Holly shite! there are numbers that cannot be counted.. — Banno

Same with the odd vs whole. Oh it's *different* with infinities of infinities? This is not established in that video

Strictly, when properly stated, that's undecidable. But as I say, you need a mathematician.

Same with the odd vs whole — Gregory

Well, no, since for every whole there is an odd, as has been shown.

That you misunderstand something does not make it wrong.

It's useful, not true. — Gregory

Not true, but useful?

Ok, that really doesn't make any sense. Calculus is a part of mathematics and totally accepted. Please don't start to argue that Calculus is not true.

I've presented at least 5 cogent arguments against infinity — Gregory

No, you haven't been at all convincing. I'm afraid that you don't simply get it.

You can't seem to recognise that the responses you are receiving actually answer your questions. It's odd. But it's not about maths, it's about you. — Banno

I think I have to agree here with @Banno. Don't want to be harsh here.

Please don't start to argue that Calculus is not true — ssu

It's very first principles are wrong. Like in history when guys started questioning Euclidean postulates? Just because you misunderstood my arguments do not make them wrong. Have a good day

You are not the first to mistake their misunderstandings for something profound. I suggest you read and try to understand Cantor's diagonal argument - you should be able to Google a version you find agreeable. See if you can read it sympathetically, rather the deciding that it is wrong from the outset; that way you can avoid confirmation bias. it's hard stuff, and if you manage to see how it works you may find you have done something quite satisfying.

It's very first principles are wrong. Like in history when guys started questioning Euclidean postulates? — Gregory

How are the very first principles of calculus wrong? What are you talking about?

Actually you give a perfect example of something not being wrong, but simply limited. It's not that Euclid was wrong, it simply was the case that not everything fell into his understanding of geometry. Root cause was that geometry on a plane and on a sphere are simply different. And you might have to think of geometry of a sphere. That's it. Yet the geometry on a plane is still correct. Hence the error is if one thinks that all geometry happens on a plane. Thus there is Euclidian geometry and non-euclidian geometry (spherical or hyperbolic etc).



Actually, what example would really be false was the Greek idea that "All numbers are rational". And the idea why people believed it was so was because... math is so beautiful. Well, there are irrational numbers. The Greeks found them, and they weren't happy about it. Yet that idea really was a genuine error.

And I think that the idea that "there is no infinity in mathematics" is simply wrong. Similar to the latter example "all numbers are rational". That you only stick to finite mathematics is another thing. Ok, do that. But then what you can do in mathematics is limited.

This hoary false paradox certainly says nothing about the actual nature of space or anything.

In this exercise you are imagining the state of the tortoise/hare at a time closer and closer to the time that the hare catches up. But never reaching that time.

You can use this method to approximate this meeting time. No one does, since obviously it can be exactly solved. But if you perform enough iterations of the hare catching up and the tortoise moving on, you will arrive at the effectively exact time and distance that they meet.

The hare never reaches the tortoise, because time, in the thought experiment, never reaches the moment that the hare does pass. As soon as you imagine time proceeding beyond this meeting time, you must imagine the hare passing, for your thought experiment to tension consistent.

there are numbers that cannot be counted... — Banno

Only if you are a Platonic realist. Metaphysically, that's an issue with set theory in general, Platonism is presupposed.

And when abstractions such as numbers, are assumed to have independent existence just like physical objects, with no principles to differentiate between the abstract and the physical, we have the problem 180 mentioned:

confusing the physical and abstract. — 180 Proof

This is why the law of identity was imposed, as a principle of differentiation between physical objects and abstract objects. A physical object has an identity unique to itself, an abstract object has no such identity. Therefore all those assumed numbers which cannot be counted, have no identity.

...just like there are numbers that are even and numbers that are prime.

More or less in the case of Zeno. Mathematics is often said to resolve the paradox in terms of the topological continuity of the continuum, by treating the open sets of the real line as solid lines and by forgetting the fact that continuum has points, meaning that the paradox resurfaces when the continuum is deconstructed in terms of points. — sime

The paradox between discrete and continuous seems a good example, to me on the outside, as something to treat Zeno's paradoxes as genuine paradoxes.

The way I put it to make it make sense to me: I can say there are "more" rational numbers than there are even numbers. Both sets are infinite, but it seems to me that the Rational Numbers > the Even Rational Numbers, as I understand the notions.

But that there can be "larger" infinites is a paradox to my mind.

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?

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

Actual measurements fail beyond Planck's constants. These paradoxes are all hypothetical involving motions of dimensionless points along rational number scales.

