What does it mean for something to be "logically possible"?

[Phil Scott]

Actually, it was Bolzano who first gave the formal epsilon-delta definition of limit and then it was Weierstrass who finished the formalization of calculus. Cauchy was an intermediary who had the right notions, but whose work still utilized the infinitesimal.

I don't think so. I've read both in original sources. Cauchy still uses the term "infinitesimally small", but gives epsilon-delta definitions. Bolzano only gets half way, but deserves the partial credit.

 

[beero1000]

Actually, it was Bolzano who first gave the formal epsilon-delta definition of limit and then it was Weierstrass who finished the formalization of calculus. Cauchy was an intermediary who had the right notions, but whose work still utilized the infinitesimal.

I don't think so. I've read both in original sources. Cauchy still uses the term "infinitesimally small", but gives epsilon-delta definitions. Bolzano only gets half way, but deserves the partial credit.

I have an annotated translation, but I can't imagine it differs from the original so drastically. Cauchy constantly talks about 'infinitely small quantities', 'adding an infinitely small quantity', 'infinitely small increments', etc. It's the one of the first things he defines, and the first theorems are about comparing infinitesimals. For sure he had the right ideas, and used arguments in the epsilon-delta style, but he never explicitly wrote the definitions down, instead favoring the infinitesimal approach. His analysis is explicitly based on infinitesimals, even if he had the basics of the ideas that would eventually get rid of them.

My money's on Weierstrass.

 

[bilby]

I don't think so. I've read both in original sources. Cauchy still uses the term "infinitesimally small", but gives epsilon-delta definitions. Bolzano only gets half way, but deserves the partial credit.

I have an annotated translation, but I can't imagine it differs from the original so drastically. Cauchy constantly talks about 'infinitely small quantities', 'adding an infinitely small quantity', 'infinitely small increments', etc. It's the one of the first things he defines, and the first theorems are about comparing infinitesimals. For sure he had the right ideas, and used arguments in the epsilon-delta style, but he never explicitly wrote the definitions down, instead favoring the infinitesimal approach. His analysis is explicitly based on infinitesimals, even if he had the basics of the ideas that would eventually get rid of them.

My money's on Weierstrass.

So neither of you is prepared to bet on untermensche overturning centuries of work by some of the finest mathematicians in history, with his simple-but-ineffective approach of declaring anything he cannot understand to be nonsensical?

Sad.

 

[Phil Scott]

I don't think so. I've read both in original sources. Cauchy still uses the term "infinitesimally small", but gives epsilon-delta definitions. Bolzano only gets half way, but deserves the partial credit.

I have an annotated translation, but I can't imagine it differs from the original so drastically. Cauchy constantly talks about 'infinitely small quantities', 'adding an infinitely small quantity', 'infinitely small increments', etc. It's the one of the first things he defines, and the first theorems are about comparing infinitesimals. For sure he had the right ideas, and used arguments in the epsilon-delta style, but he never explicitly wrote the definitions down, instead favoring the infinitesimal approach. His analysis is explicitly based on infinitesimals, even if he had the basics of the ideas that would eventually get rid of them.

My money's on Weierstrass.

Cauchy tries to stay to the popular language of the time, but I'm near certain he had epsilon-delta formally defined, and so he properly banished "infinitesimal". That he keeps saying "infinitesimally small" is hardly a crime. Modern analysts still say "tends to infinity", even though infinity isn't an object in real analysis.

More impressively, Cauchy rewrote his Calculus course on the epsilon-delta foundation, which is more than can be said for Bolzano. Even then, he was told by the university to go back to the informal stuff because it was confusing the students, suggesting that he was working under the constraint of trying to stay close to the old vocabulary.

 

[Phil Scott]

So neither of you is prepared to bet on untermensche overturning centuries of work by some of the finest mathematicians in history, with his simple-but-ineffective approach of declaring anything he cannot understand to be nonsensical?

Sad.

I'm afraid I haven't read any of untermensche's posts in this thread. It doesn't look to me, based on other threads, like he has a mathematical background.

 

[untermensche]

So neither of you is prepared to bet on untermensche overturning centuries of work by some of the finest mathematicians in history, with his simple-but-ineffective approach of declaring anything he cannot understand to be nonsensical?

Sad.

I'm afraid I haven't read any of untermensche's posts in this thread. It doesn't look to me, based on other threads, like he has a mathematical background.

Where in all your knowledge of mathematics do we find the logic to apply infinity to the real universe?

Does the field of mathematics speak about the real universe?

This is not a matter of mathematics.

