I can't. All I can do is lead the donkey to the water. I can't make him drink.

Can you show me a physics text that does not use time?

'cause, you see, as has been mentioned before, your grasp of physics is, shall we say, eccentric?

So better to pay it no attention. — Banno

@Banno explains science and math.

Cheers! I guess I am a bit like Charlie Brown to Meta's Lucy...

Cheers! I guess I am a bit like Charlie Brown to Meta's Lucy... — Banno

Yes, that’s just what I was thinking.

Can you show me a physics text that does not use time? — Banno

How's that relevant? Physics uses mathematics, but that doesn't mean mathematics is physical.

Here I would side with Moliere. It is a logical problem. Or basically that the measurement problem is a logical problem, hence you cannot just suppose there to be "an adequate way of measurement". — ssu

Whether or not it is possible to devise an adequate way is irrelevant. The problem is that we do not have an adequate way. And the lack of an adequate way produces the use of an inadequate way. Therefore the problem is not a logical problem, it is a problem in the application of logic. Principles are applied where they are not suitable for the task which they are applied to. That is a measurement problem.

The problem is infinity itself. And that is a logical problem for us. — ssu

Infinity itself is not the problem. The problem is how the concept of infinity is developed and employed. In its basic form "infinity" allows that principles of measurement such as numbers, can be extended indefinitely so that in principle anything and everything can be measured. That is beneficial, it is not a problem. The problem is that there are many misuses of infinity, such as the idea that there is some type of thing which can be infinitely divided. That is not a problem with infinity, but a problem with its application, a problem of applying the wrong principles to the task, a measurement problem.

The problem is that there are many misuses of infinity, such as the idea that there is some type of thing which can be infinitely divided. — Metaphysician Undercover

Is this a "misuse" in mathematics? We are talking about mathematics.

Pick two real numbers, and it can be shown that there are real numbers between them. Pick even two rational numbers, and you have rational numbers between them.

You would wander to the illogical, if you would to start to argue that it isn't so, that it's misuse or something.

Is this a "misuse" in mathematics? We are talking about mathematics.

Pick two real numbers, and it can be shown that there are real numbers between them. Pick even two rational numbers, and you have rational numbers between them.

You would wander to the illogical, if you would to start to argue that it isn't so, that it's misuse or something. — ssu

That's not misuse, nor is it a problem.

The problem is in Zeno's application, when things like distance, and time, are assumed to be infinitely divisible. It is a measurement problem because instead of determining the natural constraints on such divisions (these constraints are unknown), it is simply assumed that divisibility is infinite.

, it is simply assumed that divisibility is infinite. — Metaphysician Undercover

An obvious wrong assumption?

That's not misuse, nor is it a problem. — Metaphysician Undercover

But it does puzzle us still. Because if you think that we know everything about mathematical infinity, then I guess there should be an answer to the Continuum Hypothesis.

The problem is in Zeno's application, when things like distance, and time, are assumed to be infinitely divisible. — Metaphysician Undercover

I do get that point, sure. But do actually notice that Zeno belonged to the Eleatic School. Platonists were on the camp of infinite divisibility. The Eleatic School was different.

Ok.

If you think so, then wouldn't there be more natural numbers (1,2,3,...) than numbers that are millions? Isn't there 999 999 between every million?

No, similar amount, because
(1,2,3,....) can be all multiplied by million
(1000 000, 2 000 000, 3 000 000,...)

And because you can make a list of all rational numbers (as above), the you can fit that line with the (1,2,3,...) line in similar fashion. That's the bijection, 1-to-1 correspondence. — ssu

If you think so, then wouldn't there be more natural numbers (1,2,3,...) than numbers that are millions? Isn't there 999 999 between every million? — ssu

I'm not sure, actually...

That makes sense by what I said, but then your objection also hits home -- if there's a 1-to-1 mapping then really all I'm doing is performing some operation on one set to get to the other set, which is pretty much all a function is (by my understanding).

But then the set "Even numbers" is defined by whether or not they are divisible by 2. So if we have the set of all natural numbers and the set of all even natural numbers then, if there's a 1-to-1 correspondence, we ought be able to lay out a function like the above -- such as f(x) = x * 1 million

But suppose the number 3 -- does it yield 2 or does it yield 4? Once we decide that then we can say there's a function which maps, but before that it seems to me we have to make a decision.

