a question about infinity / countability

[kirby urner]

In higher mathematics we encounter Cantor's proof that the real numbers are uncountable and therefore there are "more" elements in R than in N.  The two sets get different "aleph numbers".

https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

However I don't quite understand how the proof proves what it says it proves.

What's provided is an algorithm for always obtaining a next sequence of 1s and 0s that is not in any given enumeration.

However, any sequence of 1s and 0s is already a binary number and therefore a member of N.

For N to be infinite, as advertised, it must have numbers with infinite digits, just like Pi, Phi, e and so on. 

Just remove the decimal point from 3.14159... to get 314159... and you have a member of N. Any member of R may be made a member of N by removing the decimal point.

So how is N any less infinite than R again?  How is N any more "countable" than R?  

Maybe Andrius or Ted have a good way to dispel my confusion?  There's probably a FAQ somewhere.  I'll keep noodling.

Kirby

[Andrius Kulikauskas]

Kirby, I will try.... :)

You write:
However, any sequence of 1s and 0s is already a binary number and
therefore a member of N.

Any FINITE sequence of 1s and 0s (written to the left of the "decimal
point") is a member of N.

Whereas real numbers from 0 to 1 are given by INFINITE sequences of 1s
and 0s (that are to the right of the "decimal" point).

Is that enough to clarify it for you?

Andrius

Andrius Kulikauskas
m...@ms.lt

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[kirby urner]

On Sat, Jul 9, 2016 at 12:32 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

Kirby, I will try.... :)

You write:
However, any sequence of 1s and 0s is already a binary number and therefore a member of N.

Any FINITE sequence of 1s and 0s (written to the left of the "decimal point") is a member of N.

Yes, I see that N is by definition forced to have only a finite number of digits.

This means set N is finite by definition, not infinite. 

I wonder how many digits an N is allowed to have before it hits the limit.

I was confused into thinking we could have an infinite set N, but there's an upper limit to the number of digits.

Real numbers are not confined in this way and are permitted to have infinite digits.

So yeah, it's really no wonder about the different aleph numbers.  N has its hand tied behind its back.  R wins.

I'd say the definition of N to excluded infinite-digit numbers is rather limiting and so I would define N+ to be the enhanced superset of N that includes the positive integers and 0, plus all these infinite-digit numbers that were previously excluded.

Elsewhere I have define the set N+ as the "nonsense numbers".

http://coffeeshopsnet.blogspot.com/2016/04/nonsense-numbers.html

http://math.stackexchange.com/questions/58085/a-number-with-an-infinite-number-of-digits-is-a-natural-number

Kirby

[kirby urner]

On Sat, Jul 9, 2016 at 12:32 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

Kirby, I will try.... :)

You write:
However, any sequence of 1s and 0s is already a binary number and therefore a member of N.

Any FINITE sequence of 1s and 0s (written to the left of the "decimal point") is a member of N.

Whereas real numbers from 0 to 1 are given by INFINITE sequences of 1s and 0s (that are to the right of the "decimal" point).

I don't think real numbers are restricted from having infinite numbers of digits to both the left and the right of the decimal.

You can hold a mirror up to Pi and have the digits go the other direction, to the left instead of to the right.

N+ would therefore be a subset of R as it excludes: negatives of any type, plus any positive non-integers.

Kirby

[Anna Roys]

Infinity confuses me. Anyone willing to shine a flash light to help me see?

So, in elementary school in Alaska, I was given the idea that infinity encompasses all numbers and goes on forever and is uncountable.

Then,  many years later, while listening to my  college sons' mathematical discussions, I became aware of the idea of multiple infinities, based on specific types and sets of numbers which are also infinitely expandable.

So it seems my elementary Math teacher led me astray?

Can anyone help me make sense?

Anna

[kirby urner]

I'm not exactly answering your question Anna, which I think leads to "deep answers".

I just want to say that when it comes to CS-friendly algebra, we do find it very useful to 

(A) distinguish between "types of number"

(B) discuss how they're fully specify them in terms of standards

For example, most computer languages have integer and float as distinct types, complex as another type, and rational numbers as yet another type.  Some languages of several flavors of integer such as unsigned or singly byte (8-bit ints are strictly bounded from decimal 0 to 255).

http://kias.dyndns.org/comath/13.html

JavaScript is somewhat of an exception in having only one number type under the hood (not counting boolean as a number type -- which we could).

https://javascriptweblog.wordpress.com/2010/09/27/the-secret-life-of-javascript-primitives/

Note that characters and character-strings are a type as well, right along with numbers.

CS-friendly math introduces functions that operate with non-number types, such as strings.  

Searching for the letter "A" in a string, or returning a substring, or a string matching some pattern, are mathematical operations.

However none of these number types mentioned precisely match what the 1800s mathematicians were talking about. 

Some have an arbitrary number of digits, true, such as the int type in Python, but there's still something called "memory" that creates an upper bound on the number of digits.

The floating point specification, on the other hand, has strict upper and lower bounds and finite precision, owing to the specific number of bits allocated to such a number.

The usual thing to tell students is that computer languages "approximate" the corresponding mathematical concepts, which set the bar.

From a more evolutionary point of view, the existence of number types pre digital computer helped us create the kind of closure machines needed, meaning every computation stays discrete and finite, quantum, with specific conditions for when an overflow symbol or Not-a-Number (NaN) is the result of an operation.

http://stackoverflow.com/questions/2618059/in-java-what-does-nan-mean

Two floating point numbers cannot be arbitrarily close together.  They're not even guaranteed to be associative when added:

http://www.walkingrandomly.com/?p=5380

Given the above, I would encourage elementary school teachers to emphasize the concept of "types of number" and their properties.  

Rationals have a distinguishable numerator and denominator, both integers.  

Complex numbers have a real and imaginary part (so are not unlike rationals in having component parts).

Your sons would likely disagree with me, but I'd say don't put a lot of eggs in the "aleph number" basket i.e. leave that to college math majors.  

Keep the waters clean by sticking to the relatively concrete definitions used in modern computer languages and leave 1800s metaphysics about infinity to future (possibly elective and/or not-for-credit) courses.

If your student demonstrates a penchant for philosophy, well then of course you might mention Cantor.

Kirby

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[Oleg Gleizer]

Dear Anna,

A set is infinite if there exists a bijection between the set and its proper subset. For example, the function n -> 2n gives a bijection between the set of all the natural numbers 1,2,3,4,5,... and the set of all the positive even integers, 2,4,6,8,10... Therefore, the former set is infinite. The cardinality of the set of natural numbers is called aleph zero.

The number of all the subsets of a finite set of cardinality n is 2^n - an element is either included or not included in a subset. It is not hard to construct a bijection between the set of all the subsets of the set of natural numbers and the set of all the real numbers. The cardinality of the latter set is called aleph one. Therefore, 2^{aleph zero} = aleph one. It is hypothesized that there is no cardinality in between aleph zero and aleph one. If true, we have the following sequence of cardinalities, aleph_{n+1} = 2^{aleph_n}, n = 0,1,2...

Very Truly Yours,

Oleg Gleizer.

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[kirby urner]

On Sat, Jul 9, 2016 at 1:54 PM, kirby urner <kirby...@gmail.com> wrote:

 

I don't think real numbers are restricted from having infinite numbers of digits to both the left and the right of the decimal.

Actually, on further checking, the real numbers are not allowed to have infinite digits to the left, only to the right.

http://math.stackexchange.com/questions/944284/can-a-number-have-infinitely-many-digits-before-the-decimal-point

If you hold up a mirror to Pi and write   ...951413 you will be writing a member of N+ (a "nonsense number") but it will not be real.  Of course we can't write "all the digits" any more than we can in the case of Pi itself.

The reason for these restrictions is to keep the game playable.  

We can argue about whether the axioms are "self evident" -- they certainly don't need to be, and in contemporary math you should not expect them to be.

Again, I would not want elementary level topics to be polluted with all this philosophy.  Not required.

To condense Andrej's remarks, Wittgenstein said something like: "Logic is the foundation of mathematics only as the painted rock is the support of the painted tower." – Alexander Woo Oct 25 '12 at 20:05

I find myself agreeing with Wittgenstein a bit more every year. – Andrej Bauer Oct 25 '12 at 20:21

http://mathoverflow.net/questions/110628/are-the-foundations-of-mathematical-logic-shaky

The idea that we're paralyzed (unable to do math) unless we have "logical foundations" is a myth.

Kirby

[kirby urner]

If you hold up a mirror to Pi and write   ...951413 you will be writing a member of N+ (a "nonsense number") but it will not be real.  Of course we can't write "all the digits" any more than we can in the case of Pi itself.

Put another way, if you think of pi:

3.14159265358979323846264338327950288419716939937510
  58209749445923078164062862089986280348253421170679
  82148086513282306647093844609550582231725359408128
  48111745028410270193852110555964462294895493038196
  44288109756659334461284756482337867831652712019091
  45648566923460348610454326648213393607260249141273
  72458700660631558817488152092096282925409171536436
  78925903600113305305488204665213841469519415116094
  33057270365759591953092186117381932611793105118548
  07446237996274956735188575272489122793818301194912
  98336733624406566430860213949463952247371907021798
  60943702770539217176293176752384674818467669405132
  00056812714526356082778577134275778960917363717872
  14684409012249534301465495853710507922796892589235
  42019956112129021960864034418159813629774771309960
  51870721134999999837297804995105973173281609631859
  50244594553469083026425223082533446850352619311881
  71010003137838752886587533208381420617177669147303...

  

and imagine the decimal point moving to the right at the speed of light:  31.4... 314.1... 3141.5... you will never escape the set of real numbers. 

However you will not be allowed to stop the process at any point and say "and so on, forever" as we're only allowed to zoom in infinitely, not zoom out.  

"Infinitely tiny" (zooming in) is like getting closer and closer to the true head of the pin, more and more precise.  Pi is always coming closer.

Zooming out on the other hand just makes our heads hurt, as we don't sense we're getting closer to anything, so we forbid doing so.

In computer science, we replace these numbers with different definitions.

Kirby

[michel paul]

> 

​

I don't quite understand how the proof proves what it says it proves.

I like the suggestion to consider a real as simply an infinite sequence of binary digits.

What Cantor's proof proves is that a list of all possible infinite sequences of binary digits cannot be defined.

If there did exist a list of all possible infinite sequences, then each sequence in that list would have an index.

Suppose we believe that S is that list. 

S[n] represents the nth sequence in S, and S[n][k] represents the kth digit of that sequence.

We can define a sequence x such that x[k] = complement(S[k][k]) for all k.

It turns out that x is not in S.

If you argue that x is in S at index r, that cannot be, because by definition x[r] == complement(S[r][r]), so x != S[r].

The interesting thing here is that we do not even need to refer to the 'real numbers'. We are simply talking about infinite lists of binary symbols.

We could frame the problem as this - create a list of strings beginning with 'ppp...' and ending with 'qqq...'. In between these strings your list must contain all possible infinite strings consisting of p and q.

If you say you have such a list, I can construct a sequence that ain't in your list.

So Cantor's proof ends up being a statement about the set of infinite lists. You can't define a list containing all infinite lists. It's a different kind of set. Not listable.

Somehow I think that this kind of discussion actually can be relevant at a high school level. I've seen it. There are kids who can get into that kind of stuff. I think it's relevant for mathematical/computational literacy.

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[kirby urner]

On Sun, Jul 10, 2016 at 12:30 PM, michel paul <python...@gmail.com> wrote:

> 

​

I don't quite understand how the proof proves what it says it proves.

I like the suggestion to consider a real as simply an infinite sequence of binary digits.

What Cantor's proof proves is that a list of all possible infinite sequences of binary digits cannot be defined.

True, one could never list them all.  Even writing Phi or Pi out completely is a faux undertaking (doomed from the beginning).

What one can do, short of listing, is define filters or criteria.

314159... with no decimal point, just matching the digits of Pi one for one (a bijection) is NOT a real number and NOT a natural number.

It is, however, a member of N+, as I've defined it.

 

If there did exist a list of all possible infinite sequences, then each sequence in that list would have an index.

Like an IP number.  Every bee, every grain of sand, could in theory have an IP number, although these would recycle as bees die and grains dissolve.

 

Suppose we believe that S is that list. 

S[n] represents the nth sequence in S, and S[n][k] represents the kth digit of that sequence.

We can define a sequence x such that x[k] = complement(S[k][k]) for all k.

It turns out that x is not in S.

If you argue that x is in S at index r, that cannot be, because by definition x[r] == complement(S[r][r]), so x != S[r].

The interesting thing here is that we do not even need to refer to the 'real numbers'. We are simply talking about infinite lists of binary symbols.

I think it's both a tautology and part of the definition of "infinite" that anything innumerable cannot be numerated (counted).  QED.  

The proof is in the definition.

The more interesting questions involve criteria I think i.e. can a number with an infinite number of 3s and no decimal point be considered a number.

Today's philosophers mostly say no, that "nonesense numbers" are precisely that.

 

We could frame the problem as this - create a list of strings beginning with 'ppp...' and ending with 'qqq...'. In between these strings your list must contain all possible infinite strings consisting of p and q.

If you say you have such a list, I can construct a sequence that ain't in your list.

So Cantor's proof ends up being a statement about the set of infinite lists. You can't define a list containing all infinite lists. It's a different kind of set. Not listable.

It ends up proving that what's innumerable cannot be enumerated, yes.

 

Somehow I think that this kind of discussion actually can be relevant at a high school level. I've seen it. There are kids who can get into that kind of stuff. I think it's relevant for mathematical/computational literacy.

As long as it's not a pseudo-debate where there winners are all determined in advance.

A kid needs to be free to say:  this is all 99% BS and not budge.  If it's all about caving to "authorities" in the end, I'll not agree it's worthwhile.

If a kid really does have a strong suspicion of this "head on a pin" type "infinity stuff", I'll tell him he's in good company and fortunately math does not depend on these ugly so-called "foundations" which are completely inessential to our getting work done, especially in computer science.

Kirby

[Joseph Austin]

Kriby, 

I'm encouraged to learn that I'm not the only one who is suspicious of Cantor's argument.

For my part, it's clear that the list of "infinite" numbers is growing much faster than the number of digits in the  "new" number created from the diagonal digits,  so it seems to me it's hard to argue that the "diagonal" number does not in fact occur farther down in the list!

In fact, consider a list of all numerals of N digits in some numeral system. Produce the "diagonal" numeral per the Cantor algorithm.

That numeral does in fact occur elsewhere in the list. That is true for any finite N.  So why not for "infinite"  N?

And note that I said numeral instead of number.  There are various ways of "numeralizing" a number.

The multiplicative inverse of three can be represented as 1/3, or in base three as 0.1

But it cannot be represented in binary or decimal by any finite numeral. Or I would say, even by an infinite numeral.

No number of factors of 2 or 5 will ever give a multiple of 3, even and infinite number!

So Cantor's list is not even complete, so it should be no surprise to find numbers not in it!

So what is a "real" number?  It is the solution to an equation involving operators beyond add and multiply and their inverses,

all of which are countable (rational).

So pick your operator.  Exponential?  There are two inverses. I can represent the possibilities by nth root m or log n (m).

If n and m are countable, are not all pairs countable? So there are only a countable number of roots and logs of rationals.

So close the set including those inverses.  Can the closure of a countable set become uncountable???

Trig? The trig functions of angles are defined as ratios of sides of triangles. If the lengths of sides are countable, their ratios (pairs) are countable.

What about calculus limits?  If the functions are countable, the limits are countable.

There are a countable number of functions that can be described in mathematical symbols.

If  the variable values themselves are countable, and the combinations of their variable values are countable, how can the number of the function values not be countable?

I can see that there are "irrational" numbers, but I don't see how an "uncountable" number of them can arise in the first place.

I think it's intuitively obvious that there are "more" integers than naturals, approximately twice as many.

And "more" rationals than integers. And more "irrationals" than rationals.

Rather than considering orders of infinities, I would rather study things like the Riemann sphere that exposes "infinity" as an artifact of imposing an "unnatural" geometry or numerology on a simple closed space.

For example, any schoolboy studying trig notices that the tangent curve "wraps around" from the "top" of the paper to the bottom,

in spite of fact that we are told it approaches + infinity from one side and - infinity from the other. But "obviously" plus and minus infinity are a point, and  the same point, because they are the image of the same point of a unit vector rotating around the origin of a Cartesian grid.

Seems to me the problem is that the Cartesian grid isn't "big" enough (actually, not "closed" enough) to give us names for all the points we need to talk about.

Joe Austin

On Jul 9, 2016, at 12:30 PM, kirby urner <kirby...@gmail.com> wrote:

In higher mathematics we encounter Cantor's proof that the real numbers are uncountable and therefore there are "more" elements in R than in N.  The two sets get different "aleph numbers".

https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

However I don't quite understand how the proof proves what it says it proves.

<snip>

[michel paul]

On Sun, Jul 10, 2016 at 12:54 PM, kirby urner <kirby...@gmail.com> wrote:

> ​

​

I think it's both a tautology and part of the definition of "infinite" that anything innumerable cannot be numerated (counted).  QED.  

​False.

A set can be both countable and infinite. Or, an infinite set can be listable.

Some infinite sets are listable, and some are not.

The issue is one-to-one correspondence. To say that an infinite set can be listed is exactly the same thing as saying that the set that can be placed in one-to-one correspondence with N.

An infinite set that is listable can be coded in Python using a generator. 

Some infinite sets can be coded as generators, and others cannot be. Python provides a nice way to think about these things.

​> A kid needs to be free to say:  this is all 99% BS and not budge. 

Sure, and many do. Many think programming is 99% BS and don't want to touch it. Kids think all kinds of things. And they should be encouraged to explore the implications of their ideas fully.

And there are kids who really do enjoy exploring ideas like this. Why not let them?

 

> If it's all about caving to "authorities" in the end, I'll not agree it's worthwhile.

"Authorities"?

What authorities are we addressing here?

 

> If a kid really does have a strong suspicion of this "head on a pin" type "infinity stuff", I'll tell him he's in good company and fortunately math does not depend on these ugly so-called "foundations" which are completely inessential to our getting work done, especially in computer science.

I agree that the study of mathematical foundations is not essential for 'getting work done', but so what? That does not at all constitute a thoughtful reason for NOT studying foundations.

That's the same kind of attitude that used to consider the study of number theory as 'useless'. Things do change.

If the work of Voevodsky in foundations pans out, if type theory rather than set theory provides the deeper grounding, it will turn out that mathematics and CS are the same thing! I think that's worthwhile.

​Why not do both? Why not 'get work done' AND study foundations?

[michel paul]

On Sun, Jul 10, 2016 at 12:54 PM, kirby urner <kirby...@gmail.com> wrote:

> ​

​

I think it's both a tautology and part of the definition of "infinite" that anything innumerable cannot be numerated (counted).  QED.  

​False.

A set can be both countable and infinite. Or, an infinite set can be listable.

Some infinite sets are listable, and some are not.

The issue is one-to-one correspondence. To say that an infinite set can be listed is exactly the same thing as saying that the set that can be placed in one-to-one correspondence with N.

An infinite set that is listable can be coded in Python using a generator. 

Some infinite sets can be coded as generators, and others cannot be. Python provides a nice way to think about these things.

​> A kid needs to be free to say:  this is all 99% BS and not budge. 

Sure, and many do. Many think programming is 99% BS and don't want to touch it. Kids think all kinds of things. And they should be encouraged to explore the implications of their ideas fully.

And there are kids who really do enjoy exploring ideas like this. Why not let them?

 

> If it's all about caving to "authorities" in the end, I'll not agree it's worthwhile.

"Authorities"?

What authorities are we addressing here?

 

> If a kid really does have a strong suspicion of this "head on a pin" type "infinity stuff", I'll tell him he's in good company and fortunately math does not depend on these ugly so-called "foundations" which are completely inessential to our getting work done, especially in computer science.

I agree that the study of mathematical foundations is not essential for 'getting work done', but so what? That does not at all constitute a thoughtful reason for NOT studying foundations.

That's the same kind of attitude that used to say that the study of number theory was 

If the work of Voevodsky in foundations pans out, if type theory rather than set theory provides the deeper grounding, it will turn out that mathematics and CS are the same thing! I think that's worthwhile.

​Why not do both? Why not 'get work done' AND study foundations?

[michel paul]

On Sun, Jul 10, 2016 at 8:31 PM, Joseph Austin <drtec...@gmail.com> wrote:

 

​> ​

the list of "infinite" numbers is growing much faster than the number of digits in the  "new" number created from the diagonal digits,  so it seems to me it's hard to argue that the "diagonal" number does not in fact occur farther down in the list!

​Suppose we say that the diagonal number occurs in the list at index k.

Let's call our list L, and let's call our diagonal number d.

If we say that the diagonal number occurs at index k, then we are saying L[k] == d.

However, according to the definition of d, the kth digit of d must be different than the kth digit of L[k].

Since d[k] != L[k][k], it cannot be the case that L[k] == d.

The concern that d might be further down the list does not apply. Just change k to a new index, and the same argument applies.

But again, to look at this in a completely fresh way, suppose you want to create a list starting with the infinite string "ppp..." and ending with the infinite string "qqq...".

In between you'd like to create a systematic list of all possible strings composed of 'p's and 'q's.

If the strings were to be of finite length, then creating a systematic list would be easy. It is tempting to think we could also create a systematic list with infinite strings, but it turns out we cannot.

​

​> 

Cantor's list is not even complete, so it should be no surprise to find numbers not in it!​

​Yes, Cantor's list is not complete, but that's the whole point of his proof!  : )

The point of his proof is that you CANNOT create a complete list, and that reason is NOT simply because it is infinite!

Again, there are infinite sets that are listable, and there are infinite sets that are not listable.​

Cantor's point is that the real numbers are not listable. If you think you've created a complete list, the diagonal argument shows that the supposed list is incomplete. The reals cannot be put into a sequence.

​> I think it's intuitively obvious that there are "more" integers than naturals, approximately twice as many. And "more" rationals than integers. And more "irrationals" than rationals.

False.

A very important principle --> one-to-one correspondence!

There are just as many naturals as there are integers, and there are just as many naturals as there are rationals.

These facts can be shown by setting up a one-to-one correspondence that can not be set up between the naturals and the reals.

The naturals, the integers, and the rationals can be listed. The reals cannot be listed.

[kirby urner]

On Sun, Jul 10, 2016 at 9:46 PM, michel paul <python...@gmail.com> wrote:

<< SNIP >>

 

Again, there are infinite sets that are listable, and there are infinite sets that are not listable.​

This is true using "listable" to mean "countable" in Cantor's sense, not "listable" in the sense of using enumerate() or list() in Python.

In a finite discrete state machine, we have shifted computation off the hand-wavy set theory of the Frege + Bertrand Russell era.

We don't have to use "listable" to mean the same thing as "countable in a Cantorian sense", but we're free to.  Namespaces matter.

 

Cantor's point is that the real numbers are not listable. If you think you've created a complete list, the diagonal argument shows that the supposed list is incomplete. The reals cannot be put into a sequence.

My point was Natural Numbers would not be listable either, if we didn't make them so wimpy on purpose.  '

Allow infinite digits and we're able to make that one-to-one bijection to the Reals.

But we disallow that, thanks to Peano and ZFC and other important axiomatic systems you need a PhD to discern.

I'd rather not drag kids into this kicking and screaming, but yeah, if they're wanting to be scholastics of this type, I'd say "be my guest, read more Cantor and his disciples".  I wouldn't expect every kid interested in a CS career to have the time of day for Aleph stuff, an acquired taste.

 

​> I think it's intuitively obvious that there are "more" integers than naturals, approximately twice as many. And "more" rationals than integers. And more "irrationals" than rationals.

False.

A very important principle --> one-to-one correspondence!

314159... ---> 3.14159...

Verboten!

Dang.

 

There are just as many naturals as there are integers, and there are just as many naturals as there are rationals.

"As many" in a special sense, as when we're comparing infinities to infinities, you have to know more about Aleph and all that.

