countably infinite nodes with uncountably infinite paths!!

[michel paul]

OK everybody! 

I think this will be very interesting pertaining to our discussion of countability and infinity. 

Get comfortable, and get ready to visualize and think.

We are going to visualize an infinite complete binary tree where the root node, 

row 0, 

consists of a decimal point, 

and every other row consists of alternating 0s and 1s:

.

0               1

0       1       0       1

0   1   0   1   0   1   0   1

That's kind of a clunky text version, but 

imagine from each element 

a left and a right connector to the two elements directly below.

Allow that pattern to continue indefinitely.

Each row n will contain 2^n elements.

You can see that our row 3 contains 8 elements.

Following all pathways top-down will yield the 8 binary decimals from .000 through .111.

As n increases indefinitely

every conceivable infinite string of 0s and 1s will appear in that tree.

Isn't that interesting?

Every single infinite string from

.0000...  to  .1111...

will appear in that tree.

OK?

Now, get ready ...

Though n is countably infinite,

the number of pathways from top-down is uncountably infinite!

The following paragraph is from types of binary trees:

• In the infinite complete binary tree, every node has two children (and so the set of levels is countably infinite). The set of all nodes is countably infinite, but the set of all infinite paths from the root is uncountable, having the cardinality of the continuum. These paths correspond by an order-preserving bijection to the points of the Cantor set, or (using the example of a Stern–Brocot tree) to the set of positive irrational numbers.

What's worth noting here - this simply seems to follow from the structure of the tree.

-- Michel

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

- Richard Feynman

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

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

On Fri, Jul 22, 2016 at 8:05 PM, michel paul <python...@gmail.com> wrote:

Here is another Rational Number Generator coded according to the structure of the Stern-Brocot Tree.

Back when I was exploring these ideas for my computational math class I first used mediants along with Farey sequences, then a few years later I stumbled on the Stern-Brocot tree.

Turns out it's actually more efficient than finding mediants.

--------------------------------

​

michel:

​ ​

Does it make sense to talk about a generator of all possible rational numbers? Yes.

​

​

kirby

​

:​

​ ​

I'm not so sure.

​michel: ​

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.

​

​

--------------------------------

​

--

​Michel​

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

- Richard Feynman

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

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

[Mike South]

This is great!  At first you're thinking, 'wait, I can enumerate every node...' but, yeah, it's that you have so many decisions available that "does the trick" of putting you out of countable infinity.

I'm pretty sure this is equivalent to constructing the "power set"  ( which means "the set of all subsets" see https://www.mathsisfun.com/sets/power-set.html ) of the naturals.

If you think of .1111111... as "include the first natural, include the second natural" etc and then .0111111111 means "not the first natural , but include all the others" and .0000000... is the empty set (none of the naturals is included).

So if you look at all the paths down your tree, you're making a decision each time you pick the left or right branch to include the corresponding natural or not in your subset, and the total of all the different paths is you having made every possible decision to include or exclude each natural.

This one way to see that the power set of a countably infinite set is uncountable, in other words.

That's beautiful.

mike

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[kirby urner]

On Sun, Jul 24, 2016 at 12:48 AM, michel paul <python...@gmail.com> wrote:

What's worth noting here - this simply seems to follow from the structure of the tree.

-- Michel

This seems about the same as my algorithm except I was writing the permutations of 1s and 0s down the left and

then juggling the "dot" as one more symbol to permute, going across so that, so getting all numbers we know how to represent with these three symbols {"0","1", "."} given n slots to fill.

Each path through your tree corresponds to the index into a row in my table

Path:  .0011001010101

Row:  .0011001010101, 0.011001010101,  00.11001010101, 001.1001010101.... 0011001010101.

The permutations of 1s and 0s may be described as a growing tree. 

They may also be written out going down the left.

When you do it Mike says it's great because you're "doing the trick" whereas when I do it it's in need of refutation because my motives are suspect.

So how would the tree look if counting all the rational numbers Q between .0 and 0.1111... ? 

