ranges" -- open intervals)

Can you say a bit more about this? I don't see how a point can not be discrete. Surely a point has a unique set of coordinates, and likewise the points adjacent to it. Bear in mind that I'm talking about reality, not pure mathematics.

 

[Hurkyl]

You have realized it's important to speak of neighborhoods (you called them ranges), and the open sets that are made of neighborhoods.

These certainly are core concepts of topology. For example, the definition of "continuous function" can be described entirely in terms of open sets:

Def'n: a function f is continuous if and only if the inverse image of an open set is an open set. That is, if U is an open set, then f^-1(U) is an open set.

And certainly the concept of a continuum depends on this notion of neighborhoods.



But it's a mistake to conclude that points are neighborhoods.

For example, take your words: you called them ranges. Ranges of what?



Now, there are different ways to define a topology on a set of points. You could assign the real line the discrete topology, which can be defined by saying the neighborhoods are points.


Anyways, a summary: you're confusing the notion of "point" with the notion of "nearness".

 

[Canute]

You have realized it's important to speak of neighborhoods (you called them ranges),

By 'range' I meant a point defined to some finite degree of tolerance. So if, for instance, I was an engineer a 'point' would always be a (notional) point plus or minus some specified amount, the amount being determined by the degree of accuracy required for the task. Is this in line with the notion of 'neighbourhoods?

But it's a mistake to conclude that points are neighborhoods.

I understand that conceptually-speaking points are not neighbourhoods, but literally dimensionless. However when it comes to reality rather than mathematical thought experiments a point does not exist unless it has extension. If it has extension then it it not a point but a neighbourhood (what I called a range of points).

For instance, when tuning a piano A is usually tuned at 440 cps. However A=440 is an idealised point in a continuum. In fact A is tuned to 440 cps plus or minus some small (but always non-zero) amount. Thus A=440 is a range (neighbourhood?) from, say, 440- 0.01 cps to 440+0.01 cps.

I was trying, in my mathematically inept language, to say that points do not exist, only neighbourhoods exist, and that therefore the idea of infinitessimals did not help us address the paradoxicality of motion within a medium that is quantised.

Is this reasonable?

 

[matt grime]

You are contradicting yourself in your anaology by adopting two distinct meanings for the same symbol.

First you say A is tuned to 440 exactly, and then say this is impossible, thus you are contending it is both an exact number and some range of numbers too.

A is the note that an oscillation of exactly 440 cycles per second would generate. We can't do that in practice, but that doesn't make A, an idealized concept, into an indeterminate thing. It just means no practical model of A is exact.

 

[Canute]

First you say A is tuned to 440 exactly, and then say this is impossible, thus you are contending it is both an exact number and some range of numbers too.

I suppose you could read it like that. I was saying that it is often thought to be exact but never is.

A is the note that an oscillation of exactly 440 cycles per second would generate. We can't do that in practice, but that doesn't make A, an idealized concept, into an indeterminate thing. It just means no practical model of A is exact.

I don't quite see why it's not an idealised concept. Can one tune a note to 440 cps with infinite precision? Isn't A=440 a limit.

 

[matt grime]

A=440 is an equality.

 

[arivero]

Going back to Zeno, also Democritus solved a related paradox: if an slice of a cone, cut in parallel and as near as possible to the basis, is equal or different from the corresponding slice of a cylinder.

Note we are speaking geometrical figures here, no real nor imperfect constructions.

 

[Canute]

A=440 is an equality.

Well yes. I'll put it another way then. Is it not the case that A=440 is an unachievable ambition for a piano tuner? 'A' can be arbitrarily close to 440 cps, but never precisely at 440 cps. Just as a specified position on the number line can be arbitrarily close to a point but never at one. (Since A=440 cps, and the number 5.5 (say), are points, and points have no extension).

 

[arildno]

A specified position is that position.
Don't confuse maths with "real-life" applications of maths.

 

[matt grime]

Canute, you appear to have said that (real) numbers do not exist in the mathematical sense, which is obivously nonsense. We know *exactly* what sqrt(2) is - it is the (positive) number whose sqaure is 2 etc. Measuring stuff in "real life" is neither here nor there. A is the note that cycles at 440Htz (with a perfect sinusoidal wave form on the "idealized" oscilloscope), sqrt(2) is the length of the hypoteneuse of a right angle triangle of sides 1. Nothing to do with making a real life version of anyything.

 

[Canute]

You're missing my point, but never mind.

 

[Castilla]

Info: Adolf Grumbaum: Philosophical problems of space and time. A hard book with chapters about Zeno paradoxes.