The vocabulary of sets lets you phrase all these concepts already. "Parts of a line? They're finite intersections of its interval subsets which have cardinality greater than than 1". — fdrake

A set is a bottom-up conception, assembling a whole from discrete parts. True continuity is a top-down conception, such that the whole is more fundamental than the parts.

some argue that the real numbers are not truly continuous — aletheist

Who argues that, exactly, besides the Peirceans on this forum? — fishfry

The Peirceans who are not on this forum, for starters; but it goes back at least as far as Aristotle, who recognized that numbers of any kind are intrinsically discrete, rather than continuous. The key word here is "truly"; I have acknowledged that the real numbers are an adequate model of continuity for most mathematical and practical purposes. Nevertheless, conceptually a line is not composed of points, a surface is not composed of lines or points, and a solid is not composed of surfaces or lines or points. Instead, the parts of a line are one-dimensional lines, the parts of a surface are two-dimensional surfaces, and the parts of a solid are three-dimensional solids. Anything of lesser dimensionality is not itself a part (or portion) of that which is truly continuous, but rather a connection (or limit) between its parts.

The standard mathematical view is that "the continuum," "the real line," and "the set of real numbers" are synonymous. Philosophical considerations do not alter the conventional mathematical meanings. — fishfry

Yes, and I acknowledged as much.

This was Cantor's view, which is fairly standard among mathematicians today. However, there is a power set for the real numbers, and a power set for that power set, and so on ad infinitum. That being the case, some argue that the real numbers are not truly continuous, despite comprising what is conventionally called the analytical continuum. — aletheist

2) Is there an error in thinking of a representation of a powerset as all the permutations of the elements of the original set? — tim wood

Combinations, not permutations; i.e., the different proper subsets, and the order of the members does not matter. For a set with n members, its power set has 2^n members.

4) But if 3, and there is no such point on the line, then (it appears to me) that c = P(N). — tim wood

This was Cantor's view, which is fairly standard among mathematicians today. However, there is a power set for the real numbers, and a power set for that power set, and so on ad infinitum. That being the case, some argue that the real numbers are not truly continuous, despite comprising what is conventionally called the analytical continuum.

Is 5 the true statement, that there are points on the line to which no real number can be applied? — tim wood

There are no points in a truly continuous line, period. As a one-dimensional continuum, its parts are all likewise one-dimensional, rather than dimensionless points. We could hypothetically mark points on a line of any multitude--including that of the real numbers and that of their power set--or even beyond all multitude.

The constraints of my language are the fundamental laws of logic, identity, non-contradiction, excluded middle. — Metaphysician Undercover

None of those dictate the peculiar metaphysical definitions that you insist on imposing for terms like "existence" and "object," even in the context of non-platonist mathematics where they entail nothing ontological whatsoever.

... I see no point in continuing. — Metaphysician Undercover

On this we agree.

I cannot communicate with someone who doesn't speak my language. — Metaphysician Undercover

That explains a lot. Why should I (or anyone else) accept the constraints of your peculiar language?

It strikes me that you have disregard for the fundamental rules of logic ... — Metaphysician Undercover

How could you ever make such a determination, given your admission that you are unwilling even to try to understand my (or others') usage of the terms, simply because it is different from yours?

Therefore the dualism of Platonic realism. — Metaphysician Undercover

No, I am not a platonist; I am not claiming that abstractions exist in the ontological sense. Why keep insisting otherwise?

And symbols represent subjects, so there's a double layer of representation, exactly what Plato warned us against, what he called "narrative", which allows falsity into logic, sophistry. — Metaphysician Undercover

No, some symbols are subjects, while others are predicates, although the predicate of a proposition can also be embodied in the syntax rather than the symbols.

I'm really tired of your unsupported assertions. — Metaphysician Undercover

I'm really tired of your willful obtuseness, insisting on your peculiar metaphysical terminology despite the fact that the same word often has different meanings in different contexts. "Existence" in mathematics is not the same as "existence" in ontology. An "object" in mathematics is different from an "object" in semeiotic, and both are different from an "object" in ontology. A "subject" in semeiotic can be an "object" (direct or indirect) in grammar. And so on.