You're not allowed to stop adding rows as Q is an infinite set as well.  I'd say at first glance there'd be no way to tell which set you were counting (or proving uncountable), if just given the graphic of the tree with 1s and 0s.

Kirby

[michel paul]

On Sun, Jul 24, 2016 at 8:39 AM, kirby urner <kirby...@gmail.com> wrote:

> ​

So how would the tree look if counting all the rational numbers Q between .0 and 0.1111... ?  

​The tree itself would look the same. The difference would lie only in what we were interested in counting.

The tree would have the same structure, but only a subset of the strings would correspond to what we call the rationals.

Every string that ended in an infinite string of 0s or that repeated a cycle of digits for forever would correspond to a rational, and everything else would not.

Any attempt on our part to establish a one-to-one correspondence between the strings embedded in our tree and N or Q would fail. 

Any systematic linear listing of the infinite strings we could find in this tree would end up being incomplete.

​> ​

You're not allowed to stop adding rows as Q is an infinite set as well.  I'd say at first glance there'd be no way to tell which set you were counting (or proving uncountable), if just given the graphic of the tree with 1s and 0s.

​Yeah, the tree is just a tree. It is simply a structure that happens to contain any conceivable infinite string of binary digits ​you might want. If you can imagine it, it is in there.

The strings are just there waiting to be teased out. We can then categorize them according to their properties.

This is what I love about math - it always turns out to be more than what we assumed!   : )

Math is surprising. It is the way it is in spite of us. We pride ourselves in being in control, but that is just an illusion. We create our little representations of mathematical structures and play with them, and sometimes we even do useful things with them, but the deep stuff comes up every once in awhile and slaps us in the face.

--

​ Michel

[kirby urner]

On Sun, Jul 24, 2016 at 9:09 AM, michel paul <python...@gmail.com> wrote:

On Sun, Jul 24, 2016 at 8:39 AM, kirby urner <kirby...@gmail.com> wrote:

> ​

So how would the tree look if counting all the rational numbers Q between .0 and 0.1111... ?  

​The tree itself would look the same. The difference would lie only in what we were interested in counting.

Or not counting. 

Q is listable (using your nomenclature), R is not. 

So in the one Q-tree you're arguing for countability, in the other case (R-tree) not.

Different hand-waving gestures might pertain. 

Stomp foot, say in certain terms:  with this tree, we have just proved the countability of all Q between 0.0 and 0.1111...

or...

... make shrugging gesture, speak more softly in awed tone:  R is forever beyond our ability to count, whole different Aleph number.

Math is beautiful, right kids?  The Aleph Stuff is just brilliant!  (clearing throat noise, ahem ahem).

Anyway, that's pretty high level metaphysics for 6th graders and I'd recommend against spending more than a few minutes on the Aleph Stuff at that grade level (even for adults as ACPs pertain) -- they can spiral back through in college if choosing to become philosophy majors. ACPs = Adult Career Paths, e.g. returning vets who didn't get through high school.

In the meantime, I'm tweeting up a storm about #3Dprinting A and B modules, T, E and S modules, not in @Thingiverse yet but with @TinkerCad or one of those we'll have @OMSI and @MultcoLib giving kids the opportunity to learn about the concentric hierarchy and the space-filling rhombic dodeca of volume 6, in a summer school context, maybe in just a few days. #CodeCastle. 

Python comes in here too, when sharing #3Dprinting & Geometry, as a volumes-computing machine. 

Remember: Tetrahedron = Unit Volume in this particular 4D Geometry (different meaning of 4D).

Kirby

[michel paul]

On Sun, Jul 24, 2016 at 9:47 AM, kirby urner <kirby...@gmail.com> wrote:

​> ​

So in the one Q-tree you're arguing for countability, in the other case (R-tree) not.

​No, there is only one tree. 

This one tree has many different types of strings that we can find.

This one tree, an infinite complete binary tree, contains every infinite binary character string you can imagine.