Are there as many digits of Pi as Real numbers?  No, the digits of Pi are listable as we can go:

1 --> 3

2 --> 1

3 --> 4

4 --> 1

5 --> 5  # the last time that happens!

6 --> 9

7 --> 2

8 --> 5

9 --> 6

A --> 5

B --> 3

C --> 5

D --> 8

E --> 9

F --> 7

...

(and so on -- I'm sure no one minds if I switch to hex representations for all my N, I just feel more at home in that base)

The digits of Pi are clearly a listable set, so same Aleph as N. So very like a Python list, just computers have this thing called "memory" which has some petabyte limit.

Usually in ordinary language, "as many" is with reference to finite sets.  If you say "there's as many grains of sand in the Atlantic as the Pacific" I'd know for a fact you'd be wrong, as that'd be too great a coincidence.

 

These facts can be shown by setting up a one-to-one correspondence that can not be set up between the naturals and the reals.

"Facts" in a special sense understood by a special branch.  Like cosmologies, logics come and go.  Axiomatic systems are not always "for the ages".

 

The naturals, the integers, and the rationals can be listed. The reals cannot be listed.

Not really talking Python here  (10 foot pole materializes).  

Cantor also helped pave the way for fractional dimensions right?  Not directly, however he was certainly a critic of the "it takes three numbers therefore space is 3D" (he's an ally there).  

He poked holes in that argument by showing he could serialize any-D data structures into 1-D structures, kind of like an increasing windy path will take us from 1D to almost 2.  Anyway, various patches were applied in light of his hacks.  Again, it takes a lot of study and background to get that specialized.

Kirby

[michel paul]

On Sat, Jul 9, 2016 at 2:40 PM, Anna Roys <roys...@gmail.com> wrote:

​​

So, in elementary school in Alaska, I was given the idea that infinity encompasses all numbers and goes on forever and is uncountable.

Then,  many years later, while listening to my  college sons' mathematical discussions, I became aware of the idea of multiple infinities, based on specific types and sets of numbers which are also infinitely expandable.

So it seems my elementary Math teacher led me astray?

​

Hi Anna,​

​Based on what you're saying, your elementary math teacher simply did not make clear that even though an infinite set 'goes on forever', some infinite sets can be organized sequentially, but other infinite sets cannot be.

​Of course, it is quite understandable that your elementary math teacher might not have been aware of this distinction.

Your son's discussions of the different types of infinities all boil down to this fact that some infinities can be organized sequentially in a list and others cannot be.

Although both types of infinity 'go on forever', the ones that can be organized into a list will eventually show each value in the list, but the ones that cannot be so organized will always be missing some values, no matter how far they are carried out.

[kirby urner]

Some sets are inherently unordered, whether finite or infinite, however we're free to create an order with our set N, i.e. we can count "the things" in any order we like.  

One operators like > and < do not apply, as when dealing with apples and oranges, i.e when orders are arbitrary, then it's more like using dict( ) than list( ), what we're doing.  N as keys.  The order is all in the keys.  

The values, at the other side of the bijection, have no order on their own, independently of our numbering them.  This is very common in lists too, such as job queues.  The job objects enter the queue in what might as well be random order, based on whatever is going on in user space.  However, once in the queue, they're enumerated, given keys.

I lot of modern technologies depend on "tokens" which are far from infinitely long, but long enough to be considered unique for business purposes. Yesterday I was working with the graph,facebook api and had an extremely long token deemed unique in all the universe.

Theoretically countable yet inexhaustible is an industry standard in may applications.

Kirby

[michel paul]

On Sun, Jul 10, 2016 at 9:46 PM, michel paul <python...@gmail.com> wrote:

 

​>> ​

Again, there are infinite sets that are listable, and there are infinite sets that are not listable.​

 

​On Sun, Jul 10, 2016 at 10:32 PM, kirby urner <kirby...@gmail.com> wrote:​

​> ​

This is true using "listable" to mean "countable" in Cantor's sense, not "listable" in the sense of using enumerate() or list() in Python.

​Oh no! It IS true in Python as well!

Just think in terms of generators. A listable infinite set can be coded as a generator in Python.

So "listable" turns out to be "countable" in both Cantor's sense and Python's.

When I first realized that generators could model countable infinite sequences I was thrilled for days.

 

​> ​

My point was Natural Numbers would not be listable either, if we didn't make them so wimpy on purpose.  '

​We DON'T make them wimpy on purpose. We don't 'make' them anything at all on purpose. We just count.

We just add another pebble to the pile.

Drop a rock in a pile for every sheep that goes through the gate, and you have a count, even if you don't have a name or a symbol for it.​

Counting is establishing a one-to-one correspondence between sets.

I don't think anyone purposely restricts N to only finite sequences of digits. If the process of 'getting work done' leads to the need for infinite sequences of digits, I don't think there would be any problem.

Something to consider here - I believe there is a confusion occurring between number and its representation.

It is not the case that pi consists of an infinite number of digits!

It is the DECIMAL REPRESENTATION of pi that consists of an infinite number of digits.

Pi itself has a specific location on the number line. It is not a smudge.

Kids actually think that. I would specifically ask them - does pi have a specific location on the number line, or is it more like a smudge, indefinite, because the digits 'go on forever'? 

Lots of kids believed that pi was more like a smudge on the number line than a precise location.

 

​> ​

Allow infinite digits and we're able to make that one-to-one bijection to the Reals.

​> ​

But we disallow that, thanks to Peano and ZFC and other important axiomatic systems you need a PhD to discern.

​Again, I think there is a confusion here between numbers and their representation.​

We don't 'decide' because of some set of axioms to allow R but not allow N to be represented by infinite decimal strings. 

  

​>> ​

There are just as many naturals as there are integers, and there are just as many naturals as there are rationals.

​> ​

"As many" in a special sense, as when we're comparing infinities to infinities, you have to know more about Aleph and all that.

​No, all you need to know about is one-to-one correspondence.

That's it.

OTOC, dropping pebbles in a pile, putting items in a list, is the conceptual tool by which we establish that the cardinalities of N, Z, and Q are equivalent.​

​> ​

Usually in ordinary language, "as many" is with reference to finite sets.

​"As many as" is established through OTOC, and this can occur in ordinary language.​

Again, drop a pebble in a pile for each sheep that goes through the gate, and you have "as many" pebbles as sheep.

This very same principle allows us to say that we have "as many" naturals as rationals.

​>> 

The reals cannot be listed.

​> ​

Not really talking Python here 

 

​No, we are.

You cannot create a generator in Python that will yield all of the reals.

However, you CAN create a generator in Python that will yield all of the rationals.

I've done it.​

[kirby urner]

On Mon, Jul 11, 2016 at 12:45 AM, michel paul <python...@gmail.com> wrote:

On Sun, Jul 10, 2016 at 9:46 PM, michel paul <python...@gmail.com> wrote:

 

​>> ​

Again, there are infinite sets that are listable, and there are infinite sets that are not listable.​

 

​On Sun, Jul 10, 2016 at 10:32 PM, kirby urner <kirby...@gmail.com> wrote:​

​> ​

This is true using "listable" to mean "countable" in Cantor's sense, not "listable" in the sense of using enumerate() or list() in Python.

​Oh no! It IS true in Python as well!

Just think in terms of generators. A listable infinite set can be coded as a generator in Python.

In that sense, of an "infinite loop" yes, we can code them.  

while True:

    print(1)

    if input("Next (Y/n)" == "Y":

        continue

    else:

        break

is also a way to code for infinite 1s.  Same idea as a generator.

 

So "listable" turns out to be "countable" in both Cantor's sense and Python's.

Except if you go:

>>> pidigits = genpi()

>>> list(pidgits)

you run out of memory pretty quick, and that's what makes finite state machines unable to deal with infinity.  It's defined out of the picture.

People with brains, on the other hand, are able to imagine "infinite memory" without needing to know what that means, other than it sounds good in science fiction.

 

When I first realized that generators could model countable infinite sequences I was thrilled for days.

 

​> ​

My point was Natural Numbers would not be listable either, if we didn't make them so wimpy on purpose.  '

​We DON'T make them wimpy on purpose. We don't 'make' them anything at all on purpose. We just count.

I misspoke in a way, as we insist N < R i.e. N is a subset of R.  

If N were permitted to include representations with infinite digits it would be outside the reals.  

The reals are not allowed to have infinite digits to the left of the decimal point, only to the right.  That's the law.

 

We just add another pebble to the pile.

Drop a rock in a pile for every sheep that goes through the gate, and you have a count, even if you don't have a name or a symbol for it.​

Just the ticking of a clock against the backdrop of the proleptic Gregorian calendar is in principle infinite, even if the sun blows up and the whole Universe ends (sometime later).  Infinitely long number lines were never so constrained (they pop up in classrooms by the millions per day).  

Neither space nor time are a barrier, when it comes to counting, 1, 2, 3....

Barriers to perpetual counting such as "finite memory" in a state machine, is a constraint the "infinity minded" don't need to care about.

The clock itself, even if built for 10,000 years, will not last forever, nor will any Python generator running on any machine, but that's a minor matter when you have a good imagination.

 

Counting is establishing a one-to-one correspondence between sets.

We're not talking about the Python set type if the set is infinite.  Python has not infinite sets.

Python lives in a finite state machine and we have overflow conditions. That's intrinsic to the language, part of its math.

 

I don't think anyone purposely restricts N to only finite sequences of digits. If the process of 'getting work done' leads to the need for infinite sequences of digits, I don't think there would be any problem.

Mathematics does not like to limit itself to either 'getting work done' or 'need'.  A lot of math is infotainment, empty calories.  

Its main business, in some branches, is to keep people out of trouble, give 'em something to noodle about.  Like chess and Go.

 

Something to consider here - I believe there is a confusion occurring between number and its representation.

This is deep though (grammatically), gets into philosophy (deep grammar).  

There's a school of thought that says "we have things in themselves on the one hand, and our representations of those things on the other, with a 'name --> object correspondence'; for any name to have meaning, it must point to some object in the 'domain of objects'."

That actually is the mental model encouraged by Python, plus we get inheritance and all of that.

However, not every philosopher who explores the foundations of mathematics believes in name -> object nominalism as a worthwhile beginning.  

What does the pawn in chess corresponding to?  If we picture a pawn in our mind's eye, then I guess it refers to some actual pawn on an actual chess table (why should it, which one?). But anyway that's like saying an imaginary 2 points to a sculpture of the numeral 2 and that's not what we're trying to get at is it?

Those who think there's a "thing" that has to be "out there" (where?) that *is* the number 2, whereas our symbol '2' is just a name for it, are not the only players on the field, when it comes to in foundation work.  

The meaning of a pawn in chess stems from its role in a game, according to rules.  The meaning of a screwdriver:  same thing, what do we do with it, how is it used?  There is no "object out there" that is "the meaning" of "screwdriver".  Meaning boils down to use cases, not to pointing.

So now we're doing philosophy.  

I'm saying we don't need name -> object nominalism to govern our sense of "what 2 means" and that when elementary school teachers say "the numeral is the representation of the actual thing, which is the number", they're just communicating one of many philosophies we've seen written down over the years.  They're taking a position in a long-running debate.

 

It is not the case that pi consists of an infinite number of digits!

OK then, the "representation of pi" is what I'm interested in.  I could have picked sqrt(2) or phi or just any random sequence of single digits.

All I need is a generator of objects, they don't even need to be digits.  Any type will do.  

I can count them, one by one, as you say, in an infinite loop.  The digits of pi are like this.  I'm agreeing with Cantor and orthodox Aleph stuff.  The digits of pi are countable, one to one mappable, with a dict as well.  We agree.

 

It is the DECIMAL REPRESENTATION of pi that consists of an infinite number of digits.

Pi itself has a specific location on the number line. It is not a smudge.

What is "the number line"?  Does it "actually" exist?  Anything drawn with chalk is not "really" a number line right?  Too thick, for one thing.

In Python, in our discussion of floats, ints, strings, generators, lists, is it ever necessary to think of a number line?  Certainly not an infinite one as floating point numbers crap out at MAX and MIN.  ints bang into the wall of "out of memory".

So I'd say we have no need for an infinite number line with infinite granularity in Python, but we do have meaning for > and <.  We want to be able to say 3 < pi < 4 and that's no problem, as pi is the name of a float or other object that plays well with 3 and 4.

In thinking how Python gets its results, we get by fine without trying to picture such "things" (as infinite anything) in the mind's eye and indeed I don't consider such "infinity-based" concepts necessary for work done with energy (the physics meaning of work).  A finite state machine is "plugged in" (burns calories) and obeys conservation laws.

No number line is required in physics, not a "pure" one (we need rulers of course).  

The idea of an infinite number line is a human artifact, a line in the sand, with some sage saying "now imagine this goes on forever and is infinitely thin so even a microscope wouldn't show it". Cartoons. Day dreams.  Take 'em or leave 'em.  All math is ethno-math.

 

Kids actually think that. I would specifically ask them - does pi have a specific location on the number line, or is it more like a smudge, indefinite, because the digits 'go on forever'? 

Lots of kids believed that pi was more like a smudge on the number line than a precise location.

I'd say the whole idea of an infinite number line, with infinity points between any two points, no gaps, no thickness,  is "like a smudge" in the imagination.  A culture that never imagined such a thing might still go to the moon and back.

I'm thinking the logic behind finite state machines has outgrown the more puerile logic of the superstitious 1900s, wherein people still believed in the ancient name -> object nominalism of St. Augustine and thought of "actual number objects" out there in Platonic space somewhere, along with "ideal circles" and all that stuff we imagine we can imagine.

I'm mixing up terms here because actually Nominalism and Platonism are considered opposite in that nominalists insist the objects be in physical memory i.e. the universe (more like Python) whereas Platonists thinks the objects live in some "realm" distinct from energetic time-bound special-case events (more like Penrose?).

I'm conflating both of those and suggesting the whole idea of words deriving meaning from "correspondence" is superstition and myth more than validated anthropology.  I point to Wittgenstein's work as persuasive, at least in my case, that Nominalism and Platonism are not the only choices.

 

​> ​

Allow infinite digits and we're able to make that one-to-one bijection to the Reals.

​> ​

But we disallow that, thanks to Peano and ZFC and other important axiomatic systems you need a PhD to discern.

​Again, I think there is a confusion here between numbers and their representation.​

Lets just focus on a sequence of digits I can count in the Cantorian sense.  The digits of pi make a great example.  

I can use that pi generator we both know about and enumerate the digits, one by one, until I run out of memory.

In the Cantorian sense, the digits of pi are numerable, listable, countable.  Whatever the shoptalk we wanna use.  

As such, we're giving the set N and the digits of Pi the same aleph number.  Yes, Pi's digits 0-9 keep re-occurring and pure sets don't allow the "same element" more than once, so lets remember to use our index number to help tell them apart (the subscript is a distinguishing feature).  

We could also keep writing the digits a little bigger each time, so that a 3 early in the sequence would be smaller than a 3 later on, reminding us they're different objects.  We have infinite room so no worries about "bigger", could be a logarithmic curve (we have infinite precision to detect height differences).

We don't 'decide' because of some set of axioms to allow R but not allow N to be represented by infinite decimal strings. 

I don't think "God tells us (dictates)" about the reals, and how we must define them, so yes, there are human deciders in this picture.  

Remember R is allowed to have infinite decimal strings, just not to the left, only to the right of the decimal point (like pi does).  That's the law.  

Going infinite in the left direction (as if holding up a mirror at the decimal) is strictly verboten. Stuff would break.

Since Reals are disallowed from having infinity digits going left (right OK), so are Ns, which are a sebset of Reals.  I was confused about this earlier, not realizing how strictly the Reals are restricted.

All this sets are established by axioms and definitions.  There's nothing else to go on but the rules on the back of the box, so to speak.  Shared rules keep the game playable.

Here's the prevailing dogma (bolding added, color background removed):

Set theory fully vindicates the concept of actual infinite, as, through the very simple and intuitive notion of set, it is possible to provide a fully satisfactory theory of infinities of different sizes. After Cantor's creation of the Transfinitum, and his early naive formulation of the notion of 'set' (Menge), the axiomatisation resulting in the theory known as ZFC (due to Zermelo, Fraenkel, Skolem and von Neumann) secured the internal consistency of the early infinitary set-theoretic intuitions and methodologies.

 
http://www.math.uni-hamburg.de/home/loewe/HiPhI/abstracts.html

This is the language of vindication following debate (lots of PR and propaganda, aka "spin"), the victors writing their history in the language of "we were right all along" (a common pattern).

However, others will take refuge in a body of literature which does not bow down to these authoritative voices.  Philosophy World is in perpetual ferment.

That's what I want students in my classroom to know:  they're not required to pledge their allegiance to some current crop of meta-physicians.  They're not obligated to take a side in debates they've never had time to really explore (and probably won't in adulthood either, unless very specialized).

Wait until you're much older. Having philosophical discussions at a young age too easily becomes a form of bullying or sucking up, as the teacher already has preferences.  

If moving in that direction, best to set it up as a Lincoln-Douglas style debate maybe, with judges (including lay persons, don't stack the deck with professional mathematicians, not necessarily any good at philosophy).  The math teacher should maybe recuse herself, at least as a judge, to make it more fair.

  

​>> ​

There are just as many naturals as there are integers, and there are just as many naturals as there are rationals.

​> ​

"As many" in a special sense, as when we're comparing infinities to infinities, you have to know more about Aleph and all that.

​No, all you need to know about is one-to-one correspondence.

That's it.

You also need to know that we believe in infinite time and infinite memory.  We're not allowed to say that "counting is work" i.e. energy is involved and the laws of conservation of energy are relevant.  That's physics.  

Mathematics exempts itself from these silly physical constraints.  Superpowers pop up everywhere (everytime an infinite number line -- that's not really a number line -- is drawn on a chalkboard).

In other words, we need to believe in infinity and infinite sets.  My question is:  if we don't "believe in infinity" (because we're physicists or some weird brand of natural philosopher) might we still have finite state machines and go to the moon and back?  Mars?  

Are we really required to buy all this "logic"?  I'd say "of course not, it's inessential to getting your work done" (unless you're trying to suck up to a specific subculture and its kahunas).

 

OTOC, dropping pebbles in a pile, putting items in a list, is the conceptual tool by which we establish that the cardinalities of N, Z, and Q are equivalent.​

​> ​

Usually in ordinary language, "as many" is with reference to finite sets.

​"As many as" is established through OTOC, and this can occur in ordinary language.​

Again, drop a pebble in a pile for each sheep that goes through the gate, and you have "as many" pebbles as sheep.

There were never "infinite sheep" at any time in history.  "Infinity", like a wolf, changes everything.  What was intuitive is now not intuitive at all.

I really am not enamoured of "infinity" as a concept.  It doesn't turn me on.  If God were infinite, I'd think far less of Him than if He were not.  I'm somewhat in Donald Knuth's camp on this one.  "Infinity is for wimps" might be my motto.  

That being said, I do like "eternal" and "eternity" as concepts (not quite the same thing).  I also like fractals and subdividing, up to the limit of what memory or the canvas will show:

http://www.math.cornell.edu/~mec/Winter2009/Mihai/section3.html

Mathematics comes in many blends, like coffee.  I don't think one need cultivate a taste for all of it.  

A math major friend of mine at Princeton once came back from a one-on-one office hours session wherein his professor said he like his math to be "schmaltzy".  I have no idea exactly what that meant, but I got that math comes in flavors, just like music does.

When it comes to the aleph stuff, I have a take it or leave it attitude, much as I do towards Medieval philosophy.  I don't think of mathematics as an edifice that would crumble without the busy-work of logicians.  Our finite state machines would continue to work if we stopped caring about infinity as much as we do.

 

This very same principle allows us to say that we have "as many" naturals as rationals.

​>> 

The reals cannot be listed.

​> ​

Not really talking Python here 

 

​No, we are.

You cannot create a generator in Python that will yield all of the reals.

We don't have reals in Python.  That's not a type.  We have floating point numbers and complex numbers made from floating point numbers.  We also have extended precision Decimals and in 3rd party world we have extended precision complex as well.

https://pypi.python.org/pypi/gmpy2

The fact that we get all our work down without any real numbers is to me quite persuasive that we could retire the reals and still go to Mars.

I'm not saying we're gonna do that of course.  We'll have our true believers in "the reals".  Great religions have staying power.

 

However, you CAN create a generator in Python that will yield all of the rationals.

I've done it.​

Also you can create a generator that will yield all the digits of Pi, you shared it and I've done some digging to trace where it came from on edu-sig.

Thanks for getting that ball rolling!

Kirby

 

[kirby urner]

In that sense, of an "infinite loop" yes, we can code them.  

while True:

    print(1)

    if input("Next (Y/n)" == "Y":

        continue

    else:

        break

is also a way to code for infinite 1s.  Same idea as a generator.

Oops, unbalanced parens:

while True:

    print(1)

    if input("Next (Y/n)" == "Y":

        continue

    else:

        break

 

 

However, you CAN create a generator in Python that will yield all of the rationals.

I've done it.​

Also you can create a generator that will yield all the digits of Pi, you shared it and I've done some digging to trace where it came from on edu-sig.

Thanks for getting that ball rolling!

Kirby

 

Here's the generator I'm citing:

def pi_digits():

    k, a, b, a1, b1 = 2, 4, 1, 12, 4
    while True:
        p, q, k = k*k, 2*k+1, k+1
        a, b, a1, b1 = a1, b1, p*a+q*a1, p*b+q*b1
        d, d1 = a/b, a1/b1
        while d == d1:
            yield int(d)
            a, a1 = 10*(a%b), 10*(a1%b1)
            d, d1 = a/b, a1/b1

[  http://mail.python.org/pipermail/edu-sig/2012-December/010728.html ]

>>> pi = pi_digits()
>>> "".join([str(next(pi)) for i in range(100)]))
'3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067'

I originally learned about this generator from Michel Paul!

Kirby

[Joseph Austin]

On Jul 10, 2016, at 3:30 PM, michel paul <python...@gmail.com> wrote:

So Cantor's proof ends up being a statement about the set of infinite lists.

You can't define a list containing all infinite lists.

It's a different kind of set. Not listable.

Or might I give it another interpretation?  You can't define a "set" containing all infinite lists. 

It's not a computable set.

In other words, It's not a "different kind" of set, it's not a "set" at all, just a meaningless juxtaposition of words,

like "furiously sleeping colorless green ideas."

joe

[kirby urner]

On Sun, Jul 10, 2016 at 8:31 PM, Joseph Austin <drtec...@gmail.com> wrote:

<< SNIP >>
 

And note that I said numeral instead of number.  There are various ways of "numeralizing" a number.

As you may have seen from another post of mine to this thread, I'm not a big believer in the numeral -> number model of meaning, what I call "Nominalism" (conflated with Platonism for my purposes).

Does '2' used on a city bus line, appearing on any bus following such-and-such a route -- which routing system may have no line '1' or line '13' -- point to the same object as 2 in an elevator or 2 in a math book? 

What's with this "point to" <--- the mistake right there.  To "label" is not a simple act.  There's a whole game around what we call "naming" (many games, with a family resemblance to one another).

Language is a lot of moving parts, one could say, but that's not proving that any parts "point to" other parts.

Of course it is possible (and necessary) for one to "guide the attention" of another, by means of pointing when apropos.

What does the word "the" point to?  Answer:  it doesn't point.  Language operates, so we might say it consists of operations.  Adding two numbers is a physical activity that takes time, even in a CPU.

 

The multiplicative inverse of three can be represented as 1/3, or in base three as 0.1

But it cannot be represented in binary or decimal by any finite numeral. Or I would say, even by an infinite numeral.

No number of factors of 2 or 5 will ever give a multiple of 3, even and infinite number!

So Cantor's list is not even complete, so it should be no surprise to find numbers not in it!

So what is a "real" number?  

"And do we care?" might be a prior question.  Not trying to be flippant just life is short, and if the question hasn't been settled by now, is it worth asking?

A civilization using finite state machines, some of them steam powered, and not using much "continuous math" might have only had digital numbers from the beginning. 

