Zeno's paradox
Zeno's paradox: Anything moving from point A to pointB must first travel half of that distance. Before that it has to travel half of half of that distance and so on. It is possible to iterate this to infinity. Therefore, motion is impossible.
Yet, we can easily move from point A to B - we do it everyday.
Here math and/or logic claims motion is impossible. Reality is we can move. We have a contradiction. Therefore, either logic/math is wrong OR reality is wrong.
Possibilities:
1. Logic/math is wrong
2. Reality is wrong
Both cannot be wrong because that again leads to a contradiction.
How do we make sense of this paradox?
The key assumption is that space can be infinitely divided. Without this there is no paradox. And if space is infinitely divisible then motion would be truly impossible. However motion is possible. Therefore space is NOT infinitely divisible - the assumption that it is is false. So, logic/math is wrong - it rests on a false premise. There is no paradox; logic/math is wrong.
Paradox solved???
Yet, we can easily move from point A to B - we do it everyday.
Here math and/or logic claims motion is impossible. Reality is we can move. We have a contradiction. Therefore, either logic/math is wrong OR reality is wrong.
Possibilities:
1. Logic/math is wrong
2. Reality is wrong
Both cannot be wrong because that again leads to a contradiction.
How do we make sense of this paradox?
The key assumption is that space can be infinitely divided. Without this there is no paradox. And if space is infinitely divisible then motion would be truly impossible. However motion is possible. Therefore space is NOT infinitely divisible - the assumption that it is is false. So, logic/math is wrong - it rests on a false premise. There is no paradox; logic/math is wrong.
Paradox solved???
Comments (527)
Anything moving from point A to pointB must first travel half of that distance, which will take half the time of the whole trip. Before that it has to travel half of half of that distance, which will take a quarter of the time. Each step takes half the time of the previous step. There are an infinite number of steps, but they do not take forever.
So you left out a third possibility:
3. Zeno was wrong.
You brought time into the picture. However that is really not the issue. If it is it is secondary. The key premise is the infinite divisibility of space. If space is indeed infinitely divivisble then motion would be truly impossible. Only after this is established can time enter the picture.
Quoting Banno
I did conclude that Zeno started off with the wrong premise - that space is infinitely divisible.
½ + ¼ + ?...=1
How do you know that?
We don't know if reality is wrong, but before we can actually argue about that, we should be concerned about the logical fallacy made by Zeno.
The fact that "one must first travel half way before getting to point A", and that this applies infinitely, is true. That is, however, irrelevant to whether one can move or not. This is because the argument that one must first travel half way is simply a requirement. It has nothing to do with whether one can actually travel that distance or not.
After ¾ minute she's traveled ¾ a kilometre.
And so on.
And, although there are in infinite number of steps in the process, it only takes a minute.
No paradox.
I'm sure this argument begs the question. Zeno's tries to show that motion is impossible. You can't refute it argumentatively[sup]1[/sup] by setting up an argument in which motion being possible is a premise.
[sup]1[/sup] Although obviously we refute the conclusion experimentally every day.
How do folk get through high school without being exposed to calculus?
There's no necessary reason to think that the mathematics of limits addresses the (meta)physical problem. Plenty of philosophers think it's a mistaken solution.
For example, there's Why mathematical solutions of Zeno's paradoxes miss the point: Zeno's one and many relation and Parmentides' prohibition.
How long is the circumference of the curve? But what is its area?
SO, what is the metaphysical problem? My suspicion is that there is none, once the mathematics is understood.
If I am wrong, then set out the paradox for us, clearly.
The mathematics shows that despite there being an infinite number of steps, the result is finite. So the conclusion, that motion is impossible, is a non sequitur.
The paradox is that if distance is infinitely divisible and if instantaneous travel between one point and the next is impossible (reasonable premises, at least at the time of Zeno) then any distance travelled would take an infinite amount of time, and so motion is impossible – but motion is possible.
The mathematics only shows that one can calculate the sum of an infinite series. There's no prima facie reason that this says anything about the physics of distance and time. And the above seems to equivocate on "number of steps". The "number of steps" when it comes to motion is more akin to counting the elements of an infinite series, not summing them.
It is not a paradox, it is a question of rules, lack of adequate real-world parameters, and Infinity:
Rule: OVER-STEPPING: If you are allowed to OVER-STEP point B, then, once beyond it, it can be said that you 'reached it' at some point in time, however infinitesimally short in duration.
THE ROLE OF INFINITY: Since the 'line' between Point A and Point B is 3D (which it has to be in order to physically exist, and we ARE talking about physical reality), then you may have missed it along any dimensional axis, if your measurements get small enough. If you allow a tolerance of say 3ft, then the determination can be made.
LACK OF DEFINITIONS: First 'you' hasn't been defined, so the 'paradox' is rendered 'silly'. If the point is minuscule, say just a fraction of a millimeter, and 'you' are a circumference of two feet, then it can be easily determined if 'any of you' is 'on' the point - say if an outside observer cannot see the point through your shoes. Note that this assumes that you are allowed to 'OVER-STEP' the point (meaning part of your shoe has 'gone beyond' the point.
THE LIMITING RULE and another LACK OF DEFINITION: If, on the other hand, you are NOT allowed to over-step the point, then the statement is true (and not a paradox), given infinite regression (that you will never 'reach' Point B). If an adequate parameter is given, say a 'zone' of adequacy (for instance, if you are 'close enough' for all practical purposes), then a determination can be made. So the real problem (in the real world) is in the lack of parameters. As for the math, you are using...
THE WRONG MATHEMATICAL TOOL: It is true that by halving the distance ad infinitum you will never reach Point B (nor will any part of you ever 'cross' it), so that presents no paradox in itself.
CONCLUSION
The 'paradox' is a good example of posing a real-world question (that which is beyond pure theoretical math) with a lack of real-world parameters.
It seems to me that what you have written here is misguided - an inaccurate picture of the number line.
Since a number line is infinitely divisible, there is no "one point and the next"; between any two points there are an infinite number of points.
It follows that the notion of instantaneous travel between one point and the next is muddled.
I'm not talking about a number line. I'm talking about actual space.
If there are an infinite number of points then there are two points, so this doesn't contradict what I said.
I don't know what you mean by muddled. If you just mean impossible, I'll agree. If I set up a starting point here and an ending point 10 metres away, I can't teleport directly there. It takes a non-zero amount of time to travel the distance in between. And this is true however close the points are. That's the paradox. It takes a non-zero amount of time to travel any distance, and so given the infinite divisibility of space it should take an infinite amount of time to travel from any arbitrary point to the other.
That would be a solution, but I believe the best theories suggest that space is continuous, not discrete, and so there isn't a smallest point. Given that this seems to entail that motion is impossible, it seems to be that continuous space and motion are inconsistent, and the paradox arises because we're unwilling to reject one of them.
Although I think that if continuous space and motion really are inconsistent, and assuming that motion is more obvious than continuous space, I'd take this as proof that, contrary to our best theories, space really is discrete – even if the scale is below the measurable, and so unable to be empirically determined (unless you count the evidence of motion and the reasoning that leads to the paradox as empirical proof that space is discrete).
However the paradox is framed precisely as a purported paradox of the infinite divisibility of a line. I agree with Banno; the realization that the finite time taken to traverse the line is infinitely divisible (in principle) just as the finite line is, dissolves the supposed paradox.
A line, not a number line.
The conclusion of the paradox is that it doesn't take a finite time to traverse an infinitely divisible line. It takes an infinite amount of time, and so any such traversal is impossible.
Even if we consider the infinite series of ever-shrinking fractions that Banno provided we can see the problem. How long would it take for you to sequentially add the fractions that make up the series? An infinite amount of time, and so it can't be done. The maths that allow us to determine the sum without sequential addition is analogous to teleporting to the finishing line, which is something we claim isn't possible.
No; and that's were the limits fit into the argument. There are an infinite number of steps between 0 and 1; it takes a specific time to travel between any two points; but the sum of those specific times is not infinite. Just as the sum of ½, ¼, ?... is not infinite.
As I said before, this reasoning begs the question by assuming from the start that it takes a finite time amount of time to travel a certain distance, and then considers the infinite divisibility of that finite time. In using an infinite series of fractions you've assumed your conclusion.
The article (i'm sure you know already, having linked it, but just so you know that I know) suggested that the whole thing wasn't about denying movement in any case, but criticizing a suggested movement from the one to the many. More of a critic of metaphysical reductionism of the universe to concrete multiplicity rather than a single substance, as he believed on account of Parmenides. That aside...
This is what wikipedia says about it: "the Planck length is, in principle, within a factor of 10, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that."
And "In some forms of quantum gravity, the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it is impossible to determine the difference between two locations less than one Planck length apart." - https://en.wikipedia.org/wiki/Planck_length
So, taken together, and if right (maybe not), then there is a smallest unit, and no movement within that unit can be determined, as the whole deal counts as a single location.
That it's the smallest measurable length is not that it's the smallest length.
If you take that approach, your paradox also begs the question; it becomes "if it takes an infinite time to travel from one point to anther, then motion is impossible".
Indeed.
But measurable in principle, not just in practice, as it says that this isn't a problem of technology, or measuring ability.
That's not the argument. The argument is "if space is infinitely divisible and if no distance can be travelled instantaneously then it takes an infinite amount of time to travel from one point to another".
An infinite number of non-instantaneous movements to get from A to B makes it impossible to get from A to B. You just can't dissolve this problem by appealing to a mathematical formula that allows you to sum an infinite series of fractions. It's strange to even think that you can.
Quoting Wosret
That it's the smallest measurable length in principle is not (necessarily) that it's the smallest length.
Quoting Michael
The idea of the infinite divisibility of a line necessarily involves number; and the notion of traversing it necessarily involves time and the measurement of it; which is also number.
I do actually think that is precisely what it means. When something can't be done in principle, that means that it can't be done at all, not because of any failure to meet any conditions.
Why else would it be the smallest in principle, and why else would it be that it would be inconceivable for a better future technology to measure a smaller scale?
I won't labour it, I'm not that good at physics anyhow, but that's does seem to be the suggestion to me.
The problem here is equating continuity with infinite divisibility, as if space and time consisted of infinitely many points and instants, respectively. The reality is that there are no actual points, just continuous space; and there are no actual instants, just continuous time. An infinitesimal distance can be traveled in an infinitesimal interval of time. A finite distance can be traveled in a finite interval of time.
But this is not an argument; it is an assertion.
You will need to fill it out to turn it into an argument. It appears that the missing assumption is that the sum of an infinite series must itself be infinite; but we agree that this is not the case.
So the case rests with you: why is it impossible to get from A to B?
I put it to you that the demonstration that the sum of an infinite series can be finite kills the paradox.
Better scenery.
You are still not getting the difference. Counting each half-division of a segment takes the same time as it does to count any other. Not so with traversing them. This is because the time taken to count them is not dependent on the magnitude of the segments, whereas the time taken to traverse them is.
Who designed the garden and built the path?
Probably cobbler elves. You know... on vacation from cobbling...
Having a rest from making shoes for Planck? Must be tight work!
The posited paradox was set up using a picture of the world, and a corresponding grammar, that were incapable of complex analysis of infinity. Later, with the introduction of the maths of limits, we had to hand a grammar capable of showing us some of the detail of how infinity works. The development of that grammar leads us to reject the picture of the world that leads to the paradox.
Logic cannot be wrong; but it can be inappropriate. Given a paradox, one ought look for a better grammar, a new logic; rejecting logic as "wrong" shows a profound misunderstanding. Logic is mere grammatical structure.
For the same reason that it's impossible to sequentially count the rational numbers from 1 to 2. Does your demonstration that the sum of an infinite series can be finite "kill" the claim that this would take an infinite amount of time? Of course not. It's a non sequitur.
Quoting John
This is akin to saying that counting each half division of a segment (of an infinite series) takes half the time as counting the whole segment, and that the sum of those times adds up, not to an infinite magnitude, but to the total time it takes to count the whole series.
It just doesn't work that way. You can never count to the end of an infinite series of numbers. And so, by the same logic, you can never travel to the end of an infinite series of spatial divisions. As I said to Banno, this use of the sum of a geometric series is a non sequitur.
>:O
Is this an argument by analogy? Then it doesn't get you where you want to go.
If the length of time it took to count a number reduced as the size of the number, we would be able to count the rational numbers in a finite time.
But you can sum them. Which is what is asked in the supposed paradox.
No it isn't. The paradox is about moving from one point to another, which is analogous to counting from one number to another. It's got nothing to do with summing, which is why this use of limits is a non sequitur.
So what would be the first number we count after 1?
Nuh. The argument, on your own account, is that there are an infinite number of steps, each of finite length, and that therefore the total time taken must be infinite.
Quoting Michael
Summation is there at the start.
It's not whatever I want. The task is to count all the rational numbers between 1 and 2. I can't skip any.
I can count from 1 to 2 in a finite time (see, I just did it); there is no need to count every rational number in between. Likewise, I can move from point A to point B in a finite time; there is no need to "touch" every point in between.
But the task is to count every rational number between 1 and 2.
Quoting aletheist
There is if motion is continuous, which is a premise of the argument that gives rise to the paradox. If motion is discrete then the paradox wouldn't arise.
Or more relevantly, if I'm to pass through every rational-numbered coordinate between the start point and the end point, which coordinate do I pass through first?
You have it exactly backwards - the paradox only arises by insisting that space is made up of infinitely many points, and time is made up of infinitely many instants. When we recognize that both space and time are continuous, the paradox dissolves - there are no intermediate points that I have to "touch" while moving from defined point A to defined point B in a finite interval of time, just like I do not need to count any intermediate numbers in order to get from 1 to 2.
I think you are onto a good idea and your argument makes sense; the paradox arises because Zeno slips in between reality and the abstract language we need to use to discuss certain aspects of reality, mathematics.
If space is continuous then we can plot infinitely many points in it, so I don't understand your objection. And I don't understand your distintion between infinite divisibility and continuity. As explained here, "While it is the fundamental nature of a continuum to be undivided, it is nevertheless generally (although not invariably) held that any continuum admits of repeated or successive division without limit. This means that the process of dividing it into ever smaller parts will never terminate in an indivisible or an atom—that is, a part which, lacking proper parts itself, cannot be further divided. In a word, continua are divisible without limit or infinitely divisible."
We can plot infinitely many points, but we do not have to plot any points between the two of interest. In other words, there are infinitely many potential points between any two actual points, but the only other actual points are the ones that we arbitrarily define. We can count from 1 to 2 in one step, in two steps, or in any other discrete number of steps; it is entirely up to us, and there is certainly no requirement to count all of the rational numbers in between.
Quoting Michael
A true continuum is infinitely divisible into smaller continua; it is not infinitely divisible into discrete individuals. For example, a line is infinitely divisible into smaller lines; it is not infinitely divisible into points. There is also a distinction between being infinitely divisible (potentially) and infinitely divided (actually). We are talking about the former, not the latter.
I can't choose any I want. I have to count every rational number (in sequential order) between 1 and 2. I'm not allowed to just skip ahead to some arbitrary point.
But http://www.math-only-math.com/to-find-rational-numbers.html
The task is to count every rational number between 1 and 2.
And so by the same token, assume that I overlay a region of space with a coordinate system that contains every rational number between 0 and 1. Given an object that starts at 0 and is supposed to move towards 1, what rational-numbered coordinate does it pass through first?
You are already being arbitrary by only counting all of the rational numbers between 1 and 2. What is your excuse for not counting all of the real numbers between 1 and 2 - i.e., also including irrational numbers?
I'm not claiming anything. I'm asking you for the first rational number I would count if my task was to sequentially count every rational number between 1 and 2.
My coordinate system only uses the rational numbers. So I ask again; what coordinate does it pass through first?
No, the task is to move from point A to point B. You are mathematically modeling it as counting every rational number between 1 and 2. I am challenging the fundamental assumption of your model. I can count from 1 to 2 without counting any other numbers in between. Likewise, I can move from point A to point B without touching any points in between, because there are no (actual) points in between.
Quoting Michael
My coordinate system only uses the integers. After all, units of physical measurement are completely arbitrary.
No, when I say that the task is to count the rational numbers between 1 and 2 I'm not talking about the task to move from point A to point B; I'm talking about the task to count the rational numbers between 1 and 2.
Quoting aletheist
OK, but I'm asking about my coordinate system. What rationally-numbered coordinate does the object pass through first?
There is no first rational between 1 and 2. But there are exactly a denumerably infinite number of rationals between one and two.
To count the rationals in sequential order I first have to count the first rational.
If there's no first rational between 1 and 2 then how do I sequentially count the rationals between 1 and 2? Are you admitting then that such a task is impossible, and so that your claim that "if the length of time it took to count a number reduced as the size of the number, we would be able to count the rational numbers in a finite time" is premised on a flawed understanding of what it means to count the rational numbers?
The only reason you brought this into the conversation was as a (mistaken) model of moving from point A to point B, which is the subject of the thread. I frankly have no interest in counting the rational numbers between 1 and 2. It reflects the misconception that a true continuum is made up of infinitely many individuals, which is not the case; even the real numbers do not exhaust it.
You dropped the italicised bit in.
One can count the rational numbers without putting them in sequence.
Just list the fractions between one and two; 3/2, 4/3, 5/3, 5/4, 7/4...
The whole purpose of any discrete coordinate system is to facilitate measurement. The smallest rational number that is greater than 1 cannot be identified unless you specify a finite tolerance, so a viable coordinate system using (all of) the rational numbers is impossible. Yet we can and do routinely create coordinate systems using integers, fractions, and decimals down to whatever small (but still finite) increment suits the data. Again, the only actual points on a line are the ones that we define.
How do you get =1 here Banno? It appears to me, like no matter how far you go you'll always be a fraction short of 1. Have you got a cheat?
The story in a nutshell. Points are a fiction here. The reality being modelled is the usual irreducibly complex thing of a vector - a composite of the ideas of a location and a motion...
Quoting Michael
....and the corollary is that what is being counted is not points but (Dedekind) cuts. The numbers count the infinite possibility for creating localised and non-moving discontinua.
Quoting Michael
The cut bounds the continua in question. So the continua has already been "traversed" in the fact there is this first cut. You are then asking how near the other end of the cut continua can be brought in the direction of the first cut in question. The answer is that it can be brought arbitrarily close. Infinitesimally near.
So you are creating difficulties by demanding that continua be constructed by sticking together a sequence of points. However there is no reason the whole story can't be flipped so that we are talking about relative states of constraint on a continuity - or indeed, an uncertainty - when it comes to the possibility of some motion, action, or degree of freedom.
Convert 1/3 to a decimal, then multiply it by 3. Is the result 1, or an infinitesimal fraction short of 1?
The only reason we cannot do it is because we could never count infinitely fast which is what we would need to be able to do to complete the series. When it comes to traversing though, luckily we don't have to take account of each segment as we traverse it. You are confusing analysis with actuality; there is no actual infinite number of segments that we need to traverse when we move from one place to another.
Can't be done. What does that have to do with my question?
Of course it can, students have to do it in math class all the time. You can also do it on a calculator.
Quoting Metaphysician Undercover
@Banno's example was an infinite series, so you have to keep adding smaller and smaller fractions. When you have done so infinitely many times, you get the result of 1. Likewise, if you carry the outcome of my example out to infinitely many decimal places, you get the result of 1. If his sum is always a fraction short of 1, then my product is always a fraction short of 1. Yet everyone agrees that 1/3 x 3 = 1 (exactly).
https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_?
Not at all. We can reason about infinity without actually doing anything an infinite number of times. If someone (God, perhaps) were to add up @Banno's infinitely many fractions or carry out my multiplication to infinitely many decimal places, then the result would be 1 in either case.
Quoting aletheist
The problem is, that this feat of adding infinitely many fractions would never be finished, so 1 would never be reached. Therefore it is false to say that it equals one. It does not. You can claim Banno's cheat all you want, nevertheless you're still claiming that a falsity is true.
Mathematics is entirely a matter of necessary reasoning about hypothetical states of affairs. There is no falsity whatsoever in saying that if someone were to add infinitely many fractions in a particular series, then the result would be 1. The fact that no one can actually add infinitely many fractions is completely irrelevant.
No, the fact that it is impossible to add infinitely many fractions is relevant. Because your premise is "if someone were to add infinitely many fractions...". But this is an impossibility. Therefore your premise is false. It is stating "if someone were to do something impossible..." Your premise is false therefore your conclusion is equally false.
It is really no different from philosophy in this regard; it all boils down to one's assumptions. To get us back on topic, Zeno's alleged paradox exploits this by smuggling in the idea that any finite interval of space consists of infinitely many individual points, such that one must somehow pass through them all in order to get from one place to another. Once we dispense with that misconception and recognize that space is continuous, and the only actual points are the ones that we arbitrarily define, the paradox dissolves.
You are confusing actual possibility with logical possibility. Mathematics deals with the latter, not the former. It is indeed actually impossible to add infinitely many fractions, but it is not logically impossible.
Especially if every such recitation takes the same amount of time (or longer than some specific non-zero amount of time).
Of course, if every such recitation took a duration proportional to the corresponding distance, then the moving and the reciting would be more alike.
Anyway, either both distance and duration are discrete, or both are not.
Modeling both with the continuum works, and has the bonus of numbers like ? and e. (Y)
Even the real numbers do not constitute a true continuum, because they still amount to an aggregate of discrete individuals. However, I agree that this psuedo-continuum is an adequate mathematical model of continuous phenomena (like space and time) for most practical purposes.
(Y) Though the proof is a bit short. :D
I don't think there is any issue with points in this paradox. I believe the problem is quite similar to how TheMadFool states it. The issue is the assumption that space is infinitely divisible. Zeno assumes that in theory, space is infinitely divisible. However, in practise space is not infinitely divisible. The conclusion makes a statement about what can and can't be done in practise. The theory is wrong, and therefore cannot be successfully applied in practise.
Quoting aletheist
Sure, mathematics is logic, but valid logic does not necessitate a true conclusion. The premises must be judged. So I am judging the premise with respect to what is actually possible. Your premise states "if someone does a thing which is actually impossible to do...". So I judge it as an impossibility and therefore a falsity. And so I judge your conclusion as a falsity as well.
What Meta does is simply refuse to accept the grammatical structure that allows the dissolution of the problem.
Good for him. But then the problem becomes his, not ours.
Flies and fly bottles and such. Or leaving the troll to look elsewhere for sustenance. Some conversations can only serve as examples of how not to have a conversation.
You keep confusing potentiality with actuality. Either space is infinitely divisible or it is not. Whether anyone can actually divide space into infinitely many parts is completely irrelevant - only whether it could potentially be divided into infinitely many parts. Zeno's clever strategy was to exploit this confusion. Although space is indeed infinitely divisible, it is not infinitely divided. Although there are infinitely many potential locations between two defined locations A and B, there is no need to account for all of those intermediate points in actually moving from point A to point B, because they are not actual points.