The strings come in different types. The different structures have different properties that we can study.​

​> ​

Different hand-waving gestures might pertain. 

​Well, only if that's what you want to do, or if you feel so inspired, but hand waving isn't necessary.​

​> ​

Stomp foot, say in certain terms:  with this tree, we have just proved the countability of all Q between 0.0 and 0.1111...

or...

... make shrugging gesture, speak more softly in awed tone:  R is forever beyond our ability to count, whole different Aleph number.

Math is beautiful, right kids?  The Aleph Stuff is just brilliant!  (clearing throat noise, ahem ahem).

​???

I'm not quite sure what your point is here ...

You seem to be making a caricature of authoritarian behavior, and I am not sure what that is proving.

Facts are simply facts.

The structure of the tree is simply the structure of the tree.

​> ​

Anyway, that's pretty high level metaphysics for 6th graders and I'd recommend against spending more than a few minutes on the Aleph Stuff

Again ... ??? There's no need to even mention the term 'Aleph'.​ 

Fortunately when I was in the 8th grade I had a teacher who did spend time on this kind of stuff, and he inspired me deeply. 

Very deeply.

He had a lot to do with why I wanted to become a math teacher.

He would walk into class and draw on the board a line segment and a square and ask us which contained more points.

We would spend the whole class period arguing about that. Literally the whole period, and I loved it.​

​One day he put on the board: "1/9 = 0.1111..., 2/9 = 0.2222..., 3/9 = 0.3333..., ..., 8/9 = 0.8888..., 9/9 = 1."

He asked us in a very puzzled way, "What happened to the 0.9999...? Shouldn't it be there? Why did the pattern break?"

He didn't just 'tell' us that 0.9999... = 1, and he also didn't just 'prove' it for us.

He also didn't stomp his foot or speak in softly awed tones. He just kept asking really good questions and encouraging us to figure it out.​

We spent the whole period trying to figure it out, and I loved it.

He also never used the term 'Aleph'. Never. Not once. He simply asked questions, and we tried to figure things out.

He recommended that we read certain books, one of which was called One, Two, Three, ... Infinity by George Gamow. It is a classic. I did read it during the summer of my 8th grade year, and I loved it.

--

​Michel

[michel paul]

On Sun, Jul 24, 2016 at 10:27 AM, michel paul <python...@gmail.com> wrote:

​> ​

Facts are simply facts.

​I must correct myself here.

Facts are not 'simply' facts!

: )

I also love Wittgenstein.​

 

--

​Michel

[kirby urner]

On Sun, Jul 24, 2016 at 10:34 AM, michel paul <python...@gmail.com> wrote:

On Sun, Jul 24, 2016 at 10:27 AM, michel paul <python...@gmail.com> wrote:

​> ​

Facts are simply facts.

​I must correct myself here.

Facts are not 'simply' facts!

: )

Did your teacher talk about the Infinite Monkey Theorem I wonder?  I brought that up earlier as a connected topic that usually surfaces in these kinds of debates.

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

I also love Wittgenstein.​

 

--

​Michel

Excellent.

I see his work cited thrice under References on this Wikipedia page (cited earlier) about the controversy over Cantor's theory:

https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory

Perhaps you'd care to elucidate on what Wittgenstein's views were, and why they'd be germane to this thread?

Kirby

[kirby urner]

On Sun, Jul 24, 2016 at 10:27 AM, michel paul <python...@gmail.com> wrote:

On Sun, Jul 24, 2016 at 9:47 AM, kirby urner <kirby...@gmail.com> wrote:

​> ​

So in the one Q-tree you're arguing for countability, in the other case (R-tree) not.

​No, there is only one tree. 

Right, same tree, same printed graphic.

Whether it's used to prove the countability of 0.0 < Q < 0.111... or the uncountability of 0.0 < R < 0.111... is in the eye of the beholder.  Either way, we never stop adding rows.

But in the latter case it "does the trick" (as Mike put it) whereas in the former case we stay grounded at Aleph-0.  I think in the latter case it might help to squint.