The analog numbers we remember from our textbooks are something we have in common owing to shared ethnicity, but I'd not surmise that "real numbers" are necessary to life on Mars, even hominid life on a terraformed Mars v 3.1. 

Depends how long we take to get there and how much we forget this period of history in the meantime. 

In this time period, we still put a lot of eggs in the "real numbers" basket.  Why not?  They've served us well (provided lots of employment).

I would argue that we didn't really have real numbers in the current sense until Cantor.  Lets say 1871.  So that means civilization got along for thousands of years without any concept of real numbers.  What does that tell you about how vital they are to our continued survival?

Will we still be thinking in terms of "real numbers" in 3871?  Given no computer really needs them, I'm not so sure.  Probably we'll still have access to the current literature via Bing. :-D

 

I think it's intuitively obvious that there are "more" integers than naturals, approximately twice as many.

And "more" rationals than integers. And more "irrationals" than rationals.

That's thinking more like a computer scientists.  Integers are on a giant gear of many notches. 

Floats are on two giant gears: one for significant digits, the other for where the decimal point goes (exponent of 10).

Clearly, if you're allocated n bits, there are only so many different patterns available.  ints and floats are strictly finite sets given their bitwise definitions.  Enumerating them is not a problem.

Except in some languages (e.g. Python) the integer is allowed to consume as many bytes as the operating system allocates to the process, ditto some other exotic types of number (the number of bytes is open ended, but constrained by the laws of physics in some dimensions).

In that case, overflow conditions apply.  There's no "infinite memory" one needs to learn about.

 

Rather than considering orders of infinities, I would rather study things like the Riemann sphere that exposes "infinity" as an artifact of imposing an "unnatural" geometry or numerology on a simple closed space.

I'm fine with algorithms that have no defined stopping point, wherein any stopping point is arbitrary.  You can always get the next digit of pi.

I think it was Poincaire who said "infinity is a direction" but I haven't been able to Google it up in the last few sweeps of the net.

Kirby

[Joseph Austin]

I 

I'm really more concerned with where the reals come from.

My gut feel is that "reals" as defined are "overkill"--a hypothesized set beyond what is needed for "real" computations.

I suppose mathematicians would disagree with me, but I'm not convinced the set is "well defined".

At least, none of my math professors ever took the trouble to define them.

Any "real" that I need to have can be computed.

Or perhaps I should say, can be approximated "as close as desired" by a computable number.

So let the code for the Turning machine that computes the sequence that converges to the real number be it's list index.

(It may be that the same number has multiple distinct indices, but that won't change the cardinality of the set.)

Since all of the infinite number of possible Turing machines are individually finite, the set of computable reals must be countable.

So what do you call the subset of "reals" that is computable?

Any why should one be concerned with the rest?

Show me a math problem whose solution is one of the non-computable reals.

I'm suspecting they are made of the same stuff as the Emperor's New Clothes.

To a wise mathematician, they are a set of exquisite beauty,

but to an engineer, they are absolutely invisible. :-)

Joe

[Joseph Austin]

On Jul 11, 2016, at 3:45 AM, michel paul <python...@gmail.com> wrote:

Pi itself has a specific location on the number line. It is not a smudge.

For example, consider a circle of unit radius, with circumference measured in radians. Pi occurs on the circumference exactly where the circle crosses the negative x axis.

In similar vein, the square root of 2, measured along a line through the origin at angle 45 degrees, occurs exactly where the line crosses the coordinates (1,1).

I used to have a poster in my office, with lines of a cartesian grid vertical and horizontal in red, and diagonal lines thru the intersections in blue.

Underneath was the question:  Which grid is rational and which is irrational?

(Does the answer change if you rotate the figure by 45 degrees?)

Moral of the story: there are no irrational numbers, only irrational pairs.

Joe Austin

[kirby urner]

On Tue, Jul 12, 2016 at 8:30 AM, Joseph Austin <drtec...@gmail.com> wrote:

<< SNIP >>
 

Show me a math problem whose solution is one of the non-computable reals.

I'm suspecting they are made of the same stuff as the Emperor's New Clothes.

To a wise mathematician, they are a set of exquisite beauty,

but to an engineer, they are absolutely invisible. :-)

Joe

I like where you talk about "computer numbers" as a type of number.

I'm glad we're having these philosophical discussions, as in my view

a healthy Math of the Future does encourage philosophical debate,

including among children and newcomers to the subject.

But "debate" does not mean "reading the FAQ" and learning to

stop questioning the infallible authorities.  That's not the right spirit.

Philosophy is nowhere near as strong, in a departmental sense, as

in the old days, and it shows in the degradation of debate about
"the basics".  We're more into "reciting the dogmas" in this day and

age.

I have a concern that voices such as yours, piping up for "computer

numbers" as a subset of the reals (?), will be dismissed out of hand

as "hopelessly naive" to start, the "defiant" if you stick to your guns.

You'll be shut down by teachers pre-programmed to uphold some
"party line" with the imprimatur of some Math Establishment, and
with the philosophers side-lined or the silence won through intimidation. 

You won't lean about any unit-volume tetrahedron either, given the

"not invented here" syndrome of certain cliques.

My concern is you'll be branded "obstreperous" by the authorities
(good SAT word) and your academic prospects may be diminished
accordingly.  It's the school-to-prison pipeline for ol' Dr. Tech Daddy,
who won't kowtow to the Real Numbers Commission. :-(

What I tend to do in my camp is chip away at name -> object
nominalism, i.e. the doctrine that "meaning comes from pointing",

in order to free us from "existence claims" of the type: "higher

dimensions really exist".

Existence claims are typically followed by arm-twisting attempts
to persuade, with citations to such political tracts as Abbott's
Flatland.  We're told we can imagine what it's like to be 2D
beings (not buying) and so therefore, by analogy, a 4D being
could dumb itself down to imagine being us, beings trapped

at the 3D level of the matrix.

Note that in this context (namespace) we're not thinking about
Time.  This is extended Euclideanism I'm talking about, with
no dimension for mass either, or really anything to do with
energy really, as that's physics.  Polytopes are purely mental.

By extension, goes the Abbott story, the really gifted amongst
us will be able to commune with the 4D Martians or whatever,
the higher dimensional beings. 

Here's a typical passage reflecting this standard doctrine
(bolding added):

This paper postulates a multidimensional model of the universe, based on recent developments in physics and biology. We cannot see the multidimensional reality because our senses are limited to three dimensions, yet the higher dimensional environment has a more substantial reality than our world. This is so because our three-dimensional world is only a subset of the multidimensional system. An interrelated set of holistic principles is developed. The multidimensional world is then explored with this holistic logic system.

[ https://web.archive.org/web/20000916035057/http://home.sprynet.com/~jowolf/ ]

It all gets to be about subjective "beings" with higher dimensionality
becoming something to introspect about, a window to God as it were,
and so a flavor of mysticism.  We're now in the realm of theology and

religion (the T in PATH, which crosses STEAM at the A).

People who can "see in higher dimensions" get to boast of a third eye.

But how is "really exists" different from "is really useful"? 

Higher dimensional data structures and the algorithms for working
with them are definitely useful.  The N-dimensional polytope is a data
structure with an applied metric (Pythagorean distance). 

This added gesture, the "really exists" locution, derives in part from
name->object nominalism. 

Wittgenstein tilts us towards pragmatism, as we break the hold of
referents "out there" as N-D "beings" (the things that polytopes "really

are").

Kirby

[kirby urner]

Moral of the story: there are no irrational numbers, only irrational pairs.

Joe Austin

I like the words "incommensurable" and "incommensurability" in this context.
 

Kirby

[michel paul]

On Mon, Jul 11, 2016 at 10:33 AM, kirby urner <kirby...@gmail.com> wrote:

​> ​

The reals are not allowed to have infinite digits to the left of the decimal point, only to the right. That's the law.

I have never heard of such a law.​ 

I think it's more

​a matter of ​

what

​would such a notation

 

​help us ​

accomplish?​

If it has a useful interpretation, if it has practical consequences or if that notation clarifies an idea, there really is no ultimate math authority to prohibit anyone's use of that notation.

Like tau, if people like it, it might even catch on.

The reason that it makes sense to have an infinite decimal extension to the right of the decimal point is that it is a series that converges. An infinite extension to the left does not.

Now does the fact that it does not converge mean that we are not allowed to experiment with it? Not at all. It's just like dealing with aleph stuff. It can be fun. If we find a consistent way to think about it and utilize it, great.

​> ​

OK then, the "representation of pi" is what I'm interested in.

​Precisely!    : )

And then consider - there are many possible representations! Which one do we want?

There are many ways to represent the same thing ... and that is significant.

​>> ​

Pi itself has a specific location on the number line. It is not a smudge.

​> ​

What is "the number line"?  Does it "actually" exist? 

​The number line is a useful pedagogical device for visualizing certain kinds of relations.

Mathematics consists of relations. We try to understand these relations by visualizing them in various ways, and we try to describe them by working out consistent consequences in the notations we develop.

​The important point is that pi is a precise value that can be expressed in various ways. Pi itself doesn't 'go on forever'. Its decimal representation does. 

Or ... any series that converges to pi will apparently 'go on forever', but pi is a completely determined value. 

Otherwise, coding a generator to list its digits to any desired length would make no sense. 

The digit string for pi is determined completely. Change any one of those digits, and you no longer have pi. You have something close enough that would work just as well for practical matters, but it is just a matter of fact that that value is different from pi.

​> 

 All math is ethno-math.

​False. Animals have been demonstrated to possess number sense.

Just as mathematics transcends culture, it also transcends species.​

We used to think of ourselves as the only 'tool using' animals, the only animals with 'language', the only animals with 'culture', the only animals with 'number sense'.

That's all history. Every one of those assumptions is false.

Patterns occur in nature, and these patterns created our brains.

We sometimes see patterns that are not actually 'there' in the sense of being pertinent to a situation, but the fact that our brains are capable of perceiving the patterns at all comes from the way our brains are already constructed.

I tend to agree with Heisenberg that QM vindicates Plato and Pythagoras.

 

​>> ​

You cannot create a generator in Python that will yield all of the reals.

​> ​

We don't have reals in Python.  That's not a type.

​Yes, I understand. However, my point is that it could not be done in principle.

Let's just imagine that there were somehow a data type called Real.​ My point is that even if it existed, it would be impossible to code a generator that would eventually yield all of the Reals.

However, in the case of a data type called Rational, it is quite possible to code a generator that will eventually yield all of the Rationals.

The reals cannot in principle be listed.

​The really important point that I've tried to make regarding all of this is that you can scrap all discussion of number and number lines and just look at this whole matter in terms of character strings.

Consider a character set, any set. Is it possible in principle to create an algorithm that would systematically list ​all possible finite strings composed from these characters?

The answer is yes. First list all the single characters. Then list all possible pairs of characters. Then list all the possible triads, etc. Could be done. It would in principle be possible to code a generator that would list, one by one, every possible finite character string. Any string you could think of would eventually appear.

Now - consider the possibility of a generator of generators. Doesn't have to be in Python. Forget current technical issues. Just take the pure concept of a generator and consider a generator that yields generators of infinite text strings.

​Would it be possible in principle for such a generator of generators to eventually yield all possible ​infinite text strings if left to run long enough with a clever enough algorithm?

The answer is no - a diagonal argument proves there would always be some possible string left out. 

Again, we're not talking about Romantic era ideas of real numbers here.

[kirby urner]

On Tue, Jul 12, 2016 at 12:47 PM, michel paul <python...@gmail.com> wrote:

On Mon, Jul 11, 2016 at 10:33 AM, kirby urner <kirby...@gmail.com> wrote:

​> ​

The reals are not allowed to have infinite digits to the left of the decimal point, only to the right. That's the law.

I have never heard of such a law.​ 

I think it's more

​a matter of ​

what

​would such a notation

 

​help us ​

accomplish?​

The reason R numbers are forbidden to infinitely extend to the left in their decimal or other base representations, is that if they did, we could take all the left side numbers with no digits after the decimal, and call those the set N.

314159... (digits of pi, no decimal) could be one of those numbers.

But if we did that, then simply removing the decimal point from Pi would map it to one of these left siders and indeed all real numbers would map to precisely one N:  the same digits with no decimal point.

This would place N in level playing field relationship to R and break the aleph stuff, which we don't want to break.  QED.

 

If it has a useful interpretation, if it has practical consequences or if that notation clarifies an idea, there really is no ultimate math authority to prohibit anyone's use of that notation.

Like tau, if people like it, it might even catch on.

I think we must continue to forbid infinite digits to the left.  That would destroy R as we know it.  Western Civilization would crumble. Planes would fall from the sky (just kidding).
 

The reason that it makes sense to have an infinite decimal extension to the right of the decimal point is that it is a series that converges. An infinite extension to the left does not.

Here's a question:  does a jumble of random digits going to the right "converge" to something? 

Say:  0.485100932093419382719239492813401239518364182364871...

Without an algorithm behind those digits, I'd say we have "the idea of convergence" but "no idea of what we're converging to" in terms of having some other way to say it.

We're bordering on nonsense again.  However there might be an algorithm, we just have no idea what it is.

Don't take away the decimal point though, as then we won't even have convergence and that's a last straw.

 

Now does the fact that it does not converge mean that we are not allowed to experiment with it? Not at all. It's just like dealing with aleph stuff. It can be fun. If we find a consistent way to think about it and utilize it, great.

​> ​

OK then, the "representation of pi" is what I'm interested in.

​Precisely!    : )

And then consider - there are many possible representations! Which one do we want?

I was OK with decimal digits.

Hey, here's another question: 

Sometimes we hear people say:  given the digits of pi go on and on with no end, every pattern must occur in principle, i.e. somewhere along that infinite string will be the complete works of Hamlet per some encryption scheme we might introduce.

To my ears, such claims are utter nonsense.  Nor must infinite monkeys typing on infinite keyboards ever come up with the works of Hamlet, or even Finnegans Wake.

When people talk about infinity, they let their imaginations run wild sometimes.  It gets really silly pretty quickly.  Language goes off the rails and we're left in the swamp.
 

There are many ways to represent the same thing ... and that is significant.

I think the decimal representation of the reals provides a proof of why N will never be able to number them.

Given:  reals may be represented with infinite digits to the right.

Given:  reals may not be represented with infinite digits to the left.

Given:  natural numbers have digits only to the left

It follows:  natural numbers are hopelessly outnumbered, given the constraint on the left that does not apply to the right.  This asymmetry alone is sufficient to make all the Rs unpairable with Ns.  We run out of Ns sooner, thanks to the left side being asymmetrically constrained.

A restricted subset of the reals deprived of the same privilege of an infinite digit representation, has no chance of enumerating its parent class with such a decisive advantage right out of the starting gate.  QED.

I don't really see the need for any diagonals to prove the two sets are not one-to-one comparable.

 

​>> ​

Pi itself has a specific location on the number line. It is not a smudge.

​> ​

What is "the number line"?  Does it "actually" exist? 

​The number line is a useful pedagogical device for visualizing certain kinds of relations.

Such pedagogical device may also, in addition, service to bewitch us with imagery that helps keep us confused. 

What does the Mandelbrot Set converge to, at the end of the day?  Is that a point or a smudge? 

Are we getting any closer to a precise value at the end of the convergence rainbow, as we zoom in on a specific patch?

It's like the tunnel keeps getting narrower, as if to a tip, but the observer traveling through the tunnel shrinks in proportion, meaning the tunnel always seems to stay exactly as wide.

Complex numbers are not orderable at least i.e. < and > are undefined.  At least we have precedent for number sets not being orderable.

I'd like to say the number line goes to "infinity" in both directions, with scare quotes.

That's to say "infinity" is a direction, not a place at the end of some rainbow.  If we want to use only finite number lines of indeterminate length (but never infinite length), that'd work for some maths.

 

Mathematics consists of relations. We try to understand these relations by visualizing them in various ways, and we try to describe them by working out consistent consequences in the notations we develop.

I want to have multiple "we" camps inside of mathematics.  Any discipline with only one "we" is probably corrupt.  And in truth, that's what we've had throughout history.  Mathematics evolves by debate. 

We're not gradually unveiling a grand ediface, slowly pulling back the curtain.  That's how people used to think of science until Kuhn blew that idea out of the water.  Maths are likewise evolving the way music does, branching this way and that. 

The idea of "the one true math that we'll eventually get to" is a rather silly notion no?  Sounds like Bourbaki.

So my "we" doesn't necessarily buy all the same mathematics another "we" does. 

There's no compelling divinity that makes us ("we who do mathematics") all swallow the same red pill, or drink the same kool-aid.

We don't need the aleph stuff to get to Mars.  But I'm thinking the aleph stuff has staying power, lots of inertia. 

But there's a half-life.  Sense leaks away without adequate maintenance.

 

​The important point is that pi is a precise value that can be expressed in various ways. Pi itself doesn't 'go on forever'. Its decimal representation does. 

It's the decimal representation that interests me, more than any "thing in itself" which would not exist in nature necessarily, as we know of no energetic phenomena to which that many significant figures attach.

Given I'm not a nominalist, I don't require such phenomena, either "in nature" or "in the mind" for pi to have meaning.  I don't think of pi as a pointer to a "thing".  Pi is useful and meaningful without a referent.  As long as we have many true things we can say about Pi, it'll be in our meme-plex.

I think of Pi more as an institution, a set of algorithms that confirm each other, check each other, in a mix with other imagery, standardized diagrams.

0.485100932093419382719239492813401239518364182364871...

on the other hand has no such lore or apparatus to back it up and is verging on nonsense and/or is nonsense already.  Is it real?

 

Or ... any series that converges to pi will apparently 'go on forever', but pi is a completely determined value. 

Right, otherwise we couldn't check it, one algorithm vs. another (validation is important).
 

Otherwise, coding a generator to list its digits to any desired length would make no sense. 

So you're saying infinitely long random digit sequences to the right of the decimal are *not* real numbers?

Isn't that excluding the vast majority of them?  We have algorithms for just a few values we care about. 

The reals backed by algorithms I can count on one hand so to speak (relative to numbers "in the wild").

 

The digit string for pi is determined completely. Change any one of those digits, and you no longer have pi.

But it's still a real, right? 

If I take pi and change just one single digit (not saying which one), surely that doesn't get it kick out of the set! 

It's not pi, but it's still real.  3.13159.... (note 2nd digit after decimal, but the rest of the digits same as pi). 

That's real, right?  So then keep making changes to whatever digits.  So what? 

Call it "noisy pi" or "almost pi" or "degraded pi".

 

You have something close enough that would work just as well for practical matters, but it is just a matter of fact that that value is different from pi.

​> 

 All math is ethno-math.

​False. Animals have been demonstrated to possess number sense.

I would say animals in being "forms of life" extend the intra-human idea of ethnicity.

The salmon people have their ethnicity.  The bear people....

That's sorta how N8V Americans talk -- animals get to be people too. 

Not so in standard English, a very buggy language that also sustains memes like "race" (a nasty feature of English though not exclusively of English).
 

Just as mathematics transcends culture, it also transcends species.​

That's the PR.  "Math is the universal language". 

I'm not buying it (for one thing it keeps changing). 

I'd rather say mathematicians form a somewhat ethnocentric group that could benefit from more anthropology / philosophy.

When people see "2 to the 3rd power" and automatically say "two cubed", that grates on my sensibilities. 

Why do they insist on seeing a cube as their only model of 3rd powering? 

I see blinders (ignorance) and popular culture, nothing "universal" in such speech.
 

We used to think of ourselves as the only 'tool using' animals, the only animals with 'language', the only animals with 'culture', the only animals with 'number sense'.

I think of language as just more processing, really inseparable from everything else. 

Driving a car is a kind of language game (following painted lines, reading signs and signals). 

Where language leaves off and not-language begins is highly arbitrary -- not sure why we need the line at all.
 

That's all history. Every one of those assumptions is false.

Right.  We used to think there was a "vital force" separating "living" from "non-living" physical systems. 

But then we discovered the virus as a "living machine" and that line started to blur.

How we think and speak is fluid. 

A lot of people still think in terms of "races" of human being (with "pures" and "impures") whereas to my ears that's ignorant nonsense, empty reflex-conditioning of no real value or worth.  Sloppy.  Lazy.  Has no place in serious schools (except as an access point in studying intellectual history -- one mental illness after another, to paraphrase Henry Ford).
 

Patterns occur in nature, and these patterns created our brains.

We sometimes see patterns that are not actually 'there' in the sense of being pertinent to a situation,

Like "races"....
 

but the fact that our brains are capable of perceiving the patterns at all comes from the way our brains are already constructed.

More than brains go into thinking -- the whole body is engaged, with neurons going everywhere, mixing with other systems (endocrine). 

Localizing all thinking "in the brain" is not so much science as superstition, how we like to talk in this day and age.  We're really into "brains" as the seat of everything we hope to someday understand.

Our brains would find themselves at a real disadvantage if we took away recorded media such as books. 

So much of our thinking is encoded in the environment and accessed there. 

Brains-alone thinking would be pretty stupid, pretty lame, compared to what we do today with Google and all that.

What we call thinking in 2016 is very unlike what they called thinking in the Bible, or back in the dark ages, when hardly anyone could encode / decode text formats, let alone access them with http / https. 

"That which thinks" has altered a lot since the days of Descartes' imagined "cogito".  He imagined his cogito was in the pineal gland (count me a skeptic).

I tend to agree with Heisenberg that QM vindicates Plato and Pythagoras.

 

​>> ​

You cannot create a generator in Python that will yield all of the reals.

​> ​

We don't have reals in Python.  That's not a type.

​Yes, I understand. However, my point is that it could not be done in principle.

Yes, per my proof above.  Left-handed numbers (N) are precluded from infinite digit expression, Right-handed numbers (R - N) are not.  Finity versus infinity.  Infinity wins.  QED.

 

Let's just imagine that there were somehow a data type called Real.​ My point is that even if it existed, it would be impossible to code a generator that would eventually yield all of the Reals.

However, in the case of a data type called Rational, it is quite possible to code a generator that will eventually yield all of the Rationals.

The reals cannot in principle be listed.

​The really important point that I've tried to make regarding all of this is that you can scrap all discussion of number and number lines and just look at this whole matter in terms of character strings.

Consider a character set, any set. Is it possible in principle to create an algorithm that would systematically list ​all possible finite strings composed from these characters?

I notice you say finite strings, which to me means "finite set".  So yeah, countable.

But now consider this approach:

Enumerate all possible strings consisting of 1 and 0 with only 1 slot :  1, 0
Enumerate all possible strings consisting of 1 and 0 with only 2 slots: 00, 01, 10, 11
Enumerate all possible strings consisting of 1 and 0 with only 3 slots: 001, 010, 011
...

we're never going to stop with this process, just we're going to take care of every possibility before adding a next slot.

I can see getting to all the rationals in this way as every rational will have a binary representation.  We'll eventually get there, whatever the number.  Q is "listable".  QED.

We can't get to Pi though per the earlier proof.  Enumeration is done in N and left-side numbers (per the decimal system) are in principle denied infinite-digit representations.  So we're doomed.  QED.

http://math.stackexchange.com/questions/944284/can-a-number-have-infinitely-many-digits-before-the-decimal-point

If left-side numbers *could* have infinite digits, that'd break the aleph stuff and we can't allow that now can we.

I think we've reached the same conclusions by different means.

Kirby

[kirby urner]

On Tue, Jul 12, 2016 at 3:37 PM, kirby urner <kirby...@gmail.com> wrote:

On Tue, Jul 12, 2016 at 12:47 PM, michel paul <python...@gmail.com> wrote:

On Mon, Jul 11, 2016 at 10:33 AM, kirby urner <kirby...@gmail.com> wrote:

​> ​

The reals are not allowed to have infinite digits to the left of the decimal point, only to the right. That's the law.

I have never heard of such a law.​ 

I think it's more

​a matter of ​

what

​would such a notation

 

​help us ​

accomplish?​

The reason R numbers are forbidden to infinitely extend to the left in their decimal or other base representations, is that if they did, we could take all the left side numbers with no digits after the decimal, and call those the set N.

314159... (digits of pi, no decimal) could be one of those numbers.