This is a defect of the theory, it gives us a so-called theoretical "object" which cannot be measured. — Metaphysician Undercover

It is a feature, not a bug--it reveals a real limitation on our ability to measure things.

Why would we produce a theory which presents us with an object that cannot be measured, when the theory is created for the purpose of measuring objects? — Metaphysician Undercover

What is the basis for the claim that mathematics is created for the purpose of measuring objects? On the contrary, the purpose of mathematics is to draw necessary inferences about hypothetical states of things. One such inference is that in accordance with the postulates of Euclidean geometry, the length of a square's diagonal is incommensurable with the length of its sides.

Creating "impossible" abstract objects is nothing new, it is easily done through the use of contradiction. — Metaphysician Undercover

"Impossible to measure" does not entail "impossible," full stop. There is nothing logically impossible about the diagonal of a unit square or the circumference of a unit circle. Again, given how those figures are defined, it is logically necessary for them to be incommensurable.

The problem I've been discussing is that whatever it is which is expressed as "a square" does not actually exist in "concept-space" because the perpendicular sides are incommensurable ... The figure is impossible, just like the irrational nature of pi tells us that a circle is impossible. — Metaphysician Undercover

Incommensurability does not preclude (mathematical) existence. Our inability to measure two different objects (abstractions) relative to the same arbitrary unit with infinite precision does not entail that one of them is (logically) impossible.

The figures defined by the theory are impossible, according to the theory, just like a square circle is impossible. — Metaphysician Undercover

Only according to your peculiar theory, not the well-known and well-established theory in question.

Then your beliefs are irrelevant to my concerns with algebra and set theory, which hold that the symbols represent objects. — Metaphysician Undercover

They do represent objects--abstractions, not existents.

I'm afraid you have things backward. — Metaphysician Undercover

On the contrary, this is Semeiotic 101--in a proposition, the subjects denote objects, and the predicate signifies the interpretant.

Where is your demonstration of an abstraction existing as an object, which is not a demonstration of Platonism? — Metaphysician Undercover

Again, I do not hold than there is such a thing as "an abstraction existing as an object." I reject your peculiar terminological stipulation that an "object" can only be something that ontologically exists.

... I was talking about what is represented by the symbol in logic, and that is a subject, not an object. — Metaphysician Undercover

No, a symbol in logic is itself either a subject or the predicate within a proposition. If it is a subject, then it denotes an object, which can be an abstraction or an existent. If it is the predicate, then it signifies the interpretant, which is a relation among the objects denoted by the subjects.

But "object" refers to a very specific type of thing, a unique individual, a particular, having an identity as described by the law of identity. — Metaphysician Undercover

Again, your peculiar metaphysical terminology is not binding on the rest of us.

That I reject the notion that properties which are described by concepts like "force" "mass" and "acceleration" are themselves objects, doesn't make me nihilist. It just means that I understand the difference between an object and a logical subject. — Metaphysician Undercover

Apparently not--an object is whatever a logical subject denotes, which can be an abstraction or a concrete existent.

Yes, I agree, in mathematics some people make the unsubstantiated claim that the symbols represent existent objects. This is called Platonic realism — Metaphysician Undercover

Indeed, that would be mathematical platonism, as I have acknowledged. However, I am not a mathematical platonist--I have quite explicitly denied that the symbols represent existent objects in the ontological sense.

You seem to believe that there is some other form of ontology, some other universe of discourse, which allows that abstractions have "mathematical existence", as objects, which is not Platonism. — Metaphysician Undercover

Platonism is by no means the only philosophy of mathematics that employs the well-established term "existence" when referring to abstract objects. As I have clearly and repeatedly stated, for those of us who are not mathematical platonists, ontology has nothing whatsoever to do with the "existence" of such objects.

All you have done is stated Platonist principles and lied in asserting that no one is assuming Platonism. — Metaphysician Undercover

All you have done is obtusely stuck to your rigid terminology, refusing to pay any heed to the multiple explanations that I and others have offered to correct your evidently willful misunderstanding. I see no point in wasting my time any further.