Mathematics has nothing to do with trying to apply infinity to the universe.

 

[Phil Scott]

I'm afraid I haven't read any of untermensche's posts in this thread. It doesn't look to me, based on other threads, like he has a mathematical background.

Where in all your knowledge of mathematics do we find the logic to apply infinity to the real universe?

Does the field of mathematics speak about the real universe?

This is not a matter of mathematics.

Mathematics has nothing to do with trying to apply infinity to the universe.

Agreed. You're not talking about mathematics, which is why I haven't read any of your posts. No offence to you.

 

[untermensche]

Where in all your knowledge of mathematics do we find the logic to apply infinity to the real universe?

Does the field of mathematics speak about the real universe?

This is not a matter of mathematics.

Mathematics has nothing to do with trying to apply infinity to the universe.

Agreed. You're not talking about mathematics, which is why I haven't read any of your posts. No offence to you.

I am not offended. I am not claiming to be an expert in mathematics. Although you begin to use infinity combined with limits in the first semester of calculus. So on a very introductory level it has to be defined in some way.

I am merely trying to apply very simple logic here. The consequences of trying to divide some real entity infinitely.

None of it will intrude on the field of mathematics.

If mathematics has moved past infinitesimals then the concept is not only imaginary but it has become useless.

 

[beero1000]

I have an annotated translation, but I can't imagine it differs from the original so drastically. Cauchy constantly talks about 'infinitely small quantities', 'adding an infinitely small quantity', 'infinitely small increments', etc. It's the one of the first things he defines, and the first theorems are about comparing infinitesimals. For sure he had the right ideas, and used arguments in the epsilon-delta style, but he never explicitly wrote the definitions down, instead favoring the infinitesimal approach. His analysis is explicitly based on infinitesimals, even if he had the basics of the ideas that would eventually get rid of them.

My money's on Weierstrass.

Cauchy tries to stay to the popular language of the time, but I'm near certain he had epsilon-delta formally defined, and so he properly banished "infinitesimal". That he keeps saying "infinitesimally small" is hardly a crime. Modern analysts still say "tends to infinity", even though infinity isn't an object in real analysis.

More impressively, Cauchy rewrote his Calculus course on the epsilon-delta foundation, which is more than can be said for Bolzano. Even then, he was told by the university to go back to the informal stuff because it was confusing the students, suggesting that he was working under the constraint of trying to stay close to the old vocabulary.

Using terms like "tends to infinity" doesn't compare because they are defined formally without reference to infinitesimals/infinity - Cauchy's infinitesimals ARE the standard objects from which he derives his results. He definitely did have the right ideas, but I find it a little problematic to say that he eliminated the infinitesimal when his analysis is chock full of them, and while that may be because of cultural pressures, it still doesn't sit right with me. He certainly planted the seed, but it was others who did the actual banishing.

 

[beero1000]

I have an annotated translation, but I can't imagine it differs from the original so drastically. Cauchy constantly talks about 'infinitely small quantities', 'adding an infinitely small quantity', 'infinitely small increments', etc. It's the one of the first things he defines, and the first theorems are about comparing infinitesimals. For sure he had the right ideas, and used arguments in the epsilon-delta style, but he never explicitly wrote the definitions down, instead favoring the infinitesimal approach. His analysis is explicitly based on infinitesimals, even if he had the basics of the ideas that would eventually get rid of them.

My money's on Weierstrass.

So neither of you is prepared to bet on untermensche overturning centuries of work by some of the finest mathematicians in history, with his simple-but-ineffective approach of declaring anything he cannot understand to be nonsensical?

Sad.

To be honest, I find our derail much more entertaining.

 

[Juma]

I'm afraid I haven't read any of untermensche's posts in this thread. It doesn't look to me, based on other threads, like he has a mathematical background.

Where in all your knowledge of mathematics do we find the logic to apply infinity to the real universe?

Does the field of mathematics speak about the real universe?

This is not a matter of mathematics.

Mathematics has nothing to do with trying to apply infinity to the universe.

Noone is "applying infinity to the universe".
If there was no beginning of time the time is limitless in that direction.
If you want anyone believe you that time cannot be limitless in that direction then provide a solid argument for that.
You have written thousands of posts on the matter and yet no such argument.

Sad.

 

[untermensche]

Where in all your knowledge of mathematics do we find the logic to apply infinity to the real universe?

Does the field of mathematics speak about the real universe?

This is not a matter of mathematics.

Mathematics has nothing to do with trying to apply infinity to the universe.

