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. — Michael

Which is absurd.

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. — Michael

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 don't see how it follows that they would be the same location. — Arkady

Well, I don't see how two locations separated by zero distance can be different locations.

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. — Michael

This right here is the mistake that you keep making. There are two statements here.

• An object must have passed through an infinite number of prior coordinates to get to any arbitrary point.
• A person must have counted an infinite number of prior numbers to get to an arbitrary point.

You treat these as equivalent, or at least analogous. They are not.

You're saying that an infinite series of events has been completed. But an infinite series of events cannot be completed, by definition. — 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.

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. — Michael

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.

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)? — Arkady

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 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). — Michael

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).

Presumably there is no time in between. First it's at this discrete location and then it's at that discrete location. — Michael

So the movement from one to the other is somehow instantaneous?

The logic still shows that continuous motion is impossible. — Michael

It should be no surprise to anyone that assuming motion to be discrete (like counting) renders continuous motion impossible.

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. — 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.

What's the difference between moving from one coordinate to the next and counting from one coordinate to the next? — Michael

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.

Even if we only define three coordinates between A and B it must still pass through the space between those coordinates. — Michael

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.

And so the reason motion is possible is because it's discrete. — 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?

And so what is the first potential rationally-numbered coordinate that an object must pass through in its movement from A to B? — Michael

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.

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). — Michael

Agreed, counting is discrete. That is precisely why it is a false analogy to motion, which is continuous.

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. — 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?

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. — Metaphysician Undercover

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.

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. — 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.

Like in my example a deer may be saved by committing the fallacy of affirming the consequent thereby endorsing it as a valid type of reasoning. — TheMadFool

And like I said, it is a valid type of reasoning - retroductive reasoning, rather than deductive reasoning. The conclusion is thus merely plausible at best, rather than certain.

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 π and e. — jorndoe

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.

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.

I already know you can do whatever you want with maths, just make it up as you go, and prove whatever you want to prove. — Metaphysician Undercover

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.

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. — Metaphysician Undercover

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.

So you are assuming that you can be done doing something an infinite amount of times? — Metaphysician Undercover

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.

Can't be done. — Metaphysician Undercover

Of course it can, students have to do it in math class all the time. You can also do it on a calculator.

What does that have to do with my question? — 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).

It appears to me, like no matter how far you go you'll always be a fraction short of 1. — Metaphysician Undercover

Convert 1/3 to a decimal, then multiply it by 3. Is the result 1, or an infinitesimal fraction short of 1?

My coordinate system only uses the rational numbers. So I ask again; what coordinate does it pass through first? — Michael

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.

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. — Michael

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.

The task is to count every rational number between 1 and 2. — Michael

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.

My coordinate system only uses the rational numbers. So I ask again; what coordinate does it pass through first? — Michael

My coordinate system only uses the integers. After all, units of physical measurement are completely arbitrary.

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. — Michael

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?

If space is continuous then we can plot infinitely many points in it, so I don't understand your objection. — Michael

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.

And I don't understand your distintion between infinite divisibility and continuity. — 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.

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. — Michael

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.

The paradox is about moving from one point to another, which is analogous to counting from one number to another. — Michael

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.

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". — Michael

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.

It's a strange thing that Supreme Court nominees are a partisan issue. It should be that judges just determine what the law (and Constitution) is, not what it should be, in which case whether or not you're a Democrat or a Republican or whatever shouldn't matter. — Michael

Yes, that is how it should be; but over the last several decades, the Supreme Court has gone in a much more activist direction, in many cases determining what the law should be rather than what it is, resulting in its unfortunate politicization. Even so, Republican vs. Democrat is not the issue, but whether a potential justice is more likely to interpret law vs. make law. Our societal impatience with legislative gridlock has led to the concentration of power in the other two branches - i.e., unelected bureaucrats and judges - contrary to the Founders' wise design that was intended to preclude massive national policy changes in the absence of broad public consensus.

Qualitative comparisons are subjective. Quantitative comaprisons are objective. — TheMadFool

Nonsense. My son is objectively taller than my daughter. Yellow is objectively lighter in color than indigo. A pillow is objectively softer than a stone. There are ways to assign quantities to each of these qualities, but the unit of measurement in each case will be arbitrary.

On the other hand, people make "quantitative" comparisons that are subjective all the time. This movie is four stars, that one is only two. This essay gets a grade 93%, that one gets only 88%. This soccer player has an 83 rating in the video game, that one only has a 79.

Mathematical models are an approximation yes but they work - they grasp at the key players in any situation, sweeping aside the irrelevant, the redundant, the red herrings, etc. — TheMadFool

They can be used successfully, but not always or by just anyone. It takes good judgment developed through experience to ascertain the significant aspects of an actual situation that warrant being included in a mathematical model.

In your example of the height comparison of boy and girl math helps us to answer who is and by how much taller between the two. — TheMadFool

But that is a different question than the merely qualitative comparison of which one is taller than the other, which requires no measurement - and therefore no math - as long as they are standing together.

Doesn't the exactitude of math help us fine-tune our knowledge of our universe? — TheMadFool

The exactitude of math is only possible because it deals with purely hypothetical or ideal states of things. Exact analyses of mathematical models can only serve as approximate analyses of actual situations.