But if we did that, then simply removing the decimal point from Pi would map it to one of these left siders and indeed all real numbers would map to precisely one N:  the same digits with no decimal point.

Some will argue that 3.14159... and 31.4159... and 314.159... would all map to the same N, 314159...., so it's not a bijection. 

I'd argue we can treat all placements of the decimal as an equivalence class i.e. powers of 10 (... -3, -2, 1, 0, 1, 2, 3...) are internally numerable so lets just pick n.nnnn.... as a the canonical representation for that equivalence class, for the purposes of pairing with N

Sometimes we hear people say:  given the digits of pi go on and on with no end, every pattern must occur in principle, i.e. somewhere along that infinite string will be the complete works of Hamlet per some encryption scheme we might introduce.

I understand it's the works of Shakespeare we're usually looking for, but lets leave room for Hamlet to have done some writings too.  We're giving in to the imagination, so why not?

To my ears, such claims are utter nonsense.  Nor must infinite monkeys typing on infinite keyboards ever come up with the works of Hamlet, or even Finnegans Wake.

When people talk about infinity, they let their imaginations run wild sometimes.  It gets really silly pretty quickly.  Language goes off the rails and we're left in the swamp.
 

There are many ways to represent the same thing ... and that is significant.

I think the decimal representation of the reals provides a proof of why N will never be able to number them.

Given:  reals may be represented with infinite digits to the right.

Given:  reals may not be represented with infinite digits to the left.

Given:  natural numbers have digits only to the left

It follows:  natural numbers are hopelessly outnumbered, given the constraint on the left that does not apply to the right.  This asymmetry alone is sufficient to make all the Rs unpairable with Ns.  We run out of Ns sooner, thanks to the left side being asymmetrically constrained.

A restricted subset of the reals deprived of the same privilege of an infinite digit representation, has no chance of enumerating its parent class with such a decisive advantage right out of the starting gate.  QED.

I don't really see the need for any diagonals to prove the two sets are not one-to-one comparable.

 

​>> ​

Pi itself has a specific location on the number line. It is not a smudge.

​> ​

What is "the number line"?  Does it "actually" exist? 

​The number line is a useful pedagogical device for visualizing certain kinds of relations.

Such pedagogical device may also, in addition, service to bewitch us with imagery that helps keep us confused. 

"... serve to bewitch" -- an allusion to Wittgenstein who claimed philosophy was the process of fighting to get free of the power of language (including pictures and images) to keep us buzzing around in confusion.
 

I notice you say finite strings, which to me means "finite set".  So yeah, countable.

But now consider this approach:

Enumerate all possible strings consisting of 1 and 0 with only 1 slot :  1, 0
Enumerate all possible strings consisting of 1 and 0 with only 2 slots: 00, 01, 10, 11
Enumerate all possible strings consisting of 1 and 0 with only 3 slots: 001, 010, 011
...

we're never going to stop with this process, just we're going to take care of every possibility before adding a next slot.

I can see getting to all the rationals in this way as every rational will have a binary representation.  We'll eventually get there, whatever the number.  Q is "listable".  QED.

To those who point out even members of Q have non-terminating binary representations, I say:  every Q is some p / q where p, q are both integers and we have a terminating binary representation for every p and q.

So getting to all the rationals is a matter of following the above procedure and then enumerating all pairs of p and q so far generated, i.e. once we have 0,1,2,3,4 we have (0, any), (1,1) (1,2) (1,3) (1,4) (2, 1) (2,2) (2,3) (2, 4) (3, 1) and so on.

Fine to cross out rationals that are canonically the same i.e. (1,1) (2,2) (3,3) (4,4) form an equivalence class as do all (0, n) -- with (n, 0) undefined. QED.

Kirby

[michel paul]

On Tue, Jul 12, 2016 at 3:37 PM, kirby urner <kirby...@gmail.com> wrote:

​> ​

Here's a question:  does a jumble of random digits going to the right "converge" to something?  

​> ​

Say:  0.485100932093419382719239492813401239518364182364871...

 

​Yes. If x = the something, then we know:

0.4 < x < 0.5

0.48 < x < 0.49

0.485 < x < 0.486

0.4851 < x < 0.4852

0.48510 < x < 0.48511

0.485100 < x < 0.485101

0.4851009 < x < 0.4851010

etc.

​> ​

I'd say we have "the idea of convergence" but "no idea of what we're converging to"

​Right, it's random. However, we are still narrowing in on some actual value of x. After awhile it will be 'close enough'.

This is why it makes sense to have an infinite number of decimals to the right.​ They do have a specific sum. 

Infinite decimals to the left however do not have a specific sum. 

​> ​

Hey, here's another question: 

​> ​

Sometimes we hear people say:  given the digits of pi go on and on with no end, every pattern must occur in principle, i.e. somewhere along that infinite string will be the complete works of Hamlet per some encryption scheme we might introduce.

​> ​

To my ears, such claims are utter nonsense.  Nor must infinite monkeys typing on infinite keyboards ever come up with the works of Hamlet, or even Finnegans Wake.

​> ​

When people talk about infinity, they let their imaginations run wild sometimes.  It gets really silly pretty quickly.  Language goes off the rails and we're left in the swamp.

​I'm not sure what the question is, but I think you have a point. If we WERE to discover Shakespeare in pi, it would be AMAZING. It would shock us deeply.

 

​> ​

Mathematics evolves by debate.  

​Yep.

As do philosophy, art, and science.

But you can prove things in math effectively and decisively in a way that is unique in the disciplines.​

 

​

​>> ​

The important point is that pi is a precise value that can be expressed in various ways. Pi itself doesn't 'go on forever'. Its decimal representation does. 

​> ​

It's the decimal representation that interests me, more than any "thing in itself" which would not exist in nature necessarily

​But why the decimal representation? Why not continued fraction or ​series representations?

Or is it just representation itself?

Sure, that would make sense, because representation is all we have to work with.

The deeply cool thing is how we can have all these wildly varied representations of the same value.

​>> ​

any series that converges to pi will apparently 'go on forever', but pi is a completely determined value. 

​> ​

Right, otherwise we couldn't check it, one algorithm vs. another (validation is important).
 

​>> ​

Otherwise, coding a generator to list its digits to any desired length would make no sense. 

> ​​

So you're saying infinitely long random digit sequences to the right of the decimal are *not* real numbers?

​No. 

What I am saying is what you are saying.

You said, "Right, otherwise we couldn't check it, one algorithm vs. another (validation is important).".​

 

​>> ​

The digit string for pi is determined completely. Change any one of those digits, and you no longer have pi.

​> ​

But it's still a real, right? 

​Of course.​

​> ​

That's the PR.  "Math is the universal language". 

​> ​

I'm not buying it (for one thing it keeps changing). 

​The prime numbers seem pretty stable.

Math is more than a language. 

It is a language, yes, but it is also the study of abstract objects.

Mathematics is the study of patterns. Patterns precede people.​

​> ​

Localizing all thinking "in the brain" is not so much science as superstition, how we like to talk in this day and age. 

​Yes, and that's changing. Our neural structure permeate the body. Thoughts are not just 'in the head'.

The patterns that preceded people are evident in early living forms. They were beautifully geometric.​

​> ​

I think we've reached the same conclusions by different means.

Well, my conclusions are:

• Cantor's diagonal proof ​is valid - a listing of the reals is impossible.
• Interestingly, this proof also applies to text strings independent of numerical reference.

--

​Michel

[michel paul]

On Tue, Jul 12, 2016 at 11:13 PM, michel paul <python...@gmail.com> wrote:

​

​> ​

The prime numbers seem pretty stable.

​However, it is also the case that a debate occurred over whether or not to consider '1' as a prime.​

 

--

[michel paul]

On Mon, Jul 11, 2016 at 2:46 PM, Joseph Austin <drtec...@gmail.com> wrote:

​

Or might I give it another interpretation?  You can't define a "set" containing all infinite lists.

​ ​

​

It's not a computable set.

In other words, It's not a "different kind" of set, it's not a "set" at all, just a meaningless juxtaposition of words,

like "furiously sleeping colorless green ideas."

joe 

​

Well, no, it does make sense to define a set of all infinite lists of a given character set, but it does not make sense to define a list of all infinite lists of a given character set, because some will inevitably be left out.

Consider some character set. For the sake of simplicity make it binary. Just two choices, let's say a and b.

Can you define a set containing all infinite lists created from these two characters?

Yes.

There is no problem in considering a set whose members are all possible infinite lists composed of a and b.

But can you make a systematic list of that set of infinite lists?

No.​

 

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[michel paul]

On Tue, Jul 12, 2016 at 8:30 AM, Joseph Austin <drtec...@gmail.com> wrote:

​> ​

I'm really more concerned with where the reals come from.

​They came from the Pythagoreans, reluctantly.

They killed the guy who discovered it, but they ultimately could not deny that the diagonal of a square is incommensurable with the side.

​> ​

My gut feel is that "reals" as defined are "overkill"--a hypothesized set beyond what is needed for "real" computations.

​Right, you really don't need the reals for computations. The rationals are good enough.

But when you realize that sqrt(2) is in principle not rational, it makes you curious.​

 

--

​ Michel

[Joseph Austin]

Michel,

Square root 2 comes from one of the  the inverses of exponentiation,

so some set of irrationals comes from closure of root over the naturals and positive rationals.

Similarly, imaginary (complex) numbers come from closure of root over the negatives.

But how do you get from roots to uncountability?

If I can count the rationals, why can't I count the roots of rationals?

What operation generates an uncountably infinite set?

Joe Austin

[Joseph Austin]

As you probably realize by now, I'm a computer guy, not a mathematician.

You can "define" anything you want, but that doesn't endow it with existence or "meaning".

Or as I would say, decidability.

So you define 

a set containing all infinite lists created from these two characters?

Now tell me this: how can you claim the set contains "all" such lists?

If I propose a list, you can say that that list is "in" the set "by definition".

What argument could I give to show that some list in NOT in the set?

If your hypothesis is not falsifiable, I will claim it is "meaningless",

arguing from information theory which, to oversimplify, says that "meaning" is the answer to a yes/no question.

So suppose that "by definition" is a fair argument.

Let's work out Cantor's argument in detail.

As you propose, our list will start will all zeros and end with all ones:

0000...

...

1111...

Now let's fill in the middle:

0000...

1000...

0100...

1100....

0010...

1010...

1100...

1010...

1110...

...

1111...

Now let's generate Cantor's diagonal D:

It will be, by inverting successive digit positions from each number of the list above:

1111....

All digits of D will be 1, because the ones will be moving right at logarithmic speed while the selected digit is moving right a linear speed.

but, by definition, 1111.... occurs at the end of the  list.

QED

But, you say, Cantor would invert the last digit of the last number.

But Cantor's number will never reach the last digit of any number, much less the last number itself, in order to invert it.

So the "construction" Cantor proposes is not in fact a computable function, so the number that supposedly offers the contradiction does not in fact exist,

but the number that it is "converging" toward does in fact exist in the list "further down", in fact, at the very end.  By Definition.

So I say Cantor's alleged proof is defective.

Joe Austin

[z...@unizor.com]

Closure of rational numbers with roots does not produce all the irrational numbers. Irrational numbers are much more numerable than just roots of rational ones.

One of the closures that produces all real numbers might be all the limits of rational sequences that satisfy certain simple criteria of tending. The reason for not countability should lie in some closure that produces all the real numbers.

Zor Shekhtman
Founder of Unizor Education 
Creative Mind through Art of Mathematics
http://www.unizor.com 

-------- Original Message --------
Subject: Re: [Math Future] a question about infinity / countability
From: Joseph Austin <drtec...@gmail.com>
Date: Wed, July 13, 2016 9:43 am
To: mathf...@googlegroups.com

Michel,

Square root 2 comes from one of the  the inverses of exponentiation,

so some set of irrationals comes from closure of root over the naturals and positive rationals.

Similarly, imaginary (complex) numbers come from closure of root over the negatives.

But how do you get from roots to uncountability?

If I can count the rationals, why can't I count the roots of rationals?

What operation generates an uncountably infinite set?

Joe Austin

On Jul 13, 2016, at 3:45 AM, michel paul <python...@gmail.com> wrote:

On Tue, Jul 12, 2016 at 8:30 AM, Joseph Austin <drtec...@gmail.com> wrote:

​> ​

I'm really more concerned with where the reals come from.

​They came from the Pythagoreans, reluctantly.

They killed the guy who discovered it, but they ultimately could not deny that the diagonal of a square is incommensurable with the side.

​> ​

My gut feel is that "reals" as defined are "overkill"--a hypothesized set beyond what is needed for "real" computations.

​Right, you really don't need the reals for computations. The rationals are good enough.

But when you realize that sqrt(2) is in principle not rational, it makes you curious.​

 

--

​ Michel

​

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

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[michel paul]

On Wed, Jul 13, 2016 at 6:43 AM, Joseph Austin <drtec...@gmail.com> wrote:

​​

 

​> ​

how do you get from roots to uncountability?

The nature of certain roots, like sqrt(2), pointed to the existence of a disturbing type of quantity that no one really wanted. It flew in the face of rationality and commensurability with unity.

You can't pin it down, and that really bothered the Pythagoreans.

It is not an integer, and you can't express it as a ratio of integers, and so ... what's left?

It took a long time for humans to make progress in coming to terms with what they were experiencing here.

The fact of incommensurability had to be accepted and dealt with, just like any obstinate fact in the physical sciences. 

You can't just define these things away, you have to understand them and come to terms with them.

Numbers and patterns have properties that we didn't intentionally give them. We had to discover them, just like we discover the properties of the physical objects in our experience.

The fact of incommensurability catalyzed lots of thought that eventually turned into our present understanding of the reals.

The essential nature of what we call 'irrational' numbers is that they are incommensurable with unity.

People wanted everything to be commensurable with unity, it sounds nice and cozy, but sorry humans, you are not in charge of mathematics. You just think you are. 

 

​> ​

If I can count the rationals, why can't I count the roots of rationals?

​You can count the roots of the rationals.

 

​What you are thinking here is of course true - if I can list the rationals, then right beside them I should be able to list their roots. Yeah, that makes total sense.

But roots are not the whole picture regarding the reals. The nature of the reals as a whole is such that you cannot list them.​

There's more to the reals than just roots.

​> ​

What operation generates an uncountably infinite set?

​​None.

We do not create an uncountably infinite set through any kind of arithmetic operation.

We first discovered the fact of incommensurability. We then had to come to terms with this fact and learn how to think about incommensurable values as co-existing with rational values. We later discovered that the incommensurable values are way more numerous than rational ones.

--

​Michel

[kirby urner]

SYNERGETICS:

OCTAPHASE EVERYDAY MIND

by Kirby Urner

In one segment of his TV show Cosmos, Carl Sagan tells a story about how science became differentiated from the metaphysical hocus pocus of Pythagorean numero-magic. One day, the Pythagoreans, playing with their new Law about the sum of the squares giving the square of the hypotenuse, discovered irrational numbers. The right triangle formed by the diagonal of a square of edge-length one has a "square root of two" length (in Synergetics we say "second root" since squares have been booted from center stage) and this strange length does not commensurate with any ordinary fraction derived from simple multiplication only by division of the original unit control length's segmented "number line." In other words, a thermometer that uses a root-of-2 control length is inherently incommensurable with one using a standard unit divider. No conversion formula using only rational ratios is extant. Then, the story goes, the Pythagoreans retreated into their caves in perplexity and hid their mathematics out of shame. They were ashamed to admit that nature permitted the irrational, the incommensurable, the eternal turmoil that appears to set up camp in the mind when all hope of a crystal clear omni-rational simple number mathematics is vanquished. The evil principle has won. Pythagoreans could not stomach this evil and so retreated into mysticism. Scientists, on the other hand, bravely set forth to explore the empirical domain of butterflies and atoms, star systems and computers. Scientists were not phased by the prospect of eternal turmoil. Life on Earth looked promising and still does to this day. Mystics still wrestling with suppressed experiences of mortification afraid to venture into the light of day grew up to become religious fanatics and hair-shirts, archetypally opposed to scientific reason. Their scholastic phase reached its peak during the Dark Ages when the light of Science was all but extinguished. A good story. Carl Sagan tells it well.

http://www.4dsolutions.net/synergetica/invisible3.html#top

[michel paul]

On Wed, Jul 13, 2016 at 7:40 AM, Joseph Austin <drtec...@gmail.com> wrote:

​> ​

You can "define" anything you want, but that doesn't endow it with existence or "meaning".

​Correct.​

​> ​

So you define 

a set containing all infinite lists created from these two characters?

Now tell me this: how can you claim the set contains "all" such lists?

​By definition.    : )​

​I know that sounds contradictory, but here's how it works - you can 'define' any set you like, and the number of members in your set may be 0.

That's what you were intuitively saying with, "You can 'define' anything you want, but that doesn't endow it with existence".

The definition of a set determines its members, and once defined, that set contains ALL items that meet the definition.

Now if you're not careful with your definition, it might become undecidable in some situations whether or not something counts as a member.​ 

Here's a concrete example I liked to use in class - consider Books In Print.

Books in Print is a book. Is it in print? 

Yes. 

Does it list itself? 

Yes, it does. And it would make sense for it to do so. No problem.

Now imagine another publication called Books That List Themselves. Should it list itself?

Sure, and same thing, no problem.

Now imagine a truly subversive publication called Books That Do Not List Themselves. Should it list itself? It should, but if it does, then it shouldn't.

Now we have a problem. We cannot decide whether or not to include it without creating a paradox, and this is what led to people to add axioms to naive set theory.

But that solution is still a little unsatisfactory for many. People have gotten used to it, but it is still bothersome.

Voevodsky's work in type theory may lead to something much more satisfying, and I'm very hopeful he is successful, because it thinks of mathematics and CS as the same thing.

 

​> ​

If I propose a list, you can say that that list is "in" the set "by definition".

​Correct. As long as it meets the criteria in the definition, it is in the set.​

​> ​

What argument could I give to show that some list in NOT in the set?

​Just point out how it does not meet the criteria for being in the set.

​> ​

So suppose that "by definition" is a fair argument.

Let's work out Cantor's argument in detail.

As you propose, our list will start will all zeros and end with all ones:

0000...

...

1111...

Now let's fill in the middle:

​...​

Now let's generate Cantor's diagonal D:

....

All digits of D will be 1, because the ones will be moving right at logarithmic speed while the selected digit is moving right a linear speed.

but, by definition, 1111.... occurs at the end of the  list.

QED

​Yes, 1111... does occur at the end of your list. The problem is, you never actually got there.    : )

When you say "fill in the middle" ... just try.

I noticed that what you did to create your list was to list the binary numbers with reversed digits, ala Kirby, and yeah, that makes a lot of sense.

Look what happens when you start at the bottom of the list and work upwards:

0011...

1011...

0111...

1111...

​Again, I am doing nothing more than reversing the digits of the binaries, just starting at the 'end' and following the sequence in reverse.

We are creating strings here that will have 1's in them along the diagonal, and these strings are necessarily part of your list.

--

​Michel

[Joseph Austin]

On Jul 12, 2016, at 12:39 PM, kirby urner <kirby...@gmail.com> wrote:

My concern is you'll be branded "obstreperous" by the authorities 
(good SAT word) and your academic prospects may be diminished 
accordingly. 

Kirby,

It's really amazing how retirement can free one from concerns about one's career!

Joe

[Joseph Austin]

On Jul 11, 2016, at 1:33 PM, kirby urner <kirby...@gmail.com> wrote:

Here's the prevailing dogma (bolding added, color background removed):

Set theory fully vindicates the concept of actual infinite, as, through the very simple and intuitive notion of set, it is possible to provide a fully satisfactory theory of infinities of different sizes. After Cantor's creation of the Transfinitum, and his early naive formulation of the notion of 'set' (Menge), the axiomatisation resulting in the theory known as ZFC (due to Zermelo, Fraenkel, Skolem and von Neumann) secured the internal consistency of the early infinitary set-theoretic intuitions and methodologies.

I've been reading so much about ZFC set theory that I decided to look it up.

It seems that the existence of infinity (an infinite set) is an axiom.

So, infinity exists by definition, and once defined, we can study it's properties.

So suppose we omit that axiom. What do we lose?

For example, do we lose Pi?  Do we lose circles? Do we lose the known physical universe?

Joe Austin

[michel paul]

Awhile ago I posted this: https://www.quantamagazine.org/20160524-mathematicians-bridge-finite-infinite-divide/

It seems appropriate for a revisit.

- Michel

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[kirby urner]

On Wed, Jul 13, 2016 at 12:53 PM, Joseph Austin <drtec...@gmail.com> wrote:

I've been reading so much about ZFC set theory that I decided to look it up.

It seems that the existence of infinity (an infinite set) is an axiom.

So, infinity exists by definition, and once defined, we can study it's properties.

So suppose we omit that axiom. What do we lose?

For example, do we lose Pi?  Do we lose circles? Do we lose the known physical universe?

Joe Austin

When I opened for Mandelbrot one time, small morning session at the local college, I talked about four different types of number sequence:

Convergent

Divergent

Oscillating

Chaotic

The rational numbers give us repeating sequences of digits, or just one digit, so are periodic, but we also say convergent as we're looking at rational numbers as points on a number line (I have no problem with that).

Let's say our primary goal is to find deterministic algorithms that are aperiodic in output, i.e. chaotic.  I sometimes say "phi is the phirst phractal" (a misspelling).  Phi is as easy to understand as Pi.  Sure, it's not transcendental, but so what, it's chaotic.

I think it's somewhat miraculous that the Pi generator I cited earlier (first brought to my attention by Michel Paul), just a few lines of Python, spits out the same chaotic sequence of digits as one of Ramanujan's.  

Actually the one of Ramanujan's I'm thinking about gives 1/pi, so we need to flip the final result, then compare.

https://github.com/4dsolutions/Python5/blob/master/Pi%20Day%20Fun.ipynb

Lets think of a mathematics that begins with algorithms as the primitives, little machines that put out sequences.  Deterministic machines always put out the same sequence.  But if it's chaotic, then we can't say what the next digits will be unless we actually compute them.  

We can't just say 33333.... and mean "threes forever".  Or point to some pattern like 31313131... and so on.

314159... (the digits of pi) is a published sequence we can check against.  We know there are many simple machines that create these same digits.

On the other hand, if I hand you a book of 10,000 digits then dot dot dot, you might not know if this is "purely random" or "machine driven" or what.

So we get into information theory and the difference between signal and noise, entropy and so on.  There's an observer, not just an observed.

In this math, we might say the chaotic sequence we label Pi "has applications in geometry" but we're not primarily interested in circles so much as finite state machines, especially the ones that never repeat themselves.

Given our finite state machines simply publish digits, with no need for a decimal, we might not really care if these sequences (published in books) are really "numbers" or not.  

We might have algorithms that publish Chinese characters instead.  There's no "number line" in this picture.

No one consults a book of random numbers and takes the whole book to be just one big number.  But we could look at it that way.  

Books giving Pi to a million digits are about what we call "single numbers" but "sequences of digits" is just as precise.

They're sequences of symbols, and we can count them (1,2,3,4....) like the digits of pi.

We could go with a slightly different taxonomy:

Repeating

Terminating

Chaotic 

The primitives are rule-based games that we "operate" or "run" to get our sequences. 

Perhaps the Hindu-Arabic numbers, present in the early stages of this branch of math, are long gone in a 100 years.  

The algorithms are implemented in bytecode (hex), but the source code is not in any language we'd recognize.

Kirby

[kirby urner]

I picked up on this part especially:

And yet, Yokoyama and Patey’s proof shows that mathematicians are free to use this infinite apparatus to prove statements in finitistic mathematics — including the rules of numbers and arithmetic, which arguably underlie all the math that is required in science — without fear that the resulting theorems rest upon the logically shaky notion of infinity. 

I like they admit it's an argument, not a "settled" issue, as to whether what science requires are precisely these rules and not other ones.  

What does it means to "follow the rules" especially when so few are prepared to say what they are.

Few scientists have training in logic, in any formal sense.  

But then as they say "ignorance of the law is no excuse".

