
Human beings are born with numbers in the brain  - makimaki
http://www.economist.com/science/displaystory.cfm?story_id=12847128
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biohacker42
_In one test Charles was shown a pair of digits and asked to name the larger
number. The bigger the gap, the faster most people can do this: they say
“nine” faster when shown 9 and 2 than when shown 9 and 7. But with Charles,
the reverse was the case—and the researchers could see why. Rather than
telling the answer directly, he was counting on from one number (on his
fingers) until he got to the other, which meant he must have started at the
smaller, or he got to ten, in which case he must have started at the bigger.
Most strikingly, he lacked the fundamental numerical ability possessed by most
newborns: being able to tell the number of objects in a small group simply by
looking. When asked how many dots were on a sheet of paper, he counted on his
fingers—even when there were only two.

Charles’s deficit, though severe, seemed to affect his numerical abilities
alone. Numerical deficits in people of otherwise normal abilities can be even
more striking in cases of brain damage. Lisa Cipolotti, a neuropsychologist,
studied a Signora Gaddi, who used to run a hotel and keep its accounts. After
a stroke she could find the number of things in a small group only by
counting—when asked how many arms a crucifix had, she got Dr Cipolotti to hold
out her arms so she could count them. Signora Gaddi’s problems seemed to
affect only numbers. She could still read, speak and reason, remember
historical and geographical facts, and order objects by their physical size.

In fact, Signora Gaddi’s difficulties went even deeper than Charles’s. The
stroke which damaged her innate understanding of small numbers also robbed her
of the entire numerical edifice built on that foundation. For her, numbers
stopped at four. When asked to count up from one, she got to four and no
further. If there were more than four dots on a page she could not count them.
She could not say how old she was or how many days were in a week, or even
tell the time._

Scary.

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tokenadult
The article includes the interesting comment that a fairly eminent
geometrician had little number sense of the kind described in the article. One
of the wonderful things about mathematics is that it combines reasoning about
discrete, countable things, that is numbers, with reasoning about shapes, that
is geometry. An especially good book about both aspects of mathematics is John
Stillwell's Numbers and Geometry.

[http://www.amazon.com/Numbers-Geometry-John-
Stillwell/dp/038...](http://www.amazon.com/Numbers-Geometry-John-
Stillwell/dp/0387982892/)

I'll quote a bit of Stillwell's preface to give the flavor of the book:

"What should every aspiring mathematician know? The answer for most of the
20th century has been: calculus. . . . Mathematics today is . . . much more
than calculus; and the calculus now taught is, sadly, much less than it used
to be. Little by little, calculus has been deprived of the algebra, geometry,
and logic it needs to sustain it, until many institutions have had to put it
on high-tech life-support systems. A subject struggling to survive is hardly a
good introduction to the vigor of real mathematics.

". . . . In the current situation, we need to revive not only calculus, but
also algebra, geometry, and the whole idea that mathematics is a rigorous,
cumulative discipline in which each mathematician stands on the shoulders of
giants.

"The best way to teach real mathematics, I believe, is to start deeper down,
with the elementary ideas of number and space. Everyone concedes that these
are fundamental, but they have been scandalously neglected, perhaps in the
naive belief that anyone learning calculus has outgrown them. In fact,
arithmetic, algebra, and geometry can never be outgrown, and the most
rewarding path to higher mathematics sustains their development alongside the
'advanced' branches such as calculus. Also, by maintaining ties between these
disciplines, it is possible to present a more unified view of mathematics, yet
at the same time to include more spice and variety."

A more controversial, and plenty spicy, comment on the algebraic approach to
mathematics as contrasted with the geometric approach to mathematics is Paul
Halmos's article, "Applied Mathematics Is Bad Mathematics,"

[http://books.google.com/books?id=rQ6FAAAAIAAJ&q=Halmos+A...](http://books.google.com/books?id=rQ6FAAAAIAAJ&q=Halmos+Applied+Mathematics+Is+Bad+Mathematics&dq=Halmos+Applied+Mathematics+Is+Bad+Mathematics&ei=DKJaSZT8F5GoM8mguN8B&client=firefox-a&pgis=1)

(that's not a full-text view of the article)

which I disagree with, as maybe Halmos intended his provocative title to be
disagreed with.

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wensing
Meta-comment: Could we please fix the quoted-text-not-wrapping 'problem' on
this site? I hate having to scroll x,000 pixels to read a comment just because
someone is using <pre> or whatever it is.

FF3 on OS X.

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icey
It's intentional; people just abuse it (myself included).

Indented text is intended to display code, which should not be wrapped by the
browser's width.

It's up to the poster to format their text accordingly.

~~~
Raphael
In that case, we need

    
    
        overflow: auto;

~~~
rms
I think it's there but doesn't work with FF3.

~~~
jerf
Doesn't _seem_ to be: <http://ycombinator.com/news.css>

Although if they're serving a custom .css file to my Firefox 3 that doesn't
include that precisely because it doesn't work, I wouldn't know. wget got the
same pre stuff in the last two lines, though.