Any claim of "existence" is validated (substantiated) with substance. — Metaphysician Undercover

Perhaps in metaphysics/ontology, but definitely not in mathematics.

If you think that there is a type of existence which is not substantial then please explain. — Metaphysician Undercover

I have already done so, repeatedly.

Do you not recognize that "possible" refers to what may or may not be, so it is contradictory to say that possible things are existing things. — Metaphysician Undercover

Perhaps in metaphysics/ontology, but definitely not in mathematics.

What I deny is that prediction is the goal of the scientific method. — Metaphysician Undercover

I never claimed that it is. Prediction enables us to evaluate whether our hypotheses hold up to further experimental and observational scrutiny. The goal is knowledge, which consists of beliefs (i.e., habits) that would never be confounded by subsequent experience.

It's easy to assert "no one is claiming that an abstraction is an existent object", yet everyone backs up set theory which clearly assumes that the abstraction which a symbol represents, is an object. — Metaphysician Undercover

You keep imposing your peculiar metaphysical terminology, as if everyone else is obliged to conform to it regardless of the context. In this case, you seem to be insisting that only an ontological existent can be the object of a symbol. In mathematics, and even in ordinary language, an abstraction can also be the object of a symbol, as long as the universe of discourse is established. The objects of the names "Pequod" and "Ahab" are a boat and its captain in the fictional world of Melville's novel. The object of the word "unicorn" is a horse-like animal with one horn; the fact that no such animal exists in the ontological sense does not preclude the word from having an object at all.

Due to this behaviour of yours, I can find nothing else to say other than you are boldly lying. — Metaphysician Undercover

Seriously? Due to this behavior of yours, I can find nothing else to say other than you are boldly ignorant (of mathematics and its terminology) and stubborn (about your rigid definitions).

OK, I'll assume for the sake of argument that there is a type of existence, "mathematical existence", which is a different type of existence from "ontological existence". I'll assume two different types of existing substance, like substance dualism. — Metaphysician Undercover

Wow, this keeps getting more and more ridiculous. No one is claiming that mathematical existence has anything to do with "existing substance." In mathematics--again, except for platonism--the term "existence" does not imply anything ontological whatsoever.

How would I define "mathematical existence"? — Metaphysician Undercover

Something exists mathematically if it is logically possible in accordance with an established set of definitions and axioms. The natural numbers, integers, rational numbers, real numbers (including the square root of two), and complex numbers all exist mathematically, in this context-specific sense.

A significant aspect of the "scientific method" involves "observation", and observation is meant to be objective. The goal of "prediction" introduces a bias into observation. — Metaphysician Undercover

Nonsense, prediction is just as much a significant aspect of the scientific method as observation. Why do we have theories? How do we come up with them? Our observations prompt us to formulate hypotheses that would explain them; this is retroduction (sometimes called abduction). We make predictions of what else we would observe, if those hypotheses were correct; this is deduction. We then conduct experiments to determine whether our predictions are corroborated or falsified; this is induction.

... I've come to the conclusion that abstractions are not existent objects. — Metaphysician Undercover

One more time: No one is claiming otherwise.

Ontology is the study of existence. Isn't it? — Metaphysician Undercover

No, it is the study of being, which is not necessarily synonymous with existence. For example, one view is that ontological existence (i.e., actuality) is a subset of reality (which also encompasses some possibilities and some necessities), which is a subset of being (which also encompasses fictions).

How could there be a form of existence which isn't ontological existence? — Metaphysician Undercover

By defining "existence" in another context-specific way, obviously. There are plenty of other terms that mean something different in mathematics than in metaphysics or in other sciences.

Again, we encounter the problem of pragmatism. If prediction is all that is required, then we gear our epistemology toward giving us just that, predictability. — Metaphysician Undercover

That is not just pragmatism, it is the scientific method. How else would you propose that we evaluate our hypotheses to ascertain whether they accurately represent reality?