Quoting Metaphysician Undercover
You are misstating my premiss. It is not, "if someone does a thing which is actually impossible to do ..." It is, rather, "if someone were to do a thing which is actually (but not logically) impossible to do ..." The conditional nature of the whole proposition is key here.
The half (1/2) is an arbitrary choice. I don't know if Zeno actually said ''half'' or not. Anyway the sum of the sequence 1/2, 1/4, 1/8,... IS 1.
However let us choose another fraction instead of half. Say one-third. That is to say before we reach from point A to pointB we have to reach one-third(1/3) of the way and before that 1/3 of 1/3 and so one.
The sum of 1/3, 1/9, 1/27, 1/81...sequence is NOT 1. It is half(1/2).
You can try that with other fractions too. The sum doesn't equal 1.
Therefore the paradox remains unresolved as far as math is concerned.
8-)
But we can move. We do it all the time.
But I'm summing the entire distance from A to B. The only difference is I'm moving a third of the distance now.
How do you explain that walking in thirds takes me only to the half-way point?
It was explicitly mentioned several times, and implied any time it wasn't, that the counting is sequential, given that it's an analogy to the movement between two points, which would involve an object passing sequentially through each rationally-numbered coordinate between them.
And so by the same logic, if we can't even start sequentially counting every rational number, because there isn't a first number, an object can't even start sequentially moving through each rationally-numbered coordinate, because there isn't a first coordinate.
And so what is the first potential rationally-numbered coordinate that an object must pass through in its movement from A to B?
Your talk about how actually plotting the coordinates is an arbitrary decision which will use some decided-upon minimum unit completely misses the point of the paradox. Although we might actually only plot every 1/18th of a unit in our coordinate system (and so count 1/18 as the first rational number), the object must still pass through what would have been the 1/36th unit had we plotted that, and so on.
This is why continuous movement in continuous space should be impossible. An object would have to sequentially pass through every potential rationally- (even real-) numbered coordinate to get from A to B, just as continuous counting in a continuous number line is impossible, as we'd have to sequentially count every potential rationally- (even real-) numbered coordinate between A and B.
If the counting is to be possible it must be that the number line (or just the counting) is discrete; we have some minimum fraction to work with (say, 1/18). And so if movement is possible it must be that space (or just the movement) is discrete; the object has some minimum fraction to work with. But it's because we want to maintain both continuous space and continuous movement that the paradox arises.
Then leaving aside the fundamental logical problem as explained above, this would entail that the only way an object can sequentially move through every rationally-numbered coordinate between two points is if it could move infinitely fast. But, of course, objects don't move infinitely fast.
The article suggested that it was a matter of metaphysics, and identity. One is one, two is two, just tautologically. They're crisp and well defined. One cannot derive two from one, and if you could it would always lead to an indeterminate, like zero or infinity.
This view seems very strange if taken at face value - after all, we can obviously divide space and time into measurable quantities. The only way I have been able to make sense of it is by viewing it as an imprecise conception of the Primordial Existential Question, or why anything exists at all. For Parmenides, the appearance of the world is an illusion hiding the underlying singularity of all things. So viewing it in this way, Zeno's paradox can be reformulated as something like: "How can we proceed from timelessness to time, or from dimensionlessness to dimension, if we view reality as truly timeless and dimensionless?" The point is not to demand an explanation of how one moment becomes another or how space can be divided, but rather to fundamentally question time and space. In other words, we cannot use a yardstick when the concept of space is not yet defined.
An analogy can only be useful for illustrating an argument, and you have yet to offer an argument. You assert that moving from place to place is possible if and only if one can utter all rational numbers between 1 and 2 in sequence and in finite time, but you haven't offered an argument for this assertion.
An object cannot move sequentially through every rationally-numbered coordinate between two points for the same reason that we cannot count sequentially every rationally-numbered coordinate between two points. There's no first coordinate and so no first thing to count/move through.
I will quote several posts here but I will start with the only thing where I think Michael is wrong. Every other posts he writes is spot on.
The Planck length actually is the smallest possible length/size of anything in this universe. It is so much the smallest possible length, that the very moment universe came into existence, its size was exactly equal to Planck length.
I think that pretty much everyone with 5+ posts in this thread is correct about few things and incorrect about others. The main problem in my opinion is that everyone is talking about a different thing.
Quoting Metaphysician Undercover
It actually does mean that, yes. It's exactly one. But this is surprisingly irrelevant in this case, because the problem doesn't seem to be mathematical at all.
The problem of the "paradox" is the way it's constructed.
Quoting aletheist
You are of course correct that in real life we could just run and beat the turtle, but this trivial solution is not what drove all the philosophers/logicians to it for such a long time. Instead the most common interpretation of the paradox is that the runner MUST touch every point, as you worded it. So that's the version of the paradox that people try to discuss, including majority of posters in this thread. It is assumed that that's what Zeno was thinking.
Here is the problem in different words. Suppose Achilles is twice as fast as the turtle and turtle starts with 50m advantage. Achilles needs to "touch" the 50m point. At the moment of him touching the point, the turtle will be at 75m point. So the next point he needs to touch is the 75m. He always needs to touch the point where turtle is at the moment of him reaching the previous point. Since mathematically it's obvious that at 100m point Achilles will catch the turtle, we can just say "Screw the turtle, let's ask ourselves can Achilles actually reach 100m mark?" We can simplify the problem like this, because if he does reach 100m mark, then he had caught the turtle and if he doesn't, then turtle will always stay ahead of him. So that's where we get 1/2+1/4+...
Again, of course it's silly, but if we don't construct the problem this way, it becomes trivial and not worth a single keyboard press, let alone hundreds of books and articles written by some very smart people.
Quoting TheMadFool
You misunderstand the problem and the reason why it's worded the way it is. Length of 1 is defined here at whatever point Achilles would mathematically catch the turtle. You want his first point to be at 1/3? No problem. So let's see... Turtle has 50m advantage and you want that to be 1/3 of the total length. Fine, so that means that turtle's speed is exactly 2/3 of Achille's speed. By the time Achilles reaches the 1/3 point, the turtle will be at 1/3 + 2/3*1/3 = 5/9, covering a distance of 2/9. Let's see the series:
1/3 + 2/9 + 4/27 + 8/81 + ... + 2^n / 3^(n+1) + ... = 1
There you go. Still 1. Let's move on.
Unfortunately if we want to look at the problem the way it was constructed such "mathematical" proof will not work. The rules made for Achilles are not fair. He can never catch the turtle, he will never reach the point 1. But since the construction itself assumes that he will never reach it, we don't have a paradox. It's all as expected.
Achilles is moving in steps. But he is moving in very special steps. If step n has length L, then step n+1 has length L/2. Again, not because of his choice, but because the rules are unfair. It's a fixed sport event. If I try to use a bit more mathematical language...
Edit: Initially made a terminology error, I am correcting this part.
The sequence 1/2, 1/4, 1/8, ... is a sequence with infinite number of terms. Each term corresponds to one step length. Let's make another sequence S, a sequence of partial sums:
S1 = 1/2
S2 = 3/4
S3 = 7/8
...
The reason why Achilles will never reach point 1 is because 1 is not a term of sequence S. 1 is the limit of the sequence, yes, but in order for Achilles to reach the 1, point 1 would actually have to one of the terms of the sequence.
Aside from that, the idea that space is anything like an independent substance that is infinitely divisible is wrong.
Ok, I actually forgot to reply to OP. Let's see... So it's a different Zeno's paradox, but logically it's almost identical to the turtle vs Achilles paradox. I would actually say that this one is even easier. Why? Well, in the Achilles-turtle problem, every step has a clear length, both in time and distance travelled from the last point. In the OP version, however, the first step is infinitesimally long and can be traversed in an infinitesimally short time interval. So the motion occurs. No paradox.
Another way to tackle it is also very simple and is pretty much what aletheist has been repeating in this thread over and over again. In fact maybe it is that this what caused all this misunderstanding - aletheist discussing the problem from original post and the rest of us the Achilles vs turtle version. If so, I was of course wrong to say that aletheist was not following the normal interpretation of the turtle-Achilles paradox. He was correct all the way.
Infinitesimal concept is something that Ancient Greeks had a lot of problems with and something that was discussed seemingly in half of their philosophical works. :) Well, to be honest, even in more "modern" times the infinitesimal concept was causing trouble and was even banned by Papal state in 17th century. :-O Dangerous stuff, be careful!
That makes no sense.
What I'm saying is that continuous motion between one place and another is possible if and only if it is possible to sequentially pass through each coordinate between them (and for the number of coordinates to be infinite, of course). It seems to be that this is what it means for motion to be continuous (rather than discrete). But sequentially passing through each coordinate doesn't make sense if there's not even a first coordinate to move to (or a second coordinate after that, or a third after that).
It's the exact same reason why we can't sequentially count each coordinate. There's no first coordinate for us to start our count. Where do we go from the starting point of 0? Not 0.1, not 0.01, not 0.001, ad infinitum.
I did clarify that I was talking about the Achilles racing turtle paradox, which is not the one from the OP. Are you still claiming that it makes no sense?
Edit: Oops, even when talking about the race paradox it really does not make sense. Lapsus, I see now. I will correct, thank you. :-*
The superfluous assumption here is sequentially. It would be reasonable to say that for motion to be continuous the position of the body must pass every rational (or real for that matter) coordinate in order. But you demand something on top of that: that all of these coordinates form an ordered sequence. That demand is not motivated by any reasoning (indeed, you will necessarily run into a contradiction if you try).
Yes. That "1 must be part of the sequence" came out of nowhere.
Of course it has to pass through them in order. It doesn't pass the half way point then the quarter way point then the three-quarter way point and then reach the end, making discrete jumps back and forth.
Not really. I'm using the term "sequentially" in the sense that before it can reach the half way point in must reach the quarter point, and so on. And when it comes to counting, we can't do as Banno suggested and start our counting as "3/2, 4/3, 5/3, 5/4, 7/4..."
This is a nonsensical question. The only discrete coordinates that an object must actually pass through are those that we arbitrarily establish. Spatial coordinates do not exist apart from our construction of them for specific purposes. What you have identified is the reason why no one ever uses the rational numbers as a spatial coordinate system.
Quoting Michael
Agreed, counting is discrete. That is precisely why it is a false analogy to motion, which is continuous.
Quoting Michael
Again, this is backwards; movement is only possible because space and time are continuous. If they were discrete, then it would be impossible to traverse the finite distance between adjacent locations. Where would the object be during the finite interval of time between the instant when it left one point and the instant when it arrived at the other?
Even if we only define three coordinates between A and B it must still pass through the space between those coordinates. There are an infinite number of potential coordinates than an object must pass through, and this is true even if we don't actually make use of such coordinates. Your criticism makes no sense.
Continuous motion is impossible for the same reason that continuous counting is impossible. The reason counting is possible is because it is discrete. And so the reason motion is possible is because it is discrete.
What's the difference between moving from one coordinate to the next and counting from one coordinate to the next?
Saying that passing all rational coordinates in order is not a problem is akin to saying that counting all rational coordinates in order is not a problem. It is a problem. Given that there's no first coordinate to count to from a starting point, counting cannot even start. Given that there's no first coordinate to move to from a starting point, movement cannot even start.
Consider a machine that counts each coordinate as it passes through it. If it can pass through all rational coordinates then it can count all rational coordinates. It can't count all rational coordinates, therefore it can't pass through all rational coordinates.
Even if we define as many coordinates between A and B as there are rational numbers between 1 and 2, the object must still pass through the space between those coordinates. After all, there are infinitely many irrational numbers between any two rational numbers. My whole point throughout this thread is that there is always space between any two coordinates that you define. That is precisely what it means for space to be continuous; it does not consist of discrete locations. No coordinate system, no matter how finely grained, can capture every potential location.
Quoting Michael
Then please answer my question that you conveniently ignored. Where would the object be during the finite interval of time between the instant when it left one point and the instant when it arrived at the other?
This doesn't contradict my claim; it confirms it. I don't need to capture every potential location. I only need for there to be an infinite number of potential locations (e.g. the rationally-numbered coordinates).
Presumably there is no "in between". First it's at this discrete location and then it's at that discrete location. Maybe this involves discrete units of time as well. But this doesn't really matter. The logic still shows that continuous motion is impossible.
Perhaps I'm missing something, but why must there be a finite distance between adjacent locations (assuming that you mean a non-zero finite distance; if there's zero distance, then your objection is moot)?
Simple - counting is discrete by definition, because it requires explicitly recognizing every intermediate step, but motion is not. You keep insisting that motion has to be discrete like counting, but have made no argument for this assertion.
Motion cannot be continuous for the same reason that counting cannot be continuous. There cannot be a first coordinate to count to from a starting point and so there cannot be a first coordinate to move to from a starting point. Your position just seems to turn a blind eye to this.
You are begging the question. You are essentially saying that motion is just like this impossible thing, therefore motion is impossible. You must show the necessary connection between motion and counting all rational numbers in an interval in order.
Quoting Michael
When you are saying "the next" you are already implying a sequence.
Quoting Michael
Nope. Order is not the same as sequence. Ordering is not the same as counting. Until you understand this you will keep running in circles.
I'm afraid this is the end of the road, Michael. It does matter, you will have to be more precise here and make a solid counter argument. How would you define what distance is? Specifically, how would you define what distance is in a discrete topology and how would you define it in "normal" space of real numbers?
If I wanted to show that discrete motion is the case, sure. But I don't need to do this to show that continuous motion is impossible. And the issue here is whether or not Zeno's paradox shows continuous motion to be impossible, which I'm trying to show it does.
We could just go for the Planck length as you suggested earlier. This is the smallest possible unit of space. There is no half-a-Plank length of space. So space wouldn't be continuous but composed of discrete Plank-length "tiles". And the same too with time. At 1 Planck time the object is at the 1 Planck coordinate and at 2 Planck time the object is at the 2 Plank coordinate, and it doesn't make sense to talk about the half way point (whether in space or in time).
But, again, this isn't really relevant.
No, if space is discrete, then you need to capture every actual location; i.e., you need there to be an infinite number of actual locations (e.g. the rationally-numbered coordinates).
Quoting Michael
So the movement from one to the other is somehow instantaneous?
Quoting Michael
It should be no surprise to anyone that assuming motion to be discrete (like counting) renders continuous motion impossible.
Quoting Michael
No, there is absolutely no need for there to be a first coordinate in order to move from a starting point. There is only a need for there to be a first coordinate in order to measure movement; and the distance to that first coordinate is completely arbitrary, so we can use any finite interval that we choose.
Counting does not equal movement. Counting can only equal movement if you have a discrete topology (for example when you are explicitely ordered that you have to step from one point to another and nothing else exists in between). If the space is dense, however, you can still count it, but you can't order it. Rational numbers are dense (there's infinite number of them between any two rational numbers), they are countable, but they can't be ordered by size
If the distance between adjacent locations is zero, then by definition they are the same location, not adjacent locations at all. If the distance between adjacent locations is infinitesimal, then by definition space is continuous, as I have been arguing all along.
I have, with my example of a machine that counts each coordinate as it passes through them in order.
The sequence is the rational coordinates between two points.
I still think we are missing the point though, because the discussion is mostly about a space of numbers, there is nothing quantum or physical in Q or R. Planck constant is purely physical stuff, but the logical discussion (the way I see it) here is about movement in abstract space.
Also, if I am really nitpicky here, existence of Planck length doesn't necessarily make space discrete. It's just a ... line between "normal" and quantum. Experiments so far have always confirmed continuous space-time, although who knows...
I'm addressing your claim of continuous space. You seem to think that in saying that there are spaces between my rationally-numbered coordinates you somehow disprove my argument. You don't. You admit that it makes sense to consider an object passing through every rationally-numbered coordinate between two points. You're just adding to this that an even larger infinite series can be considered. But the point still stands that given the infinite number of coordinates, an object must have passed through an infinite number of prior coordinates to get to any arbitrary point, which is like saying that a person must have counted an infinite number of prior numbers to get to an arbitrary point. It doesn't make sense. You're saying that an infinite series of events has been completed. But an infinite series of events cannot be completed, by definition. And this, incidentally, is reasoning against an infinite past.
This is why I said earlier "that it's the smallest measurable length is not that it's the smallest length". If space is continuous then there's a length of space that's smaller than the Planck length. It's just that it's impossible in principle to measure (and, in the same vein, it might be that space is discrete but at a length smaller than the Planck length).
This still doesn't seem right. That there is zero distance between adjacent locations only seems to entail that there is no boundary of any breadth between them. I don't see how it follows that they would be the same location.
You're right. Conside two 1cm lengths with (hypothetically) 0 space in between. Is there 1cm length or 2cm length? 2cm. No space in between does not entail that there's just one location.
I think I need clarification, because I don't know anymore. Suppose that you are right and that motion can only be discrete (this IS what you are arguing, correct?). So assume that there was a motion achieved, from point 2 to point 3. Since motion was discrete, there should be no problem making the list of all rational numbers you passed, in order you passed them. What was the first step you made? If you are not able to make that list, what exactly is your explanation that you are not able to make the list and how does that argument prove your point?
This is simply false. In order to build a machine that counts coordinates, you have to set it up using a particular (arbitrary) coordinate system, and that coordinate system will necessarily have finite intervals between coordinates. The fact that the machine cannot count all rational coordinates has no bearing whatsoever on whether it can pass through all rational coordinates; it merely reflects the machine's inability to measure distance at such a small interval. If the machine breaks, and thus cannot count any coordinates at all, does this mean that it cannot move?
This ridiculous assumption that all motion requires counting, or that motion is directly analogous to counting, is the whole basis of your entire argument. Anyone (like me) who rejects that particular assumption has no reason to take your argument seriously.
Well, I'm saying that continuous motion is impossible and so if motion is possible then it must be discrete. It could also be (although seemingly absurd) that Zeno's conclusion is correct and motion is impossible.
Either that or we have a genuine paradox in nature where continuous motion is logically impossible but nonetheless the case, which would then suggest a fundamental flaw with logic (as the OP suggested).
If motion is discrete then the object didn't pass through every rational number. It made jumps from one coordinate to another without passing any coordinate in between. Could be Planck-length teleportation, as an example.
This right here is the mistake that you keep making. There are two statements here.
You treat these as equivalent, or at least analogous. They are not.
Quoting Michael
Here we see your mistake from a different angle. You insist on treating the passing of each coordinate as a separate (i.e., discrete) event, just like counting. It is not - motion itself is continuous; only measuring distance is discrete, like counting.
Why? What's the difference between a physical tick that is a count and a physical tick that is a movement? Counting can't simply be reduced to, say, speaking the coordinate. We can count by tapping the table, or by clicking our fingers. In this scenario, the machine performs a count by moving to a different point in space. So there is no fundamental difference between moving and counting.
Well, I don't see how two locations separated by zero distance can be different locations.
You are talking about length (i.e., measurement), rather than location. Consider two dimensionless points with zero space in between them. How can they correspond to different locations?
I suspect that there's a failure of imagination on one (or both) of our parts. Consider:
[X][Y]
[X] and [Y] are discrete regions of space. There is no boundary of any breadth between them, and no distance separating them. Does it then follow that [X] and [Y] are the same region of space? It does not appear so to me.
This is possible if we are talking about discrete sets. It's a bit cheeky definition of distance there, though.
Which is absurd.
No, all we can say is that there is no fundamental difference between measuring movement/distance and counting, which I have acknowledged all along. Measurement is not a prerequisite for motion.
Well, it does appear so to me. So we have contradictory intuitions, which just goes to show that intuitions are not infallible guides to truth.
Edit: Wait, I see it now. You said that [X] and [Y] are adjacent regions of space, not adjacent locations in space. So I understand that two regions can have zero separation between them, yet not be the same region. But what we were discussing was whether two (dimensionless) locations can have zero separation between them, yet not be the same location.
Ok... but did it pass through ANY rational number? Are you saying that it was at point 2 at the beginning and after that it ended at point 3 without passing a single rational number in between (like 5/2 for example)?
For example, the first coordinate would be the one at 1 Planck length. The second coordinate would be the one at 2 Planck length. And so on. But at no point does it pass through the coordinate at 0.5 Planck length or at 1.5 Planck length.
You don't think that we can leave Planck constant out of this trip between points 2 and 3 on the real number axis though? If not, are you basically saying that the trip didn't visit any rational number, unless it happens to be at n x Planck length? But Planck length is a physical distance, while [2,3] is just an interval. How much is one meter on real number axis?
Could you provide an actual example of that? Are you talking about 2 sets of discrete points?
I don't really understand what you're trying to get at here. The point is that if movement is discrete then one doesn't have to consider an object moving first to the half way point ad infinitum. So there are a finite number of coordinates that it must pass through, which doesn't run into the logical problem that having to have completed an infinite series does.
This is a good example to show why we are at an impasse. Your claim is that all actual objects that actually move go from one Planck length coordinate to the next without ever occupying any intermediate locations. My claim is that all actual objects that move occupy infinitely many intermediate locations between any two arbitrary coordinates, even if the interval between them is one Planck length. You thus take the Planck length to be a limit on actual events themselves, while I take it to be a limit only on our measurement of actual events.
Ok but what are these coordinates if the movement is just between point (2,0) and (3,0) in real number space. You were mentioning something about Planck lengths, but you don't have physical units of measurement in space of real numbers. If the number of coordinates it passed is finite, what are they?
Well, it's only that something like this must happen if motion is to be possible.
Yes, and this runs into logical problems. Given that it has occupied an infinite number of prior locations in succession, it has completed an infinite series of events. But an infinite series of events cannot, by definition, be completed. And given that there's no first location for it to move to (as there's always a half-way point that it must travel through first) this infinite series cannot even be started (this being Zeno's paradox).
I really don't understand your question. We just have some distance that an object is to travel and we plot a coordinate at Planck-length intervals. There are a finite number of coordinates for the object to pass through, "jumping" from one point to the next without passing through the space in between. This differs from continuous movement in which the object must also pass through the half-Plank length intervals (that we haven't plotted), and also the quarter-Planck length intervals, and so on, leading to an infinite number of coordinates for the object to pass through.
Ok I'll try one more time. I am not talking about moving through physical space, like travelling from Earth to Mars. I am only talking about moving in R:
https://en.wikipedia.org/wiki/Real_coordinate_space
This space has defined distance and an object is moving from one point to another.
1. Do you agree that this has nothing to do with Planck constant, meters, centimetres? It's completely abstract.
2.In this space, do you still think that movement can be only discrete?
Sorry, I don't know what that is.
No. The motion from one potential location to the next is not a discrete event. Only the motion from one actual location (i.e., arbitrarily defined coordinate) to the next is a discrete event. We can only define a finite number of distance coordinates, so we can only measure motion in discrete units. However, the motion itself is continuous between those discrete coordinates that we use to measure it.
Your position is the one with logical problems from my point of view. How can something "jump" from one discrete location to another without ever occupying the space in between? This is pure nonsense to me.
