Can math.atan2 return INF?

[Steven D'Aprano]

Are there any circumstances where math.atan2(a, b) can return an infinity?

I know it will return a NAN under some circumstances.



--
Steven

[Pierre-Alain Dorange]

Steven D'Aprano <st...@pearwood.info> wrote:

> Are there any circumstances where math.atan2(a, b) can return an infinity?
>
> I know it will return a NAN under some circumstances.

atan or atan2 can't return INFINITE, it was the arc tangent of a value,
then arc tangent can only be between -PI and +PI (is was an angle).

I do not know under what circumstance atan2 can return NAN, atan2 taks 2
argument (y and x) and return the angle corresponding to y/x.
If x is 0.0, atan2 return 0.0 (do not try to make the division).

atan2 was used over atan to optimize and so that it can used the correct
quadrant (for angle answer).

--
Pierre-Alain Dorange <http://microwar.sourceforge.net/>

Ce message est sous licence Creative Commons "by-nc-sa-2.0"
<http://creativecommons.org/licenses/by-nc-sa/2.0/fr/>

[Jussi Piitulainen]

pdor...@pas-de-pub-merci.mac.com (Pierre-Alain Dorange) writes:

> Steven D'Aprano <st...@pearwood.info> wrote:
>
>> Are there any circumstances where math.atan2(a, b) can return an infinity?
>>
>> I know it will return a NAN under some circumstances.
>
> atan or atan2 can't return INFINITE, it was the arc tangent of a
> value, then arc tangent can only be between -PI and +PI (is was an
> angle).
>
> I do not know under what circumstance atan2 can return NAN, atan2 taks
> 2 argument (y and x) and return the angle corresponding to y/x. If x
> is 0.0, atan2 return 0.0 (do not try to make the division).

It seems to return NaN when either argument is NaN. It seems to return
finite values when either argument is Inf or -Inf, and negative zeroes
lead to different but finite answers.

Wikipedia says ata2(0,0) is considered undefined traditionally and in
symbolic maths but otherwise most computer implementations define it.
Python seems to produce +-0.0 and +-pi for variously signed zeroes.

I didn't see any mention of it ever being infinite, which makes sense
given that arcus functions are supposed to return angles. The extreme
case seemed to be the inclusion of -pi as a possible value.

(I know nothing, I was just curious enough to play with this a bit.)

[Steven D'Aprano]

On Wed, 22 Jun 2016 04:01 am, Pierre-Alain Dorange wrote:

> Steven D'Aprano <st...@pearwood.info> wrote:
>
>> Are there any circumstances where math.atan2(a, b) can return an
>> infinity?
>>
>> I know it will return a NAN under some circumstances.
>
> atan or atan2 can't return INFINITE, it was the arc tangent of a value,
> then arc tangent can only be between -PI and +PI (is was an angle).

I should hope it wouldn't. That's why I'm asking.

> I do not know under what circumstance atan2 can return NAN, atan2 taks 2
> argument (y and x) and return the angle corresponding to y/x.
> If x is 0.0, atan2 return 0.0 (do not try to make the division).

py> math.atan2(NAN, 0)
nan

I think that the only way it will return a NAN is if passed a NAN.



--
Steven

[Steven D'Aprano]

On Wed, 22 Jun 2016 04:32 am, Jussi Piitulainen wrote:

> pdor...@pas-de-pub-merci.mac.com (Pierre-Alain Dorange) writes:
>
>> Steven D'Aprano <st...@pearwood.info> wrote:
>>
>>> Are there any circumstances where math.atan2(a, b) can return an
>>> infinity?

[...]

> I didn't see any mention of it ever being infinite, which makes sense
> given that arcus functions are supposed to return angles. The extreme
> case seemed to be the inclusion of -pi as a possible value.
>
> (I know nothing, I was just curious enough to play with this a bit.)

Thanks, I'm in the same position as you, except that I'm in the position
where I need it use the result, and if it ever returns INF my function will
blow up. But it doesn't look like that can happen.



--
Steven

[Pierre-Alain Dorange]

yes of course if you pass an invalid argument (NAN is not a real value,
atan2 except coordinate x,y), the result would be invalid...

[Ben Bacarisse]

Steven D'Aprano <st...@pearwood.info> writes:

> On Wed, 22 Jun 2016 04:01 am, Pierre-Alain Dorange wrote:

<snip>

>> I do not know under what circumstance atan2 can return NAN, atan2 taks 2
>> argument (y and x) and return the angle corresponding to y/x.
>> If x is 0.0, atan2 return 0.0 (do not try to make the division).
>
> py> math.atan2(NAN, 0)
> nan
>
> I think that the only way it will return a NAN is if passed a NAN.

That seems to be the case but I was a little surprised to find that

>>> math.atan2(INF, INF)
0.7853981633974483

I would have expected NaN since atan2(INF, INF) could be thought of as
the limit of atan2(x, y) which could be any value in the range. And I'd
have guessed atan2(0, 0) would have been NaN too but

>>> math.atan2(0, 0)
0.0

--
Ben.

[Random832]

In C, the result of atan2(0, 0) [for any sign of zero] may be
implementation-dependent: "A domain error may occur if both arguments
are zero." In CPython, though, they're explicitly handled as special
cases by the m_atan2 function. The results match that of the x87 FPATAN
instruction, and presumably the IEEE standard (the same results are
prescribed in the IEC 60559 appendix of the C standard)

http://www.charlespetzold.com/blog/2008/09/180741.html mentions Intel's
rationale (in short, it's because 0+0j is a real number).

[Pierre-Alain Dorange]

Ben Bacarisse <ben.u...@bsb.me.uk> wrote:

> >>> math.atan2(INF, INF)
> 0.7853981633974483
>
> I would have expected NaN since atan2(INF, INF) could be thought of as
> the limit of atan2(x, y) which could be any value in the range. And I'd
> have guessed atan2(0, 0) would have been NaN too but

i'm not a math expert, but the limit of atan2 would be 45°, so pi/4
radians (0,7854).
As x,y are coordinates, the both infinite would tend toward 45°.

x only infinite would be 0° (0 radians)
y only infinite woudl be 180° (pi/2 radians)

[Ben Bacarisse]

pdor...@pas-de-pub-merci.mac.com (Pierre-Alain Dorange) writes:

> Ben Bacarisse <ben.u...@bsb.me.uk> wrote:
>
>> >>> math.atan2(INF, INF)
>> 0.7853981633974483
>>
>> I would have expected NaN since atan2(INF, INF) could be thought of as
>> the limit of atan2(x, y) which could be any value in the range. And I'd
>> have guessed atan2(0, 0) would have been NaN too but
>
> i'm not a math expert, but the limit of atan2 would be 45°, so pi/4
> radians (0,7854).
> As x,y are coordinates, the both infinite would tend toward 45°.

The limit of atan2(x, x) is as you describe, but there is no reason to
pick that one case. lim{x->oo,y->oo}atan2(x, y) is undefined unless a
relationship is given between x and y and you get get any result you
like in the range of atan2 by choosing one or other relationship.

--
Ben.

[Lawrence D’Oliveiro]

On Thursday, June 23, 2016 at 7:17:37 AM UTC+12, Ben Bacarisse wrote:

> The limit of atan2(x, x) is as you describe, but there is no reason to
> pick that one case.

It’s what’s called a “non-removable discontinuity”. The value you pick at that point will be consistent with approaching it from one particular direction, not from any other.

[Steven D'Aprano]

On Thu, 23 Jun 2016 05:17 am, Ben Bacarisse wrote:

> pdor...@pas-de-pub-merci.mac.com (Pierre-Alain Dorange) writes:
>
>> Ben Bacarisse <ben.u...@bsb.me.uk> wrote:
>>
>>> >>> math.atan2(INF, INF)
>>> 0.7853981633974483
>>>
>>> I would have expected NaN since atan2(INF, INF) could be thought of as
>>> the limit of atan2(x, y) which could be any value in the range. And I'd
>>> have guessed atan2(0, 0) would have been NaN too but
>>
>> i'm not a math expert, but the limit of atan2 would be 45°, so pi/4
>> radians (0,7854).
>> As x,y are coordinates, the both infinite would tend toward 45°.
>
> The limit of atan2(x, x) is as you describe, but there is no reason to
> pick that one case.

Given:

x = INF
y = INF
assert x == y

there is a reason to pick atan2(y, x) = pi/4:

Since x == y, the answer should be the same as for any other pair of x == y.

It might not be a *great* reason, but it's a reason.



--
Steven

[Dan Sommers]

On Thu, 23 Jun 2016 13:59:46 +1000, Steven D'Aprano wrote:

> Given:
>
> x = INF
> y = INF
> assert x == y
>
> there is a reason to pick atan2(y, x) = pi/4:
>
> Since x == y, the answer should be the same as for any other pair of x == y.

When x == y == 0, then atan2(y, x) is 0.

[Steven D'Aprano]

On Thursday 23 June 2016 14:40, Dan Sommers wrote:

>> Since x == y, the answer should be the same as for any other pair of x == y.
>
> When x == y == 0, then atan2(y, x) is 0.

/s/any other pair of x == y/any other pair of x y except for zero/

:-P


Zero is exceptional in many ways.


--
Steve

[Ben Bacarisse]

Steven D'Aprano <st...@pearwood.info> writes:

...provided you consider atan2(0, 0) == 0 to be a bug!

--
Ben.