If you are handing to "existence" a definition which allows that an imaginary, fictional thing, exists, then it's not the rigorous philosophical definition which I am used to. — Metaphysician Undercover

As I have explained to you several times now, no one except a platonist would claim that mathematical existence conforms to "the rigorous philosophical definition" of (ontological) existence. Everyone else understands this, so please stop belaboring your terminological objection.

I interpret this as your "epistemic stance" requires Platonic realism as a support, a foundation. — Metaphysician Undercover

That is a misinterpretation, and you know it by now.

Now we really are going in circles. I see no point in continuing (pun intended). Cheers!

Where exactly am I assuming that time has a start? — Devans99

By insisting that moments/states are "undefined" otherwise.

Do you believe that a greater than any finite number of days has passed? — Devans99

Time is not composed of days. A day is an arbitrary unit of duration that we use to mark and measure the passage of time.

Every moment has another moment before it and there is a start of time if time is a circle. — Devans99

If time is a circle, then every moment has another moment before it, but there is no start of time unless we arbitrarily designate one. Remember, I can use an ink stamp to "create" an entire circle on a piece of paper all at once.

That is not how probability works in the mathematics of infinity. — aletheist

How does it work? — Devans99

Again, in reality there are no instantaneous states, so the "probability" of any such state occurring is meaningless. Besides, in infinite time there would be infinitely many such states, and no reason in principle to assume that any two of them are identical.

It is not question begging it is just the way reality works: — Devans99

You cannot prove that time has a start by assuming that time has a start. Besides, if every moment has a preceding moment, then time cannot have a start, because that would require a (first) moment that does not have a preceding moment.

- The probability of being in state X must be greater than 0% (because we have been in that state)
- Leading to the number of times in state X as ∞ * non-zero = ∞ times — Devans99

That is not how probability works in the mathematics of infinity.

All existence is "ontological existence" so it makes no sense to try and separate "mathematical existence" from "ontological existence". — Metaphysician Undercover

Mathematicians and philosophers of mathematics, with the presumed exception of platonists, reject the premiss that all "existence" is ontological existence. Specifically, they acknowledge that mathematical existence does not entail ontological existence.

So it is but a small step to see that any system over an 'infinite' period of time has no initial moment or state and therefore all subsequent states are undefined. — Devans99

Non sequitur; having no initial moment/state does not entail having no "defined" moments/states (whatever that means), unless we add the question-begging premiss that a first moment/state is required to "define" any other moments/states.

Everything has a start. — Devans99

More question-begging.

Name a topology for time that has no start? — Devans99

A straight line extending from the infinite past to the infinite future. A hyperbola for which the initial and final moments/states are ideal limits that never actually occur.

Circles have start points BTW. — Devans99

Only if we arbitrarily designate one; a circle in itself has no points of any kind. If I use an inked stamp, I can "create" an entire circle on a piece of paper all at once, with no start point.

The state of the universe is given by the precise positions and velocity vectors of all its particles (10^80 or so in the observable universe I read) — Devans99

Instantaneous states, positions, and velocity vectors are all abstractions that we artificially create to describe reality. They are not themselves real. Besides, our best current science indicates that it is impossible to determine both the position and the velocity of any particle at the same hypothetical instant, let alone all the particles in the universe.

How many times has the universe been in state X in the past? — Devans99

Zero. Besides wrongly treating an instantaneous state as a reality, the latest argumentation wrongly presupposes that the universe can be in the same state more than once.

If we take an example of a finite causal regress:

1. The cue hits the white ball
2. The white ball hits the black ball
3. The black ball goes in the pocket

Note that if we remove the first element of the finite causal regress ([1] above) then the rest of the regress disappears. — Devans99

Where we mark the "start" of any finite series of events, and how we parse it out into discrete steps, is completely arbitrary. Before #1, presumably a person deliberately pushes the cue such that it hits the white ball. That involves a multitude of mental decisions, nerve signals, and muscle movements. We would have to go much farther back in time before that in order to account for all the causal factors. Moreover, none of the three listed events is instantaneous--each requires a finite interval of time, during which complex physical interactions occur--and in between, each ball presumably rolls across a frictional surface, slightly slowing its velocity.

