Can math.atan2 return INF?

[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.


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