[Ben Bacarisse]

Steven D'Aprano <steve+comp....@pearwood.info> writes:

> On Thursday 23 June 2016 14:40, Dan Sommers wrote:
>
>>> Since x == y, the answer should be the same as for any other pair of x == y.
>>
>> When x == y == 0, then atan2(y, x) is 0.

I see just added noise by making the same comment before reading the
rest of the thread. Sorry.

> /s/any other pair of x == y/any other pair of x y except for zero/
>
> :-P
>
>
> Zero is exceptional in many ways.

whereas infinity... :-)

--
Ben.

[alister]

which infinity. There are many - some larger than others



--
| |-sshd---tcsh-+-dpkg-buildpacka---rules---sh---make---make---sh---
make---sh---make---sh---make---sh---make---sh---make
-- While packaging XFree86 for Debian GNU/Linux

[Steven D'Aprano]

On Fri, 24 Jun 2016 01:04 am, alister wrote:

> which infinity. There are many - some larger than others

China has just announced a new supercomputer that is so fast it can run an
infinite loop in 3.7 seconds.


--
Steven

[Pierre-Alain Dorange]

Dan Sommers <d...@tombstonezero.net> wrote:

> > Given:
> >
> > x = INF
> > y = INF
> > assert x == y
> >
> > there is a reason to pick atan2(y, x) = pi/4:
> >
> > Since x == y, the answer should be the same as for any other pair of x == y.
>
> When x == y == 0, then atan2(y, x) is 0.

This is the only solution (0) where atan2 return sometime else than
pi/4. And INF is not 0, it can be lot of values, but not zero.

Yes x=INF and y=INF do not mean x=y (in math world) but INF is a
frontier that can be reach, so it tend to the maximum value. So the
tendancy is always atan2(y,x) tend to pi/4 if you looks at lot od y and
x that will be grater and greater each time : the final frontier would
always be pi/4, even if t take a long time to reach it.

[Pierre-Alain Dorange]

Steven D'Aprano <st...@pearwood.info> wrote:

> > which infinity. There are many - some larger than others
>
> China has just announced a new supercomputer that is so fast it can run an
> infinite loop in 3.7 seconds.

Near a black hole 3.7 seconds can last an infinite time...

[Marko Rauhamaa]

pdor...@pas-de-pub-merci.mac.com (Pierre-Alain Dorange):

> Steven D'Aprano <st...@pearwood.info> wrote:
>> China has just announced a new supercomputer that is so fast it can
>> run an infinite loop in 3.7 seconds.
>
> Near a black hole 3.7 seconds can last an infinite time...

Which phenomenon prevents a black hole from ever forming. Yet
astronomers keep telling us they are all over the place.

Oppenheimer and his co-authors interpreted the singularity at the
boundary of the Schwarzschild radius as indicating that this was the
boundary of a bubble in which time stopped. This is a valid point of
view for external observers, but not for infalling observers. Because
of this property, the collapsed stars were called "frozen stars",
because an outside observer would see the surface of the star frozen
in time at the instant where its collapse takes it inside the
Schwarzschild radius.
<URL: https://en.wikipedia.org/wiki/Black_hole>

Note that the "valid point of view for external observers" is the only
valid scientific point of view.


Marko

[Ben Bacarisse]

alister <aliste...@ntlworld.com> writes:

> On Thu, 23 Jun 2016 15:39:43 +0100, Ben Bacarisse wrote:
>
>> Steven D'Aprano <steve+comp....@pearwood.info> writes:
>>
>>> On Thursday 23 June 2016 14:40, Dan Sommers wrote:
>>>
>>>>> Since x == y, the answer should be the same as for any other pair of
>>>>> x == y.
>>>>
>>>> When x == y == 0, then atan2(y, x) is 0.
>>
>> I see just added noise by making the same comment before reading the
>> rest of the thread. Sorry.
>>
>>> /s/any other pair of x == y/any other pair of x y except for zero/
>>>
>>> :-P
>>>
>>>
>>> Zero is exceptional in many ways.
>>
>> whereas infinity... :-)
>
> which infinity. There are many - some larger than others

The one in the post up thread now (sadly) snipped. It's not a
mathematical infinity at all but a particular floating point
representation that results from float("inf"). However it is still just
as "exceptional in many ways" as zero. (Sorry if you were making a joke
and I didn't get it.)

--
Ben.

[Pierre-Alain Dorange]

Marko Rauhamaa <ma...@pacujo.net> wrote:

> Note that the "valid point of view for external observers" is the only
> valid scientific point of view.

For a scientific point of view, right. But tell this to the one that
will be close to a blackhole ;-)

--
Pierre-Alain Dorange Moof <http://clarus.chez-alice.fr/>

[Marko Rauhamaa]

pdor...@pas-de-pub-merci.mac.com (Pierre-Alain Dorange):

> Marko Rauhamaa <ma...@pacujo.net> wrote:
>
>> Note that the "valid point of view for external observers" is the
>> only valid scientific point of view.
>
> For a scientific point of view, right. But tell this to the one that
> will be close to a blackhole ;-)

Then, you'd better consult a priest than a scientist.


Marko

[Gregory Ewing]

Steven D'Aprano wrote:
> China has just announced a new supercomputer that is so fast it can run an
> infinite loop in 3.7 seconds.

They're lying. It has to be NaN seconds.

--
Greg

[MRAB]

If it was an Indian supercomputer, it would be naan seconds.

Sorry. :-)

[Gregory Ewing]

Marko Rauhamaa wrote:
> pdor...@pas-de-pub-merci.mac.com (Pierre-Alain Dorange):

>
>>Near a black hole 3.7 seconds can last an infinite time...
>
> Which phenomenon prevents a black hole from ever forming. Yet
> astronomers keep telling us they are all over the place.

Astronomers have observed objects whose behaviour is
entirely consistent with the existence of black holes
as predicted by general relativity.

> Oppenheimer and his co-authors interpreted the singularity at the
> boundary of the Schwarzschild radius as indicating that this was the
> boundary of a bubble in which time stopped. This is a valid point of
> view for external observers, but not for infalling observers.
>

> Note that the "valid point of view for external observers" is the only
> valid scientific point of view.

The singularity being talked about there is an artifact
of a particular coordinate system; the theory predicts that
there is no *physical* singularity at the event horizon.

It's true that we outside can't be absolutely sure that things
are as predicted at the horizon itself, because any observer
we sent in to check would be unable to report back. But in
principle we can observe arbitrarily close to it. The
observations we've made so far all fit the theory, and the
theory doesn't present any obstacles to extrapolating those
results to the horizon and beyond, so we accept the theory
as valid.

There *is* a difficulty at the very center of the hole, where
there is a true singularity in the theory, so something
else must happen there. But for other reasons we don't
expect those effects to become important until you get
very close to the singularity -- something on the order of
the Planck length.

--
Greg

[Gregory Ewing]

Marko Rauhamaa wrote:
> pdor...@pas-de-pub-merci.mac.com (Pierre-Alain Dorange):
>

>>For a scientific point of view, right. But tell this to the one that
>>will be close to a blackhole ;-)
>
> Then, you'd better consult a priest than a scientist.

But don't worry, you'll have an infinitely long time
to make your confessions -- from our point of view,
anyway.

--
Greg

[Marko Rauhamaa]

Gregory Ewing <greg....@canterbury.ac.nz>:

> Marko Rauhamaa wrote:
>> Which phenomenon prevents a black hole from ever forming. Yet
>> astronomers keep telling us they are all over the place.
> Astronomers have observed objects whose behaviour is entirely
> consistent with the existence of black holes as predicted by general
> relativity.

As far as I understand, all we can ever observe is black holes in the
making since the making can never (seem to) finish. IOW, the event
horizon never forms.

These almost-black-holes are virtually indistinguishable from black
holes proper. However, we don't have to speculate about the physics of
the insides of the black hole.

> The singularity being talked about there is an artifact of a
> particular coordinate system; the theory predicts that there is no
> *physical* singularity at the event horizon.

That theory can't be tested even in principle, can it? Therefore, it is
not scientific.

> It's true that we outside can't be absolutely sure that things are as
> predicted at the horizon itself, because any observer we sent in to
> check would be unable to report back. But in principle we can observe
> arbitrarily close to it. The observations we've made so far all fit
> the theory, and the theory doesn't present any obstacles to
> extrapolating those results to the horizon and beyond, so we accept
> the theory as valid.

Religious theories about the afterlife face similar difficulties -- and
present similar extrapolations.

> There *is* a difficulty at the very center of the hole, where there is
> a true singularity in the theory, so something else must happen there.
> But for other reasons we don't expect those effects to become
> important until you get very close to the singularity -- something on
> the order of the Planck length.

That's my point: such speculation must remaing mere speculation. The
universe doesn't owe us an answer to a question that we can never face.


Marko

[Gregory Ewing]

Marko Rauhamaa wrote:
>>The singularity being talked about there is an artifact of a
>>particular coordinate system; the theory predicts that there is no
>>*physical* singularity at the event horizon.
>
> That theory can't be tested even in principle, can it? Therefore, it is
> not scientific.

It can in principle be tested by a scientist falling into
the hole. The only problem is that he won't be able to
tell anyone outside what he finds out, but that's a
practical difficulty, not a philosophical one.

A lot of what the early Greeks found out got lost in
various library burnings, etc. Does that mean they
weren't being scientific?

> Religious theories about the afterlife face similar difficulties -- and
> present similar extrapolations.