What caused the start of motion? — Devans99

Again, not relevant; the issue is whether time has a start, not whether motion has a start.

So we are in an infinite causal regress. The only way out of such is to posit something uncaused as the base of the regress; IE something from beyond causality; IE something from beyond time. IE time has a start. — Devans99

Again, non sequitur; even if something outside of time created time (as we both apparently believe), that would not by itself entail that time had a start.

Are you claiming the future defines the past? — Devans99

No, the question-begging claim is that time as an infinite succession of moments is impossible because it would have no first element.

If you had thought it through, you would appreciate that the impossibility of perpetual motion implies a start of motion. — Devans99

Which does not entail a start of time.

Time with the way one moment defines the next is a example of an infinite causal regress, all of which are impossible as they have no first element. — Devans99

Begging the question (again).

We can also use physics too: perpetual motion is impossible, therefore time has a start. — Devans99

Non sequitur.

What you are missing is that the past defines the future and an infinite past can never be a fully defined past because it has no initial starting state (so it must by induction be null and void all the way through). — Devans99

This straightforwardly begs the question by presupposing that being "fully defined" (whatever that means) requires an "initial starting state." Grasping at straws, really.

There are about 6 other ways to prove time has a start. I gave a couple here: — Devans99

Again, it is impossible to "prove" that time has a start with only mathematics and logic.

Your objections apparently boil down to a demand that mathematicians revise their well-established technical terminology (existence, object, etc.) because some of the definitions conflict with those employed in your peculiar metaphysics. Good luck with that!

... we know that an infinite past essentially comes with a moment removed at the start so such a construction is therefore impossible. — Devans99

This is exactly backwards. An infinite past entails that there has never been a moment that was not preceded by another moment, consistent with the continuity of time that we directly perceive in the present. A definite beginning of time entails that there was one moment in the past that was not preceded by another moment, making it a discontinuity.

Infinite past time is like a something from nothing - there is no initial state, so no subsequent states - the existence of the present would therefore be like a magic trick. — Devans99

Again, this is exactly backwards. Infinite past time entails that there was never nothing, instead always something--namely, time itself. A definite beginning of time entails that something came from nothing, or that something outside of time created it. As I said before, a strictly mathematical "proof" is insufficient to determine which hypothesis--infinite past time or a definite beginning of time--is correct.

If we remove any moment or time interval if you prefer, then all subsequent moments or time intervals become undefined. — Devans99

Since all real moments are indefinite, it is logically impossible to distinguish one from another, let alone "remove" one. We can arbitrarily designate instants to mark off intervals of time with fixed and finite duration, but we cannot "remove" those, either. It straightforwardly begs the question to treat time as if it were isomorphic to the natural numbers, which are discrete and have a first member, thus ruling out the possibility that time is truly continuous and does not have a definite beginning.

As for the thread title, Charles Sanders Peirce has an interesting take.

It may be assumed that there are two instants called the limits of all time, the one being Α, the commencement of all time and the other being Ω, the completion of all time. Whether there really are such instants or not we have no obvious means of knowing; nor is it easy to see what "really" in that question means. But it seems to me that if time is to be conceived as forming a collective whole, there either must be such limits or it must return into itself. This is an interesting question. — Peirce, NEM 3:1075, c. 1905

Real time as a whole either has first and last instants or indeed "must return into itself," but we cannot determine which is the case solely by means of a strictly mathematical "proof."

You may, for example, say that all evolution began at this instant, which you may call the infinite past, and comes to a close at that other instant, which you may call the infinite future. But all this is quite extrinsic to time itself. Let it be, if you please, that evolutionary time, our section of time, is contained between those limits. Nevertheless, it cannot be denied that time itself, unless it be discontinuous, as we have every reason to suppose it is not, stretches on beyond those limits, infinite though they be, returns into itself, and begins again. — Peirce, RLT 264, 1898

Mathematical time, conceived as truly continuous, necessarily "returns into itself, and begins again."
Actual time, in which all events occur, might correspond to only a portion of that hypothetical representation, between initial and final instants. We "reckon" actual time by arbitrarily assigning dates in accordance with the fixed and finite intervals between recurring events, such as the earth's rotation around its axis (one day) and revolution around the sun (one year).