This doesn't matter. It still occupied the infinite number of spaces that we could have plotted as coordinates.
You even admit this yourself:
Quoting aletheist
So in reaching B from A the object has completed occupying infinitely many locations in succession. This has logical problems. And, again, so does the notion of even starting the movement, as there isn't a first location (whether plotted or not) to move to.
Yes... however the time it needed to travel between those infinitesimally small distances is infinitesimally short as well. That's why when you "add" them together, they end up being less than infinity. Infinitesimally short means that whatever time you can imagine, that value is actually shorter.
Before some mathematician comes in and says I've butchered it - I know. But I'm trying my best to present it in a way that's understandable to non-mathematicians. Trying to find common ground here really.
We agree that it is not possible for us to plot actual coordinates at distances that correspond to all of the rational numbers. We thus agree that it is not possible for us to measure actual motion at that level of granularity.
You then draw the further conclusion - unwarranted, in my view - that the motion itself cannot occur at that level of granularity, such that space itself must be discrete. This has the consequence that all actual motion must involve somehow "jumping" from one discrete location to another, without ever occupying any of the space in between.
Since I find this patently absurd, I affirm instead that space must be continuous. Our inability to measure infinitesimal distances does not entail that they do not exist; objects can and do traverse infinitely many of them while moving from one arbitrarily defined coordinate to another. However, as @Svizec just explained, each such transit occurs in an infinitesimal interval of time. In the end, objects actually move finite distances in finite times, and that is what we can and do observe and measure.
This doesn't address the logical problem with an infinite succession of events having being completed, or the logical problem with the notion of there being a first location to move to.
Less absurd than the notion of having completed an infinite succession of events or the notion of a first location to move to, which are logically absurd (whereas I think the "absurdity" you find with discrete motion is a different, lesser, kind).
I know that they actually do. The problem is that the logic of continuous motion is incoherent, hence motion isn't continuous.
The problem is that the logic of discrete motion is incoherent, hence motion isn't discrete.
Like I said, we are at an impasse. Cheers.
Then motion is logically impossible.
Which then means we have a genuine paradox in nature. Motion is logically impossible but physically actual. And so the first of MadFool's suggestions seems correct; our logic is faulty.
That which is physically actual must be logically possible; and so it is only your logic that is faulty here, because you insist on applying the logic of finite/discrete mathematics to a problem that involves infinity/infinitesimals. Peirce said it well - "Of all conceptions Continuity is by far the most dif?cult for Philosophy to handle."
Such a machine would not be possible. But we are not talking about this machine specifically, we are talking about any thing that moves, so this is a red herring.
As I said, as long as you persist in conflating ordering with counting, your argument won't get off the ground. It's simply not logical, because there is no logical requirement for counting here. If ever you allow yourself to realize this (and I realize how hard it would be, given the effort you've put into defending your position), there is still an option left for you: you could try to stake out a metaphysical claim instead of a logical one. At least it wouldn't be obviously incoherent.
I don't know why you're comparing counting to ordering. The comparison is between counting and moving. And as explained here, there's no reason to suggest that they're fundamentally different.
Quoting aletheist
You're still making the same mistake. It is false to say that space is potentially infinitely divisible unless it actually is. It is false to say that an object could potentially move to coordinate A, unless the object can actually do this. You are allowing fiction into your perspective by claiming that actuality has no bearing on potential. This allows you to claim that all sorts of impossibilities are real potentials, because your notion of potential is not restricted by actuality. This indicates that you have a deep misunderstanding of the concept of "potential".
Quoting Michael
Quoting aletheist
Michael is correct here, that is simply how motion is. Consider walking, your foot is on the ground at one point, then on the ground at the next. It is not on the ground at all points in between. When we see an object moving, we assume that it must occupy every point along its course, but this is an assumption only. The assumed "course" is an oversimplification of what is really happening, and this is well known in QM.
Because I was responding to your own line of argument, e.g. here.
Quoting Michael
You haven't argued that moving is somehow related to counting, you just imagined some impossible contraption and asserted without any argument that continuous motion necessarily involves something of the sort.
You seem to just be misunderstanding. What I'm trying to say there is that you can't answer the question "if we want to count every rational number between 1 and 2, what number do we count first?" with "pick any at random, and then pick the next one at random, and so on" (as Banno suggested). Each number must be greater than the previous, and we can't count a number if we haven't counted a smaller number.
And so by the same token, each coordinate an object passes through must be closer to the target than the previous, and it can't pass through a coordinate if it hasn't passed through one that's further away.
I'm saying that the act of moving from one location to another can be considered an act of counting, like a clock counting the hours as the hand performs a rotation. Counting is just a physical act like any other. I don't know what you think it is.
So in this case, the ticks of the clock that are the count are the movements through each rational coordinate between two points (rather than just every 1/12th of a rotation as it is with a clock). It doesn't make sense for there to be a first tick, and it doesn't make sense for it to have already ticked an infinite number of times. So continuous motion doesn't make sense.
Yes, this nicely illustrates the very confusion that I've been talking about.
True enough, but this has nothing to do with counting.
Quoting Michael
You are saying this, but you are not proving this.
Quoting Michael
True, but that doesn't imply that all physical acts involve counting.
I don't know what you're talking about here. I'm just explaining what I meant by sequentially. You seemed to take issue with that word. It was simply used to preempt any attempt to weasel out of the problem (as Banno did).
[quote=SophistiCat]You are saying this, but you are not proving this.[/quote]
If I have to prove why each 12th or 60th rotation of a clock hand can be considered a count (or "tick", if you prefer) then I think the problem here is with your understanding.
It doesn't matter. The point is that, as with the example of a clock hand, the very act of moving from one point to another can be considered to be an act of counting. Therefore, if an object can move through every rationally-numbered coordinate then that object can be said to count every rationally-numbered coordinate. But as you say, it's impossible to count every rationally-numbered coordinate. Therefore it's impossible to pass through every rationally-numbered coordinate.
And to repeat what I said above, it doesn't make sense for there to be a first tick, and it doesn't make sense for it to have ticked an infinite number of times.
You're still making the same mistake. It is false to say that space is potentially infinitely divisible only if it actually is.
Quoting Metaphysician Undercover
No, it indicates that I have a different understanding of the concept of potential. We have previously established in other threads that you and I have a fundamental disagreement about this, so there is really no point in discussing it further here.
You seem to be confusing "divisibility" and "divided". Continuous space would actually be infinitely divisible.
Again from here:
It can be, but it does not have to be. Your whole argument hinges on insisting that the very act of moving from one point to another must be considered to be an act of counting, and that this counting must include every single point corresponding to a rational number in the interval. Just because we can model it that way does not entail that it actually is that way.
Well that's surely your problem not mine. You believe that something is possible (potentially doable) though it is actually impossible to do it. If you don't recognize this as a mistake, there's not much I can do to help you.
This is exactly the question brought up by the op. Is space actually infinitely divisible, or is this just a false assumption, a mistaken theory?
Sorry, see my edit. I meant to say that continuous space would be infinitely divisible.
No, you are the one with that confusion, as I have stated before. Space only has to be discrete if it is infinitely divided, not merely infinitely divisible.
Quoting Michael
This illustrates your muddled thinking perfectly. "Divisible" means "potentially divided."
No one is talking about doing anything. To say that something is infinitely divisible does not mean that a human being is actually capable of infinitely dividing it. It means that it is possible in principle to divide it infinitely.
No. Counting and moving is not the same and can not be considered the same. You can move "through" any number of uncountable points that you wouldn't be able to order at all. Obviously you can't count them. So it's not the same. It is so obvious that these two actions are not the same that it's outrageous to suggest otherwise without a proof. Basically your response to this paragraph will be "No, counting and moving is the same, you can't do the counting so you can't do moving". Without providing a shred of evidence. Basically your "argumentation" is identical to:
Travelling faster than light is the same as Travelling to China. Travelling faster than light is not possible ergo travelling to China is not possible.
When I tell you that travelling faster than light is NOT the same as travelling to China, you just say, yes it is.
Oh, and your clock analogy, it's wrong, because you are again making counting look too easy. In your example - yes it's easy. Let me spice that analogy a bit. Let's say it's exactly noon. What is the first unit of time this clock of yours will count/move into. What's the second? How about if you put liquid nitrogen on the clock and you count all the molecules that minute hand hits? How about you put all the rational numbers [1,2] between 00 and 12 and try to count those?
I will repeat one more time (probably we should just quit at this point, not sure that anyone is willing to listen to rational arguments): existence of Planck length does NOT imply discrete nature of space. Believe it or not, but it's actually possible to observe data in a way to look for evidence of space being discrete. The opposite was found - the data implies that the space is not discrete. It's called science. Experiments. Observation. Planck length is border of a resolution problem. And no, not resolution in sense of optical lenses or sensors, the resolution problem would manifest itself in quantum effects that would make anything else irrelevant, pointless.
This is the assumption that I'm showing to be false. Each movement from one point to the next is a tick. If the space between two points is infinitely divisible then it doesn't make sense for there to be a first tick and it doesn't make sense to have ticked up to the end as that would entail having completed an infinite succession of ticks. So it doesn't make sense to move through any number of uncountable points.
Good luck, because it's a fact.
Quoting Michael
No, it's not. Once you will understand that it's not, you will see the light. I can't help you anymore.
Have fun, gentlemen, don't kill each other. We all know what the truth is, what the facts are. I am too old to try to "win" debates online.
Right, and to divide something it is to do something. So to assume that it is infinitely divisible is to assume that something is capable of dividing it infinitely. If it is not possible for something to do this, then that principle is false.
Indeed.
No this is a misunderstanding; the moving object does not have to 'account for and check off' every point it moves through, not least because it does not actually move through any points.
Sure it does, as the motion is said to be continuous. It has to pass through the half way point and before that the quarter way point and before that the one eighth point, and so on ad infinitum.
That is true only of analysis, not of actuality. This seems so obvious, that it simply astounds me that you cannot see that.
No, it actually does pass through the half way point, and so on. If it didn't then it wouldn't be continuous motion, it would be discrete.
I guess you must deny, then, that the integers are countable, since nothing and no one can actually count them all. And yet it is a proven mathematical theorem that not only the integers, but also the rational numbers are countable - i.e., it is possible in principle to count them - despite the fact that they are infinitely numerous.
Here's the mistake you have been making for years, Meta.
You can't just offer discrete movement as a solution to the paradoxes associated with analog movement without also explaining how discrete movement really works. It might be there's no coherent explanation to something as basic as movement, just like there's not with causation.
Anyway, discrete movement is an obvious adoption of the computer graphics model imposed on reality. Identity of a computer graphic over time is preserved by the underlying programming, which is a quite literal deus ex machina. If we're going to insert Deus, I suppose anything is possible, including analog movement.
But as for an actual account of discrete motion, I believe there's Atomic electron transition or "quantum jump".
MU is right that it has to be more complex than that. Talk of actually counting smuggles in the necessity of the maker of the infinesimal divisions or Dedekind cuts.
For there to be observables, there has to be an observer. Or for the semiotician, for there to be the signs (the numeric ritual of giving name to the cuts), there has to be a habit of interpretance in place that allows that to be the ritualistic case. Which is why the number line itself is just a firstness or vagueness. In the ultimate analysis it is the raw possibility of continua ... or their "other", the matchingly definite thing of a discontinuity.
So infinity and infinitesimal describe complementary limits - one is the continuum limit, the other the limit on bounded discreetness, the limit of an isolate point.
Thus counting presumes an observer then able to stand inbetween. The counter can count forever because the counter also determines the cuts that pragmatically "do no violence" to the metaphysics, at least as far as the counter is concerned.
My point is thus that an observerless metaphysics is as obtuse as an observerless physics, or theory of truth, or observerless anything when it comes to fundamental thought.
Are you conceding that discrete movement is nonsensical? If so, why'd you offer it as an possible solution to Zeno?
But I'm not really interested in defending the notion of discrete motion. What I'm interesting in is showing that Zeno's paradox proves continuous motion to be illogical, and that any attempt to save continuous motion from Zeno's paradox by referring to being able to calculate the sum of a geometric series misses the point.
So if you talking about a physical continuity on the Planck scale, your attempt to mark the first location would already then have your fixed point transversing the whole distance to its resulting destination.
It is like the way a photon is said to experience no time to get where it is going. Travelling at c means the journey itself is already described by a vector - a ray rather than a succession of points.
So in the real world, locating your starting point is subject to the uncertainty relation. The Planck scale is the pivot which prevents you reaching your goal of exactitude by diverting all your measurement effort suddenly in the opposite direction. In effect you so energise the point you want to measure that it has already crossed all the space you just imagined as the context that could have confined it.
Zeno definitely does not apply in quantum physical reality.
I'm not sure there's a way to show that. A person either understands the paradox or not. It's not complicated.
All that would imply is that you chose the wrong logic.
That is true only in a formal sense; there is no actual halfway point it goes through.
I am afraid that you just can't get past the concept of counting, or rather to see it in its context. There's no point in me trying to explain it to you now, because I would just be repeating myself. But later, when you are no longer engaged in defending your position, I suggest that you acquaint yourself with the basics of set theory and calculous.
You might think that mathematics is this very specialized discipline that is only relevant to solving certain technical problems, but it's not. Mathematics is relevant to any abstract thought, metaphysics included. It expands your conceptual apparatus and gives you the tools for dealing with complex concepts in a systematic, disciplined way.
When you become familiar with the foundations of mathematics and see how concepts such as sets and numbers are built upon each other, perhaps then you will see what we have been trying to tell you. You might still resist the concept of a continuum on physical or metaphysical grounds, but at least you will be doing it with the clear understanding of its logical structure.
I think that you are making some unconscious metaphysical assumptions here. Why does continuous motion preserve identity and discrete motion does not? You can construe your idea of identity this way, but this construal doesn't have the force of logical necessity - it is just one possibility among many.
You ask how discrete motion "really works." What do you mean by this question? Do you understand how continuous motion "really works?"
An alternative view is to accept Zeno's paradoxes as lessons of the impossibility to deduce how reality behaves a priori, from mathematics or any other way.
What can happen is determined by the laws of physics alone, and if they say that an uncountable infinity of points in space will be traversed, then that is what will happen. Something mathematically infinite in the process may have occurred, but that involves nothing physically infinite.
It seems that Zeno's mistake is to assume that a particular mathematical notion of infinity somehow determines what can and cannot happen in reality.
Indeed. I do not see the relevance of being 5'11", unless as part of an argument to say that Max Planck was infinitely tall or something, or couldn't grow, and was born fully formed.
Appealing to a degree of granularity to space probably makes matters worse. How do you get from one step to the other if there is nothing in between?
That's right. It appears very obvious to me that if it is impossible to count them, then it is false to say that they are countable. Why would you accept the contradictory premise, that something which is impossible to count is countable? That makes no sense to me. This is the basic nature of infinity, that it is not countable. To believe otherwise is very clearly to believe a contradiction. The notion of infinity may be useful, but it's a fiction, a useful fiction.
Countable infinities are precisely those which can be put into one-to-one correspondence with the integers. This is a definition, and no, no one expects you to count them all.
The real numbers, or any finite interval on the real line, cannot be put into one-to-one correspondence with the integers. This type of infinity is much larger than the countable infinity, and is the first in a sequence of uncountable infinities.
While there are many cardinalities (could be infinite number of them as far as I know) the distinction the countable and uncountable infinities is the most important.
That's the point, they are not countable, so to call them "countable" is just a name, a label, it doesn't mean that they are actually countable. You might differentiate natural numbers from real numbers by saying that one is countable and the other not, but that's just a name, in actuality neither are countable.
Try counting the real numbers between 0 and 1.
It can't be done, but that doesn't mean that the natural numbers are countable. Neither real nor natural numbers are actually countable, because of the nature of infinity. One has no beginning point, the other has no ending point, but neither, as an infinity, is actually countable.
Countable and uncountable infinities are different.
You said you couldn't count a subset of the reals.
Do you think you might be able to count a subset of the integers?
Yes of course, but a subset of integers is not infinite. The difference here is with respect to the thing being counted, what is within the set, real numbers versus integers, one is assumed to be divisible, the other is not. It is not a difference in the infinity itself. With respect to the infinity itself, one is no different from the other.
OK, so you can count integers, but you cannot count the real numbers, even in a tiny subset. There is an uncountable infinity of reals within any subset - hence it is a continuum. The countable infinities do not have this property. They are different and one is at least infinitely bigger than the other.
No, to say that one is infinitely bigger than the other is nonsense, unless you are assigning spatial magnitude to what is being counted. We are referring to quantities, and each quantity is infinite, how could an infinite quantity be greater than another infinite quantity?
What is the case is that as you say, the real numbers represent a continuum, while the integers represent discrete, indivisible units. So there is a fundamental difference between what each represents. The continuum is assumed to be infinitely divisible, and also there is assumed to be an infinite number of discrete units. The meaning of "infinite" remains the same, so there is no difference between these two infinities. There is a difference between what "infinite" is being assigned to, division or addition.
Why don't you just look it up, or Google it? Plenty of stuff on cardinalities, countable and uncountable infinities, the diagonalization argument, Cantor ...
Set theory isn't the only way to think of numbers, though. Another approach is to say that, numbers don't exist per se, but there are an agreed set of rules and methods for developing relationships (what you might call "arithmetic") between a finite set of symbols (what you might call "digits") - and these symbols can potentially be arranged in an infinite number of combinations (what you might call "numbers").
So with a base 10 numeral system and the rules of arithmetic with which we're already familiar, it's not granted that the numbers 0, 0.5 and 1 'exist'. Rather, the process of dividing 1 by 2 produces the arrangement of symbols, '0.5'.
My point being, it's naive to build a whole world view around one arbitrary concept - and therefore, greater questions about infinity aren't so easily answered by set theory alone.
(I rushed this post because my food was delivered half way through; I hope it makes any sense at all)
I love the Khan Academy with pizza
It appears very obvious to me that you do not understand the accepted meaning of the word "countable" and, more fundamentally, the distinction between logical possibility and nomological possibility. It is possible in principle to count all of the integers or all of the rational numbers, even though it is not actually possible (as far as we know) for a human being, a machine, or any other physical thing to do so.
Quoting Metaphysician Undercover
No one is talking about spatial magnitude, and talking about numbers does not entail talking about quantities. Your worldview is too small because it limits the real to the actual and the finite.
I have, but you can't believe that just because a mathematician says it is so, therefore it is so. There's a lot of misunderstanding and sophistry in the world.
Quoting aletheist
As I said, it's an accepted name, "countable". But just because it's called "countable" doesn't means it's actually countable. You seem to believe that it actually does mean that it's countable. And as I explained, when talking specifically about the infinite itself, there is no difference between the countable and the uncountable. There is simply a difference between the thing which you are attempting to count.
Quoting aletheist
I'd rather a smaller world view which distinguishes fact from fiction, than a larger world view which doesn't distinguish fact from fiction.
Exactly - it actually does mean that it is countable, but it does not mean that it is actually countable. See the difference?
Quoting Metaphysician Undercover
And I would rather have a worldview that does not make the mistake of treating that which is real as fictional just because it is not actual.
A different word could have been chosen - how about "integer-like" or "zahlen", but that would change nothing. The non-zahlen infinities are vastly bigger, and that is an astronomical understatement.
No I don't see the difference, and you've already tried to explain, but all you do is contradict yourself. "Countable" means possible of being counted. To say that there is a difference between actually countable and potentially countable is nonsense. What would potentially countable mean to you, that it's not countable but could be made to be countable? That's nonsense.
Show me one genuine contradiction in any of my previous posts, without conflating "countable" (as defined in mathematics) with "actually countable." They are two different concepts.
Quoting Metaphysician Undercover
To say that there is no difference between actually countable and potentially countable is simply incorrect. Do you really not understand the distinction between the actual and the potential? between the nomologically possible and the logically possible?
Try this:
Quoting aletheist
See, you say that no one can actually count them, yet it has been proven that it is possible in principle to count them. It's not possible in principle to count them, that's the point, that's what infinite means, that it is impossible to count them. You only contradict yourself.
Quoting aletheist
I know very well the difference between potential and actual, as well as many different senses of "possible". It really appears like it's you who has no understanding of this. But if you really believe this is the case, then try to explain the difference between actually countable and potentially countable. Just don't give me contradictions or falsities. If it is impossible to count it, then it is impossible that it is "in principle" countable, because that principle would be a false principle.
Yes, and there is no contradiction at all in saying this - unless you insist on conflating "someone can actually count them" with "it is possible in principle to count them," thus refusing to acknowledge that they are NOT the same concept. Counting all of the integers is logically possible, but actually impossible. Infinitely dividing space is logically possible, but actually impossible. Creating a perfect circle is logically possible, but actually impossible. Pure mathematics is the science of drawing necessary conclusions about ideal states of affairs; the actual has nothing to do with it.
Quoting aletheist
No. counting all the integers is not logically possible, it is impossible. That's what infinite means, that it is impossible to count them all, you never reach the end. It is such by definition. To say that it is possible to count them all is contradictory. Therefore it is not logically possible.
There are really two basic principles here:
There is nothing contradictory about either of these principles; in fact, together they constitute the very definition of what it means for something to be countable within mathematics. The fact that both the natural numbers and the integers are infinite is completely irrelevant. Think of it this way - it is logically (and actually) impossible to identify a particular integer beyond which it is logically (or actually) impossible to count. If all integers up to any arbitrary finite value are countable, but there is no largest countable integer, then all of the integers must be countable.
Quoting Metaphysician Undercover
One more time: it is logically possible, but actually impossible. You claim to know the difference, but your responses keep indicating otherwise.
Quoting Metaphysician Undercover
That is obviously not what infinite means within mathematics, since the natural numbers and integers are very explicitly defined as countably infinite. You can rail against this terminology all you want, but it will not change the fact that there is no contradiction in saying that the integers are countable as that concept is defined within mathematics.
This is not true. A set is defined as countable if it can be put into bijection with the natural numbers.
By this definition we can then show that the natural numbers, the integers, and the rationals are countable; and that the reals aren't. We define "countable" as a technical term, having no meaning other than that which we've given it. We then prove that the naturals and integers are countable. Formally, having defined the technical term "countable," we then note that the identity map on the naturals, which is a bijection, proves that the natural numbers are countable. Then we prove that the integers are countable by lining them up as 0, 1, -1, 2, -2, ...
As Tom mentioned earlier, much confusion would be avoided if Cantor had picked another name. If we say a set is foozlable if it can be bijected to the natural numbers, then we can prove that the natural numbers, the integers, and the rationals are foozlable; and that the reals aren't. But nobody would have to spend any time arguing about whether you can count the elements of an infinite set.
Surely we all agree that technical terms have specific meanings in context that do not necessarily correspond to their meaning in everyday language. An engineer and a doctor give very different meanings to the word vector. Nobody gets confused, because within their respective technical disciplines the word vector has a formal definition. In the legal profession such words are called "terms of art." A term of art is a word or phrase that has a specific technical meaning within a given discipline that is unrelated to any common meaning.