I don't think they're similar at all. Show me the equations
for one of these religious theories and I might change
my mind...

--
Greg

[Steven D'Aprano]

On Mon, 27 Jun 2016 09:08 am, Gregory Ewing wrote:

> Marko Rauhamaa wrote:
>>>The singularity being talked about there is an artifact of a
>>>particular coordinate system; the theory predicts that there is no
>>>*physical* singularity at the event horizon.
>>
>> That theory can't be tested even in principle, can it? Therefore, it is
>> not scientific.
>
> It can in principle be tested by a scientist falling into
> the hole. The only problem is that he won't be able to
> tell anyone outside what he finds out, but that's a
> practical difficulty, not a philosophical one.

Marko's complaint about black holes seems to be based on a very naive
definition of "scientific", specifically Karl Popper's naive empirical
falsification theory of science. Unfortunately, falsification is not even
close to a good description of what scientists do in their day-to-day work.

Naive empirical falsification can, at best, be considered as a best-practice
rule: if you have no way of falsifying something even in principle, then
it's not scientific. But it doesn't really give you much in the way of
practical guidance. What counts as falsification? How do you falsify
historical events like "the Earth formed from a cloud of gas"? We weren't
there to observe it, we can't repeat the experiment, and the entire process
from start to finish takes too long for anyone to watch a cloud of gas
coalesce into a solid planet.

So, black holes...

We have no way of seeing what goes on past the black hole's event horizon,
since light cannot escape. But we can still see *some* properties of black
holes, even through their event horizon: their mass, any electric charge
they may hold, their angular momentum. We can test the proposition that a
black hole that forms from hydrogen is no different from one which forms
from uranium. We can look for variations in randomness in the Hawking
radiation emitted, we can test that the event horizon is where we expect,
etc. An electrically neutral black hole with a magnetic field would likely
falsify a lot of theories about what goes on inside the event horizon.

And it may be that some future advance in quantum gravity theory will
suggest a way of testing the prediction of a singularity. There are
theories of black holes that predict "ring shaped" singularities, and
suggest that in principle one might "miss the singularity" and fall out of
a worm hole at the other end, although its doubtful that this would apply
to anything bigger than an atom.

I don't think many physicists actually believe that the singularity is a
real thing, rather than just a failure of our current gravitational
theories to correctly model matter under extreme conditions. After all,
we've been here before, with the prediction that a black-body should
radiate an infinite amount of energy at a finite temperature:

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

[...]

>> Religious theories about the afterlife face similar difficulties -- and
>> present similar extrapolations.
>
> I don't think they're similar at all. Show me the equations
> for one of these religious theories and I might change
> my mind...

I don't think that the essential difference between a scientific theory and
a non-scientific one is the presence or absence of *equations*.

There's a lot of grey area between science and non-science, but I think a
good start is to ask, "how do you know?".

If the answer comes down to one of the following:


- divine revelation, including from gurus, angels and spirits;
- visions and inspiration;
- "it just stands to reason";
- "because otherwise, what would be the point?"

then its not scientific. The last means its wishful thinking. Maybe there is
no point. Perhaps things just are, and meaning is what we decide on, not an
inherent part of the universe. The third is just a cop-out. If you can't
explain the reason, there probably isn't one. And the first two are
necessarily subjective and forms of argument by authority:

All swans are white[1] because The Master said so, and who are you to
question The Master?

Whereas, I think that for a first degree approximation, we can say that
science must be *objective*. Often that does mean it involved equations,
after all the laws of mathematics are the same for all of us. But objective
fact does not necessarily require maths. Even in the ancient days of
humanity, we can be pretty sure that two Neandertals stepping out of their
cave to watch the sun rise in the east would agree on where the light was
coming from. If one faced into the sun and shaded her eyes, while the other
turned his back on the sun and shaded his eyes, we can be confident that
the second was mucking about :-)

And that's where all forms of religious revelation fail. Ultimately,
revelation divides the world into two:

- those who personally know the truth;
- and those who just have to take their word for it.


"God wants you to give me your money, honest. Oh, and he also doesn't want
you to eat carrots. Don't question the Lord!"






[1] Apart from black swans, which came as an awful shock for philosophers
when they learned of their existence.


--
Steven
“Cheer up,” they said, “things could be worse.” So I cheered up, and sure
enough, things got worse.

[Marko Rauhamaa]

Steven D'Aprano <st...@pearwood.info>:

> Naive empirical falsification can, at best, be considered as a
> best-practice rule: if you have no way of falsifying something even in
> principle, then it's not scientific. But it doesn't really give you
> much in the way of practical guidance. What counts as falsification?

We cannot get any information on black holes proper because black holes
cannot come into existence according to the very theory that predicts
black holes. It will take infinitely long for an event horizon to form.
Speculating on what happens to an astronaut falling in is not much
different from speculating what happens after the end of the world.

> We have no way of seeing what goes on past the black hole's event
> horizon, since light cannot escape. But we can still see *some*
> properties of black holes, even through their event horizon: their
> mass, any electric charge they may hold, their angular momentum.

If an event horizon cannot come into existence, you can only see
properties of almost-black-holes. Even though there probably is
virtually no difference between the two astronomically, it relieves
physicists from answering some awkward questions on the goings-on inside
an event horizon.

> We can test the proposition that a black hole that forms from hydrogen
> is no different from one which forms from uranium. We can look for
> variations in randomness in the Hawking radiation emitted, we can test
> that the event horizon is where we expect, etc. An electrically
> neutral black hole with a magnetic field would likely falsify a lot of
> theories about what goes on inside the event horizon.

If an event horizon cannot ever form, you can't really test any of that
stuff.


Marko

[Rustom Mody]

On Monday, June 27, 2016 at 12:10:21 PM UTC+5:30, Marko Rauhamaa wrote:
> Steven D'Aprano :

I am reminded of an argument I once had with a colleague about
infinite, lazy data-structures

I said that for the Haskell list [0..]

[0..] ++ [-1] == [0..]

++ is like python's list append +

This could equally apply to a Python generator like:

def nats():
i=0
while True:
yield i
i += 1

He said (in effect) yes that -1 would not be detectable but its still there!

[Nagy László Zsolt]

> Thanks, I'm in the same position as you, except that I'm in the position
> where I need it use the result, and if it ever returns INF my function will
> blow up. But it doesn't look like that can happen.
>
Doesn't atan2 relies on the C lib math floating point library? At least
in CPython. I think (but not 100% sure) that it is implementation
dependent, and returns what is returned by clib floating point lib.
Since the meaning of atan2 is an angle, it should always be between
[-pi,pi]. The only exception maybe when there is a NaN in the arguments.
But I cannot think of any interpretation where atan2 would return Inf or
-Inf.

[Marko Rauhamaa]

Rustom Mody <rusto...@gmail.com>:

> I am reminded of an argument I once had with a colleague about
> infinite, lazy data-structures
>
> I said that for the Haskell list [0..]
>
> [0..] ++ [-1] == [0..]

[...]

> He said (in effect) yes that -1 would not be detectable but its still
> there!

Georg Cantor would probably be with your colleague, but then, Georg
Cantor was not a scientist.


Marko

[Rustom Mody]

On Monday, June 27, 2016 at 7:16:03 PM UTC+5:30, Marko Rauhamaa wrote:
> Rustom Mody :

I'm mystified

Earlier (I thought) you were on the Platonist = {Cantor, Hilbert...} side
Now you sound like you are on the constructivist = {Kronecker, Brouwer } side

[Marko Rauhamaa]

Rustom Mody <rusto...@gmail.com>:

I'm a formalist.


Marko

[Rustom Mody]

On Monday, June 27, 2016 at 7:42:26 PM UTC+5:30, Marko Rauhamaa wrote:
> Rustom Mody :

Well then formalism is semantics-free: What matters it if an argument is
theological or scientific as long as it is (internally) consistent?

[Marko Rauhamaa]

Rustom Mody <rusto...@gmail.com>:

That's what I'm saying: black holes can't exist according to the very
theory that predicts their existence.

They might exist in reality, though...


Marko

[Gregory Ewing]

Marko Rauhamaa wrote:
> We cannot get any information on black holes proper because black holes
> cannot come into existence according to the very theory that predicts
> black holes. It will take infinitely long for an event horizon to form.

Only in some frames of reference.

By your reasoning, Zeno's paradox proves that a runner
can never reach the finish line in a race. But it really
only proves that if you measure time in such a way that
the finishing time is infinitely far in your future, you
will never see him finish.

That's obviously a screwy way to measure time in a
race, but something similar is happening with the black
hole. If you draw coordinate lines in a particular
way (corresponding to the inertial frame of an outside
observer stationary with respect to the hole) then
the time axis bends in such a way that it never
crosses the horizon.

But there's no reason you have to draw the coordinates
that way; there are plenty of others in which the time
axis does cross the horizon.

--
Greg

[Gregory Ewing]

Rustom Mody wrote:
> I said that for the Haskell list [0..]
>
> [0..] ++ [-1] == [0..]
>

> He said (in effect) yes that -1 would not be detectable but its still there!

The code to generate it is there, but it will never
be executed, so the compiler is entitled to optimise
it away. :-)

He may have a point though. There are avenues of
mathematics where people think about objects such
as "all the natural numbers, followed by -42", and
consider that to be something different from just
"all the natural numbers".

So, a mathematician would probably say they're not
equal. A scientist would say they may or may not be
equal, but the difference is not testable.