Kirby

[michel paul]

Another observation I find interesting -

Going back to our construct of '0000...' on top and '1111...' on bottom and 'filling in the middle':

000000...

100000...

010000...

110000...

001000...

........

110111...

001111...

101111...

011111...

111111...

The pattern from top down and the pattern from bottom up will never meet. They cannot share a common element.

The top patterns will ALWAYS contain an infinite string of 0s, and the bottom patterns will ALWAYS contain an infinite string of 1s.

In each pattern there will be some initial 'noise', an initial period where we have a mixture of 1s and 0s, but after awhile it's just an infinite string of one or the other.

These two patterns will never meet.

On Wed, Jul 13, 2016 at 12:58 PM, michel paul <python...@gmail.com> wrote:

[Andrius Kulikauskas]

Thank you, Michel.

One way to read the "Ramsey's theorem for pairs" is that it is about
simplexes, that is, an n-dimensional tetrahedron, where n is unbounded.

A simplex can be thought of as a set of vertices for which every pair of
vertices is connected by an edge.

Suppose that you have a Simplex for which the number of vertices is
unbounded. And suppose that you color every edge either red or blue.
Then "Ramsey's theorem for pairs" says that there must be within the
Simplex an infinite subsimplex that is all red or all blue.

You can also "color" each edge as "exists" or "does not exist". In other
words, you can break up the edges of the Simplex into two subsets,
Exists and NotExists. Then by "Ramsey's theorem for pairs", either
Exists or NotExists contains an infinite simplex.

I suppose the argument is that you work on building two simplexes, one
for Exists and another for NonExists, and you should succeed with one or
the other. if you start with a vertex, then you can be sure that you
can find infinitely many edges to other vertices OR you can find
infinitely many lack-of-edges to vertices. So you have a first vertex.
You have to keep linking it up with vertices of the same type. So at
this point the question is, what is the "rule" that you can leverage.

I just wanted to add that perspective.

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

2016.07.13 22:58, michel paul rašė:
> Awhile ago I posted this:
> https://www.quantamagazine.org/20160524-mathematicians-bridge-finite-infinite-divide/
>
> It seems appropriate for a revisit.
>
> - Michel
>
> On Wed, Jul 13, 2016 at 12:53 PM, Joseph Austin <drtec...@gmail.com
> <mailto:drtec...@gmail.com>> wrote:
>
>
>> On Jul 11, 2016, at 1:33 PM, kirby urner <kirby...@gmail.com

>> <mailto:kirby...@gmail.com>> wrote:
>>
>> Here's the prevailing dogma (bolding added, color background
>> removed):
>>

>> Set theory*fully vindicates*the concept of*actual*infinite,
>> as, through the*very simple and intuitive*notion of set, it
>> is possible to provide a f*ully satisfactory*theory of

>> infinities of different sizes. After Cantor's creation of the
>> Transfinitum, and his early naive formulation of the notion
>> of 'set' (Menge), the axiomatisation resulting in the theory
>> known as ZFC (due to Zermelo, Fraenkel, Skolem and von

>> Neumann)*secured the internal consistency*of the early
>> infinitary set-theoretic*intuitions*and methodologies.

>>
>>
> I've been reading so much about ZFC set theory that I decided to
> look it up.
> It seems that the existence of infinity (an infinite set) is an axiom.
> So, infinity exists by definition, and once defined, we can study
> it's properties.
> So suppose we omit that axiom. What do we lose?
> For example, do we lose Pi? Do we lose circles? Do we lose the
> known physical universe?
>
> Joe Austin
>
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>
>
>
>
> --
> ===================================
> "What I cannot create, I do not understand."
>
> - Richard Feynman
> ===================================
> "Computer science is the new mathematics."
>
> - Dr. Christos Papadimitriou
> ===================================

> --
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[Bradford Hansen-Smith]

Kirby, Andrius, and others, the number sequence seems relevant to geometric function I observe in folding circles.

Convergent

Divergent

Oscillating

Chaotic

Draw the circle, cut it out from the paper. There are now two circles, each a dual of itself. One we call whole the other call it hole. The whole when folded converges infinitely into itself by nature of concentric high frequency density that opens to what I imagine are polytopes of unseen and unknown configurations.

On the other hand the hole circle can be folded the same way and it diverges from the hole to the perimeter of the paper. Divergent movement appears fragmenting with nothing at the center. What interest me is that the folded grids are the same 3-6,4-8, 5-10 symmetry going out as they are going in, the same patterned order of organization. This suggest oscillation is inherent in the static image where we talk about inside and outside and the line of separation; individually they are confining as concepts. The circumference is only that momentary equilibrium in passing between convergent towards alignment and divergent towards potential. The state of oscillation is one allowing the other. Similar to a spiral that goes both in towards and out from at the same time.  Maybe this momentary stasis of the circle is why pie is problematic in math, by creating a chaotic state of trying to resolve by some fixed law a dynamic principle of movement.

http://wholemovement.com/blog/item/735-the-other-circle

Brad

Bradford Hansen-Smith
www.wholemovement.com

[Joseph Austin]

On Jul 13, 2016, at 12:23 PM, michel paul <python...@gmail.com> wrote:

We do not create an uncountably infinite set through any kind of arithmetic operation.

We first discovered the fact of incommensurability. We then had to come to terms with this fact and learn how to think about incommensurable values as co-existing with rational values. We later discovered that the incommensurable values are way more numerous than rational ones.

Yes.

But we dealt with inverse of addition giving negatives,

and inverse of multiplication giving rationals,

and inverse of power giving irrationals and imaginaries,

and none of those crossed the line into "uncountable"?

So where did we cross the line?

Joe

[michel paul]

On Thu, Jul 14, 2016 at 7:18 AM, Joseph Austin <drtec...@gmail.com> wrote:

​

we dealt with inverse of addition giving negatives, 

and inverse of multiplication giving rationals,

and inverse of power giving irrationals and imaginaries,

and none of those crossed the line into "uncountable"?

So where did we cross the line?

​I would say it was in providing an analytical foundation for Calculus.​

​

​ 

Calc used the real numbers without having defined them, so in a way Calculus preceded Analysis even though we teach it in the opposite order in high school.​

​In terms of 'operations', finding roots prompted the discovery of incommensurability. ​Interesting, it turns out that the concept of irrationality was known about and accepted by Indian mathematicians prior to Pythagoras. 

And then in the 1800s there was work on the idea of transcendental numbers.

But it turns out to be Cantor who gave the first rigorous definition of the reals.

So I guess the best answer to your question is the late 1800s.

--

​Michel

[kirby urner]

On Thu, Jul 14, 2016 at 8:25 AM, michel paul <python...@gmail.com> wrote:

​ In terms of 'operations', finding roots prompted the discovery of incommensurability. ​Interesting, it turns out that the concept of irrationality was known about and accepted by Indian mathematicians prior to Pythagoras. 

And then in the 1800s there was work on the idea of transcendental numbers.

But it turns out to be Cantor who gave the first rigorous definition of the reals.

So I guess the best answer to your question is the late 1800s.

--

​Michel

​

I would agree, the "real numbers" as such started to become meaningful for the first time around the 1800s.

Descartes never knew about them.  He made do with what was at hand.

I'm glad we're bringing a timeline back into the picture. 

Minus intellectual history, one may develop a view that we have this Foundation of Axioms, known since ancient times, and all we've been doing ever since "the Greeks" is building on that Foundation.

On the contrary, the Foundations are still being worked on, right up to this moment.  So it's not like some ordinary building, where you need the Foundations first!

The whole Edifice-Foundation masonry-based extended metaphor has really baked itself into the fabric of what's presented as mathematics.  We're supposed to think of it as some Building, a Gothic Cathedral perhaps.

I'd like to recast a branch to where we talk about a Tensegrity or something (ala K. Snelson), with compression rods (known truths, proved stuff) all floating in some tensive field, without touching one another. 

Why not?  Another architectural / sculptural metaphor in place of the Ediface-Foundation one might be a breath of fresh air (I'd certainly breathe easier).

Actually, I'm hardly one to criticize "Ediface-Foundation talk" given I'm out there as @4DsolutionsPDX on Twitter, drumming up interest in #CodeCastle and pointing back over my shoulder at this gigantic castle-like church the Methodists in my neighborhood can't figure out what to do with.  No parking lot.  Americans tend to turn into piles of helpless jello when you take away their parking.

That's another one of my goals: help foster literacy, reading and writing capability.  #CodeCastle is supposed to do that. It's right across from a middle school, full of hopeful kids dreaming of a better future.

In my State of Oregon, the left-behinds are deprived of being able to read and write using contemporary tools (i.e. SQL, HTML5 / CSS3, and JavaScript). 

Without these basics, one grows up illiterate in this Portlandia culture (which is why so many retire early, like at age 18 -- the schools have largely given up teaching reading and writing beyond an elementary school level).

Kirby

[Joseph Austin]

A company I worked for once hired Carl Sagan to help us pitch a super-computer. This was in the 70's.

At that time, he was predicting an imminent ice age!  So much for global warming.

But on to the irrational.

So one cannot diagonalize a square! 

If that is true, to "square" a building, such as a back-yard deck, or even the pyramids,

one cannot use a "square" but must use a rectangle, such as a 3x4x5 triangle.

(Which reminds me, my hometown's most famous modern structures, RBF's Climatron and Eero Saarinen's arch, are both based on triangles, not squares.)

But of course it is true. Lay out a "square" of pebbles (preferably square ones, say Scrabble tiles)--that is, the "square" of a number. 

Now divide it in half along the diagonal, without subdividing individual pebbles.

Of course, if you can even get equal parts, you can't get a straight line. 

And even equal parts won't make two squares.

That's essentially the same dilemma I've been wrestling with in saying one can't create a sphere, 

that is, a 3D closed symmetrical object of uniform curvature and density.  

I've also claimed that the Babylonians were wrong: one cannot create a 360º circle 

(120º is ok, but you can't divide 3º into thirds, at least not with Euclid, and I doubt the Babylonians had a more sophisticated method).

I'm now suspecting the root of the entire irrational (and hence uncountability) dilemma is Euclid's 3rd Postulate.  

But what is geometry without circles?

Can we really have both atomism and the continuum, both numbers and circles, in the same consistent mathematics?

Or put another way, can we really prove theorems about the continuum and "infinities" by considering strings of discrete symbols?

And is there perhaps a middle way--what I intuitively call a "uniform density" surface in which all "points" are "regularly" spaced?

In music, we encounter a similar challenge in "encircling" the sequence of "fifths" or 3:2 frequency ratios.

it is not possible to construct a "circle" which is simultaneously 'harmonious" (small integral ratios) and "closed",

so we must pick one or the other: equal-temperament (but not pure ratios) or "just" harmony (but not free modulation between keys.)

Joe

[michel paul]

Here's another observation -

Though the two patterns of sequences will never meet, they must both be part of the same list, and one might be tempted to think that the two patterns together might account for all possible infinite binary sequences. 

It is tempting to assume that you could, in principle, continue these patterns in both directions to eventually list all possible sequences.  

However, no, that won't happen. Note that these two sequences:

010101...

101010...

will never appear in our list.

[kirby urner]

Isn't the simplest algorithm to do all permutations of 1, 0 in one slot, two slots, three slots....

You'll not miss any possible permutations this way, so if a number has a binary representation at all, this is a countable way of getting to it.

But than it looks like we're back to square one of just counting the positive N, so there's no gain in aleph number.

Seems one needs to have that dot do show where we're adding tinier quantities.

So do a table then, with the above going down the side and across the top, the placement of the dot...

            Dot moving left... (as far as it can)  
...  

00001  00001. 0000.1 000.01 00.001 0.0001 .00001

00010  00010. 0001.0 000.10 00.010 0.0010 .00010

00011  ...
00100

... all possible binary strings (keep added slots)

Zig-zag through that one, Cantor-style (even though it stair steps out, going to the right).

The objection "you'll never reach infinite slots" never worried people counting Q the same way, you just need a way to not leave any holes.  Cover all the bases, so to speak.

Again though, if R has members with infinite digit representations to the right of the decimal, then it seems Q is going to need (p, q) where p, q are in N, that have infinite digits too, e.g. to keep up with Pi to as many digits as we go (similar to counting Pi's digits).

Back to needing N numbers with infinite digits, but those weren't going to be allowed as I recall.  So many paradoxes.  Infinity is just a lot of headaches.  Since a lot of physicists assume a Big Bang a finite number of years ago, why posit infinity anyway?  Clearly the universe is countable given conservation laws, or don't we believe in those any more?

Kirby

[Joseph Austin]

On Jul 13, 2016, at 2:43 PM, michel paul <python...@gmail.com> wrote:

On Wed, Jul 13, 2016 at 7:40 AM, Joseph Austin <drtec...@gmail.com> wrote:

​> ​

You can "define" anything you want, but that doesn't endow it with existence or "meaning".

​Correct.​

I think what I'm trying to say is the Cantor argument "fails" because the list "D" cannot actually be constructed.

Take your catalogs the don't list themselves.

A similar argument is "this statement is false".

Now is that statement true or false?

If it's false, it's true because it says is ti false, but if it's true, then is false because it states that it isn't true. but ....

So I tried "computing" it.

It goes something like this:

Replace every occurrence of "this statement" with "this statement is false"

So:

"this statement is false" =

"this statement is false" is false =

" "this statement is false" is false " is false =

" " "this statement is false" is false " is false " is false

etc an inf.

So you see, "this statement is false" isn't actually a well-formed statement, so there is no way of assigning it a true-false value.

Since the Cantor diagonal D cannot be computed from the "algorithm" specified, we cannot say whether it occurs in the list,

But I can construct a proof that shows a number containing all digits that it does compute up to any given point IS in the list "further down".

To prove by contradiction, you must actually produce the contradictory case,

not just define it!

joe

[michel paul]

On Sun, Jul 17, 2016 at 7:14 AM, kirby urner <kirby...@gmail.com> wrote:

​> ​

Isn't the simplest algorithm to do all permutations of 1, 0 in one slot, two slots, three slots....

​Yes, that is simple, and it is tempting, but note that you're only creating finite decimal strings that way.

You'll never sequentially reach the string we use to represent the binary value for 1/3 --->  0.010101... .

Every sequence in your list necessarily contains a leftmost '1'.

You will never arrive at a sequence which does not contain a leftmost '1', but you will need such sequences to represent certain values like 1/3.

This has nothing to do with being 'allowed' or not. It is simply how the pattern unfolds, despite our preferences.

​> ​

You'll not miss any possible permutations this way,

You'll nail everything that has a leftmost '1', but you'll miss everything else.

 

​> ​

if a number has a binary representation at all, this is a countable way of getting to it.

No, it is not. Again, you will miss things, like the sequence we use to represent the value of 1/3.

​> ​

But than it looks like we're back to square one of just counting the positive N

​ ... 

.

​Yeah ... looks like.

For a moment.

But then it turns out that no, this interesting principle shows up again.

The fact that this is how the pattern works is just how it works, regardless of any foundational assumptions.

If you cast that countable binary pattern as a net in the hope of catching all infinite binary strings, sorry, some will always get away.​ In fact, a whole bunch will.

--

​Michel

[michel paul]

On Sun, Jul 17, 2016 at 11:43 AM, Joseph Austin <drtec...@gmail.com> wrote:

​> ​

I think what I'm trying to say is the Cantor argument "fails" because the list "D" cannot actually be constructed.

​Correct, you can't actually construct the diagonal.

However, the reason you can't actually construct the diagonal is because you cannot actually construct the list​ in the first place, and that is Cantor's point.

​If you think you've created the list, even algorithmically, sorry, you've left something out.​

​> ​

I can construct a proof that shows a number containing all digits that it does compute up to any given point IS in the list "further down".

​You can?

--

[kirby urner]

On Jul 17, 2016 13:44, "michel paul" <python...@gmail.com> wrote:
>
> On Sun, Jul 17, 2016 at 7:14 AM, kirby urner <kirby...@gmail.com> wrote:
>>
>>
>> ​> ​
>> Isn't the simplest algorithm to do all permutations of 1, 0 in one slot, two slots, three slots....
>
>
> ​Yes, that is simple, and it is tempting, but note that you're only creating finite decimal strings that way.
>

I don't see that. I'm covering all possible binary representations of whatever base.

> You'll never sequentially reach the string we use to represent the binary value for 1/3 --->  0.010101... .
>

I don't see that either. Every time my play head sweeps around it adds the appropriate digit to the radius you're watching. Be patient. We're on our way.

> Every sequence in your list necessarily contains a leftmost '1'.
>

No. All permutations for N slots. My cross section was a little ways into it, to make the table more interesting.

> You will never arrive at a sequence which does not contain a leftmost '1', but you will need such sequences to represent certain values like 1/3.
>

0
1
00
01
10
11
001
010
011
...

> This has nothing to do with being 'allowed' or not. It is simply how the pattern unfolds, despite our preferences.
>

I'm not sure I got it across.

>> ​> ​
>> You'll not miss any possible permutations this way,
>
>
> You'll nail everything that has a leftmost '1', but you'll miss everything else.
>  

I get everything. In due time.

>>
>> ​> ​
>> if a number has a binary representation at all, this is a countable way of getting to it.
>
>
> No, it is not. Again, you will miss things, like the sequence we use to represent the value of 1/3.
>
>> ​> ​
>> But than it looks like we're back to square one of just counting the positive N
>> ​ ... 
>> .
>
>
> ​Yeah ... looks like.
>
> For a moment.
>
> But then it turns out that no, this interesting principle shows up again.
>
> The fact that this is how the pattern works is just how it works, regardless of any foundational assumptions.
>
> If you cast that countable binary pattern as a net in the hope of catching all infinite binary strings, sorry, some will always get away.​ In fact, a whole bunch will.
>
> --
> ​Michel
> ​

We're ships in the night.

I hate spellchecker.

Kirby

>
> ===================================
> "What I cannot create, I do not understand."
>
> - Richard Feynman
> ===================================
> "Computer science is the new mathematics."
>
> - Dr. Christos Papadimitriou
> ===================================
>

[kirby urner]

On Sun, Jul 17, 2016 at 2:09 PM, kirby urner <kirby...@gmail.com> wrote:

> Every sequence in your list necessarily contains a leftmost '1'.
>

No. All permutations for N slots. My cross section was a little ways into it, to make the table more interesting.

> You will never arrive at a sequence which does not contain a leftmost '1', but you will need such sequences to represent certain values like 1/3.
>

 

0
1
00
01
10
11

000 <-- all zeros we can skip.

001 
010
011

100
111  
0000

0001  --> ( full row: .0001 -> 0.001 -> 00.01 -> 000.1 -> 0001. )

0010  --> ( not all will be "new numbers")

....

We're spiraling outward with lots of repetition for good measure. 

Keeping the leading zeros is part of the algorithm, even if we cross out the dupes.

That's how we keep moving to the right of the dot in reach row, with 0s adding to significant figures.

Going left we could say there's always a leading 1, but that's saying there's always some power of 2 of significance.

If not, the number is 0, and we've already covered it.

A first 1 always has to occur somewhere in some slot, even if only after the dot, or its 0.

Any repeating pattern of 1s and 0s will be added to as slots are added.

Just keep it in base 2 for simplicity.  Skipping dupes is OK.

As the dot moves left to right across the top, we see how the dot marks where 2**n becomes 2**(-n) with n >= 0.

Yes, I've skipped the negative numbers. 

So mirror the whole table to the left for those, or interleave (shuffle together).

+0001  --> ( full row:  +.0001 -> +0.001 -> +00.01 -> +000.1 -> +0001. )

-0001  --> ( full row:  -.0001 -> -0.001 -> -00.01 -> -000.1 -> -0001. )

Kirby

[Joseph Austin]

On Jul 13, 2016, at 3:58 PM, michel paul <python...@gmail.com> wrote:

Awhile ago I posted this: https://www.quantamagazine.org/20160524-mathematicians-bridge-finite-infinite-divide/

It seems appropriate for a revisit.

- Michel

When looking up your reference, I also discovered this one, on "univalent" theory

( of which I currently know nothing, but it sounds like I should.)

https://www.quantamagazine.org/20150519-will-computers-redefine-the-roots-of-math/

As an almost-physicist, I'm accustomed to encountering "infinity" more as the infinitesimal,

in the form of differential equations.

But we know of nothing in the "actual" universe that is "really" infinite or infinitesimal.

And indeed, differential equations break down at the "atomic" level, degenerating into eigenstates and groups.

The Michelsom-Morely experiment to discover the "ether" in which the electromagnet waves "waved" failed,

and to my knowledge no other "continuum" has ever been experimentally verified,

nor has it been established that the universe is not bounded.

And the success of "digital" computers depends on the notion that computation is ultimately discrete.

So I'm thinking "infinity" is like "probability", a convenient abstract  approximation for dealing with large numbers of individual events.

It may well be "logically true" that the real numbers can't be counted,

and that there are as many primes as rationals,

but I'm still uncomfortable with those "proofs".

I suppose it all comes down to the definition of "equal" when it comes to "infinite".

I get the gut sense that it's similar to Anselm's proof for the existence of God,

or the Animal Farm dictum that all men are equal, but some are more equal than others.

It's true by virtue of having defined the terms so as to make it true,

but then we are left with matching the terms so defined to realities,

and in that context they don't make sense.

It seems to me you can't have it both ways.

On the one hand, they say the reals are "more" than the naturals because they claim one can find one number in the "reals" that is not in the naturals.

On the other hand, they say the evens are "equal" to the naturals even though one can show an infinite number of naturals NOT in the "evens".

But perhaps it's simply confusion between the everyday meaning of terms and the mathematical definitions of them.

But what I think I'm trying to say is, you can't construct a non-countable set from the mathematical operations taught in "mathematical methods of physics"--from arithmetic through calculus.

As "proof", let the symbolic representation of the mathematical formula for each such number be it's "index" in the counting set.

If the "number" can be expressed as a computable formula, it can be "counted".

This would include all the irrational numbers represented by "infinite" series expansions or "limits" in calculus.

And if it cannot be represented in symbols, how can we talk about it as a distinct, existing entity?

And I would go so far as to claim this result could be extended to all the "numbers" (points, lines, circles, angles, etc.) constructible in geometry;

if I can describe their construction sufficient to discuss them in a theorem, those construction descriptions constitute an "index".

Joe Austin

[kirby urner]

On Sun, Jul 17, 2016 at 2:09 PM, kirby urner <kirby...@gmail.com> wrote:

We're ships in the night.

I hate spellchecker.

Oops, sorry Maria, that slipped out.  I was thumbing on my phone again and the spellchecker on my phone fights me a lot more than the one on my laptop.

I actually appreciate spellchecking software as I really don't like typos (especially when I make them).  Spellcheckers are not a panacea for typos, as we all know, but they help a lot.

Speaking of which, that's something we get in software that mathematics notations were / are less helpful about.  We have program editors that are pretty good at spotting errors in code, if a certain kind, such as defining a term but never using it later, or using a term never earlier defined.

We also have "linters" and "formatters", other ways of embedding error checking in the code.

The Andrew Hacker book (The Math Myth) makes coding sound almost inhumanly meticulous, but we get a lot of help from one another, in terms of how we've worked on tools to catch a lot of bugs automatically.

I think as we shift more teachers from 1900s math teaching to a more code-friendly style, they'll appreciate how much easier it can be to go this route, and how much further one might go.

Kirby

[kirby urner]

On Thu, Jul 14, 2016 at 5:09 AM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

Kirby, Andrius, and others, the number sequence seems relevant to geometric function I observe in folding circles.

Convergent

Divergent

Oscillating

Chaotic

While still on the topic of infinity...

https://youtu.be/-EtHF5ND3_s

No mention of chaotic sequences per se, but with respect to partial sums, they'd be divergent.

Kirby
 

[kirby urner]

A flavor of infinity I enjoy:

https://youtu.be/VGpnuTJhv1U

The Mandelbulb genre is awesome!  3D fractals, one could call them.

Kirby

[Joseph Austin]

On Jul 14, 2016, at 1:24 AM, michel paul <python...@gmail.com> wrote:

Another observation I find interesting -

Going back to our construct of '0000...' on top and '1111...' on bottom and 'filling in the middle':

000000...