Actual measurements fail beyond Planck's constants. These paradoxes are all hypothetical involving motions of dimensionless points along rational number scales. — jgill

Actual measurements fail far above Planck's constants :)

One of the concepts I've found hard to teach is the difference scientists attach to "accuracy" vs "precision"

Normally we'd interchange these words, which is the reason it's hard to differentiate. But they both deal with measurement, in reality, so are needed.

What you read on your fuel gauge on your car is a measure of how much fuel you have in your tank. The accuracy and precision of that gauge can be described as such -- suppose you have a particularly imprecise but mostly accurate fuel gauge, as I suspect most of them are. Then when it reads "1/2 tank" you know it's about, in terms of 16'ths, about 6/16's to 10/16's. Precision is saying "looks, you don't know between these numbers what it actually is" and accuracy is saying "it's definitely within this range that the precision says, and the "real" number is there but this is what you get"

****

What I'm asking is more about Heisenberg's Uncertainty Principle, which as he interpreted it meant that reality itself doesn't allow for a precision of both, but rather demands aprecision, or position,* of any one particle. But due to cuz that's how nature works, not cuz how we measure it.

*Blah. Speaking from memory makes me say wrong things. The precision of position and momentum are proportional to eachother such that a greater precision of position results in a lesser precision of momentum. Einstein interpreted this in mechanical terms, but the quantum scientists, at least of the time, interpreted this in real terms -- it wasn't the apparatus measuring but rather the behavior of the quantum particles which differed from the old billiard ball model.

One thing that is very clear here is that some folk do not understand errors. Error is fundamental to physics.

See, for more, Introduction to Error and Uncertainty.

There's that, and then there's the philosophically more interesting view expressed here:

Each measurement has a certain amount of uncertainty, or wiggle room. Basically, there’s an interval surrounding your measurement where the true value is expected to lie.

...the presumption that there is a true value; that given infinite precision we could set out the actual value as a real number. There is no reason to supose this to be true.

There's that, and then there's the philosophically more interesting view expressed here:
Each measurement has a certain amount of uncertainty, or wiggle room. Basically, there’s an interval surrounding your measurement where the true value is expected to lie.
...the presumption that there is a true value; that given infinite precision we could set out the actual value as a real number. There is no reason to supose this to be true. — Banno

Yup.

The "accuracy" part of the distinction is what I consider to be a noble fib. Speaking to a person who believes that the gauge they've always used says exactly what's in there it's time to note a difference between accuracy and precision.

It's only a half-fib, because accuracy still ends up mattering. Using the fuel gauge example if you've used that fuel gauge so many times and know that when it says "a hair up from 1/4 tank" you can easily get from A to B and back to the Gas Station in time then it's accurate, if not precise. So accuracy is important -- it just has more to do with the reason things jump around. If a gauge jumps around over the usual precision limits you might have a problem with accuracy (i.e., the gauge is busted, most of the time -- or occasionally, from the history of science, you actually figure something new out)

Yes to accuracy agains precession.

But it's not a "fib" at all; the tank really is a quarter full, ±5%. It's a truth.

What I'm asking is more about Heisenberg's Uncertainty Principle, which as he interpreted it meant that reality itself doesn't allow for a precision of both, but rather demands aprecision, or position,* of any one particle. But due to cuz that's how nature works, not cuz how we measure it. — Moliere

It's due to the way that time exists, in conjunction with the limitations of our capacity to measure. We are limited in our ability to measure time by physical constraints. If we had a non-physical way to measure time we wouldn't be limited in that way.

I think it's a half-fib when I speak to people who really believe that what you measure is what you get.

You're right it's not a "fib" because measurement requires both, so as I understand it at least.

It's a half-fib because I know the person who thinks in terms of accuracy without precision will most likely not understand the difference. They'll understand that things can be uncertain, of course -- who doesn't? -- but probably doesn't understand that the reason this is uncertain is different from why the other things were uncertain.

It's due to the way that time exists, in conjunction with the limitations of our capacity to measure. We are limited in our ability to measure time by physical constraints. If we had a non-physical way to measure time we wouldn't be limited in that way. — Metaphysician Undercover

Heh.

Well, give it some time. Perhaps we'll figure out the non-physical way to measure time :D

I can't say I agree with your first statement because "the way that time exists" and "the limitations of our capacity to measure" are both things I think about with uncertainty all the time.

We're limited in terms of measuring -- but I want to say that Zeno's paradoxes are not problems of measurement at all. They are logical problems (which is why they evoke the difference between physics and logic and math, as the OP stated already)

I can't say I agree with your first statement — Moliere

Good, becasue it is nonsense. A "non- physical" measurement of a physical quantity... what would be your non-physical units for the fuel left in the tank - not litres, since they are physical.