Noone is "applying infinity to the universe".
If there was no beginning of time the time is limitless in that direction.
If you want anyone believe you that time cannot be limitless in that direction then provide a solid argument for that.
You have written thousands of posts on the matter and yet no such argument.

Sad.

By saying time is limitless you are saying it has this imaginary quality of being infinite.

Time is real.

Infinity is imaginary.

Real entities do not have imaginary qualities.

 

[bilby]

Noone is "applying infinity to the universe".
If there was no beginning of time the time is limitless in that direction.
If you want anyone believe you that time cannot be limitless in that direction then provide a solid argument for that.
You have written thousands of posts on the matter and yet no such argument.

Sad.

By saying time is limitless you are saying it has this imaginary quality of being infinite.

Time is real.

Infinity is imaginary.

Real entities do not have imaginary qualities.

This claim makes me sad.

Sadness exists only in the mind.

Therefore your claim has caused me to cease to exist.

QED.

 

[DBT]

But if it's the smallest movement possible, then it must be possible for an object to make it.

:laughing-smiley-014

It is not the smallest move possible. If movement can be divided infinitely.

It is only the smallest move that is possible to make.

It is impossible to make the smallest move possible.

No such thing exists.

The idea dissolves to irrational absurdity.

The whole notion is totally irrational.

I don't find people who defend such nonsense funny. I find them pathetically lost.

- - - Updated - - -

It is impossible for an object to make the smallest movement possible. The idea is absurd.

Given the terms and conditions being presented in the sentence, the objection is absurd. Logic doesn't seem to be the strong point here.

Prove it.

Don't just spew it like vomit.

It was proven that your logic is flawed (to say the least) when you said the ''smallest movement possible'' - ''possible'' means that it can in fact be done, that it is not impossible.

So when you say ''It is impossible for an object to make the smallest movement possible'' you are contradicting yourself.

You are presenting an oxymoron.

 

[untermensche]

It was proven that your logic is flawed (to say the least) when you said the ''smallest movement possible'' - ''possible'' means that it can in fact be done, that it is not impossible.

What I said, and what is true, is that if movement can be divided infinitely there is no smallest movement possible. You can always have a smaller move, without end.

The idea of a smallest movement possible makes no sense if movement can be divided infinitely.

For any object to move though it must first move the smallest movement possible. That is a truism and indisputable.

But if we claim that movement can be divided infinitely there is no smallest movement possible for an object to move first.

The situation instantly dissolves into absurdity. It is absurd to claim movement can be divided infinitely.

Any way you try to apply a real infinity to the real universe the situation instantly becomes an absurdity.

Some get this hint.

 

[Phil Scott]

Cauchy tries to stay to the popular language of the time, but I'm near certain he had epsilon-delta formally defined, and so he properly banished "infinitesimal". That he keeps saying "infinitesimally small" is hardly a crime. Modern analysts still say "tends to infinity", even though infinity isn't an object in real analysis.

More impressively, Cauchy rewrote his Calculus course on the epsilon-delta foundation, which is more than can be said for Bolzano. Even then, he was told by the university to go back to the informal stuff because it was confusing the students, suggesting that he was working under the constraint of trying to stay close to the old vocabulary.

Using terms like "tends to infinity" doesn't compare because they are defined formally without reference to infinitesimals/infinity - Cauchy's infinitesimals ARE the standard objects from which he derives his results. He definitely did have the right ideas, but I find it a little problematic to say that he eliminated the infinitesimal when his analysis is chock full of them, and while that may be because of cultural pressures, it still doesn't sit right with me. He certainly planted the seed, but it was others who did the actual banishing.

Ack. You're going to make me dig out some sources, which may have to wait until I have access at my current uni. I'm working from memory here, but I recall distinctly that, while Cauchy still uses terms like "infinitely small", they are mere shorthand for a statement about a value that can be made arbitrarily small, and he gives a formal definition based on epsilon. The infinitesimal is banished because Cauchy doesn't consider it to be an actual object of real analysis. It's not a real number; it's part of a circumlocution that he can unfold if needed to.

The only thing missing from Cauchy was a proof of the least upper bound property, which had to wait for an actual construction of the reals, first provided by Dedekind.

 

[DBT]

It was proven that your logic is flawed (to say the least) when you said the ''smallest movement possible'' - ''possible'' means that it can in fact be done, that it is not impossible.

What I said, and what is true, is that if movement can be divided infinitely there is no smallest movement possible. You can always have a smaller move, without end.

The idea of a smallest movement possible makes no sense if movement can be divided infinitely.