It's a mistake to think that countability has anything to do with the ability to be counted. That's way too vague. For one thing it's arguably false for the everyday meaning of the word "count." And Cantor's transfinite ordinals let you count way past the natural numbers. Better to simply realize that in set theory, "countable" means exactly one thing and one thing only: that a given set may be bijected to the natural numbers. What Cantor really meant to say is foozlable. Or in the original German, füzlich [That's a joke]. Now any semantic confusion goes away.
Fair enough, but the fact is that you can count members of a countably infinite set. You can take a subset of a countably infinite set of any number you wish. You can order the set, and you can count from one member to the next.
You can't count the members of an uncountable infinity. There is no such thing as a next member.
No, the fact is that you cannot count an infinite set, that's what "infinite" means. You can count a finite subset, but you cannot count the infinite set. "Countable" is just a name, as fishfry explained, it has no other meaning.
Quoting tom
Nor can you count the members of a countable infinity. "Countable" is just the name of the set.
The point I made earlier is that there is actually no difference between the countable infinity and the uncountable, as "infinite", they are the same. What is different is the thing which we are attempting to count, one is a continuity the other discreet units. The continuity cannot be counted, the discrete units can.
Quoting fishfry
Right, this is all that I meant when I said that the natural numbers are countable by definition. I agree with you that we can then subsequently prove that the integers are also countable.
Quoting fishfry
Right, this is all that I meant when I said that "countable" is not the same concept as "actually countable." However, I agree with @tom that "you can count members of a countably infinite set"; again, there is no largest natural number or integer beyond which it is (logically or actually) impossible to count, so all of the natural numbers and integers must be countable.
Incorrect; "uncountable" and "infinite" are not synonyms in mathematics, since there are countable infinities and uncountable infinities. This is a fact, not an opinion.
Quoting Metaphysician Undercover
All words are just names, with no other meanings than how people use and understand them. Mathematicians use and understand "countable" in a very specific way. You do not have to like it, but it is silly to continue insisting otherwise.
Quoting Metaphysician Undercover
Not quite, since even the real numbers are still discrete despite being uncountable; they thus form a pseudo-continuum. A true continuum is "that of which every part has parts of the same kind" (Peirce), so it can never be divided into discrete individuals. For example, a truly continuous line can be divided into infinitely many smaller (continuous) lines, but never into (discrete) points.
They are not synonymous, but infinite is by definition not countable. There could be something else uncountable which is not infinite. As we've already discussed, when you refer to countable and uncountable infinities, you use "countable" in a different way, with a different meaning. This way of using "countable" does not imply that a countable infinity is actually countable (according to the other sense of countable), nor does it mean that it is potentially countable, according to the other way of using countable. It is a completely different way of using "countable".
I suggest that you continue to use "countable" in your way, and I'll use "countable" in my way, the two being very obviously incompatible with each other. But you should not claim that you can make the two compatible by saying that one refers to an actuality and the other to a potentiality, because this is not the case. Your sense of "countable infinity" does not equate with "potentially countable" according to my sense of countable, because infinite is neither potentially nor actually countable according to my sense of "countable", it is absolutely uncountable.
How does this imply that all the natural numbers are countable? It actually implies the very opposite. Every number you count has a larger number, therefore it is impossible that all of the natural numbers are countable. I think you really believe that it is possible to count infinite numbers, because this statement seems to be an attempt to justify this.
Again, incorrect. You evidently have a rather idiosyncratic personal definition of "infinite." My dictionary provides several widely accepted definitions, and none of them state or imply that it means "not countable." Besides, as I keep noting, the concept of being "countably infinite" is well-established and well-understood within mathematics.
Quoting Metaphysician Undercover
I have never claimed that our different definitions of "countable" are compatible. I have simply demonstrated that my definition is not contradictory, and that yours is simply wrong, at least within mathematics. Whether something is actually possible is completely irrelevant when dealing with ideal states of affairs, which is all that pure mathematics ever does.
Quoting Metaphysician Undercover
Because you can always keep counting beyond any arbitrary finite value; i.e., you cannot identify a single natural number or integer that is uncountable. Obviously, if there are no uncountable natural numbers or integers, then all of the natural numbers and integers are countable.
Quoting Metaphysician Undercover
Now you seem to be confusing "countable" with the idea of being finished counting. This is not what "countable" means within mathematics, either. That larger number is just as countable as the one that you already counted; and so is the next larger number; and so on, ad infinitum - which is the whole point.
Quoting Metaphysician Undercover
I have stated plainly (and repeatedly) that I believe this to be logically possible, but not actually possible.
Quoting Rich
What exactly do you mean when you assert that "space is indivisible"? Are you merely saying (as I do) that space is continuous, rather than discrete - i.e., it cannot be divided into dimensionless points, only smaller and smaller three-dimensional spaces? Or do you have something else in mind?
Bohm pointed out that where there are paradoxes there are some really nasty problems with assumptions.
It sounds like we are on the same page here. As Charles Sanders Peirce put it, citing his father:
You can certainly well-order an uncountable set. You need the Axiom of Choice to well-order the real numbers, but you do not need Choice to show the existence of the first uncountable ordinal. That Wiki page is light on detail but the idea is that the set of all countable ordinals is an ordinal (needs proof of course), and it can't be a countable one (because a set can't be a member of itself), hence it must be an uncountable ordinal. Such a thing is impossible to visualize but it exists.
An ordinal is an order type of a well-ordered set. A set is well-ordered if every nonempty subset has a smallest member. There's a first, then a second, then a third, etc. Clearly the natural numbers are well-ordered. Now to get to larger ordinals you have to allow limit ordinals, which are unions of upward chains of ordinals. I don't want to get technical, which is why in my earlier post I just wrote
Quoting fishfry
That's why the usage of the everyday meaning of counting is totally out of place here. It's vague, and mathematicians can indeed well-order uncountable sets.
Quoting aletheist
No I'm afraid you are still missing my point. I defined a set as countable if it can be put into bijection with the naturals. You claim this "defines" the naturals as countable but I say, "I don't believe you. Prove it." And you say: "Aha, the identity function on the naturals is a bijection." You have PROVED directly from the definition that the naturals are countable. It's a theorem (admittedly so easy it's never stated explicitly) and not a definition.
No, I get it, you just stated more accurately what I meant all along. :)
Yes, we must always keep in mind that such equations are models - or in Peirce's terminology, diagrams - which embody only the parts and relations within the actual situation that someone has deemed to be significant. Consequently, they are only as "accurate" as this judgment on the part of the modeler and the underlying assumptions of the selected representational system, including its transformation rules.
I went back through this thread from the beginning. Finally on page 11, this quote is the first mention of mathematical countability. The above quote is simply flat out wrong. It commits the fallacy (does it have a formal name?) of confusing a term of art with its everyday meaning. Countability as defined in mathematics simply has nothing at all to do with the everyday meaning of the ability to be counted. I already made this point but now I found the source of the recent confusion in this thread.
A child learning to count, "one, two three, four, ..." has absolutely nothing to do with mathematical countability. Saying that a set is countable does NOT mean "it is possible in principle to count them." It means exactly that there exists a bijection from the natural numbers to the set. Nothing more and nothing less.
You know the old joke. "Why can't you cross a mountain climber with a mosquito? Because you can't cross a scaler with a vector." That joke depends on conflating the engineering definitions of scalar, vector and cross (as in cross product) with the common English meaning of a climber -- a "scaler" -- and the medical meaning of vector -- a means of disease transmission, and the biological meaning of cross, as to cross-breed living things based on their genetic makeup.
But this is a JOKE, not something you can take seriously in a philosophical discussion. You can not, unless you being disingenuous, say that "The rational numbers are countable" and then say this shows that a child could count them in the every day sense of the word.
If you counted, in the sense of saying out loud "one, two, three ..." the natural numbers, starting at the moment of the Big Bang, at the rate of a number per second; or ten numbers, or a trillion -- you would not finish before the heat death of the universe.
You are simply conflating a term of art -- a technical term used with a specific meaning in a specific context by specialists -- with the everyday meaning of the term.
Sorry to be ranting now but really, the quote above is terribly wrong. You can't count the natural numbers in the every day meaning of the word. There are infinitely many of them. The natural numbers are countable, in the technical sense that there exists a bijection between the natural numbers and themselves. If you think to yourself, "The natural numbers, the integers, and the rational numbers are examples of foozlable sets," you will not confuse yourself or others by shifting the meaning of a technical term to its everyday meaning.
I'll bet that it's a big hit with the ladies on the bar scene, though. :D
I have acknowledged this repeatedly - the natural numbers (and integers) are not actually countable, in the sense that someone or something could ever finish counting them. However, they are all countable in principle, in the sense that there are no natural numbers (or integers) that are uncountable; given enough time, someone or something could count up to and beyond any arbitrarily specified value. As I said before (with sincere gratitude), you have stated more accurately what I meant all along.
Which has absolutely nothing to do with the question of whether a given set is foozlable -- able to be bijected to the natural numbers. I don't want to pile on so I'll just refer to what I've already said. The reason it's important is because this thread was percolating along on the usual lines -- nature of time and space, whether the mathematical solution of Zeno in terms of infinite series is also the physical solution, etc. Once you conflated the technical meaning of countable with its every day meaning -- a logical fallacy -- the thread lurched off on a very unproductive tangent IMO. It is far from clear that "given enough time" you could count to any specified value. If time itself is part of the universe, then you will run out of time between the Big Bang and the heat death of the universe.
Besides: What you just said is that for any given number N, you can count up to N. That is manifestly false for the reason I gave. But even if I grant you that point you still can't count ALL of the natural numbers. You have just conflated counting up to some big finite number with counting ALL the natural numbers. A big logical fallacy. Consider that the statement "X is a finite set" is true for each natural number, but not for the collection of ALL natural numbers.
So even if you can count to a zillion, and a gazilloin, and a googolty-googol-gazillion, you can't count ALL the natural numbers using that same meaning of counting. Right?
Really? Cantor proved the reals constitute a continuum. Whatever they are, they are certainly not discrete.
The reals in their usual order are a continuum. They can be reordered to be discrete. Counterintuitive but set-theoretically true. Order is important in this discussion. The rationals in their usual order aren't discrete, but we can line them up in bijection with the natural numbers and thereby identify the first, second, third, fourth, etc.
Also (not to hijack this thread further, but this relates to Zeno) what do we mean when we say the real numbers are a continuum? If by continuum we mean a particular philosophical idea of a continuous space, then the mathematical real numbers may or may not satisfy a philosopher. If by continuum we mean the standard mathematical real numbers, then we are being circular. Certainly the standard real numbers are not a proper model of the intuitionistic continuum. These are murky philosophical waters.
I think the issue here has been metaphysical - so neither everyday, nor mathematical. Although the mathematics of course has to have some grounds for finding its own axiomatic base "reasonable".
So the Zeno paradox is about a particular difficulty between a mathematical operation and the world we might want it to describe. The math seems to say one thing, our experience of the world another.
Bijection is great. It replaces the need for a global act of quantification (demonstrating an example of infinity by showing a sequence is measurably unbounded) with a local demonstration of a quality (if bijection works for this little bit of a sequence, then that property ensures the infinite nature of the whole). So bijection doesn't do away with the notion of counting or a syntactic sequence. But it does extract a local property that rationally speaks for the whole.
No problems there.
And then we get back to the metaphysics on which even the mathematical intuitions are founded. Which was the issue the OP broaches and which you are side-tracking.
Infinite: endless.
Quoting aletheist
Countable means capable of being counted. If it cannot be counted, as is the case with something infinite, or endless, it is not capable of being counted. Therefore the infinite is not countable.
Quoting aletheist
Back to your contradictory notions "the natural numbers(and integers) are not actually countable ... However, they are countable in principle... "
You refuse to face the facts of the situation, the entire set of natural numbers is, in principle, not countable. That's what Infinite means, endless, so no matter how hard you try the infinite set is not countable. What is countable in principle, is any finite set of natural numbers. But it is false to claim that the entire infinite set is countable in principle, what is countable is finite subsets.
You do agree they're foozlable, right? I just want to make sure I'm understanding you.
Quoting apokrisis
I really miss your bug-eyed avatar from the old forum :-)
Yeah sure, that's the name you gave instead of the name "countable". But I'm not sure that I would agree with the assumption that there is a substantial difference between a foozlable infinity, and an unfoozlable infinity. We can call them countable and uncountable infinities if that's easier.
I don't know if it's a "substantial" difference. It's certainly a difference. The rationals are foozlable and the reals aren't. Even in countable models of the real numbers, and yes such things exist, the reals are not foozlable. So yes it's a pretty important difference in math.
But if you insist that "countable" is to be used with its everyday meaning, then we should be careful not to confuse this with foozlability, which is a technical condition used by specialists in set theory.
If you think you can disorder the reals, then pleas indicate the number following this one, and suggest between which two numbers you might place it:
0.999... What comes next?
Quoting fishfry
Yes they are. They are countable, therefore discrete (and of measure zero on the reals).
Quoting fishfry
Only if you make them so.
If I give you a natural number, you can count it, and the next ...
Try 999 for size.
If I give you a real number, 0.999... where do you go next?
I am not the one who took us down that road by repeatedly insisting that "countable" must always and only mean the same thing as "actually countable." On the contrary, I carefully maintained the distinction between these two concepts throughout.
Quoting fishfry
Now you are the one conflating the mathematical with the actual. Any natural limitations on my ability as a human being to count up to very large numbers, whether there was a Big Bang, whether there will be a heat death of the universe, and how much time will have passed in between these two posited events are all completely irrelevant to the discussion. We are drawing necessary conclusions about an ideal state of affairs, so actuality (including time) has nothing to do with it.
Quoting fishfry
I have done no such thing. I have noted, rather, that no matter how big a finite number you specify, it is possible in principle to count up to and beyond that number. In other words, you cannot identify a largest natural number (or integer) beyond which it is impossible in principle to count. If it is possible in principle to count up to any particular natural number (or integer), then it is possible in principle to count all of the natural numbers (and integers).
Your point is that this is not part of the relevant technical definition of "countable." I have accepted that clarification, and even thanked you for it, so I do not understand why you keep harping on it. Even so, there is presumably a reason why the standard term is "countable" and not "foozlable" or anything else.
I agree that there is an important difference between natural integers and real numbers, and even an important difference between rational numbers and real numbers, what I disagree with is that there is a difference in the infinities which arise in all these different situations. I think that the infinite itself is the same in each situation, but it is applied differently.
So here's an example. We could take a point like zero, or any other integer, and count the integers toward the positive and toward the negative, from that point, and assume two distinct infinite quantities, one negative and one positive from that point. Or we could take two points, like 1 and 2, 3 and 5, or 6 and 10, and assume an infinite quantity of real numbers in between. In these two different ways of using "infinite quantity", there is no difference in the meaning of "infinite". One is not is not a larger quantity than the other, just like the infinite quantity of real numbers between 6 and 10 is no bigger than the infinite quantity between 1 and 2. We could then take an irrational ratio, like pi, and say that it extends to an infinite quantity of digits, and this use of "infinite" is still the same.
This is a textbook case of the fallacy of composition. And, you've also forgotten one premise here, that any particular natural number has numbers higher than it. And, this is the premise which makes it impossible, in principle to count all of the natural numbers.
I made the mistake of stating, way back in the thread, that the DEFINITION of a countable set is that it can be put in one-to-one relation with the Integers, when I should have said the Natural Numbers, sorry!
This does not change the fact that, you can sit down and count members of a countable set, but you cannot do that with an uncountable set.
No one claimed that. You can count as many members as you like. Given an uncountable set, you can't count any members.
I recall doing this at school at the age ~16.
Aletheist claimed that.
Quoting aletheist
Fallacy of composition.
As he defined "continuum," yes. However, Peirce argued (and I agree) that the real numbers still do not conform to our common-sense notion of a true continuum as "that of which every part has parts of the same kind," such that it cannot be understood as a collection of individuals, no matter how dense. A truly continuous line can be infinitely divided into smaller and smaller lines, but never into points. As you have noted, the real numbers form a multitude larger than that of the rational numbers; but a true continuum exceeds all multitude.
The real numbers have been proved to for a continuum, even in the Peirceian sense.
Just keep counting.
Now list the natural numbers between 0 and 0.00000000000000000001 so we can count them.
Again this is an example of rationally seeking a way for the part to speak for the whole. What can't be achieved via actualisation can be supported by appeal to the existence of a local property - in this case, not bijection but a quick demonstration that any nameable number implies in its own syntactic construction a number immediately larger (or immediately smaller).
Tom is also employing this local syntactic property.
So yes, bijection seems more abstract a level of definition because it maps maths to maths rather than maths to physics (ie: syntactic spaces where time is still part of the deal - as in saying any time you name a number, the next higher number awaits). But still, the general mathematical tactic is the same - seek a local property that constructive principles will guarantee stands for the truth of the whole. And thus, the very nature of this tactic reveals the deeper questionable presumptions that metaphysics would be interested in.
It is the idea that reality is perfectly constructible that is questioned by a synechetic or holistic point of view.
But then even a simple holism falters - the idea of the continuum being instead " the foundational". The continuum is that to which an infinity of cuts can be made. If a division is possible, another one right next to it ... but spaced by the infinitesimal of some continua ... must be possible. So simple holism is simply the inverse problem. Although - like division as an arithmetic operation - there is an advantage that at least it is being flagged that there is a more primitive presumption about there being in fact a pre-existent whole (that gets cut or divided).
So simple holism brings out the fact that simple constructionism is presuming an infinite empty space that can be filled by an unbounded act of counting. The standard atomistic approach presumes its numerical void waiting to be filled. And even bijection just illustrates the presumed existence of this numerical void as a waste disposal system that can swallow all arithmetical sequences. You can toss anything into the black hole that is infinity and it will disappear without a splash.
So the simplest view treats infinity as the void required by atomic construction. The next simplest view treats infinity as a continuum - a whole that is in fact an everything, and so able to be infinitely divided.
Then obviously - as usual - there is the properly complex view where instead of an atomistic metaphysics of nothingness, or even the partial holism of a reciprocal everythingness, we arrive at the foundational thing of a vagueness as that deepest ground which can be divided towards this reciprocal deal of numerical construction vs numerical constraint, the filling of a numberless void vs the breaking of a numberful continuum.
Of course, none of this deep metaphysics need trouble those only concerned with ordinary maths. They can believe that Cantor fixed everything for atomistic construction and the story ends there.
But deep metaphysics makes the argument that the very act of trying to cut is what produces the divided that appears to either side. The continuum arises because it is cuttable. Which like the Chesire Cat's grin, sounds really weird to those only used to everyday notions of logic or causality where something - either everything or nothing - has to be the starting point or prime mover for any chain of events.
Please identify a natural number or integer that is not capable of being counted. You are still conflating the notion of counting with the notion of being finished counting.
Quoting Metaphysician Undercover
Only if "countable" means "capable of being finished counting," which is not how the term is defined within mathematics, nor how I have ever used it in this thread.
Quoting Metaphysician Undercover
Only if I were arguing that it is possible in principle to count all of the natural numbers (and integers) merely because it is possible in principle to count some of the natural numbers (and integers). Instead, what I am arguing is that it is possible in principle to count all of the natural numbers (and integers) because it is possible in principle to count up to and beyond any particular natural number (or integer). Again, there is no largest value that is countable in this sense, so they must all be countable in this sense.
However, note that - per @fishfry's helpful clarification - this argument of mine has nothing to do with the technical definition of "countable" (or "foozlable"), which pertains only to sets and is easily proved to be a property of the natural numbers (and integers), but not the real numbers.
He certainly did not think so. Could you please point me to the proof? Note, I acknowledge that the real numbers serve as a useful mathematical model of a continuum.
Does mathematics actual model a continuum? I don't think so. If it did, it wouldn't lead to do many paradoxes and incorrect descriptions of experiences. Mathematics, I believe, provides a rough models of discrete, measurable actions , which in themselves are practical for certain applications, but are also quite distant from experiences. For example, it is impossible to divide space or time (duration). This simple observation, which addresses the OP, makes all the difference in the world. Far from being parenthetical, it totally changes the way one views the nature of life and the universe.
Understanding this, addresses the question of why Einstein was incapable of understanding what Bergson was presenting and why Bergson understood Einstein but disagreed. And as the OP points out, with this understanding Zeno's paradox is dissolved.
We can well-order the reals. https://en.wikipedia.org/wiki/Well-order#Reals
I mention this because it's a counterexample to the intuition that a set can be "counted" if its members can be lined up so that there's one after another. The real numbers can be well-ordered. That means that they can be ordered such that every nonempty subset has a least element. So there's a first real, a second real, and so forth. Now when you run through the natural numbers, you take a limit ordinal and keep on going. Cantor worked all this out.
As I say, the only relevance of this example is to attack the argument that only the natural numbers may be well-ordered. In fact any set may be well-ordered. And even if you don't believe that because you reject Choice, we can still find at least one uncountable set that can be well-ordered without Choice. So the idea of "lining things up one after another" is far stranger than it first appears, and is not a reliable intuition for what can be "counted," whatever that might mean. I don't think it means anything at all, which is why there are now several pages of confusion derailing the original topic of this thread.
No, you claimed the reals can be disordered and made discrete.
You are wrong on both counts, but of course if you would like do demonstrate?
Take the number 0.999... and place it between two adjacent reals out of order.
Please locate that quote of mine, I can't find it and don't remember saying it. I probably said reordered and definitely well-ordered, but not disordered.
I may have misspoken myself to say that a well-order would be discrete. I don't believe this is true.
However you can certainly put a discrete topology on the reals, even in their usual order. Just define the discrete metric d(x,y) = 1 if x ? y and d(x,x) = 0. This metric induces the discrete topology on any set. In the discrete topology every point is an isolated point.
Of course mathematics can and does model a continuum. However, the accuracy and usefulness of such a model depend entirely on its purpose, and that is what guides the modeler's judgments about which parts and relations within the actual situation are significant enough to include.
Here is what he actually said.
Quoting fishfry
He then added this, which is basically the same point that I made.
Quoting fishfry
So I would still like to see the alleged proof that the real numbers form a true continuum as Peirce defined it, which (as I understand it) is similar but not identical to the intuitionistic continuum. I highly doubt this - especially if, in fact, the real numbers can be reordered to be discrete.
I am not convinced that this is true. Two of Peirce's major objectives for philosophy were to make it more mathematical (by which he meant diagrammatic) and to "insist upon the idea of continuity as of prime importance." Surely he must have considered these efforts to be complementary, rather than contradictory.
I think you have presented to most essential issue. I understand the dilemma, but understand that Bergson had truly mastered all mathematics of his time but never sought to use it in any of his writings. One cannot use a fatally flawed approach to arrive at more knowledge no matter how difficult it is to admit to these flaws.
