The meaning of infinity

[bilby]

It means there is nothing you can tell me about the real practice of pharmacy.

Working pharmacists, clinical pharmacists never use it.

Only researchers use it. And only a fraction of them.

Are those researchers the REAL pharmacists?

Don't worry; There is nothing ANYONE can tell untermenshe about ANYTHING.

Never have I encountered before a person so completely incapable of even considering the possibility that he might be mistaken about anything. I doubt he has learned anything new since he left school; He is unshakably certain that he knows everything there is to know about any topic, and refuses to even consider the possibility that anyone else might have something to teach him. I haven't seen him change a single position on any topic in ten years. That's why I have him on ignore - discussion with him is completely futile.

When he says "there is nothing you can tell me about..." it isn't a figure of speech.

 

[untermensche]

When all people can do is talk about me they are a bore.

I make point after point.

I wonder why it is so hard to address them?

As far as calculus it is not something the majority of pharmacists use in their work.

It is not something I ever had to use in pharmacy school.

But it is a weeder course all pharmacists are required to take.

 

[Juma]

When all people can do is talk about me they are a bore.

I make point after point.

I wonder why it is so hard to address them?

As far as calculus it is not something the majority of pharmacists use in their work.

It is not something I ever had to use in pharmacy school.

But it is a weeder course all pharmacists are required to take.

Its not hard to adress your points. Its the fact that you are incapable of adressing our responses so it becomes a very weird ”discussion”.

 

[Phil Scott]

As I understand it, transcendental numbers can only be defined as limits of infinite series in the Reals. If the Hyperreals can do away with limits in the case of differential calculus, can they do the same trick for transcendentals? Including perhaps the non-computable ones?
EB

Thanks for the question.

It's not right to say that the transcendentals can only be defined as limits of infinite series. Everyone's two favourite transcendentals, π and e, are defined mosty elegantly as, respectively, the ratio of the circumference to the diameter, and the base of the natural logarithm. It turns out that these numbers can be given by infinite series but then, so can every real (every decimal expresses an infinite geometric series).

The transcendentals are defined by what they are not: they are not algebraic. That is, they are not the roots of a polynomial with integral coefficients. An example of such a number is the square root of 2, whose expression is literally saying that it is algebraic: it is the principal root of the equation x² = 2.

The irrationals are similarly defined by what they are not: they are not rational. That is, they are not the ratios of two integers.

As I understand (I haven't gone through the details this far), the hyperreals afford a novel way of dealing with infinite series. First, you take the function which inputs a natural number n and gives you the sum of the first n terms. The real analyst wants to take this function and take its limit as n becomes arbitrarily large. The hyperreal analyst instead takes its hypernatural extension, turning it into a function that works on both naturals and infinite hypernaturals. They then feed it an infinite hypernatural, drop off any infinitesimal value from the result, and obtain the same answer as their real analyst colleague. Thus, where the real analyst asks "what is the infinite sum" but really means "what is the limit of finite sums", the hyperreal analyst asks "what is the infinite sum" and means "what is the sum when the number of terms is infinite"! Real analysis banished infinity in favour of limits. Hyperreal analysis restores it.

Now since infinite series are ultimately how you do integration, this is how the hyperreal analyst gets hold of the ratio of the circumference of a circle and its diameter, and how they define the natural logarithm.

For differentiation, the situation is similar. Here, we use Newton's quotient. The real analyst obtains derivatives by figuring out the limit of this quotient's value as the denominator drops to 0. Instead, the hyperreal analyst asks what value the function has at an infinitesimal, drops any infinitesimal off the result, and obtains the same result as their colleague. The fact that the hyperreals form a field means that, if the function is defined by simple algebraic laws (as instances of Newton's quotient often are), you can always do this, without messing about proving a bunch of lifting theorems for limits.

So the answer to your original question is: yes. The hyperreal analyst can do away with limits, but no, they don't do away with infinite series. They also don't have anything new to say about non-computable numbers. That's a fundamental distinction in logic that no-one can get round.

 

[untermensche]

When all people can do is talk about me they are a bore.

I make point after point.

I wonder why it is so hard to address them?

As far as calculus it is not something the majority of pharmacists use in their work.

It is not something I ever had to use in pharmacy school.

But it is a weeder course all pharmacists are required to take.

Its not hard to adress your points. Its the fact that you are incapable of adressing our responses so it becomes a very weird ”discussion”.

You just don't like the responses.

Make better responses. I deal with what I get. Most days it is pretty bad.

I don't know how many lost souls have told me the negative integers END at -1.

When that is where they begin.