How can we compare two or more things without quantification (use of math) knowing that quantification is necessary in that arena? — TheMadFool

Quantification is NOT necessary for comparison. If my son and daughter stand next to each other, anyone can observe that my son is taller than my daughter, without quantifying anything at all. We make these kinds of qualitative comparisons all the time, and mathematics plays no role in them whatsoever.

I'll say, mathematics is useful in a given context insofar as counting and measuring and analyzing quantitative relationships are useful in that context. — Cabbage Farmer

Although mathematics is commonly associated with quantity, it is more broadly the application of necessary reasoning to hypothetical or ideal states of affairs. As such, the usefulness of its conclusions is entirely dependent on how well its initial assumptions capture the significant aspects of reality - not just the model itself, but the representational system that governs its subsequent transformations. It is thus highly suitable for analyzing natural phenomena, since the habits of matter are largely entrenched; but not so much for analyzing human behavior, since the habits of mind are much more malleable.

Forgive me, but I can't take any argument for a divine creating intelligence seriously. — apokrisis

I am sorry that you see it that way, but I do forgive you.

Theism (of the first cause type) is simply contradictory of Peicean semiosis ... — apokrisis

Obviously Peirce himself did not think so - not even remotely - since he explicitly affirmed the Reality of God as Ens necessarium.

You left out my first two statements ...

I am not seeing the connection between this comment and the notion that "mathematical symmetries" somehow limited God's options. For one thing, Peirce consistently held that mathematics deals only with hypothetical states of affairs, not actual ones. — aletheist

... and I am still not following you here:

So either God is constrained Himself by the general principle of intelligibility - existence as the universal growth of reasonableness - or the whole of Peirce's metaphysics collapses for a far more serious reason. — apokrisis

If God is constrained by "existence as the universal growth of reasonableness," it is only because He chose to create existence that way. In fact, Peirce characterized this as God's purpose.

Semiotics just doesn't exist unless the sign relation is in fact a sign of something. — apokrisis

And Peirce called our existing universe God's argument, a symbol whose object is Himself and whose interpretant consists of the living realities that it is constantly working out as its conclusions.

But it is one thing saying God could choose to create a world in which 1+1=3, quite another to believe it in your heart. Do you think Peirce would have gone along with such a frontal assault on natural reason? — apokrisis

I am not seeing the connection between this comment and the notion that "mathematical symmetries" somehow limited God's options. For one thing, Peirce consistently held that mathematics deals only with hypothetical states of affairs, not actual ones. He also insisted that we cannot be absolutely certain that 2+2=4, since human fallibility entails that it is possible - even if very unlikely - that every single person who ever performed this addition made the same mistake. Presumably he would have taken the same position on 1+1=2.

God conceived an inexhaustible continuum of possibilities, and then chose which of them to actualize. — aletheist

But could God have had a choice if mathematical symmetries limited His options rather rigorously? — apokrisis

Assuming omnipotence, as Peirce did, the only thing that could have limited God's options were God's own previous choices, including the creation of those mathematical symmetries.

The standard interpretation of Peirce's cosmology is that the initial state was a chaotic mix of chance and reaction in which anything was possible but nothing persisted, hence nothing was actual. The tendency to take habits was one of those spontaneous occurrences at first, but its very nature was to persist and reinforce itself, so it did. Then other things began to take habits, and that is how matter eventually came about, with the "laws of nature" serving as its habits. These are much more consistent than habits of mind - hence why nature is often much more predictable than human behavior - but they are not completely exceptionless, since objective chance is still active in the world. Peirce was not convinced that all of the discrepancies in our measurements are due to error; rather, things really are not quite as exact as our equations seem to indicate.

Toward the end of his life, Peirce seems to have adopted a more theistic cosmology in which God is indispensable as Ens necessarium, necessary Being, to account for the order that we now observe. God conceived an inexhaustible continuum of possibilities, and then chose which of them to actualize. Spontaneity is thus a manifestation of divine freedom, rather than objective chance.

For the record, I am skeptical of laws of nature. I prefer dispositions and powers. Laws of nature are mathematical abstractions based upon these things. — darthbarracuda

I think that what you call dispositions and powers - i.e., what I call tendencies and habits - are the laws of nature. Mathematical abstractions are what we use to represent them, perform calculations in accordance with them, etc.

Because the fundamentals are, by definition, fundamental. — Michael

But how do you know that the observed behaviors themselves are fundamental, rather than manifestations of something else that is even more fundamental? In other words, how can you tell when to stop looking for a further explanation?

... I think it far simpler to just accept the behaviour itself as fundamental. — Michael

Of course it is simpler to settle for calling it a brute fact, but that does not make it rationally justifiable, let alone correct. Where would science be today if Newton had shrugged his shoulders because falling from trees is merely what apples happen to do?

The fundamental behaviour of things is, by definition, fundamental. There is no further explanation. — Michael

You do not think that the remarkably consistent behavior of things calls for an explanation? If not, why not?

As I said in my first post, there is just the behaviour of things and our descriptions of such behaviour. There's no need to posit some extra thing which is the "law". If something is to be called a "law", then it's one of these things. — Michael

You do not think that questions like why things behave as they do and why this behavior is so consistent are worth exploring? We should just accept them as brute facts and not inquire further?