So concretely, a discrete approach cannot uncover the nature of a continuous ontological reality. Other approaches must be used and unfortunately current mathematics is simply not equipped. It is only adequate for discrete approximate measurements and predictions of non-living matter. It cannot be used to understand the nature of a continuous universe.
I think Rich is right that maths is generally premised on the notion of atomistic constructability and so is anti-continuity in that sense. (And that is not a bad thing in itself as constructionist models - even of continuity - have a useful simplicity. Indeed, arguably, it is only by a system of discrete signs that one can calculate. And signs are themselves premised on understanding the world in terms of symbolic discontinuities of course - signs are no use if they are vague.)
So then the holistic reply to this routine mathematical atomism would be a countering mathematics of constraints - of pattern formation calculated via notions of top-down formal and final cause. And that is damn difficult, if not actually impossible.
This would be why Peirce felt his diagrammatic logic was so important. Like geometry and symmetry maths, it tries to argue from constraints. Once you fence in the possibilities by drawing a picture with boundaries, then this is a way to "calculate" mathematical-strength outcomes.
So yes, there is no reason why a construction-based maths should not be complemented by a constraints-based maths. And arguably, geometry illustrates how maths did start in that fashion. Symmetry maths is another such exercise.
However to progress, even these beginnings had to give way to thoroughly analytic or constructive techniques. Topology had to admit surgery - ways that cut apart spaces could be glued back together in composite fashion - to advance.
So that is at the heart of things here. For a holist, it is obvious reality is constraints-based. So regular maths is "wrong" in always framing reality in constructivist terms. And yet in the end maths is a tool for modelling. We actually have to be able to calculate something with it. And calculation is inherently a constructive activity.
So while we can sketch a picture of systems of constraints - like Peirce's diagrammatical reasoning - that is too cumbersome to turn into an everyday kind of tool that can be used by any schoolkid or universal turing machine to mechanically grind out results.
Of course, that kind of holistic reasoning is also then absolutely necessary for proper metaphysical level thinking, and diagrammatical reasoning can be used to advance formal arguments in that way. You have probably seen the way Louis Kauffman has brought together these kinds of thoughts, recognising the connections with knot theory, as well as Spencer-Brown's laws of forms. And I would toss hierarchy theory into that mix too.
So construction rules the mathematical imagination as tools of calculation are the desired outcome of mathematical effort.
While that doesn't make such maths wrong (hey, within its limits, it works I keep saying), it does mean that one should never take too much notice of a mathematician making extrapolations of a metaphysical nature. They are bound to be misguided just because they hold in their hands a very impreessive hammer and so are looking about for some new annoying nail to bang flat.
I agree with this; what I question is whether mathematics is totally reliant on the discrete. As I have indicated previously, I have in mind Peirce's broad definition of mathematics as the science of drawing necessary conclusions about ideal states of affairs. In fact, toward the end of his life, he largely moved away from describing continuity using set theory - although he employed the term "collection" - and embraced topology instead.
Quoting Rich
You will likely appreciate this quote that I just came across from Philip Ehrlich, referring to Paul duBois-Reymond, another late-19th-century mathematician who wrestled with the concept of continuity.
I suspect that Peirce would have endorsed this criticism wholeheartedly.
Agreed. This is what I have in mind when I cite the famous quote by George Box: "All models are wrong, but some are useful." In my own field of structural engineering, Mete Sozen has posed the question: “Is an exact analysis of the approximate model an approximate analysis of the exact structure?” An affirmative answer is a fundamental yet subtle presupposition of modern practice - one that is easily (and often) overlooked, and not always correct.
Quoting apokrisis
Right - because although diagrammatic reasoning is deductive, it requires the retroductive steps of creating, augmenting, and manipulating an icon that embodies the significant relations among its parts within a suitable representational system. This can only be done successfully by a well-prepared mind that has developed the right kind of judgment through (mathematical) experience.
I'll certainly take that to heart :-)
Can you (or anyone) supply some of relevant Bergson and Pierce links that would shed light on the relation between the mathematical real numbers and the philosophical idea of the continuum?
Quoting aletheist
Oh I'm in no position to do that. I'm not familiar with the philosophical thinking on the continuum. You did note that I questioned whether philosophers accept the real numbers as the correct model of the continuum. I didn't claim that the real numbers form a true continuum, someone else did. I'm aware that there are philosophical objections but I don't know much about them.
The OP substantially calls to question the whole premise of using discrete to describe the continuous. As the OP describes, by discarding the discrete, one quickly resolves the paradox. Similar paradoxes can be similarly addressed with this approach. But first, discrete must be discarded in the realm of ontology.
That's a very simple question to answer. The highest number is the one that's not capable of being counted.
Quoting aletheist
Again, you are saying that because it is possible to count any particular number, it is therefore possible to count all the particular numbers, and this is known as the fallacy of composition.
This is exactly the problem with the Zeno paradox of the op. Zeno's premise is that space is continuous. Then he introduces mathematics to deal with this assumed continuity. It fails. The conclusion which should be drawn, is that mathematics is incapable of dealing with the continuous.
If we move to the ontological implications, then if there is no such thing as a real continuity there is no problem. But if there is a real continuity then we have a problem, if we are trying to understand that continuity mathematically. So either space and motion are discrete, and there will be no problem to understand them with mathematics, or they are continuous, in which case they cannot be understood with mathematics.
And if we assume a real continuity of any sort, then we should not expect to be able to understand it mathematically.
You are better off asking aletheist that as that is his argument. And I am certainly no Bergsonite.
I am not familiar with Bergson, and Peirce (not Pierce) is notoriously difficult to get one's arms around, because he never managed to write a book on philosophy - just lots of articles, and tens of thousands of pages of unpublished manuscripts. I usually recommend the two volumes of The Essential Peirce, but for this particular subject, several chapters in Philosophy of Mathematics: Selected Writings are more pertinent; likewise his 1898 lecture on "The Logic of Continuity," which is the last chapter in Reasoning and the Logic of Things.
As for secondary literature, chapter 4 of Kelly Parker's book, The Continuity of Peirce's Thought, might be a good place to start. Benjamin Lee Buckley's book, The Continuity Debate, based on his dissertation (as Lisa Keele), summarizes and compares the views of Cantor, Dedekind, duBois-Reymond, and Peirce; however, I take exception to some aspects of how Peirce's position is described.
I personally am spending my later years of life immersed in music, art, dance, Eastern meditative practices such as Tai Chi, to expand my toolset for greater knowledge. Without experience in the arts, I do not believe one is properly equipped to understand life.
Just curious, on this basis do you reject nominalism - the view that reality consists entirely of singulars - in favor of realism? Peirce did; he eventually described himself as an "extreme scholastic realist," affirming the reality of generals as continua.
https://en.m.wikipedia.org/wiki/Duration_(philosophy)
It is a reasonable starting point for further study.
This is questionable though. We can understand time as discrete units, or we can understand time as a continuity. We can also understand it as some kind of composition of both. What if real time, which we are experiencing, consists of discrete units, and it is just the brain and living systems which are creating the illusion of continuity? I tend to think that the only real continuity is the existence of the soul itself, and the soul, during the act of experiencing, renders the appearance of time as continuous, to make it compatible with its own existence, and therefore intelligible to the lower level living systems. Now, as highly developed life forms, we have developed mathematics, which will allow us to understand the true nature of time, as discrete, but we must get beyond the way that time is presented to us by our lower level living systems, (i.e, that intuitive impression of time) to be able to understand time mathematically.
And which number would that be? I asked you to identify it, not describe it.
Look, we have been using at least three different definitions of "countable" in this thread:
You have made it quite clear by now that you reject the first two, but that does not render them false or contradictory - just different from yours.
There is no highest number, that's what makes the set of natural numbers uncountable. If I could identify the highest number, we could count to it, and count all the numbers. I cannot, and nor can you, or anyone else, and so the natural integers remain uncountable, as they always will be.
Quoting aletheist
I accept the first one, but that defines "countable" relative to the natural numbers, so it is insufficient to tell us whether or not the natural numbers, are countable. It is a definition used to judge things in relation to the natural numbers so it cannot be used to judge the set of natural numbers itself. We need a definition which we can apply to see whether or not the set of natural numbers itself is countable.
The second definition of countable, your definition, makes no sense. "There is no particular largest value beyond which it is logically impossible to count." If this were the case, then no subsets of integers would be countable, because each of these has a particular largest value.
The third is the obvious choice as a definition to apply in order to determine whether the natural numbers are countable. It would be false to say that something which is not capable of being counted is countable. Therefore we can conclude that the set of natural numbers is not countable.
I am inclined to subscribe to how Peirce addressed this.
Only if it were true that we must be able to finish counting something in order to call it "countable." Your definition requires this; mine does not.
Quoting Metaphysician Undercover
And yet set theory explicitly says otherwise.
Some theories are false. It's very hard to convince the people who believe in false theories, that they are false. That's life. It appears like your set theory, if it really is as you describe, relies on the fallacy of composition. You should investigate this, and if the theory is as you describe, quit believing in it so strongly, because it's false. Or else it isn't as you describe, then you should develop a better understanding of what the theory really says.
You are each using different definitions. This is the fallacy of ambiguity. Surely we need not argue about this any more. I have humbly offered the word foozlable as standing in for the set theoretic definition, because it carries no semantic baggage from any common meaning.
It's no different than a doctor examining you and finding your condition "unremarkable." That might be an insult in daily life but it's the best news you can get in a doctor's office. Surely you understand this. It's a medical term of art.
That said, the question of whether the natural numbers can be "counted" in any meaningful sense of the word -- stipulating that technical conditions in formal set theory are not necessarily meaningful -- is a good one.
I don't think you can count past 200. You'd get bored. You can't count to a zillion. You just can't. It couldn't be done in the age of the universe. If counting is an activity that takes place in time, then a finite universe doesn't give you enough time to count any more than some finite number. There are 10^80 hydrogen atoms in the universe. That's a very small natural number. You can't count it.
What can it mean to count to 10^80? Mathematically you can count a set if you can order-biject it to a natural number, or in a more general context to some ordinal. Sure, you can count 10^80 that way. But that's just set theory. [Note that counting via numbers involves finding an order-preserving bijection, not just a bijection].
If you want to say that it means something other than formal set theory to count to 10^80, I'd like you to tell me what it is.
Myself I tend to be a formalist. I like set theory but I don't think it actually means anything. I don't think there are sets in the real world. It's the famous singleton problem. There is no such thing as an apple and "the set containing an apple." There's madness down that road.
I'm fascinated to read a little on Wiki about Bergson. "Bergson is known for his influential arguments that processes of immediate experience and intuition are more significant than abstract rationalism and science for understanding reality." I actually agree with that point of view. I'm opposed to scientism. I've got a lot of reading to do. I'm afraid I can't pick up those many volumes that have been suggested, but I will definitely Google around.
Apparently you are in such a big hurry to reply that you are not even bothering to pay attention to what I actually post. In this case, you are mixing up the first two definitions that I so carefully spelled out. The first one, which directly quoted @fishfry, is the one from set theory - not my set theory, but standard set theory - and if it helps, we can substitute "foozlable" as he just suggested (again). The second one - the one that I assume you are still criticizing - has nothing to do with set theory at all, as @fishfry helpfully pointed out a while ago. We simply disagree on whether "countable" always and only entails the ability to finish counting; you say yes, I say no.
I appreciate the sensibility that you have tried to bring to this discussion, but I still have to comment on this (again). I agree that neither you, nor I, nor anyone else can actually count that many things. However, this is irrelevant to what we have been discussing, since mathematics is the science of drawing necessary conclusions about ideal states of affairs. An immortal being in an eternal universe could actually count that many things (and more), and this is basically what I mean when I say that they are countable in principle.
Quoting fishfry
Yeah, I kind of opened the fire hose on you there; sorry about that. There is a handy online dictionary of Peirce quotes on a lot of different topics; going through a few of the most relevant terms might at least give you a taste of his thought on these matters.
Quoting fishfry
If the reals can be disordered, just do it. Can't be more than a few lines?
Take any real number, leave a gap, state where the gap is, and place it between any two other real numbers.
That's not right.
Everyone who is not being deliberately obtuse understands what countable means - it means you can count elements of the set. No one, unless they are being deliberately obtuse, thinks that this fact has any bearing on whether anyone would be willing to embark on counting all the members of a very large or even infinite set.
Now, will someone pleas count the number of reals that exist in the range (0, 0.0000000000000001)?
My point is that "the natural numbers" is defined in such a way that is impossible to count them all. No matter how many you count, there will always be more. The set of natural numbers is infinite and this means that it is impossible to count, uncountable, by definition.
Aletheist claims that it is logically possible to count them, therefore they are countable in principle. I don't agree because I see that they are uncountable, by definition, and therefore the idea that they are countable is contradictory.
Quoting aletheist
As I said, I accept the first definition, so long as we adhrere to the principles offered by fishfry. This definition has nothing to do with counting whatsoever, it is completely unrelated, that's why fishfry offered a completely different word. I have no problem, as long as we don't ambiguate between the first and third. You, for some reason want to create a bridge of relationship between the first and third, so you've offered the second. I see this as an attempt to ambiguate, to create the means for equivocation. Also, you've already argued that foozlable means "potentially countable", and "countable in principle". And you've continued to argue this long after fishfry offered the means for complete separation. So you continue to make efforts for equivocation.
Now, if we maintain this separation, and put foozlable aside for now, It is apparent to me, that you do not actually believe that the set of natural numbers, as infinite, is uncountable in an absolute sense. You keep wanting to say that in some sense it is countable. But it is defined in such a way as to be uncountable, absolutely. You keep implying that you believe that the infinity of natural numbers is logically countable, or countable in principle.
Quoting aletheist
I really have no idea of what this means, what you are proposing here. As far as I understand what counting is, something must be counted in order to be counted, If you do not finish counting something then it is not counted. If you cannot finish counting it then it cannot be counted. If it cannot be counted, then it is uncountable. Are you proposing some type of partial counting? If so, then even if I accept this principle that something might be "partially countable", and that this is a meaningful principle, then you still need to produce an argument to support your claim that "partially countable" means the same as "countable".
This right here is where we disagree. To count something is not the same as to finish counting it. Being able to count something is not the same as being able to finish counting something. In other words ...
Quoting tom
We simply have different non-technical definitions of "countable." It is not the case that yours is true and mine is false, or vice-versa; they are just different.
Given the Naturals, I can count some of them, in order e.g. 999 1000 1001
Given the reals, please count the three members that come after 0.999... and tell me what they are, or even what the next number is so we know we have only two of them.
Anyway, there need be no counting in Zeno's Paradox. If you say the golf ball must arrive at the center point in order to make it to the hole, you just bought the whole enchilada.
As I said, your definition appears like nonsense to me. To be able to do something, is to be able to complete that task. Being incapable of completing that task, is failure. When failure is guaranteed, then the claim of being able to do that task is completely unjustified.
You claim to be able to count something, when failure is guaranteed, and this is an unjustified claim of being able. The fact that you can attempt a task does not justify the claim that you are able to do the task.
But we've already solved the paradox: it is merely a confusion between an abstract attribute and a physical attribute of the same name.
Since it is possible to prove theorems about abstract mathematical attributes, which have the status of necessary truths, we are misled into assuming we have a priori knowledge of the real physical attribute of the same name. We don't.
Can you count the number of real numbers between 0 and 0.1? If so, how many are there?
Can you count the number of naturals between 1 and 50,000,000, if not, how many are there?
I already answered this days ago, I don't see the relevance. Why do you keep asking?
How many are there? Can you count them? Or is it impossible to count the real numbers, making them uncountable?
Contrast that with the Naturals, which, by definition you can count. Just try it 1, 2, 3, 4, 5. How many was that?
As I said to you days ago, it's impossible.
Quoting tom
No, you can't count the natural numbers either, because they're infinite. That's the point I'm arguing with aletheist, they are by definition uncountable, because by definition they are infinite, and infinite is by definition endless, which is by definition uncountable.
The fact that you can be counting the natural numbers does not prove that they are countable. Does the fact that a person is walking on the earth, and claims to be walking around the earth, prove that the earth is walkable? We can only get to the conclusion which you and aletheist desire, through the fallacy of composition.
I don't think so. It's just a simple question: does the golfball have to arrive at the center point before it can make it to its destination? Common sense says yes. Infinite regress appears.
Note that the regress is headed back to the starting point, not the destination.
If one takes the position that duration (real time) is consciousness that endures - which is precisely what we experience - then it is difficult to explain the notion of discrete. Are we constantly dying and being reborn in some discrete firm of unknowable duration? It would seem that continuity more accurately reflects our actual experience.
There are experiences (memories) that are shared and those that are not. It is quite a task to separate those that are Real from those that are not. Best to just accept them as all being Real, with different attributes of firm. My dreams are very real to me. If I relate them, then they also become real to others though in a different qualitative sense. What all experiences share is the essential quality that they are memory of some sort.
Yet nothing physically infinite happens, and what motion is possible is determined by the laws of physics alone, and not by the necessary truths about an abstraction that bears the same name.
Common sense dictates that Zeno's mistake was to PRESUME that a certain mathematical notion called "infinity" is physically relevant.
Why are you asking me? You and I agree that the real numbers are not countable.
Quoting Metaphysician Undercover
See, this appears like nonsense to me. One can be able to do something on an ongoing basis, such that whether one is able to complete that task is irrelevant. I am able to be thinking about elephants, so elephants are thinkable. I am able to be breathing earth's atmosphere, so earth's atmosphere is breathable. I am able to to be walking on the earth, so the earth is walkable. And I am able to be counting the natural numbers, so the natural numbers are countable.
Zeno's intentions aside, you aren't solving the paradox by saying this. You're merely restating it. Infinite regresses appear from time to time. Philosophers usually take them as a sign that something's wrong. Consider Frege's and Quine's reactions to the regresses they discovered. No one says, "Oh that's just a fluke of the mind... I'll proceed on as if I never noticed that."
No. We pay attention to regresses because philosophy is the domain where we're free to take note of such impractical doo-dads.
If you're planning a trip to the Grand Canyon, feel free to ignore Zeno's Paradox. It has nothing to do with your trip. And by the way, why are you going this time of year? Don't you know the road to the North Rim is probably closed?
So, you agree you can't count any interval of the real numbers.
So, you agree you can count any (not too big or you'll get bored) interval of the natural numbers.
Given any interval of the natural numbers, you can calculate how many natural numbers are in that interval, even if the interval is too big to actually count in a lifetime.
One of these infinities is bigger than the other, much bigger. In fact the measure of the natural numbers on the continuum is zero.
In my view (and Peirce's), they are either real (full stop), or they are not. Something is real if and only if has properties that do not depend on what any one person or finite group of people think about it. The properties of your dreams depend entirely on what you think about them, so they are not real by this definition.
The issue is not defining properties independent of a person or group of people, it is things having properties independent of what any person or group of people thinks about it.
This then goes into defining what is a property that is independent of that which is defining the property. From my position, everything is simply "fields" given definition by consciousness. In this regard, the internal dream field shares similarities with external fields but are different in how they are shared.
What I think is that it is necessary to assume that the entire physical world is reborn, comes into existence anew, at each moment in time, and this is discrete existence. But as I said, the soul provides continuity, so it is not the case that we are constantly dying and being reborn, the soul is immaterial and not part of this discrete material existence. So as living souls, continuity is our actual experience. But when we deny dualism we suffer from the illusion that the physical world is continuous as well as our own existence as living beings.
Quoting aletheist
You appear to be making a category error. "Counting" is an activity of the subject, "countable" is a property of the object. In order to deduce from the activity of counting, what it is that is countable, requires that you identify what it is that is being counted. If; it is just a part of the set of natural numbers which is being counted, then it is that part which is countable. If it is part of the earth's atmosphere that you are breathing, then it is that part which is breathable. If it is a part of the earth's surface that you are walking on, then it is that part of the earth's surface which is walkable. If you proceed from what is known about a part, to make a conclusion about the whole, then you commit the fallacy of composition.
Quoting tom
You did not describe the infinity of the natural numbers, which is that they continue forever, endlessly. And no, the infinity between two real numbers, (no matter how large or small those numbers might be), is no bigger than this infinity. They are both infinite. One is not a bigger infinite than the other, that it nonsense.
More nonsense. You are the one who wants to define "countable" entirely on the basis of whether it is actually possible for a subject to finish "counting" the object.
Quoting Metaphysician Undercover
When I say that elephants are thinkable, or that earth's atmosphere is breathable, or that earth's surface is walkable, or that the natural numbers are countable, I am not reasoning from part to whole. I am not referring to any particular part of each thing, I am stating a general property of each thing. Elephants in general are thinkable, earth's atmosphere in general is breathable, earth's surface in general is walkable, and the natural numbers in general are countable. This is a perfectly legitimate and common use of language.
That's not true, because I've claimed that the object, being the set of natural numbers is uncountable by definition, that means nothing, not even God could count it.
Quoting aletheist
You are making unjustified assertions, and this is perfectly legitimate, common use of language. But if you want to prove any of these assertions, you need to justify them. And you cannot prove a general conclusion about the whole, by demonstrating that it is true of a part. So if you prove that a part of the set of natural numbers is countable, this does not prove that the whole is.
And if you make the unjustified assertion that the natural numbers are countable, we have to juxtapose this to the contrary, and justified claim that the natural numbers are uncountable.
Why would I want or need to "prove" or "justify" a generic definition of "x-able" that I have (repeatedly) stipulated?
Quoting aletheist
But this renders all sets with a largest value as uncountable.
Your opinion is duly noted. :-}
It's a very important result in mathematics. The continuum has the cardinality of the power set or the natural numbers. It's a much bigger infinity.
You are of course free to deny knowledge and maintain your willful ignorance.
What do you mean by "bigger?" Bearing in mind that there are countable models of the reals? https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem
As is the unfortunate custom in this thread, a word with a highly technical meaning in mathematics is being conflated with the same word in its everyday usage. In math one set is "bigger" than another if the smaller set can be injected but not surjected to the larger. But that does not actually correspond to the everyday notion of "bigger," which the Lowenheim-Skolem result shows.
Along these lines ...
Quoting tom
What do you make of the Cantor set, an uncountable set of measure zero?
https://en.wikipedia.org/wiki/Cantor_set
Bijection does not preserve measure. You can see that by simply multiplying each element of the unit interval by 2. Now you have a bijection between sets of measure 1 and 2, respectively.
Actually, the set of natural numbers is countable by definition, as in mathematics a countable set is defined as a set with the same cardinality as some subset of the set of natural numbers.
Of course, if by "countable set" you mean something else, then the disagreements here are just a case of talking past each other.
OK, if that's what you think, then maybe you could explain how one boundless or endless (infinite) thing is bigger than another. I'd be very interested to see this explanation.