Up-side-down thinking like this is what I have to deal with.

And I know it is upsetting to some when I try to straighten them out.

 

[Speakpigeon]

The irrationals are similarly defined by what they are not: they are not rational. That is, they are not the ratios of two integers.

And how do you prove that a number is not the ratio of two integers? How do you prove that a transcendental number is not the ratio of two integers?
EB

 

[Phil Scott]

The irrationals are similarly defined by what they are not: they are not rational. That is, they are not the ratios of two integers.

And how do you prove that a number is not the ratio of two integers? How do you prove that a transcendental number is not the ratio of two integers?
EB

Uh, is this rhetorical or do you expect me to go through the proofs?

The proof that the square root of 2 is not rational has been known since antiquity, and is a pretty elegant proof that relies on basic number theory.

Proofs that pi and e are not algebraic are not even 150 years old and are fairly involved (consider that, for centuries, mathematicians were trying to square the circle, which would require that pi is algebraic). The proof that pi is irrational, by the way, is often taken as a corollary of it being transcendental.

Google is your friend.

 

[Speakpigeon]

The irrationals are similarly defined by what they are not: they are not rational. That is, they are not the ratios of two integers.

And how do you prove that a number is not the ratio of two integers? How do you prove that a transcendental number is not the ratio of two integers?
EB

Do you want me to go through some proofs? The proof that the square root of 2 is not rational has been known since antiquity, and is a pretty elegant proof that relies on basic number theory.

Any way would do as long as it's regarded as standard.

Proofs that pi and e are transcendental came much later and are fairly involved (consider that, for centuries, mathematicians were trying to square the circle, which would require that pi is algebraic). The proof that pi is irrational, by the way, is often taken as a corollary of it being transcendental.

Any website where I could look? Is there a standard vocabulary used for that sort of problem? Non-computability of pi?
EB

 

[Phil Scott]

Any way would do as long as it's regarded as standard.

Here is the classic proof, known since antiquity as the proof that the side of a square is incommensurable with the diagonal. It also appears in A Mathematician's Apology as an example to celebrate proof-by-contradiction

If the square root of 2 were rational, then it could be written p/q for integers p and q. Then (p/q)² = 2 and so p² = 2q². That means that p's square is even, but only even numbers have even squares. So consider p' such that p = 2p' giving 4p'²=2q² and thus 2p'²=q². We can now repeat the argument with q being even and so we can produce two endless sequences of non-zero integers, each term halving the previous. Such sequences cannot exist, so we arrive at an absurdity.

It follows that the square root of 2 is irrational.

Proofs that pi and e are transcendental came much later and are fairly involved (consider that, for centuries, mathematicians were trying to square the circle, which would require that pi is algebraic). The proof that pi is irrational, by the way, is often taken as a corollary of it being transcendental.

Any website where I could look? Is there a standard vocabulary used for that sort of problem? Non-computability of pi?
EB

Google "proof that pi is transcendental".

Pi is definitely computable. Archimedes had approximations to it based on a rock solid integration proof that could, with some (admittedly Herculean) effort, yield approximations of arbitrary effort.

What did you think uncomputable numbers were?

 

[Juma]

When all people can do is talk about me they are a bore.

I make point after point.

I wonder why it is so hard to address them?

As far as calculus it is not something the majority of pharmacists use in their work.

It is not something I ever had to use in pharmacy school.

But it is a weeder course all pharmacists are required to take.

Its not hard to adress your points. Its the fact that you are incapable of adressing our responses so it becomes a very weird ”discussion”.

You just don't like the responses.

Make better responses. I deal with what I get. Most days it is pretty bad.

I don't know how many lost souls have told me the negative integers END at -1.

When that is where they begin.

Up-side-down thinking like this is what I have to deal with.

And I know it is upsetting to some when I try to straighten them out.

Numbers are ordinals. A bigger number comes after a smaller. 1 comes before 2. -2 comes before -1.
Its kindergarten maths...
I are more and more suspecting that unter is a bot. No human can be so thickheaded.

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The irrationals are similarly defined by what they are not: they are not rational. That is, they are not the ratios of two integers.

And how do you prove that a number is not the ratio of two integers? How do you prove that a transcendental number is not the ratio of two integers?
EB

I’m amazed. How come you dont know this? Did you not go to school?

- - - Updated - - -

The irrationals are similarly defined by what they are not: they are not rational. That is, they are not the ratios of two integers.

And how do you prove that a number is not the ratio of two integers? How do you prove that a transcendental number is not the ratio of two integers?
EB

I’m amazed. How come you dont know this? Did you not go to school?