Those are the facts.

Kirby

[Mike South]

On Sun, Jul 24, 2016 at 10:39 AM, kirby urner <kirby...@gmail.com> wrote:

On Sun, Jul 24, 2016 at 12:48 AM, michel paul <python...@gmail.com> wrote:

What's worth noting here - this simply seems to follow from the structure of the tree.

-- Michel

This seems about the same as my algorithm except I was writing the permutations of 1s and 0s down the left and

then juggling the "dot" as one more symbol to permute, going across so that, so getting all numbers we know how to represent with these three symbols {"0","1", "."} given n slots to fill.

Each path through your tree corresponds to the index into a row in my table

Path:  .0011001010101

Row:  .0011001010101, 0.011001010101,  00.11001010101, 001.1001010101.... 0011001010101.

The permutations of 1s and 0s may be described as a growing tree. 

They may also be written out going down the left.

When you do it Mike says it's great because you're "doing the trick" whereas when I do it it's in need of refutation because my motives are suspect.

So how would the tree look if counting all the rational numbers Q between .0 and 0.1111... ? 

It's the same tree, you're just choosing to take a subset of the paths down it. 

You're not allowed to stop adding rows as Q is an infinite set as well.  I'd say at first glance there'd be no way to tell which set you were counting (or proving uncountable), if just given the graphic of the tree with 1s and 0s.

The graphic isn't proving anything about countability or uncountability.  The graphic is most definitely countable.  (That's what I liked about it, btw--the counterintuitive fact that it's countable, but it has uncountably many paths through it--not to mention that other awesome thing about it showing you the uncountability of the power set of N).  Anywhere you cut off the bottom of the tree would, in fact, be "too soon" to get all the rationals.  Like 1/3, which is .101010101...  You have that exactly right.

Just because people are more used to it, let's make it a ten-ary tree to talk about the rationals.  It will be harder to draw:

.

| \ \ \ \ \ \ \ \ \

0 1 2 3 4 5 6 7 8 9

[and I can't draw the lines here in ascii, but imagine 0 having ten lines to the first set of 0 1 2 3 4 5 6 7 8 9 in the next line down, and 1 with ten lines to the next set, etc]

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

 

So, if you want the rationals, you have to be very restrictive--infinitely restrictive, you might say--in how you travel down that tree.  A rational is a ratio of two integers, and the decimal expansion therefore always ends in a repeating pattern--either all zeros after some point, like 1/4 = .2500000 or some other repeating string like 1/7 = 0.142857142857142857... or 1/3 = .33333.

So, if you are traversing all the paths in the tree that take you to rational numbers, you decide, after some finite set of steps, to stop making new decisions about which path to take, and from that point you just repeat some string of those decisions infinitely, like "I'm going from 1 to 4 to 2 to 8 to 5 to 7 to 1 to 2 to 4 to 8 to 5 to 7..." in the case of 1/7th.  It's *because* you stop breaking new ground after a finite number of steps that we can safely state that, yes, I can number all the paths that do that, no problem.

Just to be clear, though, I don't think the diagram is meant in any way to show either the uncountability of the reals or the countability of the rationals.  At least, I don't see it doing either of those things.  But it shows you that *each individual real* that is expressed as a binary expansion can be arrived at in a countably infinite number of steps, and it sort of gives you an organized way of visualizing all the strings of the type .011010000101... , which is all the more fascinating because we know we can't make a list of them.

It's a countably-sized diagram which contains an uncountable number of paths through it.  That's pretty cool.

mike

 

Kirby

[kirby urner]

On Sun, Jul 24, 2016 at 12:31 PM, kirby urner <kirby...@gmail.com> wrote:

On Sun, Jul 24, 2016 at 10:34 AM, michel paul <python...@gmail.com> wrote:

On Sun, Jul 24, 2016 at 10:27 AM, michel paul <python...@gmail.com> wrote:

​> ​

Facts are simply facts.

​I must correct myself here.