Observation leads us to suppose that changing things tend toward a state in the immeasurably distant future different from the state of things in the immeasurably distant past ... It is an important, though extrinsic, property of time that no such reckoning brings us round to the same time again. — Peirce, NEM 2:249-250, 1895

Peirce's own cosmology is not "elliptical" (or "circular"), but "hyperbolic," positing that the states corresponding to the initial and final instants are different from each other as ideal limits, rather than actual events--complete chaos in the infinite past, and complete regularity in the infinite future.

But at any assignable date in the past, however early, there was already some tendency toward uniformity; and at any assignable date in the future there will be some slight aberrancy from law. — Peirce, CP 1.409, c. 1888

But the mass of the sequence must have changed. — Devans99

A sequence has no mass, since it is a mathematical concept, not anything physical. An actual collection of bananas would have mass, but it would necessarily be finite, such that adding a banana would indeed add mass.

But the maths says the sequence is identical so it has the same mass. So its a contradiction. — Devans99

You have made it clear that you reject the established mathematics of (hypothetical) infinite collections, but please stop pretending that there is no such mathematics, or that it cannot be different from the more familiar mathematics of (actual) finite collections.

A peeled banana is no longer identical to a non-peeled banana. — Devans99

It is no longer identical in one respect--whether it is peeled--but it is still identical in others. For example, it is still a banana, and most people would even say that it is still the same banana.

I assume by hypothetical you mean the imaginary structure of actual infinity in our minds? — Devans99

"Imaginary structure of actual infinity"? Now that is a contradiction.

But hypothetical means it might or might not be true. — Devans99

Hypothetical means logically possible, not necessarily true or even metaphysically possible. A hypothetically infinite collection or sequence is logically possible, while an actually infinite collection or sequence is metaphysically impossible.

If I add one banana to the sequence, I should get back something that is someway different from the original sequence. — Devans99

Not when all the bananas are stipulated as identical.

You seem to be saying it is possible logically and/or in reality to change something and it does not change. — Devans99

It is possible to change something in one respect without changing it in another respect. If I peel a banana, it is still the same banana, even though I have changed it in one respect.

Can you explain the difference between hypothetical and actual infinity? — Devans99

I honestly do not see what there is to explain. Do you not know the difference between the hypothetical and the actual?

So we have changed the sequence and it has not changed - contradiction. — Devans99

Again, we have changed it in one respect but not in another - no contradiction.

I think you are trying to defend the indefensible; actual infinity is a logical impossibility. — Devans99

We are discussing hypothetical infinity, not actual infinity. We do not have an actually infinite sequence of actually identical bananas, let alone two such sequences.

So both the sequence itself and the cardinality remain unchanged despite us adding one banana. — Devans99

The only basis for claiming that the two infinite sequences are "identical" initially is that they allegedly consist of "identical" bananas in "identical" order. Accordingly, adding another "identical" banana to the beginning of one of them is not really a change, since the new "first" banana is indistinguishable from any other.

How is 'we change it and it is not changed' not a contradiction? — Devans99

We change it in one respect (whether it includes this particular individual member), but it is not changed in another respect (its cardinality as an infinite set). Not a contradiction.

But in relating ontology to mathematics, aletheist employs intentional vaguery and ambiguity in terms, as well as outright contradiction to support unreasonable mathematical principles. — Metaphysician Undercover

LOLOL.

Anyway, this is one of the cool features of negation - allowing us to affirm by twice negating. — TheMadFool

Only in classical logic, thanks to the law of excluded middle. Not in intuitionistic logic. :cool:

Speak to metaphysics, please. Define "actual" in that context. — jgill

The actual is that which acts on and reacts with other things.