Physical measurements are not infinitely precise, nor is such precision needed.

Physical measurements are not infinitely precise, nor is such precision needed. — Banno

Oh, definitely.

Or accurately? Precisely? :D

I think this lays out a good difference between truth and measurement -- we have to be able to say that the fuel gauge is precise, or accurate, in such and such a way in order to do the things we do. Thereby accuracy and precision get relegated to truth -- as the philosopher should want -- but then the truth of truth becomes wildly different from what the philosopher wanted.

I think this lays out a good difference between truth and measurement — Moliere

No. The measurement is true. Specifying the degree of error does not render the measurement untrue. The tank really does contain 25±1 litres.

No. The measurement is true. Specifying the degree of error does not render the measurement untrue. The tank really does contain 25±1 litres. — Banno

Truth, so I'd put it, is a predicate which applies to sentences.

Measurements can be true, but it's not the same as "true" above.

I can be a true friend, and being a true friend is not the same as having a true sentence.

"The tank really does contain 25±1 liters" is true

Accuracy is the "25" and precision is the "±1 liters"

"The tank really does contain 25 liters", in this case, is true

"The tank contains '±1 liters' of what it reads" is also true

Does that confuse, or help, or do I need to say something else?

We're limited in terms of measuring -- but I want to say that Zeno's paradoxes are not problems of measurement at all. They are logical problems (which is why they evoke the difference between physics and logic and math, as the OP stated already) — Moliere

The logical problems are the result of not having an adequate way of measuring. We are reduced to logical possibility. If we had the proper way we wouldn't have to entertain those possibilities.

So for example, the true divisibility of every physical thing is determined by its physical composition. But if we do not know how it is composed we just assume the logical possibility of infinite divisibility. This is what happens with space and time, and before the atomists, matter itself. We do not know how these things are composed so we just assume the logical possibility of infinite divisibility.

A "non- physical" measurement of a physical quantity... what would be your non-physical units for the fuel left in the tank - not litres, since they are physical. — Banno

I was talking about the problems with the measurement of time (basis of the uncertainty principle), not the measurement of fuel. Show me how time is a physical quantity.

Or accurately? Precisely? — Moliere

I took a class in psychological measurement in college lo these many decades. This was how precision and accuracy were described. They are both statistical properties. The analogy that was used was to archery. If all the arrows are clustered close together in the bullseye, they are precise and accurate. If they are clustered close together but offset from the bullseye, they are precise but not accurate. If they are centered on the bullseye but are not clustered close together, they are accurate but not precise.

And that's the name of that tune.

The way I put it to make it make sense to me: I can say there are "more" rational numbers than there are even numbers. Both sets are infinite, but it seems to me that the Rational Numbers > the Even Rational Numbers, as I understand the notions. — Moliere

Umm... as the set of rational numbers is countably infinite, I would say there's as many rational numbers as there are natural numbers or "even rational numbers".

Because you can go through all rational numbers in the way Cantor showed:

...but I want to say that Zeno's paradoxes are not problems of measurement at all. — Moliere

Yep.

In that case I'd say I'm in even less understanding of the difference between the "size" of infinite sets.

The way it was explained to me was the difference between the rationals and the reals -- my thought was to extend that to the rational sets "All Rational Numbers" and "All Rational Even numbers", and note how, intuitively at least, that the first seems to contain about twice as much as the second, even though both are infinite.

That's a paradox to me.

Zeno's paradoxes are very clearly problems of measurement. Like I explain above, if we had the appropriate way of measuring things like time and space, we wouldn't have to entertain the logical possibility of infinite divisibility. Then there would be no such paradoxes. The paradoxes are due to a deficiency in measurement capacity.

Show me how time is a physical quantity. — Metaphysician Undercover

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.

We're limited in terms of measuring -- but I want to say that Zeno's paradoxes are not problems of measurement at all. They are logical problems (which is why they evoke the difference between physics and logic and math, as the OP stated already) — Moliere

The logical problems are the result of not having an adequate way of measuring. We are reduced to logical possibility. If we had the proper way we wouldn't have to entertain those possibilities. — Metaphysician Undercover

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".

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

In that case I'd say I'm in even less understanding of the difference between the "size" of infinite sets. — Moliere

Ok, it's can be difficult to understand, but I'll try to explain.

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.

my thought was to extend that to the rational sets "All Rational Numbers" and "All Rational Even numbers", and note how, intuitively at least, that the first seems to contain about twice as much as the second, even though both are infinite. — Moliere

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.