An engineer would say "Lessee, 0, 1, 2, 3, 4, 5,
6, 7... yep, they're equal to within measurement
error."

--
Greg

[Marko Rauhamaa]

Gregory Ewing <greg....@canterbury.ac.nz>:

> Marko Rauhamaa wrote:
>> We cannot get any information on black holes proper because black holes
>> cannot come into existence according to the very theory that predicts
>> black holes. It will take infinitely long for an event horizon to form.
>
> Only in some frames of reference.
>
> By your reasoning, Zeno's paradox proves that a runner can never reach
> the finish line in a race.

In Zeno's case, the limit is finite. Zeno's error is not realizing that
you can pack an infinite number of jiffies in finite time. In the black
hole case, the limit is infinite.

> But it really only proves that if you measure time in such a way that
> the finishing time is infinitely far in your future, you will never
> see him finish.

An external observer never experiences any effect whatsoever (direct or
indirect) from an event horizon or a black hole.

> But there's no reason you have to draw the coordinates that way; there
> are plenty of others in which the time axis does cross the horizon.

Not where I'm standing.


Marko

[Rustom Mody]

Yes there is a sloppiness in my statement above:

[0..] ++ [-1] == [0..]

What kind of '==' is that?
If its the Haskell (or generally, programming language implementation) version
that expression just hangs trying to find the end of the infinite lists.

If its not then a devil's advocate could well say:
"So its metaphysical, theological and can know the unknowable,
viz. that that -1 which is computationally undetectable is nevertheless present.

ie the '++' can be a lazy Haskell *implemented* function
The '==' OTOH is something at least quasi mystical

Mathematicians are more likely to say 'mathematical' than 'mystical'
Such mathematicians -- the majority -- are usually called 'Platonists'

[Steven D'Aprano]

And a programmer would write a script to compare the two, and then go to
Stackoverflow asking for help to optimize it because it takes too long to
complete.

Relevant:

https://blogs.msdn.microsoft.com/oldnewthing/20131029-00/?p=2803




--
Steve

[Random832]

On Sun, Jun 26, 2016, at 22:59, Steven D'Aprano wrote:
> We have no way of seeing what goes on past the black hole's event
> horizon, since light cannot escape. But we can still see *some*
> properties of black holes, even through their event horizon: their
> mass, any electric charge they may hold, their angular momentum.

All objects, not just black holes, have those properties. The point here
is that we are in fact observing those properties of an object that is
not yet (and never will be) a black hole in our frame of reference.

[Steven D'Aprano]

So what? You say that as if our frame of reference is privileged.

If you hop in your trusty rocket and fly towards the black hole, thinking
that it's all fine, its not "really" a black hole because "black holes
can't actually form", you'll be in for a really nasty surprise when you
cross the event horizon.

[Marko Rauhamaa]

Random832 <rand...@fastmail.com>:

> All objects, not just black holes, have those properties. The point
> here is that we are in fact observing those properties of an object
> that is not yet (and never will be) a black hole in our frame of
> reference.

A physicist once clarified to me that an almost-black-hole is
practically identical with a black hole because all information about
anything falling in is very quickly red-shifted to oblivion.

However, there is some information that (to my knowledge) is not
affected by the red shift. Here's a thought experiment:

----------
/ \
/ (almost) \ N
| black | |
| hole | S
\ /
\ /
----------

We have a stationary, uncharged (almost) black hole in our vicinity and
decide to send in a probe. We first align the probe so it is perfectly
still wrt the black hole and let it fall in. Inside the probe, we have a
powerful electrical magnet that our compass can detect from a safe
distance away. The probe is also sending us a steady ping over the
radio.

As the probe approaches the event horizon, the ping frequency falls
drastically and the signal frequency is red-shifted below our ability to
receive. However, our compass still points to the magnet and notices
that it "floats" on top of the event horizon:




----------
/ \
/ (almost) \ N
| black ||
| hole |S
\ /
\ /
----------


/
/ compass needle
/

[Marko Rauhamaa]

(sorry for the premature previous post)

Random832 <rand...@fastmail.com>:

> All objects, not just black holes, have those properties. The point
> here is that we are in fact observing those properties of an object
> that is not yet (and never will be) a black hole in our frame of
> reference.

The compass needle shows that the probe is "frozen" and won't budge no
matter how long we wait.


Marko

[Steven D'Aprano]

On Sunday 26 June 2016 09:40, Gregory Ewing wrote:

> Marko Rauhamaa wrote:
>> pdor...@pas-de-pub-merci.mac.com (Pierre-Alain Dorange):
>>
>>>Near a black hole 3.7 seconds can last an infinite time...
>>
>> Which phenomenon prevents a black hole from ever forming. Yet
>> astronomers keep telling us they are all over the place.
>
> Astronomers have observed objects whose behaviour is
> entirely consistent with the existence of black holes
> as predicted by general relativity.

Indeed.

There's a common myth going around that black holes take an infinite amount of
time to form, or another way of putting it is that it takes an infinite amount
of time for something to fall into a black hole, and therefore "black holes
can't really exist". This myth comes about because people don't fully
understand the (admittedly mind-boggling) implications of General Relativity.

First, you must accept that *your* experiences are not the only valid
experiences. Just because *you* never see the black hole form, doesn't mean it
doesn't form. You just don't get to experience it yourself.

So, let's consider a thought experiment. Imagine three astronauts, Tim, Bill
and Graham ("we do anything, anytime"). Tim and Graham are in separate space
ships, dropping straight towards a black hole under free fall. Bill is watching
them from a safely orbiting space station.

Graham drops towards the black hole, as does Tim. Neither can see the black
hole directly, or at least they can't see the *inside* of the black hole, since
no light can escape it, but they can see it by its effect on the starlight: the
black hole acts like a great big lens, bending light. If they line up their
space ships with the black hole directly between them and a distant star, they
will see one of the more amazing sights of the universe: gravitational lensing.
The (somewhat reddened) light from the star will be bent around the black hole,
so that the star will appear to be a donut of light with a black centre.

As Graham falls, he pays close attention to the distance from his ship to the
black hole. That's easy to do: he can tell how fast he is going thanks to
Bill's space station, which transmits a steady radio signal for him, a steady
clock sending one pulse per second.[1]

But as Graham gets closer and closer to the event horizon, he notices Bill's
radio signals have a higher and higher frequency, and appear to be sped up...
at the end, just before he loses his nerve and fires the retro-rockets before
crossing the event horizon, the signals are coming thousands of pulses per
second.

When Graham returns to the space station, he finds a *much* older Bill waiting
for him. Bill insists he was sending one pulse per second, as agreed, but that
Graham has been gone for many years. Graham insists that he has only been gone
a few days, and Bill has obviously been partying very hard indeed to look like
this after such a short time. But after sitting down with Albert Einstein's
head[2], they reconcile their two differing experiences:

As seen by Bill, in Bill's frame of reference far from the black hole, Graham's
experiences have been slowed down enormously. But Graham sees things
differently: he experiences his own frame of reference at the same speed he
always has, and see's *Bill's* frame of reference as being immensely sped up.
Neither is "right" and the other is "wrong", neither frame of reference is
privileged over the other. BOTH are right, even though they contradict each
other. That's the nature of the universe we live in.

What about Tim?

Tim is so engrossed by the view of the gravitational lensing that he forgets to
fire the retro-rockets, and before he knows it, he's crossed the event horizon
and there's no going back.

For a sufficiently large black hole, he might not even have noticed the
transition. From his perspective, he's still distant from the singularity
(being so small and distant, he can't quite make out what it looks like), and
space-time is still quite flat for a sufficiently large black hole. Tim can
still see out, although the incoming light is getting bluer, and he's still
receiving Bill's clock signals, though like Graham he sees them as drastically
sped up.

If Tim has sufficiently powerful rockets with enough fuel, he could *not quite*
escape: he could change his space ship's trajectory enough to avoid the
hypothetical singularity for days, weeks, years, as long as the fuel lasts. But
nothing he can do will allow him to escape the event horizon. (Well, maybe
faster-than-light travel, if his hyperdrive will still work this close to a
black hole.)

And as invariably as tomorrow follows today, he's getting closer to the
supposed singularity, and long before he reaches it, both Tim and his space
ship will be torn apart into atoms by the ever-increasing tidal forces, and
then even the atoms torn apart. And then, well we really have no idea what
happens if you try to squeeze an electron into a volume of space a trillion
times smaller than a sphere with radius equal to the Planck Length...

Should Tim realise his fate, and decide that there's no point delaying the
inevitable, his free-fall drop into the supposed singularity will be over in a
relatively short amount of time, at least according to his own frame of
reference. (For a stellar size black hole, the time from crossing the event
horizon to reaching the supposed singularity is a small fraction of a second.)

For a large black hole where the Schwartzchild Radius is 1 a.u., and ignoring
any relativistic corrections, it will take Tim six months of free fall to
collide with the singularity. During that time he will have plenty of time to
reflex on the curious way that Bill's space station is running faster and
faster, sending clock ticks millions of times a second. (At least until he is
torn apart by tidal forces.)

Meanwhile, back on Bill's space station, they're still faithfully sending out
radio pulses at the rate of one per second. By looking in their most powerful
telescopes, they can see Tim's spaceship falling into the black hole, strangely
slowing down as it approaches, the light getting redder and redder, Tim's own
radio signals getting further and further apart. Tim's spaceship appears to be
*asymptotically* approaching the event horizon, in some sort of horrible
version of Zeno's Paradoxes: each minute that goes by, Tim gets closer to the
event horizon by a *smaller* amount as time slows down for him (as seen by Bill
and Graham on the space station).