Are the natural numbers metaphysically actual? The complete set of natural numbers? The square root of 2? Chaitin's constant, which is known to be noncomputable? — fishfry

No, these are all numbers; and again, existence in mathematics entails only logical possibility, not actuality in metaphysics.

Is a brick metaphysically actual? — fishfry

Yes, in accordance with how I was using that term.

How about an electron? A quark? A string? — fishfry

That depends on whether one is a scientific realist about each of these entities. I am currently inclined to say yes, probably, and maybe.

So, combining "metaphysical" with "actual" means someone is thinking a metaphysical thought? — jgill

What does thinking have to do with anything? Metaphysics is a branch of philosophy, within which "actual" has a technical meaning that distinguishes it from "possible" and "necessary."

Or does the expression imply an interaction with physical reality? — jgill

That is closer, since whatever is physical is actual in the relevant sense. However, just to be clear, I hold that reality is not coextensive with actuality; there are also real possibilities and real necessities.

Please clarify with examples. — jgill

If I say that I have an apple, what I usually mean is that I have an actual apple. If I posit a set of apples in the strictly mathematical sense, then I am talking about something that is logically possible, but not (necessarily) actual. :smile:

So a moment of time must have a non-zero width — Devans99

Yes, assuming that you meant non-zero duration. Where we disagree is whether this non-zero duration must be finite, and therefore measurable.

The only other definition of an infinitesimal I'm aware of is a number x>0 such that x^2=0. — Devans99

The proper mathematical definition of "infinitesimal" in this context is having length that is non-zero, but shorter than any assignable value relative to an arbitrary unit interval. — aletheist

Energy and matter are discrete quantities so that means c^2 is discrete. But c is distance/time, so are these also discrete? — Devans99

This once again confuses measurement with reality.

You are claiming there are moments in the last second I did not experience. — Devans99

No, I am claiming that all the real moments that we ever experience are indefinite.

... every possible position the hand occupies must be a distinct position ... — Devans99

This is the fundamental assumption on which we disagree--that the only possible positions are distinct positions.

... and there must be a greater than any number of such distinct positions if space is a continuum. — Devans99

In fact, true continuity entails that there are possible positions exceeding all multitude, such that they cannot be distinguished; as I have been saying all along, they must be indefinite. We only distinguish, and thereby actualize, the individual positions that we deliberately mark.

No number can have a length greater than zero and less than all reals IMO. — Devans99

No number can have a length at all. In any case, we are not talking about numbers, we are talking about time.

So it must be the imaginary construct of 1/∞ that you refer to? — Devans99

No, and I have stated this plainly before. I provided the relevant definitions, so please stop trying to impose others.

How do we measure time? — Devans99

Irrelevant; how we measure time does not dictate the real nature of time.

It is worth noting that someday we maybe able to prove empirically that space/time are discrete. — Devans99

We cannot "prove" anything empirically, only gather evidence. What kinds of experiments could somehow demonstrate that time is discrete?

We will never, ever be able to empirically prove they are continuous. — Devans99

Again, we directly perceive the continuity of time. What more conclusive empirical evidence could there be?

But in the second of time that started two seconds ago and finished a second ago, I experienced all possible instances of time as distinct moments - I actualised each moment. — Devans99

No, we already agreed that "now" is not a durationless instant, and all we ever experience is "now"; so we never experience any distinct moments, let alone an actual infinity of them.

Likewise, when I move my hand, I actualise all possible intermediate positions. — Devans99

We had that conversation already.

No, the only actual intermediate positions are the ones that we individually mark. There is a potential infinity of such positions, but we can only mark (and thereby actualize) a finite quantity of them. Again, continuous motion is the reality, while discrete positions are our invention. — aletheist

Positions are artificial creations for describing motion, not real constituents of the motion itself. Likewise for instants and any "distinct moments" that they allegedly define.

That is not what I mean by "metaphysical actuality." I just mean the modal property of being actual, rather than merely possible or strictly necessary, such that something possessing it acts on and reacts with other things in the environment.