Facts are not 'simply' facts!

: )

Did your teacher talk about the Infinite Monkey Theorem I wonder?  I brought that up earlier as a connected topic that usually surfaces in these kinds of debates.

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

NOTE:  I acknowledge the case where we claim the proof is empirical, because Shakespeare fits our definition of "monkey" for all intents and purposes (so what about a few chromosomal differences, don't be so picky).  So duh, a monkey already *has* come up with Hamlet, a tale told by an idiot, that monkeys also read.

Kirby

 

[michel paul]

On Tue, Jul 26, 2016 at 11:47 AM, kirby urner <kirby...@gmail.com> wrote:

​> ​

Did your teacher talk about the Infinite Monkey Theorem I wonder? 

​I don't believe so.

It is possible that it came up in class discussion from a student, but only possibly. I don't remember it happening, but it could have. ​

The discussions were wild mixes of all kinds of things. After all, we were 8th graders.

​What I do remember happening is trying to figure out a ways to map any given point on a line segment to a determined point on a square, and vice versa. ​

It was not a formal thing - it was just this simple idea of one-to-one correspondence. Is it possible to establish it?

​Same with the 9/9 thing not yielding .999... . Hey, what happened here? Why did the pattern break?​

I have to give that guy a lot of credit. He was able to ask questions in a way to provoke us to think about some pretty deep math.

​> ​

NOTE:  I acknowledge the case where we claim the proof is empirical, because Shakespeare fits our definition of "monkey" for all intents and purposes (so what about a few chromosomal differences, don't be so picky).  So duh, a monkey already *has* come up with Hamlet, a tale told by an idiot, that monkeys also read.

​: )​

--

[michel paul]

Reflecting on the question of countability and infinity, it seems like the primary reason for uneasiness has to do with accepting a completed infinity. However, it seems that no one has a problem with the idea of 'one more'. 

I like The Math Page, and I highly recommend his The Evolution of the Real Numbers. He makes an interesting argument there against the existence of an arithmetic continuum.

It has to do with numbers and naming. We have not and will never be able to name all the numbers in the continuum. 

And that is what is interesting about the structure of this binary tree.

We have no problem considering a tree for extremely large n.

Counting the number of complete pathways from  the root is clear, 2^n, but as soon as we try to imagine an infinite n, the tree turns into a fractal, self-similarity everywhere, and evaluating the number of pathways becomes problematic.

A unique path can be found through that tree to represent any nameable value you can provide, like pi, e, phi, sqrt(2).

However, we will never be able to name them all, even algorithmically, because what can be named can be listed.

It also is the case that we will never actually construct that tree!  : )

--

[Joseph Austin]

On Aug 1, 2016, at 7:02 PM, michel paul <python...@gmail.com> wrote:

I like The Math Page, and I highly recommend his The Evolution of the Real Numbers. He makes an interesting argument there against the existence of an arithmetic continuum.

I skimmed it, and he seems to be making many of the same points I've been trying to make.

I don't deny irrational numbers. 

I don't really deny that it is impossible to list/count/name all elements of the set of all infinite sets of digits,

although I remain skeptical of the Cantor "proof" of that.

What I challenge is that we "need" numbers that we can't name/list/count/compute.

Or even that we have any "use" for them.

Joe

[Mike South]

Don't know if you guys noticed this page on that site (it was linked to on the last page of the "Evolution" series, but not linked as "next" in the navigation, so maybe some people missed it).

http://www.themathpage.com/aCalc/anumber.htm

He summarizes and, I would say, more forcefully states his position on the reals being "fantasy math" (arguably almost all math that is done fits this category, btw.  The reals just happen to hit the intersection between deep theoretical math and readily practically applicable math.  If you could see the stuff that 90% of the mathematicians are doing 90% of the time, you would think the reals were the most concrete, normal, real-world things, ever :) ).

Note that my linking to it is not an endorsement of his logic :).