 

[Speakpigeon]

Here is the classic proof, known since antiquity as the proof that the side of a square is incommensurable with the diagonal. It also appears in A Mathematician's Apology as an example to celebrate proof-by-contradiction

If the square root of 2 were rational, then it could be written p/q for integers p and q. Then (p/q)² = 2 and so p² = 2q². That means that p's square is even, but only even numbers have even squares. So consider p' such that p = 2p' giving 4p'²=2q² and thus 2p'²=q². We can now repeat the argument with q being even and so we can produce two endless sequences of non-zero integers, each term halving the previous. Such sequences cannot exist, so we arrive at an absurdity.

It follows that the square root of 2 is irrational.

Thanks. Pretty straightforward and indisputable.

Any website where I could look? Is there a standard vocabulary used for that sort of problem? Non-computability of pi?
EB

Google "proof that pi is transcendental".

Thanks.

Pi is definitely computable. Archimedes had approximations to it based on a rock solid integration proof that could, with some (admittedly Herculean) effort, yield approximations of arbitrary effort.

What did you think uncomputable numbers were?

Sorry, I wrote that a bit too fast. I knew pi is computable. What I'm not clear is how you come to decide that a number is not computable. Say, the number where each new decimal digit is decided randomly. Presumably, that's not computable yet there's an algorithm, which is that each new digit is decided randomly. What kind of proofs are used for non-computability, if any?
EB

 

[Juma]

Thanks. Pretty straightforward and indisputable.

Google "proof that pi is transcendental".

Thanks.

Pi is definitely computable. Archimedes had approximations to it based on a rock solid integration proof that could, with some (admittedly Herculean) effort, yield approximations of arbitrary effort.

What did you think uncomputable numbers were?

Sorry, I wrote that a bit too fast. I knew pi is computable. What I'm not clear is how you come to decide that a number is not computable. Say, the number where each new decimal digit is decided randomly. Presumably, that's not computable yet there's an algorithm, which is that each new digit is decided randomly. What kind of proofs are used for non-computability, if any?
EB

The definition ”A number where each decimal is decided randomly” doesnt define a number. It is just a definition of a random number.
You brought up the concept of ”not computable”. You may explain what it means.

 

[Speakpigeon]

Thanks. Pretty straightforward and indisputable.



Thanks.



Sorry, I wrote that a bit too fast. I knew pi is computable. What I'm not clear is how you come to decide that a number is not computable. Say, the number where each new decimal digit is decided randomly. Presumably, that's not computable yet there's an algorithm, which is that each new digit is decided randomly. What kind of proofs are used for non-computability, if any?
EB

The definition ”A number where each decimal is decided randomly” doesnt define a number. It is just a definition of a random number.

In my humble opinon, you don't "define" numbers. You specify them. You can also sometimes give a representation, such as a decimal representation, if you know it.

So, I did specify one number, the number that would be obtained by the process I specified. It's a physical process. It can be done. Any idiot can do it.

I grant you you'd get a different number each time you do it. So, I also specified a class of numbers. Given the context, though, it was clear I was talking about the first number to be so produced. So, there's just one such number in the whole history of God's Creation.

You brought up the concept of ”not computable”. You may explain what it means.

I'm not a mathematician. I am a professional amateur. I wouldn't know my pi from nose.

Still, I tried to find clues...

Non-computable numbers are "numbers that cannot be computed by any computer program".

I found something very elegant... It is "basically Cantor’s diagonalization argument, only recast in a different terminology:

http://igoro.com/archive/numbers-that-cannot-be-computed/
C# programs are countable. That means that we can assign a different positive integer to each program. The shortest valid C# program will be 1, the next shortest will be 2, and so forth. If there are multiple valid programs of the same length, we will sort the programs lexicographically and assign integers in that order.
(...)
Consider a number whose part before the decimal point is 0. We choose i-th digit after the decimal point to be different from the digit in the same position in the number printed by program i (by “program i”, I mean the program associated with integer i, as described earlier in the article). So, each digit after the decimal point guarantees that the constructed number will differ from the number printed by a particular program. This demonstrates that the constructed number will be different from any number printed by a computer program!

From this I conclude that we know there are non-computable numbers, an uncountable infinity of them, but we don't necessarily know which numbers would be non-computable.

My point essentially is that if we can't compute a number then we don't know its decimal expansion so we can't possibly know it's not computable.

So non-computable numbers are numbers we don't know are not computable...

I hope you like that piece of mathematical humour.
EB

 

[Phil Scott]

Yes. The idea that there are uncomputable numbers comes from classical interpretations of diagonal arguments.