Now that doesn't seem to make them uncountable, and perhaps the sizes of the sets are still the same -- the whole idea of infinite sets having different sizes is the thing that is confusing me. I'm just responding here in my own words and thinking out loud.

Let's say you have a set of numbers, let's call them Moliere-numbers. As they are numbers, you can always create larger and larger Moliere-numbers. Hence we say there's an infinite amount of these numbers. The opposite of this would be a finite number system that perhaps an animal could use: (nothing, 1, 2, 3, many) as that has five primitive "numbers".

If we then say that these Moliere-numbers are countably infinite, then it means that there's a way to put them into a line:

Moliere-1, Moliere-2, Moliere-3,.... and so on, that you can be definitely sure that you would with infinite time and infinite paper write them down without missing any.

If Moliere-numbers are uncountably infinite, then we can show that any possible attempted list of Moliere numbers doesn't have all Moliere-numbers. — ssu

Thanks for indulging my curiosity. If what I said above is entirely whack then feel free to just point me to a text ;)

But if you're willing to continue....

How could we show that Moliere-numbers are uncountably infinite?

An obvious wrong assumption? — Corvus

That assumption does create a measurement problem. So unless we think that measurement problems are good, then I'd say it's a wrong assumption.

But it does puzzle us still. Because if you think that we know everything about mathematical infinity, then I guess there should be an answer to the Continuum Hypothesis. — ssu

Again, "continuum" assumes something being divided. Simply saying that there is a number between any two numbers does not assume anything being divided, just like assuming that there is always a higher number does not assume anything being counted. These are simply pure mathematical axioms. But when we say things like "there is a continuum", "numbers are objects", then we introduce ontological premises into the mathematical axioms, which may or may not be true.

That assumption does create a measurement problem. So unless we think that measurement problems are good, then I'd say it's a wrong assumption. — Metaphysician Undercover

Correct. Wrong assumptions lead to invalid conclusions. End of the story. I think I wrote this point a while back in the thread.

But then the set "Even numbers" is defined by whether or not they are divisible by 2. So if we have the set of all natural numbers and the set of all even natural numbers then, if there's a 1-to-1 correspondence, we ought be able to lay out a function like the above -- such as f(x) = x * 1 million — Moliere

And there's a 1-to-1 correspondence:

(1, 2, 3, 4, 5, 6, 7,...)
(2, 4, 6, 8,10,12,14,...)

But if you're willing to continue....

How could we show that Moliere-numbers are uncountably infinite? — Moliere

OK, basically how Cantor showed that real numbers are uncountable is the way to do this.

Basically if you have a list where all the Moliere-numbers would be and then you show that there's a Moliere that differs from the first Moliere-number on the list, differs from the second Moliere-number on the list and so on. This way you show that there's a Moliere-number that isn't on the list. Hence there cannot be a list of all Moliere-numbers. The conclusion is a Reductio ad absurdum proof.

The precision of position and momentum are proportional to each other such that a greater precision of position results in a lesser precision of momentum. — Moliere

How does this enter into a discussion of these Zeno type paradoxes? Define the momentum of a point as it progresses to zero. Does the tortoise have momentum? Too much of a stretch for me.

How does this enter into a discussion of these Zeno type paradoxes? Define the momentum of a point as it progresses to zero. Does the tortoise have momentum? Too much of a stretch for me. — jgill

I think it's because quantum stuff is based on the notion that reality is discrete vs the continuous reality of the block universe; but, yes, if it's just a logical puzzle then the science is irrelevant.

And there's a 1-to-1 correspondence:

(1, 2, 3, 4, 5, 6, 7,...)
(2, 4, 6, 8,10,12,14,...) — ssu

OK that makes sense seeing it like that rather than the muddle I wrote. (While it's interesting to me I claim little knowledge here)

OK, basically how Cantor showed that real numbers are uncountable is the way to do this.

Basically if you have a list where all the Moliere-numbers would be and then you show that there's a Moliere that differs from the first Moliere-number on the list, differs from the second Moliere-number on the list and so on. This way you show that there's a Moliere-number that isn't on the list. Hence there cannot be a list of all Moliere-numbers. The conclusion is a Reductio ad absurdum proof. — ssu

That helps me work through the wikipedia page on Cantor's diagonal argument. Thanks!


It's still a concept that confuses the hell out of me, but this gives direction if ever I'm tempted to talk on it again ;)

Good grief.

OK that makes sense seeing it like that rather than the muddle I wrote. — Moliere

Yep, it's an easy way to understand the whole thing.