Quoting Michael
Actually we've been through all this. It's taken a few days, and numerous pages, but we haven't agreed on any conclusion. Here, you are defining "countable" in relation to any "subset of the set of natural numbers". Aletheist kept wanting to commit the fallacy of composition, assuming that what is true of the part is true of the whole. So we cannot assume that because a subset of the natural numbers is countable, then the complete set is countable. And, since the set of natural numbers is defined as infinite, boundless, endless, then by that definition, it is impossible to count, and therefore uncountable.
Indeed, but @Metaphysician Undercover only accepts this definition if we divorce it from the colloquial meaning of "countable," which according to him applies only to finite sets small enough that someone can actually finish counting all of their members. He would prefer a different term altogether for the standard set-theoretical concept, like "foozlable" as suggested by @fishfry.
Maybe we should just go with "denumerable" or "enumerable." My dictionary says that these two words both always and only mean "capable of being put into one-to-one correspondence with the positive integers." Now I wish that I had looked them up several thread pages ago ... :(
As someone already pointed out, countability is defined as the ability to place a set, whether finite or infinite, in one to one correspondence with the set of natural numbers or some portion of it. This is logically just the very same as to be able to count the elements of the set.
So no complete infinite set (if such a notion even makes sense) can be actually counted, obviously, any more than any complete infinite set can be actually placed in one to one correspondence with the complete set of natural numbers. There seems to have been much very silly confusion and argument over this very point.
I do not understand the concept of being reborn at each moment in time (exactly what is this moment and what is happening in between?), But do I understand why one should even entertain such a concept when all is well and good by just acknowledging what one is experiencing, i.e. a continuous experience of consciousness which we experience within a duration. Such a point of view helps things along mightily in understanding the nature of existence while admittedly it may be too obvious and easy to consume for those who are looking for academic debates about stuff.
If indeed we are all just accumulated memory within a universal field, with the brain acting as a reference wave that perceives the holographic-like images within this field (as opposed to someone storing images within it), then the soul is nothing more than the persistent wave pattern which we call memory coupled with the same consciousness that consumes it. Conscious, memory and the field are aspects of one. In Daoism it all begins as one, the Dao (consciousness) which becomes Yin and Yang (the wave) which then moves as Qi (energy). The model is straightforward. No need for discontinuities or paradoxes.
Everything falls into place very quickly. Traits, inherited characteristics, inborn abilities are all nothing more than persistent memory.. Nature this becomes very concrete and real and rather than discuss what is occurring between discontinuities and paradoxes, we can get down to the business of understanding life more fully in a manner that encourages full exploration of life.
By this I mean, a self-fulfilling journey into the arts, for without such a journey, all I am suggesting will seem like gibberish. One can surely understand nature without mathematics but I believe such understanding without experiencing art is not possible. I will reiterate, science confuses and muddles by utilizing symbols to replace life.
No, I think any finite set of natural numbers is in principle countable, it's the fact of being infinite which makes the whole set of natural numbers uncountable.
Quoting John
Do you think it is possible to count the elements of an infinite set?
Quoting Rich
Ever since Newton's laws, the discipline of physics has taken the continuity of physical existence for granted, it is a given. As such, continuity is apprehended as a necessity. It underlies the laws of physics. But in Aristotelian physics, continuity is assigned to matter, and matter is understood to have the nature of potential. As such, continuity is understood as possible.
When, as philosopher's, we come to understand the nature of intention and free will, we realize that the continuity of existence of any object can be interfered with, interrupted, even ended, at any random moment of the present, by means of a free will act. If any object can be annihilated at any moment of the present, then the continuity of existence of physical objects, at the present, cannot be taken to be necessary. This is why continuity must be classed in the category of potential.
Following this classification, that the continuity of existence at the present, is possible rather than necessary, we need to seek a cause of such continuity. Any potential which is actualized must have been caused to be actualized. This implies that at every moment in time, as time passes, there is a cause of existence, a becoming, or coming into being of each physical object. That is necessary to account for the assumption that the free will act can randomly annihilate the physical object at any moment of the present. The continuity of the physical object is not necessary.
Quoting Rich
I feel there is something inverted about this perspective, which I cannot quite put my finger on. Memory is a function of the continuity of the physical world. When things go to memory, they are held there by the continuity of this part of the physical world remaining the same through a duration of time. But this continuity is of the essence of potential, and must be caused to actually occur in the way that it does. So whatever type of thing, which you might infer the existence of, which must actually cause the physical continuity, it must be proper to the part. The problem is that continuity is proper to each object individually, each part, and not proper to the whole. A part may stay the same, in continuity, but the whole always changes. So continuity, and therefore its cause, must be sought by understanding the part rather than the whole. Rather than modeling the part as being derived from the continuity(as a field or such), the continuity must be derived from the part. So from this perspective, a similar thing to which causes memory, must also be the cause of continuous physical existence, but this should be found within the parts themselves, not within the "field".
You answered the question yourself in the previous paragraph. I think you are being unnecessarily pedantic about what is a self-evident and trivial point.
Of course it is not possible, either logically or actually, to finish counting an unending series. That is a matter of mere definition.
Didn't I say in my previous post that it is not possible, actually or logically (in principle in other words) to finish counting an infinite series. It seems obvious that this is the same as to say that it is not possible in principle to count all the elements of an infinite series or set.
But it is not impossible, as you say, to count the elements of an infinite set, any finite number of them can in principle be counted.
The set of natural numbers is a subset of the set of natural numbers. A countable set is a set with the same cardinality as some subset of the set of natural numbers. Therefore, the set of natural numbers is a countable set.
You seem to keep switching in some non-mathematical definition of "countable". But the term "countable" that is being used here is the mathematical term. Nothing about the term "countable" in mathematics entails the possibility that we could enumerate the complete set.
Is this the case, when physics uses discrete measurements to describe everything whether it be time, particles, etc. and then uses these discrete measurements to describe things? Physics has always taken the position of separate and measurable for practical reasons but such descriptions are just practical tools. Philosophers who take opposing views are quickly marginalized for not adhering to these mathematical descriptions. Even your description of time explicitly capitulates to discrete measurements by insisting on "moments of time". This is precisely the problem. The are no moments of time but philosophers adopt this point if view because t appears in equations. Further on you discuss parts of things, further concessions to discrete mathematical equations.
That there is no beginning, no end, no parts, no moments, cannot be represented in mathematical equations. Infinities and division by zero create havoc in mathematical equations, yet they constantly occur. Such situations should make it quickly apparent that equations are not useful for understanding nature. One must learn to use consciousness to penetrate consciousness, but when was Mozart or Van Gogh, or Eastern meditation (Tai Chi) ever part of a philosophy curriculum. All of this would provide real experience as opposed to awful, inappropriate symbols. Philosophers have been relying on tools of science which are simply incorporate for penetrating nature which is why Bergson and Bohm should be read by philosophers who are interested in understanding the nature of nature.
I never switched definitions. I maintained my non-mathematical definition, which was "capable of being counted" (#3), and this was contradictory to the mathematical definition you've provided (#.1), according to the fact that an infinite number is not capable of being counted. So long as we keep the two completely separated, and there is no ambiguity as to which one we are using, and therefore no equivocation, then there is no problem. But aletheist wanted to bridge the gap between #1 and #3 with a #2. The proposal was that we could qualify #3 in a particular way, to produce #1, such that #1 would be a particular type of #3; we could say that #1 is "#3 in principle", "#3 potentially", or " # 3 logically".
I have maintained that the two are inherently incompatible, contradictory on this point of infinity. However there is another possible route of reconciliation which we haven't explored yet. It is possible that #1 is the more general, and that #3 is a particular type of #1. This would require a description of what "countable" actually means in #1.
Here are the two dubious principles which I see are involved with your mathematical definition, which need to be justified. First, what does it mean to take a set, the set of natural numbers for example, and produce a subset which is the same as the set, and say therefore, that the set of natural numbers is a subset of the set of natural numbers. One is the "set", the other is a "subset", yet they are the same with two names. The names refer to something different. What is the reason for giving the same thing two distinct names? What I see is that the set is the whole, and a subset is a part. To represent the set as a subset is to class it as a part. But it is false to represent the whole as a part of itself, because it is not a part of itself, it is the whole of itself. So this is a category error, to make the set a subset of itself, without having some means to distinguish between the whole as whole, and the whole as part. To make them equivalent is category error.
The other dubious principle is the cardinality of the infinite set. If the set of natural numbers is a subset of the set of natural numbers, then this subset has an "infinite cardinality". Judgement of cardinality is required in order to designate a set as countable. Therefore to judge the set of natural numbers as countable, requires a judgement of its cardinality, to ensure that it is the same as itself (the set must have the same cardinality as some subset of the natural numbers). How would you judge this cardinality?
Set A is a subset of set B if every element of set A is an element of set B. Nothing about the definition requires that A containers fewer elements than B. Such a set is instead called a proper (or strict) subset.
That the set of natural numbers has the same cardinality as the set of natural numbers is a tautology. As for other sets, I believe bijective functions are used to determine if two sets have the same cardinality.
I don't think "capable of being counted" is the same as "possible to enumerate the entire set". When I say that apples are edible – and by this I mean that they are capable of being eaten – I'm not saying that it's possible to eat every apple.
Of course, if by "countable" you mean "possible to enumerate the entire set" then obviously the set of natural numbers is not countable. But I don't think aletheist (or anyone else here) is saying that it is. As I said before, there's just a lot of talking past each other.
This is so only because in principle the set of, say, even numbers, can be placed in one to one correspondence with the set of both even and odd numbers.
But 'all of them' (in itself a meaningless term) could never in practice be so placed, any more than they could in practice all be counted. There just doesn't seem to be any reason why we could not keep placing them in one to one correspondence forever just as we could keep counting them forever. So, it is a meaningless exercise to compare the "size" of infinite sets; no infinite set has a size that is what is meant by 'infinite'.
OK, thanks for that good clear definition Michael. I was wrong to think of sets and subsets as parts and wholes, they are actually completely separate entities with a describable relationship.
Quoting Michael
The referred to tautology takes for granted that the set of natural numbers has a cardinality. If it does not have a cardinality there is no such tautology. Being infinite, I do not think that it is possible that the set of natural numbers has a cardinality. To be a countable set, according to the definition you provided, a set must have a cardinality. This could be the root of my disagreement with aletheist.
Quoting Michael
To say that all apples are edible is to say that each apple may be eaten. There is no apple anywhere which cannot be eaten. In the case of the natural numbers, by definition, there are numbers which are incapable of being counted. This is because no matter how high a number you take, there are always higher numbers. There will always be, necessarily, uncounted numbers. So your analogy is not good, because saying that all apples are edible does not allow that there are always some apples which necessarily cannot be eaten.
Quoting Michael
How could a set have a cardinality if it's not possible to enumerate that entire set?
I've studied the nature of time for very many years now, and I've read a lot of related material. Here is some speculation. For the longest time, I believed as you do, that time is continuous. It appears quite obvious that all divisions in time are artificial. We derive "the point" in time from our experience of the present. We know that the present is real because of the radical difference between future and past, so we assume a point in time which separates future and past. This is an abstraction, the point is abstract. We utilize this point, by moving it around, projecting it forward and back, to mark off particular durations, periods of time. The periods of time are completely arbitrary within what appears to be one continuous time without any such points in reality.
However, there is some "part" of time which is not arbitrary, and this is the present, as the division between future and past. We must consider this reality concerning time. This difference between future and past, we respect as real within all of our activities. So the continuity of time can be described as the continuous difference between future and past. And what we can see, if we look directly at time itself, is that future time is continuously becoming past time. The time designated as tomorrow (future time, 23/02/2017) will become yesterday (past time), as the future is continuously becoming the past. What about this division between future and past, which we've represented as a dividing point, and from which we've derived the highly useful "point" in time?
Now we can ask, what is actually occurring when the future is becoming the past. With reference to my prior post, and how we experience the physical world, and free will, we can conclude that the physical world is coming into existence, "becoming", as the future becomes the past. Consider that with respect to the physical world, you are looking backward in time only. If you turn around, and look ahead, toward the future, there is no physical world there, nothing, just predictions. There is what the Neo-Platonists would call Forms there, in the future, and these Forms are what is ensuring that the physical world which you see behind you, in the past, is consistent, and lawful.
But there proves to be an issue with the assumed continuity of time. We derive the continuity by looking at how time has passed. We produce an order which would start from the furthest back in time, continuing to now, and assign continuity to this. But in reality, the order of the real physical world is such that it begins now, at the present. However, time is passing, so it must begin again, and again, and again, at each moment. This necessitates that with respect to the physical world, there are real points in time, the points at which the physical world repeatedly comes into existence. It may be the case that there is a further continuity which underlies this, but with respect to the physical world, there must be real points in time. The continuity which we look at, is created by us. We look back toward the beginning of the physical universe, and produce a continuity from there until now. But this continuity has the real existence of the physical world backwards. The real existence of the physical world is such that it begins at the present, not at the so-called "beginning of time".
This is actually a great mathematical question. If you assume the Axiom of Choice (AC), then all sets have a well-defined cardinality that is the smallest ordinal that bijects to the set. If you take AC as false, then there exist sets that can't be well ordered, hence sets that don't have well-defined cardinalities. Or if they do, they're not comparable to the standard Alephs. I'm a little fuzzy on that point.
You have good insight into this. For a set to have a sensible cardinality, it needs to be able to be enumerated.
Quoting John
I don't know what "in principle" means. If we are in standard set theory, we can biject the naturals to the evens by mapping each natural n to the natural 2n. If we are not in standard set theory, then you'd have to say what the rules are for that system.
It doesn't make any sense to say that "in principle" you can do something that's legal in set theory. If set theory says you can do it, you can do it. If you are using some other framework for talking about numbers and sets, you have to say what that is.
To make this clear, I interpret the phrase "in principle" as indicating a lack of clarity in specifying what domain we are in. If we're in set theory we can biject, and if we're not, what are the rules? It's like sitting down to play chess and they won't tell you how the pieces move. Tell me what your rules are for defining functions between sets, and I'll tell you whether there's a bijection between the naturals and the evens.
I don't really understand this objection. I can say that no matter what apple I eat, there will always be apples that I haven't eaten. Therefore apples aren't edible?
So what exactly do you mean by saying that there are numbers which are incapable of being counted?
Because cardinality is defined in such a way that to have one does not depend on it being possibile to enumerate the entire set.
That the set of natural numbers has the same cardinality as the set of natural numbers just is that the natural numbers can be placed in a one-to-one correspondence with the natural numbers. We don't just take it for granted that it can; we can mathematically show that it can.
Thanks fishfry, it's rare to see a complimentary comment here. It's only taken me days to get to this point. Notice that the recognition that something is not quite right (insight) played a very small part in getting to this point, the big part was persistence in analysis, to determine exactly where the problem is.
Quoting Michael
Natural numbers are countable, just like apples are edible, but that's a generalization. The entire set of natural numbers, being a particular defined thing, is defined as infinite, and is therefore not countable. Apples are edible. The entire set of all apples, if it is infinite, cannot be eaten.
I am making a statement about the nature of being infinite, what it means if a particular thing is defined as infinite, not an inductive generalization about a thing being counted or eaten, such as numbers or apples. However, if we assert that the thing being counted, or eaten, is infinite, then we must constrain ourselves with respect to what it means to be infinite, when we go to make other assertions about those things, in order to avoid contradiction.
When you assert that all natural numbers are countable, this is an inductive conclusion. This conclusion contradicts the defined essence of the set of natural numbers, as infinite and therefore uncountable. When you proceed in your mathematical operations, from the premise that all natural numbers are countable, you proceed from an inductive conclusion rather than from the true defined essence of the set of natural numbers. And these two premises are contradictory. The defined essence takes into account what it means to be infinite, the inductive conclusion does not. Therefore when you proceed from the inductive premise you will inevitably produce false conclusions concerning infinities. The one I've already seen on this thread is that some infinities are "bigger" than others.
Quoting Michael
I mean exactly what I said, no matter how high you count, or even how high of a number you can name, there will always be higher, unnamed or uncounted numbers. That's the nature of being infinite. It means that the set of natural numbers is uncountable.
Quoting Michael
Cardinality is defined on Wikipedia as " a measure of the 'number of elements of the set'. It seems quite obvious that it is impossible to have a measurement of the number of elements in an infinite set. "Infinite" is not a number, nor is it a measure.
Quoting Michael
That is false. Being infinite, you cannot establish a bijection, just like you cannot count them. You might assume that if you could count them, you could place them in a one to one correspondence, but you cannot count them, so such an assumption is irrelevant. It's like saying if the infinite were finite, then we could do this. It's just a contradictory assumption. I am quite convinced that such a bijection cannot be done, it is a falsity. You say it can be mathematically shown, and it is not just assumed. Let's see the demonstration then.
Sorry, I got my terms mixed up. It's surjection, not bijection when it comes to infinite sets.
It seems to me that you're trying to argue against mathematical terminology without actually understanding the mathematics involved. Seems like a layman trying to argue against the physicist's definition of "charm" or "strange" when it comes to quarks.
If you have something constructive to say, then address the issues. I didn't come this far in this thread just to have you piss me off with ad hom. The fact is, as I explained, that you are dealing with inductive conclusions concerning "the natural numbers", but these inductive conclusions are inconsistent with the defined essence of "the natural numbers", as infinite. Therefore whatever you say about infinity is completely unreliable. Referring to functions only brings you deeper into inductive territory, without first recognizing that any conclusions you make concerning infinity cannot be respected
No, it is a deductive conclusion that is necessarily true, given the standard mathematical/set-theoretic definition of countable/denumerable/enumerable/foozlable.
Quoting Metaphysician Undercover
Not if cardinality/multitude is defined in a particular way that specifically pertains to infinite sets. For any set with N members, there is a "power set" that consists of all of its subsets, and that power set has 2[sup]N[/sup] members. For any value of N whatsoever, 2[sup]N[/sup] > N. Therefore, a power set always has more members than the set from which it is derived. This is true even if N is infinite; the power set of the natural numbers must have more members than the (infinite) set of the natural numbers. That is all it means to say that the set of real numbers is "bigger" than the set of natural numbers. I will grant that it is counterintuitive, but it is defined this way in set theory, where it is not problematic at all since mathematics is the science of drawing necessary conclusions about ideal states of affairs; it has nothing to do with actual states of affairs.
Quoting Metaphysician Undercover
Being able to finish counting them has absolutely nothing to do with establishing a bijection, or one-to-one correspondence. @Michael's subsequent reply is incorrect - bijection applies to infinite sets, as well as finite sets; it is a specific type of surjection.
I knew I should have stuck to my guns. MU's comment had me doubt my interpretation. Oh well. Thanks for your clarification.
He might very well understand it, he just refuses to accept it. He is committed to the presupposition that only the actual is real, so if something is actually/nomologically impossible, it just is impossible, full stop.
Not that it matters one jot due to the level of willful ignorance on display, but I think you were quite correct to use the term "bijection" = one-to-one and onto.
I'm no longer finding it amusing to follow the obtuse denial of well established mathematical truths, so I can't say for sure because I can't be bothered to tease out the remnants of sanity in this thread, but I get the feeling that a surjection will not suffice for your purposes.
Yes, I think you're right. aletheist clarified earlier that I was wrong to admit to being wrong.
Well, perhaps we'll find that definition. So far, "cardinality" is incapable of producing your desired conclusion, because it is impossible that an infinite set has a cardinality.
Quoting aletheist
So your claim is that there is a different definition of cardinality for infinite sets then there is for finite sets? Then what is true of finite sets in relation to cardinality is not necessarily true of infinite sets, because cardinality would mean a different thing.
Quoting aletheist
This doesn't solve the problem. It assumes a set with N members. An infinite set has indefinite members.
Quoting aletheist
I am ready to accept it, as soon as all inconsistencies and contradictions are removed. As of now, there is a necessity to resolve the incompatibility between the definition of cardinality, and the definition of infinite.
Quoting tom
Those who are "in denial" will always refuse to face the fact that their "well established truths" are actually falsities.
Quoting Michael
Are you ready to address my post now, and show me how an infinite set has a cardinality?
Not when "cardinality" is defined as a specific property of infinite sets.
Quoting Metaphysician Undercover
There are no inconsistencies or contradictions within the hypothetical realm of mathematical set theory.
Quoting Metaphysician Undercover
Pot, kettle, black.
A one-to-one relationship between sets e.g. from the natural numbers onto itself (i.e. a permutation) is not a one-to-one function, but a one-to-one and onto function. An easy slip-up to make when moving from the language of relations to the language of functions, wile being brow-beaten by unreason.
As explained here, "Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets."
So, if there is a bijection between elements of two sets then these two sets have the same (and so have a) cardinality. In the case of finite sets, the cardinality is equal to the finite number of elements in the set. In the case of infinite sets, the cardinality is a stipulated aleph number, which in the case of the natural numbers is ?[sub]0[/sub].
Which is why your question doesn't make sense. Infinite sets have a cardinality by stipulation. As aletheist explains above, cardinality is defined in such a way that infinite sets have one.
And following from this, a countable set is defined as a set that has the same cardinality as [s]the set[/s] a subset of natural numbers. Which, if the notion of cardinality is so problematic for you, is just to say that a countable set is defined as a set that has an injective to [s]the set[/s] a subset of natural numbers. Which, if the notion of bijection is so problematic for you, is just to say that a countable set is defined as a set upon which we can apply a particular formula to each member and uniquely pair it to some member of [s]the set[/s] a subset of natural numbers.
OK, so there is a different definition of cardinality for finite sets then there is for infinite sets, the former relates to bijection, the latter to stipulation. Which one is used in the determination of "countable"?
Quoting Michael
Yes, well I already went through this with aletheist. Aletheist produced a definition of "countable" according to which, no finite sets are countable. And that's what you have done here. The "cardinality" of the infinite set is not the same as the "cardinality" of the finite set. If "countable" is defined relative to the cardinality of the infinite set, then by definition, the finite sets are not countable.
It appears like you are unwilling to accept the fact that there is a substantial difference between a finite set and an infinite set. So you are unwilling to accept that what is true of the finite set is not necessarily true of the infinite set, and vise versa. You desire that the same principles are true for both finite and infinite, and so you are attempting to stipulate that this is the case. However it is not the case, and this appears to be a real problem for you. It is a problem because you will proceed to produce all kinds of false conclusion concerning the infinite, such as Tom's insistence that one infinity is "bigger" than another.
No, it's the same definition and both relate to bijection. It's just that the cardinal numbers used are different. In the case of finite sets we use the natural number that is equal to the number of elements in the set and in the case of infinite sets we use stipulated aleph numbers.
Quoting Metaphysician Undercover
Sorry, I worded it wrongly above (although worded it correctly earlier). A countable set is defined as a set with the same cardinality as some subset of the set of natural numbers (which, again, includes the natural numbers themselves).
It's the same definition of cardinality: "Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets."
It's just that the value of the cardinality is different. We can't use natural numbers to denote the cardinality of infinite sets, just as we can't use natural numbers to denote the numbers greater than 0 but less than 1. So we use a different notation: in the latter case, fractions or decimals; in the former case, aleph numbers.