However, this argument, in my opinion, is a lot tighter than what I saw of the video.  Even if I find some of the assertions a bit questionable.  A strong advantage that it has, from my perspective, is that it's pretty clear that he's not making these comments due to not understanding the math.

mike

 

Joe

[michel paul]

On Tue, Aug 2, 2016 at 8:07 PM, Joseph Austin <drtec...@gmail.com> wrote:

​> ​

What I challenge is that we "need" numbers that we can't name/list/count/compute.

​> ​

Or even that we have any "use" for them.

​Well, it definitely would be hard to find a way to use a number that you can't name/list/count/compute, so it would seem to follow that we certainly don't need such a thing.​

But, ​I've actually never heard anyone or read any text that made the claim that we do need such things, so I'm not sure what the challenge is.

​Again, the fact of incommensurability provoked people. It made them wonder. A question like by how much should the linear dimensions of a statue be extended in order to double its volume? was kind of like what quantum physics is for us. It just didn't make sense. It pushed against the boundaries of thought.

Was there a need to find a rational formula for the exact amount?  No. Approximations can work just fine.

​But humans have a really deep need to understand things, and so they go exploring.​

--

​Michel

[michel paul]

On Tue, Aug 2, 2016 at 9:21 PM, Mike South <mso...@gmail.com> wrote:

​> 

this argument, in my opinion, is a lot tighter than what I saw of the video.  Even if I find some of the assertions a bit questionable.  A strong advantage that it has, from my perspective, is that it's pretty clear that he's not making these comments due to not understanding the math.

​Right. I appreciate his clarity, because it allows discussion to be focused and productive, but I also hold his conclusions in the 'very interesting but requires further attention' category.

I find the semantic rejection argument of the continuum significant, because this theme of numbering and naming points to something deep. 

It reminds me of Taoism: 

The Tao that can be spoken is not the eternal Tao

The name that can be named is not the eternal name

​.​

​I always come back to the perspective that mathematics is more than just a human creation. Humans are mathematically very creative animals, but I think it's arrogant for us to say that we are sufficient to account for its existence, and this theme of numbering and naming I think gets right to the heart of things philosophically.

Things like incommensurability and the (so far) unpredictable distribution of the primes show themselves. We do not decide that things should be this way - we cannot help but conclude that they are. History shows that people wanted things to be otherwise, but ... surprise!

: )

[michel paul]

From http://www.themathpage.com/aCalc/anumber.htm: 

Time, distance, motion are continuous.  Numbers are not. That is the tension between geometry and arithmetic, a tension realized by Pythagoras with his discovery of what we call the irrational, and he called "without a name" (alogos). 

Again, I appreciate how he clearly points out important ideas.

Yeah, it was from geometry that incommensurability was discovered, and the Pythagorean notion of number was first challenged, and it was because of this crisis that Plato based his own cosmology on geometry rather than arithmetic. That's why he had the sign "Let no one ignorant of geometry enter" over the door to his academy.

Now a very interesting thing is that time, distance, and motion might not be continuous! It might just be the best way we have of thinking about these things. It might just be the result of how we are constrained by our neural structure.

These are things humans are still working on, and its really fascinating that this is the case after thousands of years of wondering.

--

[Joseph Austin]

Mike, 

Yes, I did notice that page.

I might replace "rational" with "computable" in his discussion of limit.

In the case of irrationals like pi, 

instead of "the" number L,

might we define the "limit" as the "interval" between two sequences,

e.g. the ratios of inscribed and circumscribed polygons of a circle to the diameter,

since we cannot exhibit the "number" L for comparison with a single sequence alleged to converge toward it?

So a "limit" does not necessarily "approach" a "number" but merely describes an ever-shrinking interval, a "smudge" if you will!

If the "smudge" gets small enough, can we pretend it's not really "spread out", that there is just one "number" in there,

even though we can never compute a precise "name" for it? 

And I would argue that, since we can use an arbitrary number base, or mixed bases, for the digits, 

we could have an infinite number of distinct names (i.e. sequences with distinct values) for the same "limit".