It's still a concept that confuses the hell out of me, but this gives direction if ever I'm tempted to talk on it again ;) — Moliere

If your smart and observing, it should!!! It is confusing.

Because then you get to the really awesome question: OK, if we have countable and uncountable infinities, what is the relationship between these two infinite sets?

Cantor himself gave us the Continuum Hypothesis as an answer. But what does it mean? And what actually the whole idea of there being larger and larger infinities mean, because the only thing we have shown is this "uncountability" of the reals.

(Oh btw, I think Moliere-numbers sound very cool. You might really think that there are Moliere numbers)

Wrong assumptions lead to invalid conclusions. — Corvus

It's better to say that those conclusions are unsound rather than invalid.

OK, if we have countable and uncountable infinities, what is the relationship between these two infinite sets? — ssu

Take the rationals in [0,1] and form the union with the irrationals in [0,1] and you get a continuum. They are complementary in the complete interval - which itself is a complete metric space with the usual metric.

It's better to say that those conclusions are unsound rather than invalid. — Metaphysician Undercover

Unsound argument means the premise was false, and also invalid reasoning was applied for the conclusion. Here reasoning seems valid, but the premise was false, which led to the false conclusion. Hence the argument is invalid.

To be precise, conclusion is either true or false, but arguments could be either valid or invalid. If the conclusion was true and followed from the premise, then the argument is valid and sound. If the conclusion was false and had false premise, then the argument is invalid and unsound.

Arguments can be valid if it followed from the premise even if the conclusion is false. Argument is invalid, if it didn't follow from the premise even if the premise was true. Do you agree with these points?

The statement P -> Q was false could have been proved via MT.
P -> Q
~Q
~P

Unsound argument means the premise was false, and also invalid reasoning was applied for the conclusion. — Corvus

Replace "and" with "or" here, and you'll see that if the reasoning is valid and the premise is false, then the argument is valid but unsound. So you should conclude "Hence the argument is unsound', instead of the following:

Here reasoning seems valid, but the premise was false, which led to the false conclusion. Hence the argument is invalid. — Corvus

https://iep.utm.edu/val-snd/

t if the reasoning is valid and the premise is false, then the argument is valid but unsound. — Metaphysician Undercover

Depending on the type of two points in the distance (which is not clear in the OP), the conclusion can be true, which makes the argument invalid.

Think of the case where the two runners are running around a circle, not a straight line. When A takes over T, he is still behind T in the circle. A must run again to take T over, but when he does, he is still behind T and so on ad infinitum, which makes the conclusion true, and argument invalid.

So you should conclude "Hence the argument is unsound', instead of the following: — Metaphysician Undercover

Doing so without clear evidence or information of the type or nature of the distance in the track would be commiting a fallacy of illicit presumption. Until all is clear and evident, the argument must be judged as invalid.

In my view, Zeno's arguments pointed towards position and motion being incompatible properties, but the continuum which presumes both to coexist doesn't permit this semantic interpretation.

Is this in any way motivated by the uncertainty principle? — Moliere

If you mean the Heisenberg uncertainty principle no - although I'm tempted to think that Zeno was close to discovering a logical precursor to the Heisenberg Uncertainty Principle on the basis of a priori arguments.

The semantic problems of calculus with regards to Zeno's arguments stem from the fact that calculus isn't resource conscious. Sir Isaac Newton and Leibniz had no reason in 17th century to formulate calculus that way, given the use cases of calculus that they had in mind.

A notable feature of resource-conscious logics is how they naturally have "quantum-like" properties, due to the fact their semantic models are state spaces of decisions that are generally irreversible, thereby prohibiting the reuse of resources; indeed, the assumption that resources can be reused, is generally a cause of erroneous counterfactual reasoning, such as when arguing that a moving object must have a position because it might have been stopped.

So in the case of a resource-conscious calculus that avoids mathematical interpretations of Zeno's paradoxes (as in a function having a gradient but also consisting of points), a function must be treated as a mutable object whose topology undergoes a change of state whenever the function is projected onto a basis of functions that "measure" the function's properties -- Thus the uncertainty principle of Fourier analysis has to be part of the foundations of a resource-conscious calculus rather than a theorem derived from real-analysis of the continuum that is the cause of the semantic unsoundness of calculus with respect to the real world.

An obvious candidate for contributing to the foundations of such an alternative calculus is some variant of differential linear logic, which incidentally has many uses in quantum computing applications.