It might be worth noting that Cantor proved that any interval of the Reals [a,b] cannot be placed in one-to-one correspondence with the Naturals, before he developed the idea of cardinality.
So, instead of getting bogged down with definitions, the remarkable discovery that there are more Real numbers in any finite interval than all the Naturals, indicates a fundamental difference between the continuous and the discrete.
Of course, this difference was analysed further and more remarkable discoveries were made, but it is the distinction between the continuous and the discrete that is of fundamental importance.
Another remarkable feature of the continuum is that in any interval [a,b] there are an uncountably infinite number of transcendental numbers.
You're getting too tied up with the term "countable". As suggested above, just consider the term "floozable". A floozable set is a set with the same cardinality as some subset of the set of natural numbers.
And I'm not sure what you mean by limited infinity.
The set {1,2,3,4,5} has the same cardinality as {6,7,8,9,10}, so we should call it foozable? How do you calculate the cardinality? Do you fooze the set, or do you perhaps count it's members?
As I've said earlier, I am satisfied with two completely distinct definitions, but there are always those who what to bridge the gap. Furthermore, I believe that if we adhere to this separation, mathematics will be rendered useless, because we will not be able to use it to actually count or measure anything. If the principles of mathematics have no relation to what it means to actually count, or measure something, then what good are they?
So now you've introduced a further problem, despite your protestation, you've given "cardinality" a new definition, one quite distinct from that used when dealing with finite sets. As you recall, from Wikipedia, cardinality was defined as " a measure of the 'number of elements of the set"'. Now you "measure" the infinite set in relation to aleph numbers, instead of in relation to natural numbers. So I need a demonstration that this is a valid form of measurement. An arbitrary determination is not a valid form of measurement, so show me that you can actually measure something with aleph numbers rather than just making arbitrary judgements. We demonstrate the validity of the natural numbers by actually counting and measuring things
This right here is precisely the reason why we have been at such loggerheads throughout this discussion (and others). As I keep saying over and over, mathematics is the science of drawing necessary conclusions about ideal states of affairs; it does not pertain to anything actual, except to the extent that we use it - with varying degrees of accuracy and success - to model the actual. You have an idiosyncratic metaphysical prejudice that requires something to be actually possible in order to be considered possible in any sense. Again, your worldview is too small; there is much more to mathematics than merely counting and measuring things, and the value of pure mathematics - like that of pure science - is not limited to its practical usefulness. Do not block the way of inquiry!
The way to produce, and increase accuracy, in modeling what is real, reality, is to determine and exclude as possibilities, those "ideal states of affairs" which are actually impossible. Without moving to exclude those ideals which are impossible, we have no way to increase accuracy and success.
Quoting aletheist
If you want a metaphysics which allows that anything is possible, you go right ahead and adopt that metaphysics, but it's not for me. I would prefer to exclude things which at first glance may appear to be possible, but which are later shown not to be possible, as impossible. If that makes my "worldview too small" for your preference, then so be it. And I don't want to spoil your party, but the way to succeed in inquiry is to narrow the possibilities, by eliminating unjustified possibilities.
It is a mistake to treat accuracy and success in the actual world as the only legitimate objectives of inquiry. For one thing, it is inconsistent with what most people mean when they talk about "ideals."
Quoting Metaphysician Undercover
It is a mistake to confuse mathematics with metaphysics. Many things are possible within mathematics that are not actually possible. I also happen to believe that there are real possibilities that are not actually possible, but that is not at all the same thing as allowing that anything is possible.
Quoting Metaphysician Undercover
It is a mistake to block the way of inquiry by ruling out possibilities too hastily. I would prefer not to exclude things which at first glance may appear to be impossible, but which are later shown to be possible. The key is to formulate retroductive conjectures that are amenable to deductive explication and inductive evaluation through experimental testing - often in the actual world, but sometimes in the imagination, as for example within mathematics.
I think it is a bigger mistake to write off the great works of Georg Cantor and Nicolas Bourbaki (that makes at least 8 geniuses) on the basis of your personal inability to comprehend the first thing about it.
That does not follow. It must be proved. That's Cantor's theorem. Well worth looking at since it's a beautiful little proof that gives us an endless hierarchy of transfinite cardinalities.
What exactly is it that you think I am not comprehending? Sincere question, I am eager to learn.
Quoting fishfry
If it is a theorem that has been proved, then it follows, does it not? What am I missing?
OK, you can consider the individual who produces fictitious fantasies to be successful, I have no problem with that, it may be a pleasant and fulfilling activity. But to consider such fantasies as metaphysical successes, I will not agree with you there.
Quoting aletheist
We are talking about the nature of the infinite here, and that is a metaphysical issue. If mathematics treats the infinite in a way that is metaphysically unacceptable, then we have an epistemological problem. Either the metaphysics is wrong, or the mathematics is wrong. But I'm not about to adapt my metaphysical principles just so that mathematicians can be understood as correct in the way that they deal with the infinite. If you are convinced that mathematicians are correct, then please justify their method of measuring the infinite.
Your full quote earlier was:
Quoting aletheist
There is only one way to read this.
* For any value of N whatsoever, 2^N > N.
* Therefore, a power set always has more members than the set from which it is derived. This is true even if N is infinite.
That is flat out false. You said "therefore ..." and that's wrong. Surely you see that. The correct statement is: "And in fact we can prove that this holds for infinite sets as well." In fact the very definition of 2^N needs to be changed to make this work. Rather than talking about natural number exponentiation, we now change the meaning to redefine 2^N as the collection of subsets from a set of cardinality N to the set 2.
There's no "therefore" in this. You need to make a new definition of the notation and then prove a theorem. You started with the expression 2^N meaning natural number exponentiation, then changed definitions in midstream to redefine 2^N as the powerset of N.
Thanks for yet another helpful clarification.
I may have been responding to you, but I thought it obvious I was not referring to you!
Ah, okay; it was (obviously) not obvious to me, since I generally assume - unless there is a clear indication otherwise - that a reply to one of my comments is directed at me. Thanks for clarifying.
I would agree that the future is much different in that it is a concept rather than a concrete event in memory. In memory, we have a possiblity of an action which defines the future. In regards to past and present things get much more problematic. Both are simply memory being apprehended. There is only a qualitative immediacy difference between that which feels like now and that which feels like before. I am watching a TV show but that apprehension is really what is known in my memory add qualitatively more recent than the show I watched yesterday. There is no hard line between any aspect of memory including an action with intent into the future. There is no reason to make one, it just muddies the water and creates problems in explaining such a line.
Adopting this approach creates a single aspect of life called Memory that endures in duration (time). An utterly continuous flow pushed on by consciousness (or Bergson's Elan Vital).
I always get a little uppity when people try to dismiss Zeno's paradoxes with the fact that an infinite series can have a sum. It misses the point entirely. And the paradox can even be worded to INCLUDE the summation idea within itself. If I tell you I want to walk 10 feet and that I'm going to walk 5 feet first, and then 2.5 feet, and so on... Haven't I just implicitly stated that I believe an infinite series has a finite sum? Namely the 10 feet I talked about at the beginning? It's just lazy to think that the idea of an infinite series is actually a good response to Zeno's charge of the impossibility of motion.
Just stating that an infinite series can have a sum, as some have done in this thread is not enough to resolve the paradoxes of Zeno or the other examples on that wiki page, in fact Thomson's Lamp is a brilliant example of just how ineffective that argument is, at least in this case.
I think the question that ought to be asked, in light of Zeno's criticisms of the pervasive idea of a divisible world, is whether motion is a supertask, or not.
Is motion a supertask?
Then, if we can answer that question, we might move on to the possibility of supertasks themselves, about which there has been much disagreement.
Further, I'd like to point out that Zeno's Paradoxes can be tweaked to not only attack the continuity and coherence of space, but also time. As has been alluded to already in this thread, time and space are central to Zeno's line of dialectic, and dealing with time takes us away from the pesky and ultimately fruitless Planck length explanations that crop up just as regularly as the sum-of-series ones do.
An example: Suppose you wanted to microwave your frozen tv dinner. You set the timer for one minute (this is one of those fast cooking dinners that you love so much), and then you wait. But before you wait a full minute, you have to wait a half minute, and before a half a quarter, and so on all the way down... And in the end, of course, you end up not starting at all, because there is no smallest amount of time that you can actually wait. And then you go hungry.
I've seen it argued that Zeno's Paradoxes are an indication that space is not continuous, but I haven't seen the same said of time. Maybe I just missed it?
And finally, I think it might be worth examining two of the modern "successors" (I use the term loosely) to Zeno's paradoxes, in the Thomson's Lamp Paradox and the Ross-Littlewood Paradox, both of which can be found in the wiki link I provided.
Why is motion a supertask rather than a hypertask?
It might be indivisible at a certain scale, but it's not indivisible at every scale. There is a half way point between the start of a 100m line and the end, and this is true even if we don't plot it, which is why I don't understand aletheist's and apokrisis' objection at the start.
But as for space (or at least motion) being indivisible at some fundamental scale, I'd agree. The paradox is avoided if space (or at least motion) is discrete rather than continuous. And I believe atomic electron transition is a known example of discrete motion in nature.
While I welcome your approach and think that it is among the more promising ways of looking at the problem, I must object to the remark about "missing the point." The problem is that most statements of Zeno's so-called paradoxes not only fail to make your points, but fail to make any cogent points, as has been largely the case in this thread.
The point about the convergence of infinite series, to which you take an exception, is an effective response to those statements that boil down to the thesis that an infinite sequence of events necessarily, as a matter of logical reasoning, takes an infinite amount of time. But I agree that the response is often given reflexively to statements that either rely on different assumptions or are so vague that one cannot confidently make out their core assumptions.
So I would set aside the two questions that you formulated - is motion a supertask? and are supertasks (metaphysically?) possible? - as open questions that, prima facie at least, are not incoherent or trivial. Other things that you mention, such as Thompson's lamp, might actually be less problematic than you think, being ultimately language problems rather than problems of metaphysics.
But anyway, if you want to talk about the point, a good way to start would be to give a crisp statement of the alleged paradox.
Supertask is defined like this "a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time". Since you are asking whether a supertask is possible, I would say no. A countably infinite sequence is one which is endless, never completed, according to the definition of "infinite". Any operation requires a duration of time in order to occur. Therefore in a finite amount of time that sequence of operations would not be completed. A supertask is logically impossible.
Even if we are to assume an operation which requires a zero duration of time, then there could be an infinite sequence of these operations, but it would be in a zero amount of time. So to make an infinite sequence of operations in a finite period of time would require inconsistency within the sequence of operations, some operations taking time, to make a finite period, and some operations taking no time, to make an infinite sequence.
This is why I find the use of the aforementioned geometric series to address the paradox to be nothing more than trickery. It assumes from the start that it takes a finite amount of time to travel some finite distance (e.g. 10 seconds to reach the half way point), and then extrapolates from there. But obviously if your reasoning assumes that it takes 20 seconds to get from A to B (which you have done if you've also assumed a constant speed), then you're going to conclude that it takes a finite amount of time to get from A to B.
So instead of halving the unit of time for each successive half way point, why not double the unit of time for the previous half way point (e.g. by defining a new unit of time for each successive half-way point and considering that to be the unit that is used to measure the time spent)? The logic is the same, but the maths doesn't work out the way the "solution" wants it to.
E.g. when considering 0 - 0.5m, define the time as 1 unit. But then when considering 0.5m to 0.75m, define that time as 1 unit and so the time from 0 - 0.5m is 2 units, and so on. What's the sum then?
But even then, the "it would take an infinite amount of time" claim isn't really what the paradox is implying (contrary to my account of it at the start). It's more concerned with the logic of actually being able to make a move at all.
If you are thinking of discrete quantum states of electrons in an atom, that is not an obvious example of discrete motion (except in a generalized sense of "motion" as "change").
Why not? The electron's position is a value in its quantum state. So going from one quantum state to another involves [or can involve] going from one position to another.
No, it isn't.
We could begin with the assumption that there is no such thing as a finite amount of time. I think this is a reasonable assumption, and those who argue that time is continuous would agree. Any application of a point, to divide time, is an arbitrary application, and is not actually being applied to any real point in time, so there is no real indication as to where that point actually is.
Quoting Michael
Now I don't understand what you are doing here. A unit of time is set out, determined, by some physical activity. You cannot just randomly change your unit of time, so that the same activity which takes 1 unit of time will later take 2 units of time, and then 4, etc.. What kind of measurement of time is that? We will just end up with an infinite amount of time in a finite motion, which resolves noting.
I mean something akin to going from saying "0.001 second" to saying "1 millisecond". We just define new units of time (e.g. "nanosecond", "picosecond", etc.) so that the time taken is always considered in natural numbers and not in fractions/decimals.
What this would require is a starting point, a definable, discrete, unit of time. Where would you get that from? An arbitrary designation would not do, because it would be just like saying we're making 1/64 our smallest divisible unit, arbitrarily. We would need an actual smallest unit, demonstrably indivisible, as the starting point. The Fourier transform indicates that the smaller the period of time, the more uncertainty there is in determining it, so you would have to get beyond that problem.
This just unnecessarily maintains the paradox. There is no half at any scale. There is just a fleeting representation of such which is necessitated by a desire to measure.
Quoting Metaphysician Undercover
There is no such thing as zero duration. If there was, then the flow of duration (time) would have to stop and then what. Stop for how long? How does it restart? Duration (real time) is continuous and heterogeneous. It never stops and cannot be seen as stopping. Scientific time (clock time) is just a movement in space (not real time) that is symbolic and is used to approximately establish simultaneity. This is something different and shouldn't be given ontological significance. Doing so leads to all kinds of paradoxes such as those associated with Zeno's and Relativity's.
If there are paradoxes one must immediately look at issues with assumptions in order to resolve them. In this case the assumption is that time is divisible and homogenous and that is what Bergson challenged.
Sure there is. If the object is to travel 10 metres then it passes the half-way point after 5 metres. And this is true even if we're not measuring it.
So what are you saying? That it doesn't have to first travel 5 metres, or that it's incorrect to say that 5 metres is the half-way point unless we're measuring? Because I think both are ridiculous.
Actually, my suggestion was a change (operation) which requires zero amount of time. This implies that state A is simultaneous with state B, but are contradictory, such that state A changes to state B without any time passing. I agree that this is random nonsense, and incomprehensible, just like your description of "zero duration", but I was just trying to make sense of the proposed "supertask", which also appears to be nonsense.
Space and time must be thought of in a different way as not being divisible. An object doesn't travel half-way. It moves from here to there in one indivisible motion. There is no half in a continuously flowing and changing space.
We do not have to treat every halfway point as a discrete step in the motion from the start of that 100-m line to its end. We can traverse the one full interval (100 m) without individually traversing infinitely many half intervals (50 m, 25 m, 12.5 m, etc.).
I don't get this. You do pass the half-way point (after 50m). And you do pass the quarter way point (after 25m). And so on, ad infinitum.
Quoting Rich
As above, I don't get this. You do travel half the way before you reach the end. That's just a fact that's entailed by continuous motion. You don't just teleport from the start to the end. That would be discrete motion.
Yes, this is the whole issue. We are layering symbolic notions of divisibility onto a continuous flow, leading us to paradoxical concepts. We must stop using mathematics for describing life experiences. I saw Achilles moved from here to there. That is what happened. He didn't move half-way of anything.
But I didn't move half-way. I moved from here to there. In retrospect to may try to figure out what half-way might have been and you may be approximately correct with your measurements, but my motion was one motion as you viewed it and add I experienced it - the two being totally different.
To understand experiences one must understand from the point of consciousness, not via some mathematical symbol or equation. Observe what you see.
As you continuously moved from A to B in one smooth motion you passed the half way point, and did so even if nobody was measuring. What's so problematic about this?
Yes, you pass each of those arbitrarily identified "points"; but each instance of doing so is not a separate, discrete step in the continuous motion of traversing the entire 100-m line.
It order to relieve oneself of mathematical symbolism of life experiences, one must stop layering symbolic concepts and observe what one is experiencing. I run from here to there. I don't run half-way. This is a very important observation to make. No one runs half-way. I don't hear letters. I hear sounds. I don't hear notes. I hear sounds. Symbols are awful representation of experiences.
I still think that this is an overreaction. We can still use mathematics for describing certain life experiences, depending on our purpose in doing so.
It doesn't matter if you don't consider the movement to be in separate, discrete steps. The problem is that it has to pass through discrete points (which exist even if I'm not measuring; it would be absurd to say that there isn't a half-way point between A and B unless I'm there with a ruler).
There is a half-way point. If one object is 10 metres away from another then the half-way point is 5 metres between them, and this is true even if nobody is there to measure.
Symbols are not experiences.
It has nothing to do with how I consider it. The movement does not actually consist of an infinite series of separate, discrete steps. It is simply a single, continuous motion from the start of the 100-m line to its end. This is what I mean when I say that the line itself does not actually consist of infinitely many separate, discrete points; it is simply a single, continuous line.
It doesn't matter if the movement doesn't actually consist of an infinite series of separate, discrete steps. It still has to pass through an infinite series of separate, discrete points. There is a half-way point that must be passed before the end is reached, there is a quarter-way point that must be passed before the half-way point is reached, and so on ad infinitum.
We agree that the phenomenal experience cannot be modeled adequately by mathematics, or even by other symbols like narratives; but various other aspects of it can be - again, depending on the purpose of the model.
Only if all of those points actually exist, which is precisely what I deny. The line does not consist of separate, discrete points; it can only be modeled as having separate, discrete points.
Your claim, as I understand it, is that the line does consist of infinitely many separate, discrete points, and thus can only be modeled (or "considered') as continuous. This seems to be our basic disagreement.
Of course the points actually exist. There actually is a half way point between the start and the end of a 100m line. There actually is a quarter way point. And so on. Are you denying this?
Points and lines do not actually exist; they are mathematical abstractions that we use to model things that do actually exist, like objects moving from one place to another. A line is simply the path through space over time that an object would trace if it were to move with constant velocity. In that sense, the concept of motion is more fundamental than the concept of a line; and as such, the object's path through space over time is more accurately modeled by an unbroken continuum than by an infinite series of separate, discrete locations.
So you deny that there's an actual half way point between the start position and the end position?
I see how this makes sense with space, but I don't think it makes sense with time. With space it only makes sense to claim that there is a half distance if we can actually identify the real existence of that half distance, to say that the object travels that distance. So if we start with 100m we can mark this, and see that the object travels that spatial unity. We can mark a 50m unit, a 25m unit, and so on, and see the object travel these units. Inevitably there will be a point where we can no longer mark the distance, or observe the object travel it. So it doesn't make sense to speak of space in terms of divisibility like that.
Time however is different. Time is a concept derived from the motions of objects. It relates one motion to another. Because of this, it is not the property of any particular motion. This abstractness provides that it must be inherently divisible in order that we may apply it to ever faster and ever shorter duration of motion. So in the case of time I think we must always allow that even in the shortest identified time period, there is still a possible shorter time period, to provide us the capacity to identify even faster and shorter motions, in the future.
It depends on exactly what you mean by "an actual half way point." As I said, there are no actual points at all, if by that we mean mathematical (i.e., dimensionless) points. There obviously is a location on the continuous line that is equidistant from the start and end positions, but there is only a "point" there if we define and mark it as such for some particular purpose, such as measuring. The object would "pass through" any point that you wish to define and mark on the line - but that act of defining and marking a point does not somehow create a separate, discrete, intermediate step that the object must now take in order to get from one place to the other.
Said another way, the object's motion comes first from a logical standpoint. Drawing a line that traces the object's path, and then defining and marking whatever points on that line serve whatever purpose we may have in doing so, only comes afterward.
This seems like you're being unnecessarily pedantic. But fine, I'll use your terminology. There is an infinite series of discrete locations between A and B that an object in continuous motion must actually pass as it moves from A to B – the half-way location, and before that the quarter-way location, and before that the one-eighth location, and so on.
Quoting Rich
This, incidentally, does not appear far off from what Zeno was arguing for in the first place. Would you consider yourself a Parmenidean? Maybe a Neo-Parmenidean?
Please see my (second) previous response. You can define and mark as many discrete locations between A and B as you like, but this does not in any way affect the continuous motion of the object from A to B, which is logically prior to that mathematical exercise.
I agree that most of the time discussions on this topic tend to descend rather quickly and that’s what I was trying to point out, but you’re quite right. The actual questions raised by the paradox are rarely ever even addressed.
The language problem of Thomson’s lamp: Yes, this is exactly what I’m getting at, that the profundity of Zeno’s paradox (as well as Thomson’s) don’t lie in the realm of mathematics, but in logic/language. This is the point that I feel is often missed.
A form of the paradox that I like is this (from Wikipedia):
From this, I think it's easy to see that the issues that can be taken with the paradox are issues of logic, not of mathematics and especially not of sums of series.
What does it mean for a motion to be "complete"? Is motion made up of "steps"? These are the core issues that the paradox is getting at.
And before someone brings up physical properties of space and the Planck length, this same argument can be applied to time as well. I find that it is not as contentious a statement to say that time is continuous when compared to space.
This version takes it out of the physical realm and makes it a pure thought experiment. How would one deal with this version of the paradox?
If your statement is true, then the next question is whether motion is a supertask. And if it is, doesn't that mean motion is logically impossible?
Most of my views and approach parallel those of Bergson.
These seem to be metaphysical questions, not questions of logic or language. There's nothing logically inconsistent or ambiguous about supertasks (and this is where mathematical treatment of convergence comes in). But one can still ask the questions that you ask as questions of metaphysics (informed by physics).
Thompson's Lamp, on the other hand, as well as a number of other such paradoxes, including the Bernardete paradox that you brought up later, are just logical puzzles. The key to their solution is that their premises are either inconsistent (Bernardete) or incomplete (Thompson). In the former case we can conclude that the premises cannot describe a possible state of affairs, which dissolves the paradox. In the latter case the problem (necessarily) does not have a unique solution, which again renders the seemingly surprising result as inevitable.
True, but in order to progress to a logical analysis, I think metaphysically defining our subject is a worthy cause.
Quoting SophistiCat
But in comparison, wouldn't Zeno's paradoxes just be a version of this? Zeno just happened to target physical phenomena as his subject, but isn't his reasoning and the scenario he cooked up just as much a logic puzzle as the other two paradoxes you and I mentioned?
Can't we use the same methods of investigation, discussion, and analysis for all three?
I've always gained more insight from looking for the hints of things he sort of got right or was on the right track about, than trying to defeat the paradox and move on.
The marks in space themselves are also symbolic since nor cannot truly divide space with a mark.
Time, or Duree as Bergson called it to avoid confusion, is not created by motion (this is the scientific time of a repeatable motion in space), but is a feeling that we capture via existing. It comes from consciousness not repeatable movements. I exist and feel my existence flowing as a duration whether or not can see the sun rise and set, or hear a clock. Real time is a psychological feeling of enduring in memory.