(Digit sequences in bases 2, 3, and 5 will never have any rational "points" in common, though we might say they all converge to the "same" number.)

Joe

On Aug 3, 2016, at 12:21 AM, Mike South <mso...@gmail.com> wrote

Don't know if you guys noticed this page on that site (it was linked to on the last page of the "Evolution" series, but not linked as "next" in the navigation, so maybe some people missed it).

http://www.themathpage.com/aCalc/anumber.htm

He summarizes and, I would say, more forcefully states his position on the reals being "fantasy math" (arguably almost all math that is done fits this category, btw.  The reals just happen to hit the intersection between deep theoretical math and readily practically applicable math.  If you could see the stuff that 90% of the mathematicians are doing 90% of the time, you would think the reals were the most concrete, normal, real-world things, ever :) ).

Note that my linking to it is not an endorsement of his logic :).

However, this argument, in my opinion, is a lot tighter than what I saw of the video.  Even if I find some of the assertions a bit questionable.  A strong advantage that it has, from my perspective, is that it's pretty clear that he's not making these comments due to not understanding the math.

mike

 

[Joseph Austin]

Discussing the Cantor proof an related topics has led me to conclude that I am not alone in my skepticism of the "real" number system.

Perhaps that places me with critics of earlier ages who rejected "new things" that unsettled their orthodoxy.

Except that Cantor is the "orthodoxy" and the computer is the "new thing" that is unsettling it.

Joe

[michel paul]

On Thu, Aug 4, 2016 at 2:40 PM, Joseph Austin <drtec...@gmail.com> wrote:

​> ​

Cantor is the "orthodoxy" and the computer is the "new thing" that is unsettling it.

Well, consider that 'the computer' here would actually be 'computer science'.

It's not that computers suddenly showed up and then started ​upsetting set theorists.

It's more like computer science arose when mathematicians like Turing, Church, Von Neumann and others created ways to formalize our notions of computation.

And it is also true that with the advent of computers our ability to automize abstraction has changed what it means to do mathematics.

​Lots of things are shifting.

​Again, I keep pointing to the work of Voevodsky in shifting foundations from set theory to type theory, because that move would make mathematics and computer science the same thing.

--

​Michel

[Joseph Austin]

Type theory? One more thing to learn!

I'm beginning to understand why people retire; 

we will never "finish";

at some point, one must simply turn over the reins to the next generation!

But i do believe, at the end of the millennium, the computer will regarded be one of the crucial turning points in civilization,

right up there with fire and the wheel and writing.

Joe

[michel paul]

On Fri, Aug 5, 2016 at 10:09 AM, Joseph Austin <drtec...@gmail.com> wrote:

​> 

i do believe, at the end of the millennium, the computer will regarded be one of the crucial turning points in civilization,

​ ​

right up there with fire and the wheel and writing.

​Without a doubt. Like the printing press. Democracy became possible because of it. Already social media has unleashed a new culture that we don't understand yet.

I think the larger impact, of which the computer is a part, will come from information theory. I believe it has already changed how we understand physical reality.

I used to believe that humans created information, but then I came to understand that information creates us, and that makes you go "hmmm ...".​

 

--

​Michel

[Andrius Kulikauskas]

very nice :)

I wonder what this means for the origin of math?

> I used to believe that humans created information, but then I came to
> understand that information creates us, and that makes you go "hmmm ...".​
> --
> ​Michel
> ​






>
> ===================================
> "What I cannot create, I do not understand."
>
>
> - Richard Feynman
> ===================================
> "Computer science is the new mathematics."
>
>
> - Dr. Christos Papadimitriou
> ===================================
>
>

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--
Andrius Kulikauskas
+370 607 27 665
m...@ms.lt

[Joseph Austin]

Michel,

This may be getting far afield from math, but it occurs to me, that in the realm of intellectual-social organization, 

we are starting to see a new dimension of Evolution, one it which humans are just the cells rather than the species.