100000...

010000...

110000...

001000...

........

110111...

001111...

101111...

011111...

111111...

The pattern from top down and the pattern from bottom up will never meet. They cannot share a common element.

The top patterns will ALWAYS contain an infinite string of 0s, and the bottom patterns will ALWAYS contain an infinite string of 1s.

In each pattern there will be some initial 'noise', an initial period where we have a mixture of 1s and 0s, but after awhile it's just an infinite string of one or the other.

These two patterns will never meet.

Agreed. 

But that in itself doesn't mean we can't count them.  We can count the integers, even though they run off to infinity in two directions that never re-meet.

Your bottom patten in nothing more than the reverse digits of the negative integers in two's complement notation,

starting at the bottom with minus one and moving upward to minus two, minus three, etc.

e.g.

...

...000100 four

...000011 three

...000010 two

...000001 one

...000000 zero

...111111 neg one

...111110 neg two

...111101 neg three

...111100 neg four

...111011 neg five

...

BTW, Kirby, if you are following, this is an example of infinite digits on the left!

(Furthermore, this illustrates the infinite analog of finite-digits wraparound: in finite 2-s complement, the most-positive number wraps to the most-negative number,  and this behavior sustains as the number of digits approaches infinity, so it "makes sense" that positive infinity wraps around to negative infinity!  This is similar cyclic behavior under addition as we see in multiplication on the Riemann sphere, or we see in polar complex numbers as the "sign" cycles.

What happens to our notion of "infinity" if it is replaced by a notion of "arbitrarily large cycle"?  What if we can always "reach east by sailing west"?)

On the other hand, we might "fill in the middle" with a pattern ending in infinite repetitions of 01:  010101... that will meet neither top nor bottom.

Or take any repeating pattern of ones and zeros you may choose as the ending--or we might as well enumerate all of them.

Then we have a 2D infinite tableau of unique starting patterns and repeated ending patterns, somewhat like the rationals, which we can count.

Joe

[michel paul]

On Sun, Jul 17, 2016 at 2:09 PM, kirby urner <kirby...@gmail.com> wrote:

​> 

I'm covering all possible binary representations of whatever base.

​No, not all of them. 

Only the finite ones.

​>​

You'll never sequentially reach the string we use to represent the binary value for 1/3 --->  0.010101... .

​> ​

I don't see that either. Every time my play head sweeps around it adds the appropriate digit

​...

Since you're adding only one digit at a time, you will get to these strings:

0.01 ---> 1/4

0.0101 ---> 5/16

0.010101 ---> 21/64

etc.

​You will never get to this string:

0.010101 ... ---> 1/3

You will keep getting closer and closer to that value, but you never actually get there. For practical purposes, sure, you'll be 'close enough'.

>

0
1
00
01
10
11
001
010
011
...

​> ​

I get everything. In due time.

​No​

, sorry, you don't.

​ 

T

his​

 process of sequentially listing finite binary strings

​ ​

​

will only be able to represent those rational numbers that have a denominator that is a power of 2.​

In order for your expressions to exactly represent fractions with denominators like 3, 5, 7, etc. you need infinite binary strings, and you're never going to get there.​

​> ​

We're ships in the night.

​No, not really.​

 

--

[michel paul]

On Sun, Jul 17, 2016 at 3:34 PM, kirby urner <kirby...@gmail.com> wrote:

​

>

​

>

​

Every sequence in your list necessarily contains a leftmost '1'.

​> ​

Going left we could say there's always a leading 1, but that's saying there's always some power of 2 of significance.

​Yes, exactly. 

My point in saying that there will always be a leading 1 is that there will always be a finite number of 1s.

This leading 1 will be preceded by an infinite number of 0s.​

 

​You will never produce a string with an infinite number of 1s.

However, you need them in order to represent values like 1/3.

You need strings with an infinite number of 1s in order to represent rationals with denominators that are not powers of 2.​

--

[michel paul]

On Mon, Jul 18, 2016 at 6:42 AM, Joseph Austin <drtec...@gmail.com> wrote:

​>> ​

In each pattern there will be some initial 'noise', an initial period where we have a mixture of 1s and 0s, but after awhile it's just an infinite string of one or the other.

​>> ​

These two patterns will never meet.

​> ​

Agreed.

​​

But that in itself doesn't mean we can't count them.  

​Correct. We can combine these two patterns, in fact they already must be combined, and put that combination into a 1 to 1 correspondence with N. Yeah, that's not a problem.

But it also turns out that the combination of these two lists does not include all of the possible strings.

It doesn't include '0.010101...', the binary representation for 1/3.

That's the fascinating point that keeps returning - you can't create a complete and countable list of these things, not even in principle by some algorithm.​

[kirby urner]

On Mon, Jul 18, 2016 at 8:31 AM, michel paul <python...@gmail.com> wrote:

On Sun, Jul 17, 2016 at 3:34 PM, kirby urner <kirby...@gmail.com> wrote:

​

>

​

>

​

Every sequence in your list necessarily contains a leftmost '1'.

​> ​

Going left we could say there's always a leading 1, but that's saying there's always some power of 2 of significance.

​Yes, exactly. 

My point in saying that there will always be a leading 1 is that there will always be a finite number of 1s.

I thought the challenge was to come up with an algorithm that wouldn't have any gaps if we kept at it.

0.0101010101010101010....

will always have finite 1s in the sense that one cannot actually count to infinity. But that's what infinity means.

But in terms of aleph number, I'm not leaving any holes. 

The playhead spirals around and adds the next digit, and the next...

To say the digits of Pi are countable as in enumerate(pi) where pi is a pi generator does not imply this is a realistic program.

Are the decimal digits of pi countable (listable) or not in your view?

Seems I can draw a zig-zag line through my table like Cantor does when counting the rationals.

We can always say "but Cantor, you'll never get to 1/10101010101.... where that's a rational number with infinite digits in the denominator."

Oh, I forgot, infinite digits in the denominator or numerator are not allowed. 

Why?  Because that'd break the aleph stuff.  QED.

If members of N were allowed infinite digit representations, the way members of R are allowed, then N couldn't count itself.

Some people find all this "beautiful".  That's why I talk in terms of flavors of math. 

I'm glad I don't have to use real numbers for real.

Kirby

[kirby urner]

BTW, Kirby, if you are following, this is an example of infinite digits on the left!

Not allowed.  

Members of N must have only finite digit representations according to these authorities:

http://math.stackexchange.com/questions/58085/a-number-with-an-infinite-number-of-digits-is-a-natural-number

 

(Furthermore, this illustrates the infinite analog of finite-digits wraparound: in finite 2-s complement, the most-positive number wraps to the most-negative number,  and this behavior sustains as the number of digits approaches infinity, so it "makes sense" that positive infinity wraps around to negative infinity!  This is similar cyclic behavior under addition as we see in multiplication on the Riemann sphere, or we see in polar complex numbers as the "sign" cycles.

Every system "leaks sense" in my view. 

I picture little jets of steam escaping from some pressure cooker.

Every system has a half-life at best (others just go "pop!" in the night -- you wake up and they're gone).

Some logicians have this sense "until I plug all the holes, the entropy will increase". 

They have a finger-in-the-dike psychology.  They apply patches.

I'm not one of those who believe mathematics is an "edifice" and that when Descartes and Fermat formalized the XYZ apparatus, they were building something "shaky" that would "stabilize" once the "real numbers" were finally put on a logical footing by Cantor or whomever.

Mathematics arises in everyday chaotic experience and those branches that gain traction may attract the attention of formalizers who want to provide "logical foundations" well after the fact. 

But in what sense are these foundations actually "load bearing"?  Good question.

Newtonian calculus chugged along for a century before French formalizers felt we had a "logical basis" for it. 

There's this process of "finding anomalies" and "plugging holes" that goes on as people "psych themselves into a mindset".

"Logical foundations" help with the process of auto-brainwashing plus add a security layer. 

You get some intimidating fortifications that will keep others from questioning one's authority too easily.

Logical foundations arise from a need for "thicker skin" i.e. "shielding".

Some storytellers may resist portraying maths as evolving through debate, argument, altercation, as in their minds math is supposed to be that quiet cloistered sanctuary where we all partake of that "universal language".

People often escape into mathematics because they're frankly tired of noisy debate.  "Why can't we all just agree on something!"

But that's not how the history looks when we dig into it.  

Cantor kept ending up in the mental hospital (literally) complaining that Kronecker was trying to take him out.  Bishop Berkeley attacked Newton's calculus as "of the devil".

http://uh.edu/engines/epi2859.htm

https://thonyc.wordpress.com/2012/01/07/infinity-did-not-drive-cantor-crazy/

Such is mathematics in the rear view mirror, once we decide to stop smoothing over real differences.
 

What happens to our notion of "infinity" if it is replaced by a notion of "arbitrarily large cycle"?  What if we can always "reach east by sailing west"?)

I am not especially drawn to branches of logic that invest a lot in "infinity" myself.  The topic smacks of Medieval Scholaticism.

I'm happy to see infinity fly by now and then, as a direction or singularity. 

I love zooming in / out in the space of Mandelbulbs.  I must have spent an hour doing that last night, plus I've shared those videos in my Saturday Academy classes.

All the mathematics is done with finite no-infinity-required stateful computing equipment, a proxy for Universe itself.

 

On the other hand, we might "fill in the middle" with a pattern ending in infinite repetitions of 01:  010101... that will meet neither top nor bottom.

I'm fine with incommensurability by the way.  Algorithms with no stopping point:  not a problem.  Chaos is cool.
 

Or take any repeating pattern of ones and zeros you may choose as the ending--or we might as well enumerate all of them.

Then we have a 2D infinite tableau of unique starting patterns and repeated ending patterns, somewhat like the rationals, which we can count.

Joe

I'm reluctant to spend any more time on the aleph stuff myself.  I don't see much load-bearing happening.  

I don't think the XYZ coordinate system of Descartes and Fermat "depends" on any aleph stuff for its utility.

This metaphor of "logical foundations" is too gravitational for my taste.  I want tensegrity structures in zero gravity, floating far from any planets, to inform my mathematical metaphors.  

The more the metaphor looks like a heavy stone building, the more I want no part of it.

Kirby

[michel paul]

On Mon, Jul 18, 2016 at 9:24 AM, kirby urner <kirby...@gmail.com> wrote:

​> ​

I thought the challenge was to come up with an algorithm that wouldn't have any gaps if we kept at it.

​Correct. And it still is.   : )

 

​> ​

Are the decimal digits of pi countable (listable) or not in your view?

​They are listable and therefore countable in both of our views.

However the set of all infinite strings created from a binary character set is NOT listable.

This fact takes us by surprise. Like the Pythagoreans regarding incommensurability, we might prefer that it be otherwise.

We can code a function that will list all sequences of binary digits of a specified length, and we are tempted to generalize our method for sequences of indefinite length.

But we cannot.

We will never generate a string preceded by an infinite number of 1s.

​It would be a possible string, but we will never reach it using that algorithm.

[kirby urner]

On Mon, Jul 18, 2016 at 1:48 PM, michel paul <python...@gmail.com> wrote:

On Mon, Jul 18, 2016 at 9:24 AM, kirby urner <kirby...@gmail.com> wrote:

​> ​

I thought the challenge was to come up with an algorithm that wouldn't have any gaps if we kept at it.

​Correct. And it still is.   : )

 

​> ​

Are the decimal digits of pi countable (listable) or not in your view?

​They are listable and therefore countable in both of our views.

Except I'm worried about running out of digits while counting them. 

Only a finite number are allowed.  Pi suffers no such restrictions.
 

However the set of all infinite strings created from a binary character set is NOT listable.

The ones my algorithm misses are missed because we never get to infinity digits when counting 1, 2, 3....

"Counting" as no meaning in that context.

The set N is countable because the set N excludes infinity digit numbers.

The definitions seem circular.  I suppose that's by design.
 

This fact takes us by surprise. Like the Pythagoreans regarding incommensurability, we might prefer that it be otherwise.

I'm not all that surprised, given the stacked deck.

 

We can code a function that will list all sequences of binary digits of a specified length, and we are tempted to generalize our method for sequences of indefinite length.

But we cannot.

We will never generate a string preceded by an infinite number of 1s.

0.11111111...

is something we can write and attach meaning to (1/9).

...11111111.0

has no meaning.

 

​It would be a possible string, but we will never reach it using that algorithm.

Right.  We never reach infinity.  Why didn't I think of that?  ;-D

Kirby
 

[Joseph Austin]

On Jul 13, 2016, at 10:26 PM, kirby urner <kirby...@gmail.com> wrote:

When I opened for Mandelbrot one time, small morning session at the local college, I talked about four different types of number sequence:

Convergent

Divergent

Oscillating

Chaotic

In twos complement, what is the "sign" of a number whose LH digits infinitely oscillate between 0 and 1?

It occurs to me that my proposed sequence with fixed part and a repeating part actually encoded two integers (perhaps with some ambiguity as to the boundary between them, and the repeating "sign" portion might be interpreted more generally as a "direction" of sorts.

Of course, my tableau would not generate "chaotic" numbers, or even pi.  

But a list of all Universal Turing Machine programs should index all *computable* irrationals.

And if the number is not computable, could it exist?  Could we "know" it?

We comprehend "infinite" numbers in a finite mind by imagining a computable sequence that "converges" toward that number.

How would we conceive of an infinite, chaotic number?

Or to put it in terms I used earlier,

what mathematical operation would have such a number as it's result?

...

[Joseph Austin]

On Jul 13, 2016, at 10:26 PM, kirby urner <kirby...@gmail.com> wrote:

Lets think of a mathematics that begins with algorithms as the primitives, little machines that put out sequences. 

 

That is essentially the method of Euclid's elements.

And it is not far from the Peano approach:

We have a "successor" operation, and recursive definition of operators such as addition and multiplication.

And CAS is basically converting algebra to rewriting rules.

And Wolfram's "New Kind of Science" takes that approach with cellular automata.

I keep saying that Computing *is* Math,

it's just more advanced, or perhaps more primitive, than the syllabus most "mathematicians" have been taught.

Joe

[kirby urner]

On Tue, Jul 19, 2016 at 9:35 AM, Joseph Austin <drtec...@gmail.com> wrote:

On Jul 13, 2016, at 10:26 PM, kirby urner <kirby...@gmail.com> wrote:

Lets think of a mathematics that begins with algorithms as the primitives, little machines that put out sequences. 

 

That is essentially the method of Euclid's elements.

That's a creative way of thinking about straightedge and compass operations, as recipes, algorithms to be followed, cook books, programmes.

Indeed, how does one construct a regular pentagon?  Give us the steps.

I don't see a good reason to purge the scene of a coordinate system though, nor distance, area and volume computations, mostly missing from Euclid.

I've got this function that computes the volume of a tetrahedron based on its six edges as input, that I'm always finding new ways to use:

https://github.com/4dsolutions/BookCovers/blob/master/tetravolumes.py

It's way to complicated and tedious to use in a "manual math" class, yet perfect for "machine math".  Once we stop worrying about tortuous tedium, because the computers do the drudge work, we're free to develop far more imaginative curricula than our ancestors.  Technology opens new doors.  Fractals a good example.

 

And it is not far from the Peano approach:

We have a "successor" operation, and recursive definition of operators such as addition and multiplication.

Another place we typically introduce "machines" as an elementary topic is when discussing functions.  

Functions provide verbs, behaviors, transformations.

To "make bigger", to "make smaller" -- the scale function does this.

>>> tet1 = Tetrahedron()     # initialize new instance

>>> tet1.volume

1

>>> new_tet = tet1.scale(2)  # all linear interdistances doubled

>>> tet1.volume

8

(because of power rule).

Lambda calc with its top level functions, able to ingest and return functions as inputs and outputs, is machine-oriented in this way.

>>> compose = lambda f,g: lambda x: f(g(x))

>>> F = compose(lambda y: y+2, lambda y: y*2)

>>> F(10)

22

>>> G = compose(lambda y: y*2, lambda y: y+2)

>>> G(10)

24 

As shown above, composition is not commutative.  Stir then fry not the same as fry then stir.

And CAS is basically converting algebra to rewriting rules.

In computer science and in mathematics we talk a lot about "substitution" a lot.  

That's a kind of rewriting.

To have algebra be CS-friendly, we need to make sure the "machines" are not exclusively for doing numerical operations.  String-type operations, such as substitution, happen in algebra too.

Substituting a string into another string, or extracting a substring.... such operations are routinely absent from CS-unfriendly textbooks, or at least they're not given formal treatment as mathematical functions. 

MathPiper uses string operations to simplify expressions, making these operations explicit.  That's all part of algebra, as well as functional programming.

 

And Wolfram's "New Kind of Science" takes that approach with cellular automata.

Yes.  Strings beget strings according to rules.  input -> machine (rules) -> output

We also talk about functions as the "set of ordered pairs" i.e. all (input, output) pairs.  It's not either / or.

Sometimes we just have a "black box" and can't determine what the rules are, even if the rules appear to be deterministic.

A lot of mathematics is about "reverse engineering" various "black boxes" (found in nature, or maybe in a competitor's dumpster) in hopes of knowing ahead of time how to get desired results.  

"What makes this thing tick?" is a typical question.

Reducing the amount of trial and error required to attain a desired outcome:  that's one definition of problem solving.

 

I keep saying that Computing *is* Math,

it's just more advanced, or perhaps more primitive, than the syllabus most "mathematicians" have been taught.

Joe

We're free to innovate new syllabi, and indeed, why not encourage every student to jump into "teacher mode" and give presentations.

In my high school in Manila, we were frequently asked to come up to the board to explain to the whole class how we obtained this or that result.  Work it through with them.  Answer questions.

Any course that's all "teacher talks, students listen" is unlikely to exercise students' ability to talk, and that's detrimental. Mastering the five minute presentation format is going to give you a leg up.  

As soon as one starts to get clear on something, it's really helpful to turn around and work at passing it on.

Not only is math an outdoor sport (e.g. geocaching, PokemonGo), it's a discipline requiring public speaking.

"Teaching X" is a lot about "teaching ways to teach X" which of necessity involves learning X.

Kirby

[michel paul]

On Wed, Jul 13, 2016 at 7:26 PM, kirby urner <kirby...@gmail.com> wrote:

​> ​

Lets think of a mathematics that begins with algorithms as the primitives, little machines that put out sequences.

​Yep. And that's what a generator is.

It's a function that remembers 'where it was' or 'where it is about to go'. That's what makes it useful. It remembers its state. It is always ready to spit out the 'next' value.

The generator could be coded in all kinds of ways, but the important thing is that it is ready, on hold, to provide the 'next' value in a sequence.

This concept of 'next' is the foundation for N, and any set that can be coded as a generator has the same cardinality as N.

Now consider a generator of generators. Does that make any sense?

Yes, it does. Forgetting about language specific constraints we can conceive of a generator that yields successive generators.

Consider a generator, R, that yields generators of infinite sequences of binary digits. 

Each call of next(R) yields a generator that is itself a generator that successively yields the digits of a real number.

One of those generators will yield all the digits of sqrt(2). 

Another will yield all the digits of phi.

Another will yield all the digits of e. 

Another all the digits of pi. 

Etc.

Can we guarantee that our generator will eventually yield generators for the digits of ALL reals? If just left to run long enough? 

Nope.

Can we guarantee a generator that will eventually yield generators for the digits of ALL rationals? If just left to run long enough?

Yep.

--

​Michel

​

===================================
"What I cannot create, I do not understand."

[michel paul]

On Tue, Jul 19, 2016 at 7:54 PM, michel paul <python...@gmail.com> wrote:

​>​ 

Can we guarantee a generator that will eventually yield generators for the digits of ALL rationals? If just left to run long enough?

​> ​

Yep.

​This was done using VPython for class years ago:

​

https://docs.google.com/presentation/d/1zprypgc_1pVvChNKLTkTdw6I6ciOF6ro5K10l0ryQ2I/edit#slide=id.i0

--

[Joseph Austin]

Michel,

As a physicist, I appreciate the power of differential equations to model what behaves, in macroscopic details, as a continuum.

But I don't specifically recall any appeal to Cantor when learning how to "solve" differential equations.

To me, Calculus was the "epsilon-delta game" of taking limits, typically limits of functions, typically polynomial functions,

some infinite (but countable) series.

i could wish that differential equations were introduced earlier in the science curriculum,

say start Calculus I with  y' = ky (the growth equation) instead of the derivative of d/dx (x^2).

Joe Austin

On Jul 14, 2016, at 11:25 AM, michel paul <python...@gmail.com> wrote:

On Thu, Jul 14, 2016 at 7:18 AM, Joseph Austin <drtec...@gmail.com> wrote:

​

we dealt with inverse of addition giving negatives, 

and inverse of multiplication giving rationals,

and inverse of power giving irrationals and imaginaries,

and none of those crossed the line into "uncountable"?

So where did we cross the line?

​I would say it was in providing an analytical foundation for Calculus.​

​

​ 

Calc used the real numbers without having defined them, so in a way Calculus preceded Analysis even though we teach it in the opposite order in high school.​

​In terms of 'operations', finding roots prompted the discovery of incommensurability. ​Interesting, it turns out that the concept of irrationality was known about and accepted by Indian mathematicians prior to Pythagoras. 

And then in the 1800s there was work on the idea of transcendental numbers.

But it turns out to be Cantor who gave the first rigorous definition of the reals.

So I guess the best answer to your question is the late 1800s.

--

​Michel

​

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

[michel paul]

On Wed, Jul 20, 2016 at 6:23 PM, Joseph Austin <drtec...@gmail.com> wrote:

 

> 

I don't specifically recall any appeal to Cantor when learning how to "solve" differential equations.

​Right, there would have been no need.​

 

​> ​

To me, Calculus was the "epsilon-delta game" of taking limits

​That's because of the later efforts to provide 'proper' foundations. Previously, Newton & Leibnitz didn't bother with that stuff. 

You might have enjoyed their approach more.​​

--

​ Michel

[kirby urner]

As I was saying earlier, I'm all for encouraging philosophical debate, but in this case I'd say the Kronecker vs. Cantor controversy is not yet over, or settled.  Wikipedia seems to make that clear, and there's plenty of other reading one might dredge up.

https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory  (see Axiom of Infinity as dispensable feature)

Sure, given teacher might feel obligated to weight in and take a side, but leaving some questions open is sometimes the better approach. 

Premature optimization (reducing to supposed foundations too early) is not required.  There's no need to have permanent foundations anytime in our lifetime.  I wouldn't hold my breath in other words.  Math works great with foundations still up in the air -- providing them quickly just isn't a problem that needs solving any time soon.

In actual fact,

>>> pi = pi_digits()  # pi digits genertor
>>> listable = list(enumerate(pi))

is not going to finish the job, because "forever" is not in the cards.  The Python language specifies a 'MemoryError' exception for such situations.

Kirby

[Andrius Kulikauskas]

I look forward to catching up with letters. I just want to add a link
to Norman Wildberger's videos on "Famous Math Problems". He has a four
part series on what he calls the most fundamental math problem, the
problem of continuity. It starts here:
https://www.youtube.com/watch?v=Nu-YPJSNFpE&index=4&list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf

I've only seen the first part. He argues that the real numbers are not
well-defined. He then discusses affine geometry, and I think next,
projective geometry.

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

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[michel paul]

On Wed, Jul 20, 2016 at 7:52 PM, kirby urner <kirby...@gmail.com> wrote:

>>> pi = pi_digits()  # pi digits genertor
>>> listable = list(enumerate(pi)) 

is not going to finish the job, because "forever" is not in the cards.  The Python language specifies a 'MemoryError' exception for such situations.

​Right ... but why would you do that?  : )

Avoiding that memory error is the very reason generators exist in the first place! So you don't have to list them all at once!​

One of the practical reasons for generators is that they conserve resources. You don't have to generate a whole list of values all at once, especially if doing so is impossible. The generator can provide them one at a time, as needed.

​Yeah, we can't 'list' infinity 'all at once'. It creates big problems, both practically and philosophically. 