Graham decides to mount a rescue mission. He flies back towards the black hole,
and experiences the same speeding up of signals coming from Bill. But no matter
how closely he approaches the event horizon, Tim always appears to ahead of
him, like the turtle to Achilles (from Graham's frame of reference).

As seen by Bill, the closer Graham gets to Tim, the slower he appears to be
experiencing time, with his return clock ticks coming back one a day instead of
one per second, then one a week, one a month, ...

Unless Graham is willing to cross the event horizon too, he cannot catch up to
Tim. That's because, from Tim's perspective, Graham left the space station too
late. Tim has already experienced crossing the event horizon, so unless Graham
has a time machine, there's nothing Graham can do to reach Tim before he
crosses the event horizon. No matter how close Graham gets to the event
horizon, Tim will already be just a bit closer. Unless Graham is suicidally
willing to cross the event horizon too, he's going to have to pull out (after,
from his perspective, perhaps as little as a few hours of high-acceleration),
and return to the space station, where he will find that Bill has experienced
possibly many thousands of years.

So whose viewpoint is right? According to Bill and Graham, Tim is frozen just
before the event horizon, infinitely red-shifted, his radio signals coming in
infinitely slowly. But according to Tim, the opportunity for rescue is long
gone. He long ago watched Graham's idiotic rescue mission ("don't bother
Graham, I'm well past the event horizon, you can't save me now..."), the daring
flight almost to the event horizon, and Graham's eventual return to the space
station.

Although their perspectives are very different, neither is "more right" than
the other.

So does the black hole form? Do objects cross the event horizon? From the frame
of reference of such objects, yes, certainly, and in a fairly short period of
time too. From our frame of reference, we seem them asymptotically approaching
the event horizon, but never cross it. Both are equally correct.





[1] Or very possibly playing "A Walk In The Black Forest".

[2] In a jar.




--
Steve

[Marko Rauhamaa]

Steven D'Aprano <steve+comp....@pearwood.info>:

> There's a common myth going around that black holes take an infinite
> amount of time to form,

That appears to be the case. (Identical discussion points here: <URL:
http://astronomy.stackexchange.com/questions/2441/does-matter-accumulat
e-just-outside-the-event-horizon-of-a-black-hole>.)

> or another way of putting it is that it takes an infinite amount
> of time for something to fall into a black hole,

That's not another way of putting it. That's a completely different
story.

> and therefore "black holes can't really exist". This myth comes about
> because people don't fully understand the (admittedly mind-boggling)
> implications of General Relativity.

No, the fundamental question here is whether it makes scientific sense
to speculate about topics that are beyond the reach of science. Few
scientists speculate about what went on before the Big Bang, for
example.

> First, you must accept that *your* experiences are not the only valid
> experiences. Just because *you* never see the black hole form, doesn't
> mean it doesn't form. You just don't get to experience it yourself.

The main point: the only direct information we can ever have about black
holes is by falling into one. Since none of that information can be
communicated back, it cannot be considered any more scientific than the
religions' beliefs about life after death (you can verify, say,
Christianity by dying but that doesn't make it valid science).

anyone that asserts a singularity exists inside a black hole is
simply saying that the mathematical model they're using says there is
one

<URL: http://astronomy.stackexchange.com/questions/2441/does-matter-a
ccumulate-just-outside-the-event-horizon-of-a-black-hole#comment21507
_2448>

> Neither is "right" and the other is "wrong", neither frame of
> reference is privileged over the other. BOTH are right, even though
> they contradict each other. That's the nature of the universe we live
> in.

Nobody has claimed otherwise.

> Tim is so engrossed by the view of the gravitational lensing that he
> forgets to fire the retro-rockets, and before he knows it, he's
> crossed the event horizon and there's no going back.
>
> For a sufficiently large black hole, he might not even have noticed
> the transition. From his perspective, he's still distant from the
> singularity (being so small and distant, he can't quite make out what
> it looks like), and space-time is still quite flat for a sufficiently
> large black hole. Tim can still see out, although the incoming light
> is getting bluer, and he's still receiving Bill's clock signals,
> though like Graham he sees them as drastically sped up.

By the time the event horizon hits Tim at the speed of light, Tim will
have received all of our Universe's signals at an ever accelerating
frequency and increasing power. He will have seen the End of the World
before leaving it.

We have no way of telling if your prediction would be true for Tim
inside the black hole.

> Tim's spaceship appears to be *asymptotically* approaching the event
> horizon, in some sort of horrible version of Zeno's Paradoxes: each
> minute that goes by, Tim gets closer to the event horizon by a
> *smaller* amount as time slows down for him (as seen by Bill and
> Graham on the space station).

Correct, and very relevant. In fact, that's the reason the even horizon
never even appears to form to us outsiders. The star just keeps on
collapsing for ever. That is true even for Tim who can't experience a
true black hole before it hits him.

> Although their perspectives are very different, neither is "more
> right" than the other.

No, but only one of them can be examined scientifically.

Heisenberg's uncertainty principle was at first presented as some sort
of limitation to what we can know. Nowadays, it is viewed more
fundamentally as a law of physics; en electron cannot fall in the
nucleus of an atom because it would end up violating Heisenberg's
uncertainty principle.

Similarly, the Universe does not owe us an answer to what happens to
Tim. The Universe will come to an end (even for Tim) before the question
comes to the fore.

The cosmic teenage hacker who created our virtual world probably simply
typed this in that part of his code:

raise NotImplementedError()

thus terminating Tim's thread.

> From our frame of reference, we seem them asymptotically approaching
> the event horizon, but never cross it.

More than that, we see the star collapsing but never quite being able to
create an event horizon.


Marko

[Random832]

On Wed, Jun 29, 2016, at 05:35, Steven D'Aprano wrote:
> Although their perspectives are very different, neither is "more
> right" than the other.

I think usually the idea that there are "no privileged frames of
reference" doesn't go so far as to include ones from which it is
impossible to send information to the rest of. When it's impossible for
observations to be transmitted back to Earth (or to anywhere else),
you've crossed a line from science to philosophy.

[Lawrence D’Oliveiro]

On Wednesday, June 29, 2016 at 10:55:03 PM UTC+12, Marko Rauhamaa wrote:
> No, the fundamental question here is whether it makes scientific sense
> to speculate about topics that are beyond the reach of science. Few
> scientists speculate about what went on before the Big Bang, for
> example.

On the contrary. The cosmic microwave background is too smooth to be explained by a simple Big Bang. Hence the invention of cosmic inflation, to try to smooth it out. Trouble is, once you turn on the inflation field, how do you turn it off? But if you don’t turn it off, then you get an infinite series of Big Bangs, not just one.

So you see, like it or not, we are drawn to the conclusion that there *was* indeed something before our particular Big Bang.

Every time somebody tries to point to an example of a “topic that is beyond the reach of science”, it seems to get knocked over eventually.

[Rustom Mody]

What is the physics that you folks are talking of ... Ive no idea
OTOH Computer Science HAPPENED because mathematicians kept hotly disputing for
more than ½ a century as to what is legitimate math and what is
theology/mysticism/etc:

In particular the question: "Are real numbers really real?" is where it starts off...
http://blog.languager.org/2015/03/cs-history-0.html

[Chris Angelico]

On Thu, Jun 30, 2016 at 12:13 PM, Rustom Mody <rusto...@gmail.com> wrote:
> ... hotly disputing for more than ½ a century...

You keep using that character. Is it just to show off that you can? I
was always taught to match the style of the rest of the sentence, so
this would be "half a century". Same for most of your other uses - it
would be far more grammatically appropriate to use the word "half".

ChrisA

[Gregory Ewing]

Marko Rauhamaa wrote:
> By the time the event horizon hits Tim at the speed of light, Tim will
> have received all of our Universe's signals at an ever accelerating
> frequency and increasing power. He will have seen the End of the World
> before leaving it.

I don't think that's right. From the point of view of an
infalling observer, nothing particularly special happens
at the event horizon. Being able to see into the future
would count as rather special, I would have thought.

--
Greg

[Marko Rauhamaa]

Lawrence D’Oliveiro <lawren...@gmail.com>:

> Every time somebody tries to point to an example of a “topic that is
> beyond the reach of science”, it seems to get knocked over eventually.

Of course, an experiment trumps theory, always.


Marko

[Rustom Mody]

On Thursday, June 30, 2016 at 11:54:54 AM UTC+5:30, Marko Rauhamaa wrote:
> Lawrence D’Oliveiro :

> > Every time somebody tries to point to an example of a “topic that is
> > beyond the reach of science”, it seems to get knocked over eventually.
>
> Of course, an experiment trumps theory, always.

The scholastics would call the question: "How many angels can dance on a pin"
a 'thought-experiment'

[Marko Rauhamaa]

Gregory Ewing <greg....@canterbury.ac.nz>:

Yeah, you have a point:

IF you could stand still, you would see light from behind as
blue-shifted, precisely the flip side of gravitational redshifting
for distant observers. However, if you were traveling on a free-fall
geodesic, you would see no blueshift from light directly behind,
because you and it would be in the same inertial reference frame.

<URL: http://physics.stackexchange.com/questions/26185/what-will-th
e-universe-look-like-for-anyone-falling-into-a-black-hole>


Marko

[Paul Rubin]

> Every time somebody tries to point to an example of a “topic that is
> beyond the reach of science”, it seems to get knocked over eventually.

