Working with the set of real numbers

[Roy Smith]

In article <mailman.7792.1393994283.18130.python-list at python.org>,
 Ben Finney <ben+python at benfinney.id.au> wrote:

> Roy Smith <roy at panix.com> writes:
> 
> > I stopped paying attention to mathematicians when they tried to convince 
> > me that the sum of all natural numbers is -1/12.
> 
> I stopped paying attention to a particular person when they said “I
> stopped paying attention to an entire field of study because one
> position expressed by some practicioners was disagreeable to me”.
> 
> Would you think “I stopped listening to logicians when some of them
> expressed Zeno's paradox of the impossibility of motion” to be a good
> justification for ignoring the entire field of logic?
> 
> Rather, a more honest response is to say why that position is incorrect,
> and not dismiss the entire field of study merely for a disagreement with
> that position.

I *was* partly joking (but only partly).

Still, there's lots of stuff mathematicians do which I don't understand.  
I cannot understand, for example, Andrew's Wiles's proof of Fermat's 
Last Theorm.  I can't even get past the first few paragraphs of the 
Wikipedia article.  But, that doesn't sour me on the proof.  I can 
accept that there are things I don't understand.  I don't know how to 
speak Chinese.  I don't know how to paint a flower.  I don't know how to 
run a mile in 4 minutes.  But I accept that there are people who do know 
how to do those things.

I can watch a friend pick up a piece of paper, a brush, and some 
watercolors and 5 minutes later, she's got a painting of a flower.  I 
watched her hands hold the brush and move it over the paper.  There's 
nothing mystical about what she did.  Her hands made no motions which 
are fundamentally impossible for my hands to make, yet I know that my 
attempt at reproducing her work would not result in a painting of a 
flower.

But, as I watch the -1/12 proof unfold, I don't get the same feeling.  I 
understand every step.  I wouldn't have thought to manipulate the 
symbols that way, but once I've seen it done, I can reproduce the steps 
myself.  It's all completely understandable.  The only problem is, it 
results in a conclusion which makes no sense.  I can *prove* that it 
makes no sense, by manipulating the symbols in different ways.  The sum 
of any two positive numbers must be positive.  I can group them and add 
them up any way I want and that's still true.

But, here I've got some guy telling me it's not true.  If you just slide 
this over that way, and add these parts up this way, it's -1/12.  That 
does not compute.  But it doesn't not compute in the sense of, "that's 
so complicated, I have no idea what you did", but in the sense of "thats 
so simple, I know exactly what you did, and it's bullshit" :-)