But we can always focus on 'next'. This is where we are, and ... now there's one more.

How did Euclid prove an infinity of primes? First he assumed there was a last one, and then ... he showed that there's always one more.​

Infinity clearly did not 'begin' with Cantor. The issue is not non-terminating processes. The issue is assuming a completed infinity. Cantor suggested a way of reasoning about completed infinities, and that is what bothered people.

But I don't see that anyone has a problem with potential infinity, the concept of 'next'.

Sometimes there is no 'next', and sometimes there is no 'last'.

​There are many reasons why I fell so in love with the concept of generators when I discovered them in Python. You can code many concepts about sequences using them. Plus, a generator is a nice segue for math students into object-oriented thinking. A generator is an object with a member function 'next'.

​Notice that when discussing a generator of generators and so on, I need to make no reference to axiomatic set theory. I'm just talking about this concrete idea of a list. We can organize stuff into lists. And we can pop out the values of our list one at a time. That's a generator. That's all we're talking about.

Can we talk about a generator of a non-terminating sequence of binary characters? Of course we can.

Does it make sense to talk about a generator of such generators? Of course it does.

Does it make sense to talk about a generator of all possible rational numbers? Yes.

Does it make sense to talk about a generator of all possible generators of non-terminating digit sequences? Well, it makes sense to talk about it, but it doesn't seem possible to so, even algorithmically. So far no one has shown a way to do it.

Notice that nowhere here are we required to make reference to Cantor or axiomatic set theory. At least not explicitly.

​> ​

As I was saying earlier, I'm all for encouraging philosophical debate,

​... 

but leaving some questions open is sometimes the better approach.  

​I completely agree. That is why I would bring these things up in class in terms of Books in Print. Does it list itself? Sure, no problem. What about Books That Do Not List Themselves. Does it list itself? Uh-oh. Now we have a problem.​ What should we do?

Maybe not even try to write that book? OK, that's a suggestion ...

​I actually did have a student who refused to accept the existence of infinity. Absolutely refused. He was fine with really, really, really big quantities or really, really, really small quantities, but not an actual infinity. I loved that kid. He was a genius. Highly functioning autistic, could visualize like crazy. I had these 3d magnetic puzzles in the back of my room, and he'd put them together in ways that were actually kind of scary. Weirdly beautiful. And he'd explain to me how if he had more pieces the shape would circle back and close in on itself.​

​> 

Math works great with foundations still up in the air

​I completely agree.

And the interesting this is - the reason I agree is that I believe mathematics is something intrinsic to physical reality, not something we 'just' make up.

​​One of the reasons that biologists and computer scientists are having deep discussions these days is that we are recognizing that computational structures occur naturally in various ways. Computer science is useful to biologists for far deeper reasons than just number crunching.

​And if it turns out that type theory allows us to ground mathematical reasoning in ways that are computer scientific, and if this ties into naturally occurring biological structures ... well, how cool is that?​

--

[kirby urner]

On Thu, Jul 21, 2016 at 9:43 AM, michel paul <python...@gmail.com> wrote:

 

I really like generators too.  Nothing I'm posting should be taken as a rant against generators.
 

​Notice that when discussing a generator of generators and so on, I need to make no reference to axiomatic set theory.

Exactly. 
 

I'm just talking about this concrete idea of a list. We can organize stuff into lists. And we can pop out the values of our list one at a time. That's a generator. That's all we're talking about.

Lists and generators are different types though. 

Generators are not sequences in the sense of having an index (subscript), they're iterators and respond to __next__. 

We can turn a list into an iterator by running it through iter( ).  That's just Python though, with other executable math notations not resting on these particular concepts so clearly and obviously. 

That's one thing I like about Python: its relatively clean meta-language.

 

Can we talk about a generator of a non-terminating sequence of binary characters? Of course we can.

That's what I was doing with my "all possible binary permutations with dot" generator wherein 0.010101010101010.... would be generated as a __next__ out to any number of digits, just with may other sequences in between. 

The algorithm covers all possible permutations of {0,1,"."} before adding a next digit.  It's a 2D table with a zig-zag traversal path, similar to Cantors, but it also picks up dupes e.g. 01.0, 001.0, 0001.0 would all be generated.  We can throw them away as the equivalence class for the canonical 1. (same as 1.0, 1.00, 1.000...).

If a number has a binary representation at all, I'd get there eventually.  No holes.  But then comes the objection that no finite counting process ever gets us to "infinity digits".  Right, back to square one.

Since we can never get to infinity, this was not considered an Aleph_0 way of reaching all the reals, even though the rationals are countable with a similar zig-zig diagonal (in all the Aleph stuff text books), and even though Q is likewise considered an infinite set.
 

Does it make sense to talk about a generator of such generators? Of course it does.

Does it make sense to talk about a generator of all possible rational numbers? Yes.

I'm not so sure. 

I read a Scientific American article a long time ago about how we could measure the randomness of a digit sequence relative to the number of bytes it'd take to write like a generator for it. 

0.111111111.... may be defined to have "infinite digits" but the program would just be:  while True: print(1) or like that.  Not many bytes.  So the sequence is hardly random.

However what about completely random sequences of digits for which we have no algorithm?  That's what I was getting at in asking about Pi with digits randomly changed all along the way.  I could easily claim the majority of real and even rational numbers are "non-algorithmic" in this sense.
 

Does it make sense to talk about a generator of all possible generators of non-terminating digit sequences? Well, it makes sense to talk about it, but it doesn't seem possible to so, even algorithmically. So far no one has shown a way to do it.

My single generator was covering whatever binary digits cover and could be extended to base 10 if that were of more interest.  No holes.  Just the problem of an ever increasing perimeter as my playhead spirals around through all "N-digit plus dot" permutations. 

The radius of my spiral gets longer and longer, but yet no skipping occurs.  If this isn't a problem for counting the rationals, then I don't see why it's a problem for counting the reals. 

1/314159... is a rational number as long as we have finite digits.  But then there's always a "next digit" with the rationals too.  The point was to have an algorithm for getting a next and a proof we'd not be missing anything if allowed to "go forever".  The rational numbers are non-terminating too and we have an infinite number of them between any two.
 

Notice that nowhere here are we required to make reference to Cantor or axiomatic set theory. At least not explicitly.

Nor do we have any evidence that Python requires set theory of any kind as a basis or foundation.  The set is just one more data structure in Python, not the basis of anything.  What do we even mean by "foundation"?  It's a metaphor, and a tempting one, but should we give into it's lure?

 

​> ​

As I was saying earlier, I'm all for encouraging philosophical debate,

​... 

but leaving some questions open is sometimes the better approach.  

​I completely agree. That is why I would bring these things up in class in terms of Books in Print. Does it list itself? Sure, no problem. What about Books That Do Not List Themselves. Does it list itself? Uh-oh. Now we have a problem.​ What should we do?

Or we might turn the philosophical debate to other issues such as "why is ordinary volume considered 'three-dee' and might we really have anything geometric outside the context of volume e.g. pure width with no height or depth?"
 

Maybe not even try to write that book? OK, that's a suggestion ...

Or don't call it a book.  This Report lists all books that don't list themselves.  It doesn't list itself because it's not a Book.
 

​I actually did have a student who refused to accept the existence of infinity. 

I wonder what "existence" meant to him.  Why do we require "existence" for a concept to have utility?

I think it's because of a naive view of language which assume words have meaning because they "point" to things.  Nouns or names have meaning because of corresponding "things out there" or "things in themselves" or whatever.  So if "infinity exists" that must mean we can point to instances of it.  Dots on a chalkboard come in handy, labeled with Greek letters, other hand-waving.

I think this "words as pointers" view of language is been successfully added to the ash heap of history by now, but many people will continue to believe it for decades or centuries to come.  Disabusing people of their beliefs is not something one just goes around doing, without being considered a nasty bully.

 

Absolutely refused. He was fine with really, really, really big quantities or really, really, really small quantities, but not an actual infinity. I loved that kid. He was a genius. Highly functioning autistic, could visualize like crazy. I had these 3d magnetic puzzles in the back of my room, and he'd put them together in ways that were actually kind of scary. Weirdly beautiful. And he'd explain to me how if he had more pieces the shape would circle back and close in on itself.​

If I pick up a hammer and ask "what is the meaning of this?" a good response is to show the hammer hammering, doing useful work, pounding in nails, prying open a paint lid, whatever.

If I pick up the sideways 8 infinity symbol and ask "what is the meaning of this?" we again have many use cases in many math texts.  We also love to use "..." (dot dot dot), probably the most powerful operator in all of mathematics (with multiplication runner up).

 

​> 

Math works great with foundations still up in the air

​I completely agree.

And the interesting this is - the reason I agree is that I believe mathematics is something intrinsic to physical reality, not something we 'just' make up.

Depends how we define mathematics? 

Is driving a car engaging in a mathematical activity?  Do we have to use numbers?  Is chess mathematics?  How about Go.  How about Pokemon Go?

Do we "make up" stuff at all or are we constrained by "the rules of making sense"?  Is parsing Shakespeare mathematics?

I don't see these questions gaining a lot of traction right out of the box.  It takes a lot of stage-setting and context-defining to make philosophical debates go anywhere.  Out of the box, they're frictionless.

Some philosophy in math class makes sense, but maybe not a lot. 

Lets bring back philosophy class if we want to do philosophy? 

Don't just assume math teachers are well-trained when it comes to philosophy.  They'd need free PD, just like for Gnu Math, which most of them aren't prepared to teach yet either.

 

​​One of the reasons that biologists and computer scientists are having deep discussions these days is that we are recognizing that computational structures occur naturally in various ways. Computer science is useful to biologists for far deeper reasons than just number crunching.

Randomly find some Youtube guy saying it's all Math running on a giant Math Machine (I'm sure you'll find many).  Is that deep? 

I'd say it's grammatically interesting, this playing around with "what English one might get away with" in terms of maybe making sense (maybe not).  Close to music.

My working hypothesis is English, or any language, is a barrier as much as a means, for discerning what's so.  One should not blindly trust one's native language.  One is lucky if able to think in a minimum of two (like Maria). 

I have a similar attitude towards computer languages:  if you only know one, you haven't really learned that one yet, as it takes more than one to learn any deeply. 

Likewise number bases:  you won't know what "Base 10" means if you've never learned about other number bases.  Which is why Common Core, with its exclusive focus on "Base 10" is verging on nonsensical.  From recent reading I see alternatives are already being proposed.

 

​And if it turns out that type theory allows us to ground mathematical reasoning in ways that are computer scientific, and if this ties into naturally occurring biological structures ... well, how cool is that?​

This whole idea of "grounding" is what I'd want to study more first. 

Every generation spawns a number of thinkers into "grounding" but it's not always clear they're making much progress, or that this work is all that crucial to the continuance of humankind or any other species.

"Certainty" as a concept is certainly worthy of investigation though.  I recommend 'On Certainty' by my hero Ludwig Wittgenstein.

Kirby

[michel paul]

On Thu, Jul 21, 2016 at 10:51 AM, kirby urner <kirby...@gmail.com> wrote:

 

​>> ​

Does it make sense to talk about a generator of all possible rational numbers? Yes.

​> ​

I'm not so sure. 

​Why not?

It's what I used in creating ​

​

Generating the Rational Numbers.

I've lost the original VPython files that created those slides, but I recreated things a bit here in Sage. 

The generator uses mediants to create all rationals between 0/1 and 1/1.

> I recommend 'On Certainty' by my hero Ludwig Wittgenstein.

​Yep. I love it.​

 

[kirby urner]

On Thu, Jul 21, 2016 at 9:40 PM, michel paul <python...@gmail.com> wrote:

On Thu, Jul 21, 2016 at 10:51 AM, kirby urner <kirby...@gmail.com> wrote:

 

​>> ​

Does it make sense to talk about a generator of all possible rational numbers? Yes.

​> ​

I'm not so sure. 

​Why not?

I can't shake the idea of infinite digit members of N.

314

----  = 3.14

100

3141

----    = 3.141

1000

...

314159...

---------  = pi = 3.14159...

100000...

I have a hard time accepting that 

pi = 3.14159...

might be meaningful but 

314159...

---------  = 3.14159...

100000...

cannot be.  

We've agreed all along that allowing members of N with infinite digits breaks the aleph stuff.

Sometimes I'm in the mood to break the aleph stuff. (-;

Kirby

[Mike South]

On Fri, Jul 22, 2016 at 1:25 AM, kirby urner <kirby...@gmail.com> wrote:

On Thu, Jul 21, 2016 at 9:40 PM, michel paul <python...@gmail.com> wrote:

On Thu, Jul 21, 2016 at 10:51 AM, kirby urner <kirby...@gmail.com> wrote:

 

​>> ​

Does it make sense to talk about a generator of all possible rational numbers? Yes.

​> ​

I'm not so sure. 

​Why not?

I can't shake the idea of infinite digit members of N.

314

----  = 3.14

100

3141

----    = 3.141

1000

...

314159...

---------  = pi = 3.14159...

100000...

I have a hard time accepting that 

pi = 3.14159...

might be meaningful but 

314159...

---------  = 3.14159...

100000...

cannot be.  

pi = 3.1459... is not really even remotely like the division of N+ number problem you've written there.  It's really just an expression saying that we can represent an approximation of pi in base 10 and the ... says "by the way, this isn't all" and if someone asks further about "what do you mean it isn't all, how far do you have to go to get it exact" we say "well, actually, you'd have to go infinitely far, it never ends in a repeating sequence like 1/3 or 1/2 do". 

 

We've agreed all along that allowing members of N with infinite digits breaks the aleph stuff.

You say that as if we're clinging to the the cardinality of the naturals as a dogma.  It's really just a consequence of the way you define the naturals.

Complaining that we're not interested in N+ as you define it is not productive.  Do something with it that makes it interesting.

I can take any element of N and put a 1 in front of it and make it bigger by a known amount.  I can add 1 to it and understand what that means, and I can understand that independent of whether it's represented in binary or decimal or whatever.

Let's look at your expression:

 

314159...

---------  = 3.14159...

100000...

What does it mean to have an N+ number written above a line with another N+ number below it?  For *any pair" of numbers in N I can tell you exactly what that means.  What does it mean in N+?

In N, if I have a string of digits representing a number and then put a 1 in front of it, I understand exactly what I've done.  If it's a base ten representation, then putting that 1 on the left side of the number adds 10**int(log(the number) + 1) to it.

Now let's think about what happens when you put a 1 in front of your digits-of-pi-in-a-list member of N+:

Is it:

1314159...

---------  = 1.314159...

100000...

Looks like it reduced the value to me.  So if you take your infinitely long list and make it longer on the left side it makes the number it represents smaller?  If we were talking about N that digit on the left would be the most significant digit, but it doesn't seem to work that way in N+.  Or does it?  You would need to tell us, because N+ is something that you defined.  When people say that an infinite string of digits is not in N, there's a reason we say that--it's not just obstinance.  The finite-string-of-digits representation of each number in N is meaningful because the positions of the digits each represent something well defined--that digit multiplied by the base of the representation raised to the power of [count how many digits you are from the right-most digit, i.e., "the ones place"].

Although N is infinite, we can talk about any two arbitrary elements of N and say what happens when you add them together.  Although you're calling N+ something along the lines as "N as normally defined but with the addition of infinite-strings-of-digits", you're not mentioning the fact that suddenly you have a set where you used to be able to do operations on the elements and now you can't.  You used to be able to take any two of them and say if one is bigger than the other and now you can't.  When we say that an infinite string of digits is meaningless in N, we have very good reason for doing so.

Consider this:

314

----  = .314

1000

3141

----    = .3141

10000

314159...

---------  = .314159...

100000...

Therefore, in your system 3.14159... is equal to .314159...

Sometimes I'm in the mood to break the aleph stuff. (-;

You didn't just change the cardinality, though, you made something that has essentially nothing to do with the original set.

 

The fact that the reals are of higher cardinality than the rationals isn't any more debatable than the fact that the square root of two is irrational.  It's just a little (perhaps a lot) harder to understand the proof.  There's no aleph agenda attempting to dictate that it must be so because that's the way we like it.  As michel pointed out quite extensively, it was definitely not the way people liked it, but something they were forced to accept as true given the definitions that they happened upon but did not, at first, understand the implications of.

The naturals are easy to understand the source of, and the rationals as well.  Someone mentioned the epsilon-delta game in calculus--the reals historically took on precise definitions because (I think the following is roughly true) people were trying to work out what kinds of numbers we needed in order to not have any "holes in the number line" when the processes involved in the epsilon-delta game were taken into account.

It's clear enough that the rationals can be put into a 1-1 correspondence with the naturals, which is really an incredibly surprising result.  When you get a result like that it's natural to ask other questions.  We approximate the reals with rationals all the time.  So you very easily move on to the question "well, does the same thing happen with the reals?  If I come up with a clever arrangement of them, can I put *them* into a 1-1 correspondence with the naturals?"

This is just human curiosity, not an agenda.  It's a naturally occurring question.  And it's pretty brilliant that someone figured out a simple way to show that we can all go work on other interesting questions, because, no, you can't do that.  And when I say simple, I mean you don't need a phd-level understanding of math to get it.

Once you've proven that you can't trisect the angle with a compass and straightedge, you don't need to go through the thousands of supposed counterexamples.  That's one of the great things about math--it's designed to save you work.

Your generator is like an angle trisection algorithm.  It is going to break down on closer inspection, because the logic of the uncountability is sound.

michel did a great job of sticking with it to show you where the breakdown is, but it might bear some elucidation.

In the argument that the rationals have the same cardinality as N, we provide an algorithm that tells you "exactly when"--and what "exactly when" means is important--each rational will be arrived at.  That is essentially what it means to have the same cardinality as N--you can say "I got element X of my set at step n, when n is a finite number and thus an element of N".  With Cantor's proof of the countability of the rationals you can give me any rational number and I can tell you exactly at which step, "n", in the algorithm I get to that number.

michel pointed out to you that .1010101010.... isn't in your list.  You attempted to counter that with "well, just wait, we're getting there".  But that means that the "n" for .10101010.... is, in fact, infinity, and is therefore not in N.  N is an infinite collection *of finite things*.

If you want to say you have a set that is the same size as N, you need to associate every element of that set to one of the finite numbers in N.  

You can simplify the question of doing this with the reals by just worrying about the numbers between 0 and 1.  There are an uncountable number of reals between 0 and 1, so forget about moving the dots around or going out in both directions or whatever.  We can use the binary representation, so just give us an algorithm to list all the strings of 1's and 0's with a . in front.

I can give you one that lists all such values that either (a) end in an infinite string of zeroes or (b) at some point end up repeating the same sequence over and over (note that (a) is just a special case of (b)).  It's Cantor's algorithm!  

But if you insist that it's no problem to list them all, show us!

For any given element in the set of all 1,0 strings, you need to tell us a natural number n such that the element is included by the nth step.  That number n needs to be finite, or we wouldn't be talking about a 1-1 correspondence with the naturals any more, which is of course the whole point and meaning of countability.

In case anyone isn't following the counterexample to Kirby's listing algorithm, let's draw it out, only looking at the ones that start with a dot, as I suggested above:

 1 .0

 2 .1

 3 .00

 4 .01

 5 .10

 6 .11

 7 .000

 8 .001

 9 .010

10 .011

11 .100

12 .101

13 .110

14 .111

15 .0001

16 .0010

...

First, I listed all the permutations of one digit, that is step 1 and step 2.  Then I'm out of those possibilities, so I move on to two-digit strings--that's step 3, 4, 5, and 6.  Then I've exhausted those, so I start on all the 3-digit strings--those are items 7-14.  Then on to the four-digit ones.

Now, this is not a perfect rendering of the idea, in that it repeats some things.  The value of the item (when interpreted as a binary number with a radix point) at step 4 is the same as the value of the item at step 9.  We could filter those  duplicates out very easily, but the way I have it drawn above is to make it very clear that I'm not missing any, and to make it easy for you to state definitively what the value is at step n.

I'm pretty sure that this is, in principle, exactly Kirby's listing algorithm, just limited to the numbers between zero and one (which isn't a problem--if you can't list everything between zero and one, you certainly can't list all of the reals).

To be extra, extra clear about it we should point out that when we see ".0" we really mean ".000000000000..." and ".1" means ".100000000...".  And we could have a sister sequence where we start with ".1111111111....", then do ".011111111111..." at step 2 and so on (just the binary inversion of what we have above).  And even if we combine these lists together (which would still be countable with N), we still don't have all the reals.  As michel paul pointed out, .010101... which is rational, is not even there, so in fact it doesn't even cover all the rationals.

Note that I *could* extend it to cover the rationals, though.  The .010101...example is, in this limited sense, a bit of a red herring.  I could make a third companion sequence that listed the rationals between zero and one in their binary representation by taking that subset out of Cantor's list of the rationals.  So the specific example of .01010101...--and, indeed, *any series of digits that eventually repeats the same thing forever on the end" can be covered.  But .01010101... is still *instructive*, because it's an example of something that Kirby's sequence (without my Cantor modification) *cannot get to in a finite number of steps*.   It shows you the *idea* of what we mean by there being something missing from an infinite list.  What we mean is not "well, we looked through all infinity of them (man, it took forever!) and it wasn't there" but rather "there is no natural number n that corresponds to .01010101... showing up".  This is important to understand.  If you've listed them all, *what we mean by that*--what we mean by "I found a way to list them all!"--is that you've found a way to say "give me anything in the set.  I'll tell you what "n" in the naturals is the step in my algorithm where your particular anything shows up".

Cantor covers the hole for .010101... by telling us *at which element of N* .010101... shows up.  Cantor creates an exact, provable, algorithmic correspondence relationship between the individual members of N and the rationals, which is *what we mean* when we say that the sets are the same cardinality.

I imagine that it was after many, many, many attempts to list the reals that Cantor realized "wait, maybe you can't list them" and tried to prove that (because, as I said before, this saves you work--you both learn something about the nature of the things you are studying, and you cut off infinitely many blind alleys at the same time).  It's an incredibly brilliant leap, and arguably another incredibly brilliant leap to get to the "diagonal argument" version that can be understood without any deep reference to mathematical or set theory foundations.

Because you don't need to understand Dedekind cuts or Cauchy sequences (those are the things that "construct the reals" in such a way that you can play the epsilon delta game and always know there is somewhere to land).  You only need to intuitively understand that we can represent any number between zero and one with a dot and a sequence of digits after it.  If you can grasp that concept, then Cantor's diagonal argument shows you that you cannot list all those numbers.

Note--they don't have to even *be* numbers.  You could have all the sequences of dot followed by flower, bagel, airplane for all the different permutations of flower, bagel, and airplane without any reference to the real numbers at all.  Then you assume you listed them:

1. flower flower flower ...

2. airplane bagel flower flower ...

3. bagel flower airplane flower ...

....

and you can say "I *know* that your list is incomplete, because I can construct a sequence that *is not in your list* by changing the first value in the first list from flower to airplane and the second value in the second list from bagel to airplane and the third value in the third list from airplane to flower.  My new sequence is both well-defined--exactly as well defined as any of your listed sequences--and *not in the original list*.

You can't list all sequences of things.  There are too many sequences of things.  If you restrict your sequences of things to sequences that eventually repeat infinitely, then yes, you can list those.  But the set of *all* sequences of things is too big to fit.

Once confronted with this you can either do the pythagorean thing of stubbornly going back to your cave in shame and not accepting the fact that there are different sizes of infinities, or you can embrace the newly revealed reality and experience the joy of understanding different sizes of infinity.

For example: the real numbers aren't just a larger set than the naturals--they are so much larger that they make the size of the naturals look like zero.  If you could pick a number at random from the real number line, the probability of getting a rational would be zero because there are so many more reals (just like the probability of picking any particular number from all of N would be zero because there are infinitely many things in N--the fact that the probability of something is zero doesn't mean it can't happen, just that the chance is vanishingly small).