Generate a sequence of "random" bits from your favorite physical source
(radioactive decay, quantum entanglement, or whatever). Is the sequence
really algorithmically random (like in Kolmogorov randomness)? This is
scientifically unknowable. To check the randomness you have to solve
the halting problem.

[Andreas Röhler]

On 30.06.2016 03:33, Lawrence D’Oliveiro wrote:

> So you see, like it or not, we are drawn to the conclusion that there *was* indeed something before our particular Big Bang.

That's linear like the Big Bang theorie. What about assuming something
beyond our notion of time and space, unknown still?

[Lawrence D’Oliveiro]

On Thursday, June 30, 2016 at 6:57:41 PM UTC+12, Paul Rubin wrote:

>> Every time somebody tries to point to an example of a “topic that is
>> beyond the reach of science”, it seems to get knocked over eventually.
>
> Generate a sequence of "random" bits from your favorite physical source
> (radioactive decay, quantum entanglement, or whatever). Is the sequence
> really algorithmically random (like in Kolmogorov randomness)? This is
> scientifically unknowable.

The definition of “random” is “unknowable”. So all you are stating is a tautology.

[Lawrence D’Oliveiro]

M-theory? Branes? Multiverse? All part of the current scientific intellectual landscape.

[Paul Rubin]

Lawrence D’Oliveiro <lawren...@gmail.com> writes:
> The definition of “random” is “unknowable”. So all you are stating is
> a tautology.

What? No. You read a bunch of bits out of the device and you want to
know whether they are Kolmogorov-random (you can look up what that means
if you're not familiar with it). Quantum theory says they are, but if
it is true, there is no scientific way to confirm it.

[Lawrence D’Oliveiro]

On Thursday, June 30, 2016 at 7:32:55 PM UTC+12, Paul Rubin wrote:

Randomness is not something you can ever prove. Which is why it is the weak point of all encryption algorithms.

[Steven D'Aprano]

On Thursday 30 June 2016 12:13, Rustom Mody wrote:

> OTOH Computer Science HAPPENED because mathematicians kept hotly disputing
> for more than ½ a century as to what is legitimate math and what is
> theology/mysticism/etc:

I really don't think so. Computer science happened because people invented
computers and wanted to study them.

What people like Turing did wasn't computer science, because the subject didn't
exist yet. He was too busy creating it to do it.

And as for Kronecker, well, I suspect he objected more to Cantor's infinities
than to real numbers. After all, even the Pythogoreans managed to prove that
sqrt(2) was an irrational number more than 3000 years ago, something Kronecker
must have known.

> In particular the question: "Are real numbers really real?" is where it
> starts off... http://blog.languager.org/2015/03/cs-history-0.html

The pre-history of what later became computer science is very interesting, but
I fear that you are too focused on questions of "mysticism" and not enough on
what those people actually did and said.

For example, you state that Turing "believes in souls" and that he "wishes to
put the soul into the machine" -- what do his religious beliefs have to do with
his work? What evidence do you have for the second claim? What does it even
mean to put "the" soul (is there only one?) into "the" machine?


Besides, the whole point of science is to develop objective, rational reasons
to believe things. The chemist Friedrich Kekulé was inspired to think of
benzene's molecular structure as a ring through a dream in which a snake bit
its own tail, but that's not why we believe benzene is a ring-shaped molecule.
No chemist says "Kekulé dreamed this, therefore it must be true."

The irrational and emotional psychological forces that inspire mathematicians
can make interesting reading, but they have no relevance in deciding who is
write or wrong. No numbers are real. All numbers are abstractions, not concrete
things. If there is a universe of Platonic forms -- highly unlikely, as the
concept is intellectually simplistic and implausible -- we don't live in it.
Since all numbers are abstractions, the Real sqrt(2) is no more, or less,
"real" than the integer 2.



--
Steve

[Steven D'Aprano]

On Thursday 30 June 2016 17:16, Lawrence D’Oliveiro wrote:

> The definition of “random” is “unknowable”.

It really isn't.

What Julius Caesar had for breakfast on the day after his 15th birthday is
unknowable.

To the best of our knowledge, the collapse of a quantum wave function is
random.


--
Steve

[Andreas Röhler]

On 30.06.2016 10:24, Steven D'Aprano wrote:
> On Thursday 30 June 2016 12:13, Rustom Mody wrote:
>

> [ ... ]

> Besides, the whole point of science is to develop objective, rational reasons
> to believe things.

Science is not about believing, but about models.
Believing is important to make the career of a scientist maybe, it's for
people outside.

[Lawrence D’Oliveiro]

On Thursday, June 30, 2016 at 9:31:29 PM UTC+12, Andreas Röhler wrote:

> Science is not about believing, but about models.

The nice thing about science is, it works even if you don’t believe in it.

[Andreas Röhler]

On 30.06.2016 10:24, Steven D'Aprano wrote:

> On Thursday 30 June 2016 12:13, Rustom Mody wrote:
>
>
> The irrational and emotional psychological forces that inspire mathematicians
> can make interesting reading, but they have no relevance in deciding who is
> write or wrong.

Hmm, so math is not inspired by solving real world problems, for example
in physics or big-data?


> No numbers are real.

Numbers express relations, which are represented as symbols. So far they
are real, as math exists, the human mind exists.

We don't have any natural numbers, assume Kronecker made a joke.
"Natural numbers" is the most misleading term in the field maybe.

[Andreas Röhler]

Thats it!

[alister]

in theory there is no difference between theory and practice
in practice there is



--
Q: What is purple and commutes?
A: An Abelian grape.

[Rustom Mody]

On Thursday, June 30, 2016 at 1:55:18 PM UTC+5:30, Steven D'Aprano wrote:

> you state that Turing "believes in souls" and that he "wishes to
> put the soul into the machine" -- what do his religious beliefs have to do with
> his work?

Bizarre question -- becomes more patently ridiculous when put into general form
"What does what I do have to do with what I believe?"

More specifically the implied suggested equation "soul = religious"
is your own belief. See particularly "Christian faith" in the quote below.

> What evidence do you have for the second claim? What does it even
> mean to put "the" soul (is there only one?) into "the" machine?

Excerpted from https://blog.sciencemuseum.org.uk/the-spirit-of-alan-turing/

=======================
Morcom was Turing’s first love, a fellow, older pupil at
Sherborne School, Dorset, who shared Turing’s passion for
mathematics (who died in 1930)

He was profoundly affected by the death of his friend... Turing
admitted that he ‘worshipped the ground he trod on’.

Morcom’s death cast a long shadow. Turing turned away from his
Christian faith towards materialism, and began a lifelong quest
to understand the tragedy. As he struggled to make sense of his
loss, Turing pondered the nature of the human mind and whether
Christopher’s was part of his dead body or somehow lived on.

The October after the loss of his friend, Turing went up to
Cambridge, where he studied mathematics. Our exhibition includes
an essay, entitled “Nature of Spirit” that Turing wrote the next
year, in 1932, in which he talked of his belief in the survival
of the spirit after death, which appealed to the relatively
recent field of quantum mechanics and reflected his yearning for
his dear friend.

Around that time he encountered the Mathematical Foundations of
Quantum Mechanics by the American computer pioneer, John von
Neumann, and the work of Bertrand Russell on mathematical
logic. THESE STREAMS OF THOUGHT WOULD FUSE when Turing imagined a
machine that would be capable of any form of computation. Today
the result – known as a universal Turing machine – still
dominates our conception of computing.
===================================

> And as for Kronecker, well, I suspect he objected more to Cantor's infinities
> than to real numbers. After all, even the Pythogoreans managed to prove that
> sqrt(2) was an irrational number more than 3000 years ago, something Kronecker
> must have known.

They -- reals and their cardinality -- are the identical problem
And no, the problem is not with √2 which is algebraic
See http://mathworld.wolfram.com/AlgebraicNumber.html
It is with the transcendentals like e and π

ℕ ⫅ ℤ ⫅ ℚ ⫅ A ⫅ ℝ

is obvious almost by definition

That upto A (algebraic numbers) they are equipotent is somewhat paradoxical but still acceptable (to all parties)
See the Cantor pairing function https://en.wikipedia.org/wiki/Countable_set#Formal_overview_without_details
parties

However between A and ℝ something strange happens (where?) and the equipotence
is lost. At least thats the traditional math/platonic view (Cantor/Hilbert etc)


Constructivist view
"Yes real numbers are not enumerable
That means any talk of THE SET ℝ is nonsense
(talk with language, symbols whatever can only be enumerable)
Therefore Equipotence is like angels on a pin"

[Steven D'Aprano]

On Thu, 30 Jun 2016 08:11 pm, Andreas Rc3b6hler wrote:

>
>
> On 30.06.2016 10:24, Steven D'Aprano wrote:
>> On Thursday 30 June 2016 12:13, Rustom Mody wrote:
>>
>>
>> The irrational and emotional psychological forces that inspire
>> mathematicians can make interesting reading, but they have no relevance
>> in deciding who is write or wrong.

"Write" or wrong? Oh the shame :-(

> Hmm, so math is not inspired by solving real world problems, for example
> in physics or big-data?

I didn't say that.

Of course the work people do is influenced by real world problems, or their
own irrational and subjective tastes. Why does one mathematician choose to
work in algebra while another chooses to work in topology? As interesting
as it may be to learn that (say) Pascal worked on probability theory in
order to help a friend who was a keen gambler, the correctness or
incorrectness of his work is independent of that fact.