You won't actually be able to compute an uncomputable number, by definition, and it's coherent for a suitably inclined mathematician to not admit of anything they cannot compute. They then draw a weaker conclusion from the diagonal arguments, saying merely that there is no way to construct an enumeration of reals.

 

[Speakpigeon]

Yes. The idea that there are uncomputable numbers comes from classical interpretations of diagonal arguments.

You won't actually be able to compute an uncomputable number, by definition, and it's coherent for a suitably inclined mathematician to not admit of anything they cannot compute. They then draw a weaker conclusion from the diagonal arguments, saying merely that there is no way to construct an enumeration of reals.

I think it would be rather a mistake to dismiss non-computable numbers. As I understand the problem, first there are way more non-computable numbers than computable ones, and second, if you were to remove all non-computables from the Reals, you'd end up with a countable set... I guess it would be something like the Rationals plus the Algebraics and maybe some. End on the continuum dream... So, I think they are here to stay. Maybe we need to improve our understanding of them.
EB

 

[Phil Scott]

I think it would be rather a mistake to dismiss non-computable numbers. As I understand the problem, first there are way more non-computable numbers than computable ones, and second, if you were to remove all non-computables from the Reals, you'd end up with a countable set... I guess it would be something like the Rationals plus the Algebraics and maybe some. End on the continuum dream... So, I think they are here to stay. Maybe we need to improve our understanding of them.
EB

All of that assumes they exist. If you instead hold that existence requires a computational construction, then what you write here becomes nonsense.

 

[Juma]

Yes. The idea that there are uncomputable numbers comes from classical interpretations of diagonal arguments.

You won't actually be able to compute an uncomputable number, by definition, and it's coherent for a suitably inclined mathematician to not admit of anything they cannot compute. They then draw a weaker conclusion from the diagonal arguments, saying merely that there is no way to construct an enumeration of reals.

I think it would be rather a mistake to dismiss non-computable numbers. As I understand the problem, first there are way more non-computable numbers than computable ones, and second, if you were to remove all non-computables from the Reals, you'd end up with a countable set... I guess it would be something like the Rationals plus the Algebraics and maybe some. End on the continuum dream... So, I think they are here to stay. Maybe we need to improve our understanding of them.
EB

There is nothing remarkable with ”uncomputable numbers”..
Its simply a result of the fact that some numbers have infinitely many decimals: you cannot print infinitely many decimals in finite time.

 

[Phil Scott]

No. Numbers can have infinite decimals and still be computable. If there is a program that can compute the decimal to arbitrary precision, then the number is said to be computable.

And to correct an earlier post of yours: 0.999... is a real number. It denotes the limit of a particular geometric series, and that limit is 1.

 

[Speakpigeon]

There is nothing remarkable with ”uncomputable numbers”.

In a world that's becoming more digital by the picosecond I think it's a very, very remarkable notion.

And, just conceivably, if ever there's anything physical that would somehow feature a non-computable quantity, then it would imply that the universe itself is not computable. Some knowledgeable folks seem to hold that this would be a real puzzler. Parochially remarkable, if you like.

Its simply a result of the fact that some numbers have infinitely many decimals: you cannot print infinitely many decimals in finite time.

That's not the point. Such trivially computable numbers as 1/3 or 17/61 do have infinite decimal expansions that you could not all print. Computable means that you can compute it, not that you can do it within a finite time span.

This is similar to the notion of countable set, which really means that you can count the elements of the set but maybe not all within a finite time span. Some sets are not only countable, but also counted, like all integers between 1 and 10 for example.

And all numbers that have been computed are computable but the reverse isn't true.

Talk of computability proves faith in the meaningfulness of the notion of the infinite. Talk of non-computability proves faith in the reality of infinities.
EB

 

[Speakpigeon]

I think it would be rather a mistake to dismiss non-computable numbers. As I understand the problem, first there are way more non-computable numbers than computable ones, and second, if you were to remove all non-computables from the Reals, you'd end up with a countable set... I guess it would be something like the Rationals plus the Algebraics and maybe some. End on the continuum dream... So, I think they are here to stay. Maybe we need to improve our understanding of them.
EB

All of that assumes they exist. If you instead hold that existence requires a computational construction, then what you write here becomes nonsense.

Sure, but let me catch up. I need first to make sense of it all before I could start to badger people about whether they exist.

Still, there can be a computable number of children in the court yard, or a countable number of quarks in a proton. So why ever not some non-computable quantity of whatever?

The ratio of the circumference to the radius in a curved space may well happen to be a non-computable number. And then what happens to quantum physics?
EB