For any object to move though it must first move the smallest movement possible. That is a truism and indisputable.

But if we claim that movement can be divided infinitely there is no smallest movement possible for an object to move first.

The situation instantly dissolves into absurdity. It is absurd to claim movement can be divided infinitely.

Any way you try to apply a real infinity to the real universe the situation instantly becomes an absurdity.

Some get this hint.

Weren't you the very one arguing strenuously against the concept of infinity. Should I quote you on the subject? It would prove quite embarrassing for you if I did.

Apart from that, you are shifting the gaol posts. You can't divide the ''smallest possible movement'' because, by self definition, it has already been reduced to its final state; the smallest possible movement. You logically you cannot reduce the 'smallest possible movement' by evoking a movement that is smaller than ''the smallest possible movement.''


That is your oxymoron. But of course you cannot admit to making an error.

 

[Phil Scott]

The position of infinity in mathematics is contestable, and there is a lot of rigorous mathematical logic that has been devoted to the subtleties of reasoning with the infinite.

Forget limits and tending to infinity for now. Every real number is an infinite object in its own right. Every infinite decimal expansion (save those that end up as just 9s) corresponds to a unique real. So each real number is a huge object.

Having so much of mathematics being about infinitely large objects was a bit contentious at first, and I'm told that Dedekind's strange "cuts" may only be strange because Dedekind was being quite careful with the logic of them. Even so, others thought that the machinery of infinite sets didn't have a place in mathematics, and the discovery of paradoxes of infinite set theory put the whole enterprise on shaky ground and led Hilbert to suggest we find a way to tame the infinite by effectively defining it out of existence. There are still such finitists alive and doing interesting mathematical logic today, while the constructivists took a more liberal route which nevertheless places heavy restrictions on how one can reason about infinite things such as real numbers.

 

[beero1000]

Using terms like "tends to infinity" doesn't compare because they are defined formally without reference to infinitesimals/infinity - Cauchy's infinitesimals ARE the standard objects from which he derives his results. He definitely did have the right ideas, but I find it a little problematic to say that he eliminated the infinitesimal when his analysis is chock full of them, and while that may be because of cultural pressures, it still doesn't sit right with me. He certainly planted the seed, but it was others who did the actual banishing.

Ack. You're going to make me dig out some sources, which may have to wait until I have access at my current uni. I'm working from memory here, but I recall distinctly that, while Cauchy still uses terms like "infinitely small", they are mere shorthand for a statement about a value that can be made arbitrarily small, and he gives a formal definition based on epsilon. The infinitesimal is banished because Cauchy doesn't consider it to be an actual object of real analysis. It's not a real number; it's part of a circumlocution that he can unfold if needed to.

The only thing missing from Cauchy was a proof of the least upper bound property, which had to wait for an actual construction of the reals, first provided by Dedekind.

I'm annoying that way sometimes.

After looking into it myself, it seems like Cauchy's actual stance might still be a bit of a contentious issue in the history of mathematics - https://arxiv.org/pdf/1108.2885.pdf, so it seems like either might be a reasonable interpretation. I guess that means we need to have the bitterest of arguments about it. To the death!

 

[DrZoidberg]

It has to do with the form of the argument. Logically possible means, if the predicates are true, then it's true.

Is it logically possible that Jesus was the son of god?

Yes. There's nothing contradictory within the statement.

Was it logically possible that the Big Bang was a miracle of some bored gods?

Sure

What does it mean for something to be logically possible? What are the objective criteria to determine such a thing?

That it follows the rules for logic as laid out by Aristotle.

https://en.wikipedia.org/wiki/Term_logic

An example of broken logic.

1) All religions in the world are equally true.
2) The Abrahamic God created the world.
3) Brahma created the world.

How does logic make things possible?

A road makes it possible for you to walk on it. But you're still doing the walking. The road doesn't walk you over it. Same with logic making things possible.

Does logic or modeling off data, or actual data, show it is possible or impossible to move faster than the speed of light?

You don't understand how logic works. Logic doesn't care about the real world. It only cares about how your argument is structured. When I studied logic my professor only used absurd examples. Like monkeys eating meat. Or Elefants driving zeppelins. Just to remind us that logic doesn't care about if it's likely in the real world.

There is no logic in saying because there was a yesterday that means there may have been infinite yesterdays. No logic to saying because the past is a mental collection of present moments those present moments were possibly infinite.

How does some idea with no evidence to support it become logically possible?

Because you don't need evidence. That's the beauty of logic. It works without requiring anything but the sentences itself you are evaluating