No, I don't think that motion is a supertask, I think a supertask is an impossibility. I do not believe that motion is impossible though. We observe motion. What I believe is that motion is not well understood. That's what Zeno's paradoxes indicate, that we do not have an adequate understanding of motion, it was this way back then, and it remains so, now. We like to think that human knowledge of motion has greatly advanced with the special and general theories of relativity, and mathematical equations but we really persist in an extremely inadequate understanding of the relationship between space and time. As is evident in modern metaphysics, we have a woefully deficient understanding of what it means to exist, and motion is what existing things do. Until we understand what existence is we will not understand what moving is.
,Quoting Rich There is always substance though, a surface which we mark, or a ruler, or some such thing. So we measure space by referring to material substance, but we can only go so small with material substance, that is the point.
Quoting Rich
I do not believe that we get a sense of time simply from existing. I think we get the experience of time from sensing things. If you go to a quiet place and meditate, you can lose track of time. I do believe that you always know that time is passing, regardless of any sensing, but this is not "time" as we know it, i.e. time as a measure of duration, and maybe this is why you are distinguishing it from "scientific time". I do believe that we must recognize two distinct concepts, but time without measure needs to be better defined if we are going to call this "time".
But if you examine it closely, you are not cutting space. The mark simply dissolves into space as more precision is required. There is no materiality but there a continuum of substantial or density of the underlying field. The is simply no way to create units within continuity and if one tries to, out pops Zeno. There is no way to say it other than mathematical ideas and symbols do not carry any underlying ontological weight. So what do you use instead? Bergson said intuition in the sense of conscious penetrating consciousness. Actually, this is the heart of meditation. One just needs to practice the art of peeling away the layers of the onion. A totally unique skill.
Quoting Metaphysician Undercover
When I examine time, all I sense is a feeling of flow memories. I don't feel and units of measurment. Time sometimes feel like it is passing slowly and sometimes quickly and sometimes it seems to disappear into something else when I am dreaming or call unconscious, this last experience being particularly interesting.
The problem though is that we actually are marking things, with a ruler and other forms of measurement. But when we think about space in our minds, we think about dividing it, making geometrical figures, and whatever, this is an imaginary space. The imaginary space, which we can divide infinitely is not consistent with the space full of substance which we work with, which we cannot divide infinitely. In other words our concept of space is inadequate, because we don't know how space is really divided. We only actually divide space by dividing up substance, and substance divides quite differently from the way that we divide space conceptually.
Quoting Rich
But don't you have this reversed? What exists is units, objects, but we want to talk about space in terms of a continuity. So it's not like we're trying to create units within a continuity, what exists is units and we are trying to make these units into a continuity. That's what Zeno shows us. It's not that motion is continuous, and we are trying to understand it as units, it's that it is not continuous, but we are trying to model it as being continuous. And this creates the paradox.
Quoting Rich
Do you think that you can sense a feeling of time when you are unconscious? I don't think so. Do you think you sense a feeling of time when you are dreaming?
If one insists on a discontinuous space and time then Zeno will always be there along with Achilles not ever reaching the finish line and arrows that are forever stopping in mid-air and restarting itself. The only way out is to challenge the assumptions and the methods used. If mathematics as a tool to reveal is too precious to give up then so be it. As for me, the idea of symbols actually being used as a placeholder for nature has passed. A piano n teacher one taught me to disregard the notes when you actually play music. The notes are inadequate.
As for what happens to time once I am unconscious or in a state of sleep is a question I have been long exploring and I come back to it now and then. As far as I can tell, it is a state where time is at its slowest, where there is some feeling of existing (via dreams) but it seems as though time isn't passing at all. Possibly an analog for the life-death cycle.
You still have it exactly backwards. Space, time, and motion are all continuous; we only model them as being discrete.
Prompted by some of the discussion in my spin-off thread on "Continuity and Mathematics," I have been reading up on category theory and one of its outcomes, synthetic differential geometry, also known as smooth infinitesimal analysis. I just came across this very pertinent passage on page 277 of John L. Bell's 2005 book, The Continuous and the Infinitesimal in Mathematics and Philosophy (emphases added).
As I said before, continuous motion is the most fundamental concept here. It is logically prior to any series of discrete locations - including an infinite one - through which an object passes while traveling from one place to another. In fact, the object's actual path that includes those identified "points" only exists as the result of the motion.
We measure space, time and motion as discrete, because that's the only way we can apply the numbers. But we tend to believe that these are continuous. It is this false belief, that space and time are continuous, which give rise to Zeno's paradoxes. So long as you hold this belief, that space, time and motion are continuous you will have paradoxes.
The concept of "infinitesimal points" is incompatible with continuous motion, it is only compatible with discrete motion. An infinitesimal point must be separate from another infinitesimal point or else it is not a point, and this negates any possibility of continuity. A series of "timelets" is a description of something discrete. Your quote from John Bell has provided a description of discrete motion, not continuous motion. He has perhaps recognized that our belief in continuous motion must be adapted to be represented as discrete.
I have no problem with measuring continuous things using discrete models; as I have acknowledged previously, they are very useful for that purpose.
Quoting Metaphysician Undercover
No, I have explained how Zeno's paradox dissolves when continuous motion is properly understood as more basic than discrete locations. Besides, a paradox by definition is only an apparent contradiction, not an actual contradiction; beliefs that are paradoxical are not necessarily false.
Quoting Metaphysician Undercover
No one is talking about "infinitesimal points" except you. Infinitesimals are not separate dimensionless points, they are lines of extremely small but non-zero length that smoothly blend together so as to be indistinct. A continuum is that which has parts, all of which have parts [I]of the same kind [/I]. A one-dimensional line [I]cannot[/I] be divided into zero-dimensional points, [I]only[/I] shorter and shorter one-dimensional lines.
Quoting Metaphysician Undercover
That would be news to him. I guess you missed the part about the timelets "smoothly overlapping" such that "time is, so to speak, still passing" within each of them, rather than being frozen in a discrete instant.
It doesn't matter how you lay the infinitesimal out, as a point, or as a line, there is still the assumed separation between it and other infinitesimals, and therefore it is necessarily a discreteness. A continuum cannot have parts, or else it is by virtue of those parts, not continuous, it is discrete.
Quoting aletheist
By saying "smoothly overlapping" you are speaking in terms of discreteness. You have identified separate parts which overlap. This is not continuity.
Wrong. There is no separation (assumed or otherwise) between infinitesimals. Neighboring infinitesimals are indistinct; the principle of excluded middle does not apply to them.
Quoting Metaphysician Undercover
Wrong. A continuum - by definition - is that which has parts, all of which have parts of the same kind. What a continuum cannot have are indivisible parts, like points.
Quoting Metaphysician Undercover
Wrong. Discreteness requires separation and distinction; infinitesimals, as defined by synthetic differential geometry (a.k.a. smooth infinitesimal analysis), are neither separate nor distinct.
Then they are not infinitesimals, are they? They are united as one large continuum and it is false to refer to them as separate infinitesimals.
Quoting aletheist
A continuum is a continuity. It is the desire to model the continuum as a serious of parts, like we do in a mathematical model, which negates the essence of the continuity, rendering it as a series of discrete units. To say that a continuum has parts is contradictory. By saying it consists of parts, you no longer describe it as continuous.
Quoting aletheist
I agree that discreteness requires separation, but what you seem to be failing to recognize is that "part" also requires separation. That is why it is contradictory to say that a continuity consists of parts. The true continuum must be indivisible, that's why it cannot be modeled mathematically.
One more time: By definition, infinitesimals are not separate.
Quoting Metaphysician Undercover
One more time: By definition, a continuum has parts, all of which have parts of the same kind.
Quoting Metaphysician Undercover
No, it does not. Once again, you are rejecting the commonly accepted definitions of terms, and imposing your own idiosyncratic ones.
Quoting Metaphysician Undercover
No, it must be infinitely divisible - i.e., there cannot be any indivisible parts - and smooth infinitesimal analysis does model this mathematically, whether you recognize it or not.
Clearly I do not accept your contradictory definitions. "Part" implies of necessity, a separation, and this negates any claim of continuity, which is a lack of such separation. You may proceed with your deluded metaphysics if you so desire.
Unless you can demonstrate that the concept of "part" necessarily involves separation, rather than just baldly asserting this over and over again as your own idiosyncratic definition, I have no reason to take it seriously. It blatantly begs the question to insist that anything with parts of any kind must be classified as "discrete," rather than "continuous."
"Part" implies some but not all of. How do you propose that we can have some but not all of, any particular whole, or supposed continuum, without a separation? "Some but not all of" implies necessarily, separation, that's what "some but not all of" means, separation.
Out of curiosity, I checked my dictionary to see how it defines "continuous." Here is what it says: "Having continuity of parts; without cessation or interruption; continued." Once again, you got it backwards - having parts is necessary for something to be continuous; otherwise, it would be "indivisible," which is a completely different concept.
You want to describe the continuous as consisting of contiguous parts. But then you are describing a contiguity rather than a continuity. Do you recognize the difference between contiguous and continuous?
Yes, you have - I just quoted one to you, verbatim, from Webster's Collegiate Dictionary, Fifth Edition (1936), which is what I happen to have on the shelf here at home. See below for a philosophical definition that explicitly refers to parts.
Quoting Metaphysician Undercover
Sources, please? On the contrary, here is what the SEP article on "Continuity and Infinitesimals" has to say (italics in original, bold mine).
In other words, being continuous is generally (although not invariably) the exact opposite of being indivisible.
Your SEP article appears to have a very shallow and unmetaphysical explanation of continuity. Suppose we assume that a continuum is in principle divisible, how do you avoid the problem of my prior post? It is necessary that the continuum does not actually consist of the parts which it will be divided into, or else it is not, at that time a continuum.
You already acknowledged that this is not a true continuum, because it has points at the ends, which are discontinuities. When you add a third point, you indeed break the continuity yet again; in fact, that is precisely the nature of all points on a line - they are discontinuities that we introduce by the very act of marking them. Before you posit point B, it does not actually exist; if anything, it is merely potential. Furthermore, the "two distinct continuities" that you get by assuming the point B are not "parts" of the original continuity in the relevant sense, since the point B itself is not part of the original continuity at all. Remember, the parts of a continuous line are not points - they are shorter lines.
By the way, according to your view, which "part" contains B - the one from A to B, or the one from B to Z?
"My" SEP article? I certainly did not write it, I just referenced it. I asked you for sources to justify your claim, "It is a well known metaphysical principle, that the continuous is indivisible"; but you provided none, which is telling. See above for my response to your example.
Quoting aletheist
I agree.
Quoting aletheist
Right, the two distinct continuities are not parts of the original continuity, they are produced by division.
Quoting aletheist
Now why do you go and contradict yourself? There are no such shorter lines until you posit some points of division. If there were such shorter lines, there would not be a continuity, because the shorter lines would be already separated out. There would be a series of shorter lines in contiguity. Do you understand the difference between continuity and contiguity? I really don't think that you do because you keep describing a contiguity, and claiming that it is a continuity. They are not the same. If the long line consists of shorter lines, then it is necessary that there is a boundary between the shorter lines, so that it actually consists of shorter lines. But these boundaries contradict "continuity", you have only a contiguity.
Quoting aletheist
This is not my view, I was offering you a compromise, to allow for your insistence that a continuity can consist of parts. Parts imply separation, so I offered points in the continuity as separations. You seem insistent that the continuity consists of such parts, without any separations, but this is purely contradictory. Without the separations there are no parts.
It's simple Aristotelian logic. Anything divisible necessarily consists of parts. Every part is individuated, or separate from every other part. A continuity has no such separations. Therefore a continuity is indivisible.
You seem to take exception to the opening premise, assuming that something which does not consist of parts (continuum) is divisible. But then you contradict yourself by describing that thing as consisting of parts.
Why is it so hard for you to understand that there are no points in a continuous line, only shorter lines? Positing points of division makes the line discontinuous.
Quoting Metaphysician Undercover
Yes, contiguity only applies to discrete things; so that is obviously not what I am describing.
Quoting Metaphysician Undercover
NO! There are no intrinsic boundaries between the parts of a continuum; in this case, between the smaller lines within a continuous line.
Quoting Metaphysician Undercover
I asked you for sources, not a rationalization; and in any case, it should be quite clear by now that I reject your unwarranted stipulation that a "part" is necessarily "individuated" or "separate." Besides, what did Aristotle himself have to say about this matter? Warning - you are not going to like it!
If you want to stick to your guns and claim that Aristotelian logic somehow contradicts Aristotle's own explicitly stated views ... well, good luck with that.
If they are shorter lines, they must end. Where they end, there must be something which signifies the end or else there is no end and therefore no shorter lines. If you don't want to acknowledge that "end" as a point, then call it something else, but the fact is that this "something else" interrupts any supposed continuity.
quote="aletheist;57607"]NO! There are no intrinsic boundaries between the parts of a continuum; in this case, between the smaller lines within a continuous line.[/quote]
Then what ends the short lines, making then short lines? How can you not see the contradiction? How can there be short lines if there is nothing to end these lines, making them short lines.
Quoting aletheist
If it's not individuated, or separated from the whole, how can you say that there's a part? All you have is a whole.
The problem with your metaphysics, is as I described earlier in this thread. You have some idea of the way things should be described, an ideal, then you define your terms according to this ideal. But this ideal is just a fantasy, a fiction, and you have no respect for reality, for the way that things actually are, according to empirical observation. So you continue your metaphysics based in some fictional ideal, rather than in solid principles of how things actually are.
Quoting aletheist
Aristotle says many different things in many different places, often contradicting himself. That's not odd, he has a lot of material. Pretty much the entirety of "On the Heavens" has been discredited, proven wrong.
Your obstinate dogmatism would be quite impressive if it were an admirable trait. You simply refuse to accept the established definitions - as quoted from a standard dictionary, an online philosophy encyclopedia, and the writings of Aristotle - of what it means for something to be continuous. Once again, engaging with you has been a waste of my time. Cheers.
Just answer a couple quick questions for me, if you really believe that you have a tenable position. How can you have short lines unless they have ends? And how can you have a continuity which has ends inherent within it?
I explained this already, multiple times and in various ways. A continuous line has no ends, so by definition its parts also have no ends. You cannot actually divide a continuous line without introducing a discontinuity (point), but it is potentially divisible without limit, as the SEP article explains. Mathematically, infinitesimals likewise have no ends; they are indistinct, such that the principle of excluded middle does not apply to them. In Aristotle's words, "the extremities of things [i.e., parts] that are continuous with one another are one [i.e., not two or more] and are in contact [i.e., not separate]."
Based on our past encounters, I expect you to respond by insisting that a line is "divisible" only if someone can actually divide it. I see no use in going back down that road, so again, cheers.
This is where you stray from observed empirical reality. Empirically proven principles demonstrate that anything which is divisible is such because it consists of parts. The points which provide for division are already existent within the divisible thing, or else it could not be divided. You are assuming that a continuous thing, a thing which exists without such points for potential division, can still be divided. This is an unsupported fantasy.
Quoting aletheist
And this is nonsense. Such entities have no individual identity.
Explain any empirical evidence that anything is truly divisible, that is stands completely separate from all that surrounds it. It must be shown at the finest granularity that has been empirically explored.
We could assume the existence of a continuous whole, but it would be false to say that this continuity consists of parts. It is equally false to say that it is divisible.
You don't think that there is evidence that the area of your field of vision is made up of separate objects, separate parts? Isn't the fact that I can pick up a chair and move it to the other side of the room, or move the dishes from the cupboard and use them, then wash them, evidence that they are individual parts, able to move independently of the others? Isn't the fact that water boils and evapourates evidence that it is made of separate parts, molecules? Aren't chemical reactions evidence that the molecules are made of parts, atoms? What more evidence do you need.
Quoting Rich
Fields are mathematical formulae. You are just entering a fictional fantasy like aletheist, referring to some ideal, a fiction of how you think reality should be, then you will describe things to match this ideal, instead of shaping your ideal to the way thins really are..
Quoting Rich
Waves are something distinct from particles, but they clearly are a pattern of movement of particles, like sound waves and water waves.
Quoting Rich
I believe the atom has been split, in the nuclear reaction. And electrons are commonly separated from atoms in electrical practises.
Quoting Rich
A wave can only exist within an assembly of parts, it is a particular type of activity of particles.
I am saying that as far as empirical evidence exists at this time, there is no evidence of full and total separation. There seems to be more evidence to the contrary. You are speaking of separation (the concept of isolated particles) for which there is no empirical evidence and never was. The idea of somehow separate particles is a belief system, which one is free to embrace, but then one must explain what is in-between. It is rather simple, on the other hand, to have a continuous fabric of waves from which substantive matter is formed (and collapses into).
There is certainly empirical evidence for fields which are continuous and stretch forever. Much more evidence for this than separate and distinct particles. Particles are probably nothing more than perturbations (spikes) in the fields that appear and disappear in the field, but there is no empirical evidence one way or another. They would be real but they would also be part of and inseparable from the field since the field would be the fabric.
Actually, the splitting is a metaphor. Energy was release as the "droplet" (as Bohr described it) was reformed. One can analog this as one massive wave being reformed into two smaller ones and in the process releasing energy, as a wave hitting a beach might.
The wave in the above description is not part of anything, it would be the fabric of the universe. Consciousness, movement (energy) and memory are all sewn into this fabric and are everywhere just as an image is sewn into every part of a hologram. It is waves that make this all happen.
I apprehend "full and total separation" as a rather useless concept. Things always exist in relation to other things. To not have a relation to something else (full and total separation) is to not exist. Unless you conceive of a whole which consists of all existing things (the universe), and this whole, by definition would not have a relation to anything else, because it is everything, full and total separation is impossible. But what kind of separation is that? It's just the logical separation between what is, and what is not. So is this the "full and total separation" you refer to, the separation of logic, between being and not being?
Quoting Rich
As I said, we move objects around, in different directions relative to each other. Does this not indicate a separation between them to you? It's nonsense to insist that this separation must be absolute, such that there is not even any relationship between the two, because then one object would have to be existing and the other not-existing. So you requirement for "separation" is to completely annihilate the object. You will not admit that one object is separable from another unless it can be completely annihilated, removed from any relationship to the other. Why do you not allow that moving one object in all sorts of different directions relative to another, constitutes a real separation between them? What we assume as "in-between", which allows for such movement, is "space". Why do you hold such a strong propensity to reject this idea?
Quoting Rich
I do not see why you claim that this idea is "rather simple". Do you not recognize that waves require a medium? So all you are doing is reducing the "substantive matter", and taking for granted a new substance which necessarily underlies the waves. I assume a "space" between substantive objects, you assume "waves", which are necessarily in a substance, then you have to account for the appearance of objects, so one is not more simple than the other. You've replaced my lack of substance, "space", with substance, "waves". Now you still must account for what I call substance, objects. The difference, is that my position allows for the real separation between objects, which we utilize daily, to move objects in different directions relative to each other. You deny this real separation. So how is it that we move objects like this then? How do you account for our capacity to freely move objects this way and that way in relation to each other, if there is not real separation (space) between them?
quote="Rich;57691"]The wave in the above description is not part of anything, it would be the fabric of the universe. Consciousness, movement (energy) and memory are all sewn into this fabric and are everywhere just as an image is sewn into every part of a hologram. It is waves that make this all happen.[/quote]
Don't get me wrong, I am not denying the need to refer to wave activity, it as well as objects, is observable, and waves are empirically verified. The point though is that it doesn't get us any further ahead, to deny the reality of the independent activity of objects, for the assumption that all reality is a "whole" consisting of waves. What is needed is to establish compatibility, not to choose one over the other by excluding the possibility of the other.
Could you point me towards the empirical evidence for these fields which are continuous and stretch forever?
It is unfortunate, but my whole model and approach really does undermine all of academic philosophy as it is instructed since it denies the use of mathematics and logic as a method for penetrating nature.
As I said earlier, for probing nature, the arts should be the major part of the philosophical academic curriculum.
What has entanglement got to do with the existence of continuous fields of infinite extent?
But it is this, the missing impetus which demonstrates that the parts are really separate. The free willing act can move the object any which way, so the wave in the ocean analogy is not really adequate to explain this motion.
Quoting Rich
So how would the conscious, free will act move one particular object independently of the other objects? It cannot be by means of the wave perturbations which you describe, because these are not independent. It's easy to make the claim that Bergson's Elan Vital solves this problem, but until you explain how one object (a living being) moves itself independently of all the surrounding objects, your description of an ocean with waves remains incompatible with this reality.
Sorry to break it to you, but entanglement has nothing to do with continuous fields that extend to infinity.
It doesn't move it independently. It is embedded within the wave form and uses Will to attempt to move in a specific direction which would change the wave form within the field. Movement is thus a change in the flux of the waveform which is pretty much what quantum probability wave is describing.
When I push something out is one wave acting upon another within the field. The mind is using a reference wave to observe this movement and creating a corresponding memory wave form that itself is constantly changing. Everything would be waveforms but referenced in different ways and some some more substantial than others. Substantiality can be analogued by observing threads being compressed into strings and strings being compressed into balls.
It is necessary to use the creative mind to form the impressions. In the same way, musical sounds (more substantial)are transformed in impressions within memory (less substantial).
Your msunderstanding of QM is not evidence for infinite continuous fields, which by-the-way can't be continuous if they are quantized.
In my model, which dovetails Bergson's, the impulse is the Elan Vital which is embedded, or at once literally the wave. The impulse can be considered a creative desire to learn, and it manifests partially as Will, partially as Creative Intuition, and partially as Memory. In Chinese metaphysics the counterparts would be the Dao, Yin/Yang, and Qi. Heraclitus called it the Logos. The image that would analogue this would be the Ocean Wave with Gravity embedded within it to create movement. However, in this case, since we are discussing life and the Elan Vital, the movement can be directed to satisfy a desire to create and learn.
Can you describe the "Ocean Wave"? In your model, is there a wave which initiates from a point, like when you drop a pebble in water, or is there just perturbations in existing waves? If there is such a wave, which initiates from a point, what would cause this wave?
We can use Bohm's own image of quantum potential and the Implicate Order.
Bohm himself suggested that one can consider consciousness being embedded within these movements creating the impulse for the movements. Another way to put it would be that all of these impulses are consciousness in action, or consciousness manifesting. Those spikes in the quantum potential would be what is modeled as particles. As you can see, with this model there is no discreteness.
I suppose this means that there can be no beginning point of a wave. Such a beginning would be a discrete occurrence.
But the particle must be within one of the discrete grooves.
One way to view the wave would be consciousness expanding over time (duration). The beginning and ending being consciousness itself.
The paradox doesn't talk about deceleration.
I am not aware of any version of Zeno's paradoxes that assumes that some distance is covered instantaneously, or that makes any extraordinary assumptions about velocity or acceleration.
This has nothing to do with the paradox.