>> No numbers are real.
>
> Numbers express relations, which are represented as symbols. So far they
> are real, as math exists, the human mind exists.

They are real in the same way that "justice" and "bravery" are real. That
is, they are *abstract concepts*, not *things*.





--
Steven
“Cheer up,” they said, “things could be worse.” So I cheered up, and sure
enough, things got worse.

[Steven D'Aprano]

On Fri, 1 Jul 2016 01:28 am, Rustom Mody wrote:

> On Thursday, June 30, 2016 at 1:55:18 PM UTC+5:30, Steven D'Aprano wrote:
>
>> you state that Turing "believes in souls" and that he "wishes to
>> put the soul into the machine" -- what do his religious beliefs have to
>> do with his work?
>
> Bizarre question -- becomes more patently ridiculous when put into general
> form "What does what I do have to do with what I believe?"

Lots of people do things that go against their beliefs, or their beliefs (at
least, their professed beliefs) go against what they do. But I'll ask
again: in what way does Turing's supposed beliefs about souls have anything
to do with his mathematical work?

Let's be concrete:

In what way does the Halting Problem depend on the existence (or
non-existence) of the soul?

How was his work on breaking German military codes during World War 2
reliant on these supposed souls? (In the sense of a separate, non-material
spirit, not in the figurative sense that all people are "souls".)

> More specifically the implied suggested equation "soul = religious"
> is your own belief. See particularly "Christian faith" in the quote
> below.

Of course belief in souls is a religious belief. It certainly isn't a
scientific belief, or a materialistic belief.

Don't make the mistake of thinking that materialism is a religious belief.
It is no more a religious belief than "bald" is a hair colour.

>> What evidence do you have for the second claim? What does it even
>> mean to put "the" soul (is there only one?) into "the" machine?
>
> Excerpted from
> https://blog.sciencemuseum.org.uk/the-spirit-of-alan-turing/

[snip irrelevancy about the death of Morcom]

> Around that time he encountered the Mathematical Foundations of
> Quantum Mechanics by the American computer pioneer, John von
> Neumann, and the work of Bertrand Russell on mathematical
> logic. THESE STREAMS OF THOUGHT WOULD FUSE when Turing imagined a
> machine that would be capable of any form of computation. Today
> the result – known as a universal Turing machine – still
> dominates our conception of computing.

"These streams of thought" being the work of von Neumann and Russell.

Perhaps, and I'll accept this as a fairly unlikely by theorectically
possible result, Turing was only capable of coming up with the concept of
the Turing Machine *because of* his rejection of Christianity and his hopes
and fears and beliefs about the supposed soul of his deceased friend
Morcom. I'll grant that as a possibility, just as it is a possibility that
had Friedrich Kekulé not eaten a late night snack of cheese[1] before going
to sleep, he never would have dreamt of a snake biting its own tail and
wouldn't have come up with the molecular structure of benzene, leaving it
to somebody else to do so. But the idiosyncratic and subjective reasons
that lead Turing to his discovery are not relevant to the truth or
otherwise of his discoveries.

>> And as for Kronecker, well, I suspect he objected more to Cantor's
>> infinities than to real numbers. After all, even the Pythogoreans managed
>> to prove that sqrt(2) was an irrational number more than 3000 years ago,
>> something Kronecker must have known.
>
> They -- reals and their cardinality -- are the identical problem
> And no, the problem is not with √2 which is algebraic
> See http://mathworld.wolfram.com/AlgebraicNumber.html

What reason do you have for claiming that Kronecker objected to
non-algebraic numbers? Nothing I have read about him suggests that he was
more accepting of algebraic reals than non-algebraic reals.

(I'm not even sure if mathematicians in Kronecker's day distinguished
between the two.)

I daresay that the famous Kronecker comment about god having created the
integers was not intended to be the thing he is remembered by. It was
probably intended as a smart-arsed quip and put-down of Cantor, not a
serious philosophical position. For is we take it *literally*, Kronecker
didn't even believe in sqrt(2), and that surely cannot be correct.

> It is with the transcendentals like e and π

Both e and π can be written as continued fractions, using nothing but ratios
of integers:

e = (2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...)

π = (3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, ...)

so surely if Kronecker accepted square root of two:

√2 = (1; 2, 2, 2, 2, ...)

he has no reason to reject the others.



I'm not entirely convinced that transcendentals deserve to be a separate
category of numbers. "Algebraic numbers" seem quite arbitrary to me: what's
so special about integer polynomials? Just because the Ancient Greeks had
some weird ideas about integers doesn't mean that we have to follow along.

I mean, sure, it's interesting to look at integer polynomials as a special
case, just as we might look at (say) Diophantine equations or Egyptian
Fractions as special cases. There might even be some really important maths
that comes from that. But that doesn't necessarily make algebraic numbers
any more *fundamental* than "Rustom numbers", the solutions to equations of
the form:

a0 + a1 * e**x + a2 + e**(x**2) + a3 * e**(x**3) + ... + an * e**(x**n)


where the a's are all rational numbers where the numerator and denominator
are co-prime.

So I accept that transcendental numbers are interesting. I don't necessarily
agree that they are fundamental in the same way integers, rationals and
reals are.




[1] Or perhaps if he *had* eaten a late snack of cheese.

[Rustom Mody]

On Thursday, June 30, 2016 at 11:33:58 PM UTC+5:30, Steven D'Aprano wrote:
> On Fri, 1 Jul 2016 01:28 am, Rustom Mody wrote:
>
> > On Thursday, June 30, 2016 at 1:55:18 PM UTC+5:30, Steven D'Aprano wrote:
> >
> >> you state that Turing "believes in souls" and that he "wishes to
> >> put the soul into the machine" -- what do his religious beliefs have to
> >> do with his work?
> >
> > Bizarre question -- becomes more patently ridiculous when put into general
> > form "What does what I do have to do with what I believe?"
>
> Lots of people do things that go against their beliefs, or their beliefs (at
> least, their professed beliefs) go against what they do. But I'll ask
> again: in what way does Turing's supposed beliefs about souls have anything
> to do with his mathematical work?
>
> Let's be concrete:
>
> In what way does the Halting Problem depend on the existence (or
> non-existence) of the soul?

Lets change your question -- ever so slightly, given the Turing-context:

In what way does the Turing Test depend on the existence (or non-existence) of the soul?

>
> How was his work on breaking German military codes during World War 2
> reliant on these supposed souls? (In the sense of a separate, non-material
> spirit, not in the figurative sense that all people are "souls".)

When you say "soul" you (seem to?) mean what has been called with justifiable
derision, "dualistic notion of soul"
- something for which Christianity gets bad press though its the invention of Descartes
http://www.iep.utm.edu/descmind/
- something which caught on because of Descartes huge reputation as a scientist

There are other more reasonable non-religious non-dualistic notions of soul possible:

http://www.bu.edu/agni/poetry/print/2002/56-szymborska.html


<snipped stuff on numbers>
You are missing the point -- its not about specific numbers its about
cardinality not adding up

- Cantor theory points to uncountably many real numbers
- All the sets upto algebraic are countable
- So the uncountable fellas need to be transcendental
- We only know two blessed guys -- π and e

Where are all the others hiding??

And dont try saying that if e is transcendental so is 2e 3e 4e...
And no use trying more such tricks -- they only multiply these two countably infinite times
And you may produce a few more, more esoteric transcendentals --- a very
finite set!

Nor is it much good saying this is not much relevance to our field:
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

So our (computer scientists') predicament in short

- our field is opposed to Cantor's non-constructivist *outlook*
- At the foundation of CS are *methods* that.... come from Cantor... OOPS!

[Marko Rauhamaa]

Rustom Mody <rusto...@gmail.com>:

> There are other more reasonable non-religious non-dualistic notions of
> soul possible:

Software engineers should have an easy time understanding what a soul
is: a sufficiently sophisticated software system in execution. I'd say
the minimum requirement for a soul is the capacity to suffer. I don't
think anything built by humans has yet reached that level of
sophistication, but insects probably can feel pain as authentically as
humans.

> - Cantor theory points to uncountably many real numbers
> - All the sets upto algebraic are countable
> - So the uncountable fellas need to be transcendental
> - We only know two blessed guys -- π and e
>
> Where are all the others hiding??

Here: <URL: https://en.wikipedia.org/wiki/Transcendental_number>.

> And dont try saying that if e is transcendental so is 2e 3e 4e... And
> no use trying more such tricks -- they only multiply these two
> countably infinite times And you may produce a few more, more esoteric
> transcendentals --- a very finite set!

The banal way of putting it that we can express only countably many
individual items.

The more philosophical point of view is that mathematics failed at
circumscribing all of philosophy and is condemned to counting beans.
Naive set theory was a Grand Unified Theory of Everything, but of course
inconsistent. The bottom-up set theories are sane but completely fail at
being the ultimate metalevel.


Marko

[Rustom Mody]

On Thursday, June 30, 2016 at 1:55:18 PM UTC+5:30, Steven D'Aprano wrote:

> On Thursday 30 June 2016 12:13, Rustom Mody wrote:
>
> > In particular the question: "Are real numbers really real?" is where it
> > starts off... http://blog.languager.org/2015/03/cs-history-0.html
>
> The pre-history of what later became computer science is very interesting, but
> I fear that you are too focused on questions of "mysticism" and not enough on
> what those people actually did and said.