Was the "king" ever in sole control of the state, or just the clearinghouse for the pressures from the population?  

--any more than a given thinker, no matter how brilliant, can control the evolution of mathematics or science.

As in the biological realm, the success of a school of thought depends on it's ability to reproduce itself in the next generation of thinkers.

I fear many a worthy advance is lost because we pay less attention to how to teach it than how to "think" it.

We teach "what" and "how", but we need to be teaching "why".  The right question is ever more important than the right answer.

Joe

[Mike South]

Joe,

You may be interested in The Evolution of Everything by Matt Ridley.  I heard about it on Econ Talk:

http://www.econtalk.org/archives/2016/02/matt_ridley_on_1.html

The idea you're expressing here is similar to some of the ideas he's working on in the book.

mike

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[jennifer kurtz]

Joe and Mike~

Another book that might be of interest....

Why Information Grows--The Evolution of Order, from Atoms to Economies by Cesar Hidalgo

http://www.rethinkeconomics.org/opinion/2015/07/from-atoms-to-economies-a-review-of-cesar-hidalgos-why-information-grows/

Peace~ Jenn

---

From: Mike South <mso...@gmail.com>
To: mathf...@googlegroups.com
Sent: Saturday, August 6, 2016 3:50 PM
Subject: Re: [Math Future] countably infinite nodes with uncountably infinite paths!!

Joe,

You may be interested in The Evolution of Everything by Matt Ridley.  I heard about it on Econ Talk:

http://www.econtalk.org/archives/2016/02/matt_ridley_on_1.html

The idea you're expressing here is similar to some of the ideas he's working on in the book.

mike

On Sat, Aug 6, 2016 at 7:51 AM, Joseph Austin <drtec...@gmail.com> wrote:

Michel,

This may be getting far afield from math, but it occurs to me, that in the realm of intellectual-social organization, 

we are starting to see a new dimension of Evolution, one it which humans are just the cells rather than the species.

Was the "king" ever in sole control of the state, or just the clearinghouse for the pressures from the population?  

--any more than a given thinker, no matter how brilliant, can control the evolution of mathematics or science.

As in the biological realm, the success of a school of thought depends on it's ability to reproduce itself in the next generation of thinkers.

I fear many a worthy advance is lost because we pay less attention to how to teach it than how to "think" it.

We teach "what" and "how", but we need to be teaching "why".  The right question is ever more important than the right answer.

Joe

On Aug 6, 2016, at 3:05 AM, michel paul <python...@gmail.com> wrote:

I think the larger impact, of which the computer is a part, will come from information theory. I believe it has already changed how we understand physical reality.

I used to believe that humans created information, but then I came to understand that information creates us, and that makes you go "hmmm ...".​

 

-- 

​Michel

​

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[michel paul]

I tend to say that patterns precede people, just like rocks do. 

In the unfolding of things rocks came before people, and so did patterns. 

Just as we learn to perceive rocks and trees and other stuff, we also learn to perceive patterns. 

In order for people to be able to evolve in the first place natural processes had to lay a foundation for the perception of things like rocks. In order to become tool using animals, people had to be able to perceive rocks and imagine them differently.

And certain animals have learned to do the same.

Same thing with patterns. People are pattern seeking animals. Sometimes they will perceive patterns that are not actually essential to the matter at hand, but the ability to perceive pattern at all has been tremendously important for people.

Of course, that doesn't necessarily make us unique. We used to think we were unique because we used tools and had language. Well ... it turns out that that kind of stuff happens with other creatures as well.

And it turns out that other creatures demonstrate number sense, quantitative reasoning, pattern recognition ...

It just seems to me that if we are going to look at the natural world as non-dogmatically as possible, it seems that other animals demonstrate ways of being that we used to assert were uniquely human.

Nature thinks. Proof: us.

Does she think in ways other than us?

Sure. Why not?

[Andrius Kulikauskas]

Michel, Joe, Thank you for this thread on infinity and foundations! I'm
very interested in these topics but will write separately. Andrius

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