Cantor's diagonal argument isn't up for debate, or dependent on controversial and esoteric set theory axioms or whatever.  It's simply a logical sequence of steps starting from an agreed-upon point where the agreement is quite easy to understand.  If you accept that there are infinite sequences of integers that don't end with a repeater--that is, if you accept the existence of such things as the decimal representation of pi and the square root of two, and extend that to not just roots and however you discovered pi but to every sequence of digits--you are stuck accepting that you have a set of things that is bigger than the naturals.

There is no cabal of aleph-authorities trying to artificially define things in such a way that N can't count all of those sequences.  The completely intuitive definition of N (start counting, and keep going) and the completely intuitive idea of a list of infinite sequences of flower, airplane, and bagel, are all you need to get there.  To reject it you have to either say "it is not meaningful to talk about all the infinite sequences of flower, airplane, and bagel.  You can only talk about lists that end in infinite repetition, or that correspond to either roots of rationals or the list of transcendentals that I am willing to accept the existence of (pi, e, and and the sine of 1)".  In short, you need to artificially define away any way of defining more sequences that will fit in N.

BTW, if you are insistent on disallowing any set that has a larger cardinality than N, you also need to say this:  "The set of all subsets of N doesn't exist, either", because that also has greater cardinality than N.

It's a pretty cool result--there are infinities that are bigger than N.  It's even cooler that Cantor found a proof of it that is so easy to follow.  I can't imagine the fun we would be having if we were trying to work with his first uncountability argument ( https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article )--count your blessings! (assuming that you have N or less of them, I mean).

mike

Kirby

[Joseph Austin]

Haven't you just defined the "computable reals" as a countable subset that includes irrational elements?

Now show me a real that is not in that set.

If you offer the output of the cantor "diagonal negating generator", how is that machine not already in the list of generators?

and if it is in the list, how is it's output not in the list of generated numbers?

Joe Austin

[Joseph Austin]

On Jul 18, 2016, at 11:31 AM, michel paul <python...@gmail.com> wrote:

​You will never produce a string with an infinite number of 1s.

When considering a Finite State Machine, we don't speak of "producing" a string but "accepting" a string.

If we want to "accept" an infinite string, we would have to say that the machine in principle will never block for the given string.

If it is a FINITE state machine, then of course the infinite string must end with infinite repetitions of some finite string,

so "acceptance" of an infinite string can be defined as cycling back to a given state (or set of states), rather than halting.

The concept can be easily related to the corresponding regular expression, which can "produce" a string.

E.g, the infinite string .010101... can be produced by the expression (01)* where * equals ∞ .

Am I missing something?

Joe Austin

[Mike South]

On Fri, Jul 22, 2016 at 10:14 AM, Joseph Austin <drtec...@gmail.com> wrote:

Haven't you just defined the "computable reals" as a countable subset that includes irrational elements?

I think the set I described was N plus Q plus the real algebraic numbers, plus a few transcendentals included because they're the ones we're familiar with.  Many (infinitely many, I'm sure) of the real algebraic numbers are irrational. It's a limited subset of the irrationals.  And, in fact, it's countable.  It's a bit harder to see that it's countable, but that is in Cantor's first paper.  It's in the wikipedia article I linked earlier:  https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article#First_theorem  and expand that "show" box

Cantor's enumeration of the real algebraic numbers
Real algebraic number	Polynomial	Height of polynomial
x1 = 0	x	1
x2 = −1	x + 1	2
x3 = 1	x − 1	2
x4 = −2	x + 2	3
x5 = −1/2	2x + 1	3
x6 = 1/2	2x − 1	3
x7 = 2	x − 2	3
x8 = −3	x + 3	4
x9 = −1 − √5/2	x2 + x − 1	4
x10 = −√2	x2 − 2	4
x11 = −1/√2	2x2 − 1	4

 

Now show me a real that is not in that set.

One real that is not in that set is 

.0123456789101112131415161718192021222324252627282930...

It's transcendental.  If you want a direct proof that there are uncountably many *other* transcendentals, here you go:

Theorem. There exist uncountable many algebraically independent real numbers. So the set of the transcendental real numbers is uncountable. 
Proof. Let B be a transcendence basis (which exists by Zorn’s lemma) of the field extension R/Q. If B were countable, then Q(B) would be countable (because its elements are of the form P( ~b)/Q(~c) with n ∈ N, P, Q ∈ Q[X1, . . . , Xn] and ~b, ~c ∈ Bn ), so R would be countable (because its elements are roots of polynomials in Q(B)[X]\{0} and there would be only countable roots), which is false. The elements of B are uncountable many algebraically independent real numbers.

I got that from http://diego.mat.unb.br/doc/directproof2.pdf and I don't know what the definition of a "transcendence basis" is.  It would probably take me a few days to work through all the definitions and then a reaaaaaaaaaaaaaaaaaaaaly long email to get it into terms that could be easily followed by someone who hasn't spent much time in theoretical math.  I doubt that many people would want to read through it, though I might have fun writing it.

 

If you offer the output of the cantor "diagonal negating generator", how is that machine not already in the list of generators?

and if it is in the list, how is it's output not in the list of generated numbers?

Keep in mind that I constructed a set by deliberately grabbing things I knew to be countable and combining a finite number of sets.  (I believe you can still remain countable with a countably infinite number of countably infinite sets, but I don't remember for sure, but the real algebraics might be an example of that very thing now that I think of it.)

So I created something by design that only included a handful of transcendentals because they were the "blessed" ones that people are already familiar with.  My point was to make it clear that it's an intellectually dubious practice to take a few transcendentals but then shut your eyes tight when someone talks about there being more than the ones you already know about.

mike 

[Joseph Austin]

I'm getting way behind, or out of sequence, but I thought I answered this.

Your list is simply reverse binary integers--ONE list

I go from one to zero to negative one to negative two by subtracting 1 (a defined binary operator) thus:

...0001

...0000

...1111

...1110

Turn it around if you wish, but we can alway use the "mirror" algorithm as a generator.

Joe

[kirby urner]

I'd do all the same hand-waving if wanting to explain:

314159...

---------  = 3.14159...

100000...

i.e. you'd have to go infinitely far, it never ends, and it converges to pi (or would if we allowed infinite-digit members of N).

Complaining that we're not interested in N+ as you define it is not productive.  Do something with it that makes it interesting.

That reminds me of "do something smart" -- a famous movie line:

https://youtu.be/belFdtYROqc

 

I can take any element of N and put a 1 in front of it and make it bigger by a known amount.  I can add 1 to it and understand what that means, and I can understand that independent of whether it's represented in binary or decimal or whatever.

Let's look at your expression:

 

314159...

---------  = 3.14159...

100000...

What does it mean to have an N+ number written above a line with another N+ number below it?  For *any pair" of numbers in N I can tell you exactly what that means.  What does it mean in N+?

It's just another way to express pi is all, no big deal.  We always have the same number of 0s below as digits above the line, to keep the decimal point where we want it.  We could write a generator.

<< SNIP >>

The fact that the reals are of higher cardinality than the rationals isn't any more debatable than the fact that the square root of two is irrational.  It's just a little (perhaps a lot) harder to understand the proof. 

The whole idea of infinite sets with different cardinality is what I call the "aleph stuff".

It was debated when it first came out.  Then debate died down.

 

There's no aleph agenda attempting to dictate that it must be so because that's the way we like it.  As michel pointed out quite extensively, it was definitely not the way people liked it, but something they were forced to accept as true given the definitions that they happened upon but did not, at first, understand the implications of.

This happens with invented board games too.  People discover unintended consequences, given the rules.

I remember when we were ahead in soccer I had my team just kick the ball out of bounds whenever we gained possession (usually just one kick does the job).  That wasn't sportsmanlike.  I was applying logic and messing up the game.  This is about the time new rules get invented.  "Oh, we didn't mean you could to *that*..."

<< SNIP >>

michel pointed out to you that .1010101010.... isn't in your list.  You attempted to counter that with "well, just wait, we're getting there".  But that means that the "n" for .10101010.... is, in fact, infinity, and is therefore not in N.  N is an infinite collection *of finite things*.

Right.  You can't count the reals because then Q and R would have the same cardinality, not to mention N, and the whole language game would break down.  Allowing N to have infinite-digit members, even as rationals converging to pi, is strictly verboten for good reason then. 

We love the aleph stuff, and can't imagine what life must have been like for poor slobs before Cantor.

<< SNIP >>

Once confronted with this you can either do the pythagorean thing of stubbornly going back to your cave in shame and not accepting the fact that there are different sizes of infinities, or you can embrace the newly revealed reality and experience the joy of understanding different sizes of infinity.

Expressing joy is one thing, whereas proselytizing, being a missionary, is something else.

I'm going to allow N to have infinite-digit members for now, in my namespace. I'll call the set N+ when it matters.  The aleph stuff all breaks but I wasn't using it anyway.

314159...

---------  = 3.14159... = pi

100000...

 

just doesn't seem that problematic to me, in terms of representing pi.

Note that I'm keeping:

Convergent / Repeating

Chaotic

Divergent

as flavors of infinite sequences, and as "numbers" in this lexicon.  

0.101010101010101.... is an oscillating convergent number whereas irrational numbers are chaotic (non-repeating) and may actually have algorithms behind them (may not).  

When it comes to computation, I only care about finite state machine renderings i.e. I do not define what it means to "compute" with anything infinite (it's never been done).  

Trying to compute with infinity digits just leads to a morass of nonsense results.  But I'm happy to have a taxonomy for sequences anyway.  The chaotic vs. repeating distinction is useful, relates to periodic / aperiodic when tiling floors or using space-filling volumes. 314159... is a divergent number.

Of course I'm free to use the aleph stuff and conform to social norms when I want a math teaching job.  Established dogmas rule the roost and all that.  I can pass as a true believer.  

And I do think N < Z < Q < R < C is useful regardless of the aleph stuff i.e. that's not about cardinality but computational properties.  int < (int, int) < float & extended precision < complex is more how it looks in computer memory i.e. the finite space of computation (not entirely equivalent to 1800s metaphysics I agree).  

Complex numbers are built from pairs of floats or extended precision decimals whereas rationals are from pairs of ints.  Pythonic Math has all these types well-defined and implemented, no need to reinvent the wheel here.  

Sets are just another data structure, not a foundation of anything, just one more tool.

<< SNIP >>

BTW, if you are insistent on disallowing any set that has a larger cardinality than N, you also need to say this:  "The set of all subsets of N doesn't exist, either", because that also has greater cardinality than N.

I think it's more a matter of disallowing any set with lower cardinality than R -- if it's infinite, it's infinite.  Infinite-digit members of N put N on a par with R.  That's why we can't allow them.  But if we're shifting to different "foundations" anyway... (for those who think in those terms).

Kirby

 

[Mike South]

First off, you might note that in my email I explicitly added the infinite number of 1's in the sister sequence.  It's still countable if you do that.

The crux of what michel was pointing out was this:

Kirby's listing does have holes.

He was just giving the examples that came to mind as he looked at that listing.

I helpfully showed how to cover every hole that michel brought up, I think, although I didn't set out to do that intentionally.

Every argument for trisection of the angle via compass and straightedge is flawed, because it's not possible to do.

In the same way, Kirby's generator is flawed, because it claims to be able to do something that is not possible to do.  While michel was helpful in pointing out specific examples that don't show up after any finite number of steps in Kirby's generator, it turns out that it's not all that hard to cover the specific holes that michel pointed out.

However, it's irrelevant that the set {all the holes our little group has noticed so far} is countable and can be added to Kirby's generator (the Kirby-South-Cantor-Paul generator, let's say, were you cover the .010101 hole with my suggestion of adding Cantor's list of rationals to it, and you cover the .11111... hole by adding that sister sequence inverting the original restricted to [0..1] Kirby sequence).

It's irrelevant because there are infinitely many other holes.  Pi, in fact, is a hole.  It's a hole because of this:

There is no step n, where n is some natural number in N, by which you can say "we got to pi at step n!  woohoo!".  

It has nothing to do with what a finite state machine accepts, or the dedekind cuts, or even the real numbers, except as a consequence.

You can't put all the infinite sequences of bagel, airplane, flower into a list.  You'll be missing one, and I can construct that one that you're missing in a very clear and well defined way.  THAT is the "QED", not "we've defined the naturals to be countable and we refuse to accept any other definition".

Any other definition would not be the naturals.  It would be the naturals plus things that break the definition of what the naturals are, and you end up with stuff like 3.14159.. being equal to .314159....  We don't include infinite sequences of digits in N because they have nothing to do with what N means to us ("start with one and keep counting").  When you say "this set B is the same size as N", you HAVE to show that you can get to every b in B with a finite number of steps, because that's what N *is*.  If you want to argue that the reals are the same size as N, you need to show me a list where every real shows up at some n for every single r in R.  Not just "the reals I've thought of so far" or "the reals I'm comfortable with".

But, like I said, just give me a list of all the airplane, bagel, flower sequences.  Forget about the reals and Dedekind and Cauchy and even Cantor.

There's a reason that Kirby had to start changing the definition of the naturals to something else very quickly in his arguments, because N is only so big, and there are infinitely many sets that are infinitely bigger.

BTW, the reals don't get some kind of pass.  If you construct the set of all the subsets of the reals, it's bigger than the reals on the same scale that the reals are bigger than N.

But forget the reals, and just give me a generator for all the bagel, airplane, flower sequences.  I know it may be maddening to hear, but you are not going to be able to do it, because you can't, because the naturals are not big enough to contain them.

Just to be clear--for each bagel, airplane, flower sequence, you must tell me the natural number at which that particular sequence was arrived at.  This was something we could have corrected very early on in Kirby's argument (sorry Kirby, I don't mean to "third person" you here, I'm just using your name as a handle for the argument you made) if we'd noticed it.  When he was saying "we will get there eventually" he was really saying "we're making a bunch of successive approximations--isn't that enough?  After all pi=3.14159... is a successive approximation, and you don't have a problem with that" (all quotes here are paraphrase) but the reals are *precisely* what we discovered when we started asking "where do all those successive approximations lead to?".  And by adding all those convergences, we found that we had waaaaaay more things than the naturals can list.

But then we realized we don't even need to talk about the reals, just all the lists of bagel, airplane, flower, and that is enough to overwhelm the natural numbers and prove to us that there are infinities that are bigger than N.

mike

[Joseph Austin]

On Jul 17, 2016, at 4:57 PM, michel paul <python...@gmail.com> wrote:

However, the reason you can't actually construct the diagonal is because you cannot actually construct the list​ in the first place, and that is Cantor's point.

​If you think you've created the list, even algorithmically, sorry, you've left something out.​

How is that not a circular argument?

Don't fractals demonstrate that you can fill a surface with a line?

My problem is, I can construct "algebraic irrationals" and e and π and all the useful transcendentals,

so other than "by definition", I haven't met a number yet that I "need" but can't compute.

So rather than stress my mind with orders of infinities, 

I would prefer to develop a consistent hierarchy of constructions for the number I can compute.

Instead of walking off a cliff into the "infinite space" of the "reals" once I get to the diagonal of a unit square or the "ratio" (isn't that a misnomer) of circumference to diameter, why not just keep pairing the numbers I already have with indicated inverse operators?

"e" (as an example of transcendentals) is definable as the limit of a computable sequence (1+1/n)^n,

which follows geometrically as the solution of the differential equation y' = y.

So "limit" is a new kind of operation, but it can be expressed in a finite symbol-string.

So we ought to be able to develop an "algebra" of limits,  just as well have an algebra of rationals.

And if we can agree on a "ratio" of two infinite sequences, what sense does it make to say the two "infinities" are "equal"?

Cantor's definition of "reals" may theoretically require uncountable numbers.

So why is that not a defect in Cantor's definition, rather than a requirement for uncountability?

Especially since we never encounter that situation in practice.

In computing, we define "uncomputable" differently, not as "too many to list" but as "too ambiguous to resolve".

Joe

[Mike South]

>The aleph stuff all breaks but I wasn't using it anyway.

One of your <SNIP>s cut out the part where I showed that it wasn't just "the aleph stuff" that breaks.  

When you put your infinite digit strings into N, you made 3.14 equal to .314.

Are you happy with the fact that in your namespace 3.14 = .314?

mike

[michel paul]

On Fri, Jul 22, 2016 at 9:42 AM, Joseph Austin <drtec...@gmail.com> wrote:

On Jul 17, 2016, at 4:57 PM, michel paul <python...@gmail.com> wrote:

However, the reason you can't actually construct the diagonal is because you cannot actually construct the list​ in the first place, and that is Cantor's point.

​If you think you've created the list, even algorithmically, sorry, you've left something out.​

How is that not a circular argument?

​It is not circular, because it assumes the opposite of what it asserts.

The diagonal argument assumes that the reals have been listed and then shows that the assumption leads to a contradiction.

If you assume that you can list the reals, the diagonal argument proves that you cannot.

​If you want to assert otherwise, fine, go ahead and create the list. That will shut everyone up.

Granted, you can't actually write an infinite list, so you're allowed to do it virtually via a clever algorithm.

However, no one has ever done it, and that has a whole lot to do with the fact that it cannot be done. 

Just like you cannot find a rational number which when squared will give you 2. Close? Yeah. Exact? No.

​> ​

I would prefer to develop a consistent hierarchy of constructions for the number I can compute.

​OK, the rationals work just fine for practical purposes.​

​> ​

So we ought to be able to develop an "algebra" of limits,  just as well have an algebra of rationals.

​Yep. That's been done. 

It's called Analysis.​

--

​Michel

[michel paul]

On Fri, Jul 22, 2016 at 8:48 AM, Joseph Austin <drtec...@gmail.com> wrote:

I'm getting way behind, or out of sequence, but I thought I answered this.

Your list is simply reverse binary integers

​ ... ​

we can alway use the "mirror" algorithm as a generator.

​Joe,

The issue we keep coming back to is that any suggested list is incomplete.

Yes, "​we can alway use the "mirror" algorithm as a generator", or we might try using something else, but it will never generate a complete list.

So - we can in fact list all the rationals, in principle, using a generator, and here is an example.

The list comprehension only lists 100 of them, but you can change that to any value you like.

I use mediants to construct the rationals between 0/1 and 1/1. Creating reciprocals and opposites from that list using further list comprehensions accounts for everything else, and that is what I did using visual Python in  ​

​

Generating the Rational Numbers.

So we have established both logically and through example that the rationals can be listed. 

I've done it.

Can the reals be listed? Or, an equivalent question, can a generator be coded that will systematically list ALL possible infinite strings?

Nope. 

At least no one has ever done it, but you're welcome to try.

​The diagonal argument suggests a strong reason for why the attempt might turn out to be futile, but again, you're welcome to try.

--

​ Michel

[kirby urner]

On Fri, Jul 22, 2016 at 10:56 AM, Mike South <mso...@gmail.com> wrote:

>The aleph stuff all breaks but I wasn't using it anyway.

One of your <SNIP>s cut out the part where I showed that it wasn't just "the aleph stuff" that breaks.  

When you put your infinite digit strings into N, you made 3.14 equal to .314.

Are you happy with the fact that in your namespace 3.14 = .314?

mike

I thought I'd addressed that by explaining the algorithm in more detail, controlling the zeros.

314/100 = 157/50 = 3.14

3141/1000 = 3141/1000 = 3.141

31415/10000 = 6283/2000 = 3.1415

314159/100000 = 314159/100000 = 3.14159

3141592/1000000 = 392699/125000 = 3.141592

31415926/10000000 = 15707963/5000000 = 3.1415926

314159265/100000000 = 62831853/20000000 = 3.14159265

3141592653/1000000000 = 3141592653/1000000000 = 3.141592653

31415926535/10000000000 = 6283185307/2000000000 = 3.1415926535

314159265358/100000000000 = 157079632679/50000000000 = 3.14159265358 

...

Here's the generator I'm using.  Do you see a bug?

from fractions import Fraction

def pi_digits():

    k, a, b, a1, b1 = 2, 4, 1, 12, 4

    while True:

        p, q, k = k*k, 2*k+1, k+1

        a, b, a1, b1 = a1, b1, p*a+q*a1, p*b+q*b1

        d, d1 = a/b, a1/b1

        while d == d1:

            yield int(d)

            a, a1 = 10*(a%b), 10*(a1%b1)

            d, d1 = a/b, a1/b1

        

def next_rational():

    pi = pi_digits()

    denom = "100"

    numer = str(next(pi)) + str(next(pi)) + str(next(pi))

    while True:

        print("{}/{} = ".format(numer, denom), end="")

        yield Fraction(int(numer), int(denom))

        denom += "0"

        numer += str(next(pi))

        

algorithmic_number = next_rational()

for _ in range(10):

    not_yet_pi = next(algorithmic_number)

    print(not_yet_pi, "=", float(not_yet_pi))

    

[Joseph Austin]

Kirby,

My example was the integers, not the natural numbers.

But there's nothing in the Peano definition about "digits", or how you lay them out in space.

(But let us agree that by "left" you meant "most significant" direction.)

So I do not concur with the verdict of your "authority".

On Jul 18, 2016, at 1:48 PM, kirby urner <kirby...@gmail.com> wrote:

BTW, Kirby, if you are following, this is an example of infinite digits on the left!

Not allowed.  

Members of N must have only finite digit representations according to these authorities:

http://math.stackexchange.com/questions/58085/a-number-with-an-infinite-number-of-digits-is-a-natural-number

 

[Joseph Austin]

Perhaps my quarrel is not with the theorem statement, but with the so-called "proof".

I think that was Kirby's original objection also.

But perhaps my quarrel is also with the definition of the "reals",

which is that there is a non-countable set of "trans-rational" numbers which are necessary for traditional mathematics.  

I'm conjecturing that the numbers we use in algebra, analytic geometry, calculus, etc are all computable,

that is, operationally definable, not just exist "by definition". 

So the statement of "all infinite strings of two characters" a "meaningless" concept.

Joe

On Jul 18, 2016, at 4:48 PM, michel paul <python...@gmail.com> wrote:

On Mon, Jul 18, 2016 at 9:24 AM, kirby urner <kirby...@gmail.com> wrote:

​> ​

I thought the challenge was to come up with an algorithm that wouldn't have any gaps if we kept at it.

​Correct. And it still is.   : )

 

​> ​

Are the decimal digits of pi countable (listable) or not in your view?

​They are listable and therefore countable in both of our views.

However the set of all infinite strings created from a binary character set is NOT listable.

This fact takes us by surprise. Like the Pythagoreans regarding incommensurability, we might prefer that it be otherwise.

We can code a function that will list all sequences of binary digits of a specified length, and we are tempted to generalize our method for sequences of indefinite length.

But we cannot.

We will never generate a string preceded by an infinite number of 1s.

​It would be a possible string, but we will never reach it using that algorithm.

--

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

[kirby urner]

On Fri, Jul 22, 2016 at 2:11 PM, Joseph Austin <drtec...@gmail.com> wrote:

Kirby,

My example was the integers, not the natural numbers.

But there's nothing in the Peano definition about "digits", or how you lay them out in space.

(But let us agree that by "left" you meant "most significant" direction.)

So I do not concur with the verdict of your "authority".

You mean my "authority" the voices I'm quoting in the URL?  Yes, I think that's what you mean, not my authority as I'm not a participant in said stackexchange

http://math.stackexchange.com/questions/58085/a-number-with-an-infinite-number-of-digits-is-a-natural-number

The math heads posting there seem pretty adamant, and the poster asking the question seems to eventually cave, though I detect notes of dissent in the comments.  

I cited this URL because it seems the common wisdom that infinity-digit numbers may extend to the right but not to the left of a dot.

You and I seem in agreement we may call this infinity-digit patterns "numbers" and characterize them in various ways (e.g. chaotic), but when it comes to computation, these are not what gets used, either in nature or anywhere else (wherever that "else place" might be).

Kirby

 

[kirby urner]

I don't think we need to fall back on the 1800s to have the last word.  They had their metaphysics, we have ours, which may include some family resemblance to theirs.  Or some of us may want to hold on to those "foundations" as they were imagined to be, lots of what today we'd call "advertising" went into those, or call it "spin".  Likewise when it comes to "dimensions", Abbott's 'Flatland' should never be bleeped over, as though it was a work of political satire, it later became a cornerstone for hypercross geometries.

Kirby