A notable feature of resource-conscious logics is how they naturally have "quantum-like" properties, due to the fact their semantic models are state spaces of decisions that are generally irreversible, thereby prohibiting the reuse of resources; indeed, the assumption that resources can be reused, is generally a cause of erroneous counterfactual reasoning, such as when arguing that a moving object must have a position because it might have been stopped. — sime

Can you explain a bit more thoroughly what you mean by "resource-conscious"?

Take the rationals in [0,1] and form the union with the irrationals in [0,1] and you get a continuum. They are complementary in the complete interval - which itself is a complete metric space with the usual metric. — jgill

But that doesn't answer the questions we have about infinity. If we have countable infinity and then uncountable infinity, Cantor argues there's this hierarchial system of larger and larger infinities (from aleph-0 to aleph-1 and higher). Is that really how it goes? And the Continuum Hypothesis is a hypothesis, it's not a theorem.

Set theory, which can be viewed as the foundations of mathematics, simply takes infinity as an axiom. That's not a proof and that's the problem. The questions that have been around for thousands of year, things like if there is either a potential or an actual infinity are still puzzling. That's why Zeno's paradoxes are talked over and over again, even if we do understand that Achilles brushes past the tortoise in reality and we do have the math to calculate it, the foundational question are still there.

As I have stated before in this thread, we don't have a similar debate about of "Are all numbers rational?". Nobody is saying that they would be so. No, we can prove that there are irrational numbers. Above all, we can understand that there are and have to be irrational numbers. There wouldn't be this kind of over and over repeating debate Zeno's paradoxes, if we fully would understand the infinite or infinity.

Can you explain a bit more thoroughly what you mean by "resource-conscious"? — Metaphysician Undercover

Resource conscious logics such as Linear Logic don't automatically assume that the premise of a conditional can be used more than once. They are extensions or refinements of relevance logic. The best article relating resource-sensitivity to the principles of quantum mechanics is probably nlabs description of linear logic

https://ncatlab.org/nlab/show/linear+logic

As for uncertainty principles:

Recall that classical logic has the propositional distributive law, that for all A, B and C

A ∧ ( B ∨ C) = (A ∧ B) ∨ ( A ∧ C)

Here, the meaning of "and" is modelled as the Set cartesian product, and the meaning of "or" by set disjunction, neither of which are resource conscious - therefore one always has the same cartesian product, even after taking an element from one of its sets. The negation of this principle is more or less a definition of the uncertainty principle and characterizes the most remarkable aspect of quantum logic, which is in fact a common-sense principle that is used extensively in ordinary life.

The connectives of Linear logic cannot be interpreted in terms of the cartesian product and set disjunction. Instead it has the tautology

A ⊗ ( B ⊕ C) ≡ ( A ⊗ B ) ⊕ ( A ⊗ C )

If this formula is interpreted to be a true conclusion that needs to be proven with respect to unknown premises , then it has the interpretation "Assume that we are sent an A i.e. an element (a : A), and that we are also sent either (b : B) or (c : C) at our opponent's discretion, neither of which consume the (a : A) (that is to say B and C are independent of A). Then we end up with either (a : A) and (b : B), or (a : A) and a (c : C)".

Likewise, our opponent's side of this interaction is then described by the tautology

¬A ⅋ ( ¬B & ¬C) ≡ ( ¬A ⅋ ¬B ) & ( ¬A ⅋ ¬C)

"If our sending of (a : A) also implies our sending of either (b : B) or (c : C), where B and C are independent of A , then we either send both (a : A) and (b : B), or we send both (a : A) and (c: C).

But there isn't the theorem

A ⊗ ( B & C) ≡ ( A ⊗ B ) & ( A ⊗ C )

The inability to derive this theorem is the common-sense uncertainty principle of linear logic: getting an A together with a choice of B or C for which this act of choosing is independent of the existence of A, isn't equivalent to the outcome of the choice being independent of the existence of A.

(Imagine winning a bag of sugar together with a choice between winning either ordinary ice cream or diet ice cream. It might be that the awarders of the prizes use the awarded bag of sugar to produce the chosen ice-cream.)

By analogy, by using a resource-conscious logic as the foundation of an alternative calculus, smoothness and pointedness can be reconciled by defining them to be opposite and incompatible extremes of the state of a mutable function that is affected by the operations that are applied to it. This is also computationally realistic.