I’ve been preparing material for some classes on Theory of Computation (ToC)

Starting from the premise that in a way Gödel is more primary to ToC than Turing

[And also been fan of Gödel-Escher-Bach from my college days]

And came across this proof (by Gödel!) that God exists <tickled> :
https://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof

[Rustom Mody]

<snip>

>
> What reason do you have for claiming that Kronecker objected to
> non-algebraic numbers? Nothing I have read about him suggests that he was
> more accepting of algebraic reals than non-algebraic reals.
>
> (I'm not even sure if mathematicians in Kronecker's day distinguished
> between the two.)
>

Lots of questions... I would guess rhetorical.

However under the assumption that they are genuine (or could be genuine for
others than you), I went back and checked.
I recollected that I started thinking along these lines — viz. that philosophical disputes led to the genesis of computers — after reading an essay
by a certain Adam Siepel... which subsequently seems to have fallen off the net

I tracked him down and re-posted his essay here:
http://blog.languager.org/2016/07/mechanism-romanticism-computers.html

Just to be clear — this is Dr. Adam Siepel's writing reposted with his permission

[Steven D'Aprano]

On Sat, 30 Jul 2016 03:46 pm, Rustom Mody wrote:

> Lots of questions... I would guess rhetorical.

They weren't rhetorical.

You've made a lot of claims about the origins of computer science, and I've
questioned some of your statements. Answers would be appreciated.

> However under the assumption that they are genuine (or could be genuine
> for others than you), I went back and checked.
> I recollected that I started thinking along these lines — viz. that
> philosophical disputes led to the genesis of computers — after reading an
> essay by a certain Adam Siepel... which subsequently seems to have fallen
> off the net
>
> I tracked him down and re-posted his essay here:
> http://blog.languager.org/2016/07/mechanism-romanticism-computers.html
>
> Just to be clear — this is Dr. Adam Siepel's writing reposted with his
> permission

The essay is not awful, but I wouldn't shout its praises either. It looks to
me like an undergraduate essay, taking a very narrow and rather naive view
of the field. There's not a lot of references (only nine), which means the
author is (in my opinion) excessively influenced by a small number of
views, and I don't see any sign that he has even made a half-hearted
attempt to seek out alternate views.

The author makes a claim:

"... Principia Mathematica, between 1910 and 1913, which in its attempt to
place mathematics squarely in the domain of logic, represented the first
new system of logic since Aristotle"

but doesn't give any justification for the claim. Why single out the
Principia and ignore the works of the Stoics, Peter Abelard, William of
Ockham, Augustus DeMorgan, Gottlob Frege and most especially George Boole
dismissed?

I would think that if anyone truly deserved credit for creating a new system
of logic, it should be Boole. But perhaps that's just a matter of opinion
on where you draw the lines.

http://www.iep.utm.edu/prop-log/#H2

That's not really central to his argument, but it does suggest that his
views are quite idiosyncratic. To my mind, that feels like someone claiming
that Stephen Hawking is the first genuinely original physicist since
Newton. Einstein? Schrödinger? Dirac? Never heard of 'em.

A rather large section of the essay is an irrelevant (and, I think,
incorrect) digression about "Mechanists" and "Romantics", neither of which
is really relevant to the philosophy of mathematics. He eventually mentions
the Intuitionists, but I don't think he understands them. By linking them
to the Romantics, he seems to think that the Intuitionist school of thought
doesn't require mathematical proofs, or that they are satisfied with
the "intuitively obvious truth" of axioms. But that's not what the
Intuitionists were about:

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

He mischaracterises and over-simplifies the argument over the foundations of
mathematics:

https://en.wikipedia.org/wiki/Brouwer–Hilbert_controversy

with at least three separate groups involved (Logicists such as Russell,
Formalists such as Hilbert, and Constructivists such as Poincaré -- four
groups if you count Intuitionism separate from Constructionism). He
exaggerates the death of the Logicist school of thought. It continues today
with Second Order Logic.

And I wonder why you are taking this essay as supporting your position.
According to this essay, the Intuitionists won in mathematics. And yet
Turing and Von Neumann (two major pioneers of computing) were "Mechanists".
If Intuitionism influenced computer science, where is the evidence of this?
Where are the Intuitionist computer scientists? On the contrary, academic
CS seems to have come from the Logicist school of thought, and practical
computer engineering from "whatever works" school of thought.

None of this even *remotely* supports your assertions such as "[Turing]
wishes to put the soul into the machine". Maybe he did. But this essay
gives no reason to think so, or any reason to think that Turing's personal
beliefs about souls is the slightest bit relevant to computer science.

[Paul Rubin]

Steven D'Aprano <st...@pearwood.info> writes:
> If Intuitionism influenced computer science, where is the evidence of this?
> Where are the Intuitionist computer scientists?

Pretty much all of them, I thought. E.g. programs in typed lambda
calculus amount to intuitionistic proofs of the propositions given in
the types (this is the Curry-Howard correspondence). Languages like Coq
and Agda use the correspondence directly so you can state a
specification as a type, then prove the type as a theorem, and then
extract the proof as executable code which is certified to implement the
specification.

The trendy area now is called "homotopy type theory", but I don't
understand even a tiny bit of it.

Intro to Coq: www.cis.upenn.edu/~bcpierce/sf/current/index.html

Proofs and types: www.paultaylor.eu/stable/Proofs+Types.html

Homotopy type theory: github.com/HoTT , homotopytypetheory.org

There's also a wiki at ncatlab.org which is in the intersection of logic
and CS, that develops a lot of this stuff.

> On the contrary, academic CS seems to have come from the Logicist
> school of thought

Do you mean formalist? I thought that logicism was the idea that
mathematics could be derived from pure logic, and thus it went away when
Gӧdel's incompletness theorems appeared.

[Rustom Mody]

On Monday, August 1, 2016 at 9:12:00 AM UTC+5:30, Paul Rubin wrote:

> Steven D'Aprano writes:
> > If Intuitionism influenced computer science, where is the evidence of this?
> > Where are the Intuitionist computer scientists?
>

> Pretty much all of them, I thought. E.g. programs <details snipped>

Yeah…
Saying what does CS have to do with Intuitionism is like asking what
what does Carnot’s (theoretical) heat-engine have to do with Toyota

[Rustom Mody]

My agreement vis-a-vis this essay is basically this:
Abstruse philosophical how-many-angels-on-a-pin type arguments amongst
philosophers of math/logic across 19-20 century gave rise to Computers and Computer Science.

Some of the other detailed points you make, I agree with, eg.

1. There were more than 2 parties in the dispute
2. There is some arbitrary line-drawing there

(Un)fortunately arbitrary line-drawing is the name of the game everywhere:
Religion→Philosophy→History→Politics

<OT-Aside>
Right now as we speak there looks like a war — maybe world-war — in offing:
https://www.thenation.com/article/the-united-states-and-nato-are-preparing-for-a-major-war-with-russia/

Possibly democrat-engineered to discredit Trump:
https://www.thenation.com/article/neo-mccarthyism-and-olympic-politics-as-more-evidence-of-a-new-cold-war/

All starts with the disorder in the middle-east and a whole lot of arbitrary lines
drawn there
[Going backward in time]
- A line drawn in space called ‘Israel’
- Based on a line drawn in time called the ‘Exodus of Moses’
- Based on the supremely authoritative history-record called ‘The Bible’
(or Torah depending on the speaker)

Leaving aside the first and the third, why specifically the Exodus as a definer?
Before Canaan they were in Egypt, so Israel=Egypt is equally legitimate
And before…before… they were in Eden.
Why not set Israel up in Eden?

May it just be the (in)convenient fact that knocking the Palestinians out of
their homes is easier than removing Mr. God from Eden?

Collapse of borders:
http://www.independent.co.uk/news/world/middle-east/isis-in-a-borderless-world-the-days-when-we-could-fight-foreign-wars-and-be-safe-at-home-may-be-long-a6741146.html
</OT-Aside>

[Ian Kelly]

On Sun, Jul 31, 2016 at 10:05 PM, Rustom Mody <rusto...@gmail.com> wrote:
> All starts with the disorder in the middle-east and a whole lot of arbitrary lines
> drawn there
> [Going backward in time]
> - A line drawn in space called ‘Israel’
> - Based on a line drawn in time called the ‘Exodus of Moses’

I'm no expert but I believe that Zionism was based upon the idea of
Negation of the Diaspora, not Exodus.

> - Based on the supremely authoritative history-record called ‘The Bible’
> (or Torah depending on the speaker)
>
> Leaving aside the first and the third, why specifically the Exodus as a definer?
> Before Canaan they were in Egypt, so Israel=Egypt is equally legitimate
> And before…before… they were in Eden.
> Why not set Israel up in Eden?

Since the Exodus story has it that the Israelites were slaves in Egypt
(despite the utter lack of archaeological evidence that they were ever
there en masse), placing Israel in Egypt would seem rather insulting.

> May it just be the (in)convenient fact that knocking the Palestinians out of
> their homes is easier than removing Mr. God from Eden?

Fun historical fact: the land of Palestine was so named by the Roman
Emperor Hadrian after he kicked the Jews out of their homes and forced
them to leave. The name was a Romanization of the ancient land of
Philistia, home of the Philistines, despite that they were long gone
by this point. So the entire existence of Palestine is basically one
big fat Roman middle finger to the Jews.

Not that any of this is meant to condone any of the violence in the Middle East.