> It is time to reign in the imagination. Numbers are

Not to human intellect. We know today that the chimpanzee is able to compute

> `out there' but only in relation to the human intellect.

> [...]

basic operations and has the knowledge of small numbers. I would not be

surprised if my cats have also a representation of the number 2 in their

mind. The number is a step in (known) intelligent life, and we are not the

only animal that already have the knowledge of numbers.

There are also some people who don't know what a number is and are not able

to compute basic operations. I read a paper about it... I may be able to

find it for people who are interested and read French. Often, it is because

of a tumour that was removed. They are able to compute simple operations

with their memory but cannot estimate or "see" the result of the operation.

For example, a guy explained that to compute 5-2, he should use his memory

to think on one hand that the successor of 2 is 3, the successor of 3 is 4,

... and on the other hand (and simultaneously!) that the successor of 0 is

1, of 1 is 2, ... It was observed that the part of the brain that is

activated when we compute is never activated for him. Surprisingly, it was

not a real problem for this guy who is a psychologist and learnt to live

without numbers. Then I am not sure that we can say that numbers are needed

for intellect.

To me point many 'philosophies' about numbers are just an old fashion point

of view of the world. You cannot speak about philosophy today and ignore

neurobiology and neuroscience. Science totally changed philosophy during the

20th century. Of course, they are still people who don't know anything about

science and continue to write some books of 'philosophy', but there are also

some people who continue to think that their life depends on the position of

planets!

I don't think that we can today consider that it is possible to know

anything about the 'reality' independently of our mind. We only know the

reality through the filter of our mind. The faith is the fact to think that

you know something about reality independently of your mind. Someone who

thinks that numbers exits independently of his mind has the faith of

numbers. This may be sociologically and historically interesting, but it has

nothing to do with philosophy.

Paul, it's funny when you say that David Underbakke is Realist: David

discovered the largest known twin primes, the largest known Sophie Germain

prime and the largest known prime which is not a Mersenne prime: of course

he is realist! :o)

Yves

Paul, note also that 12345*2^n + 1 is not a GFN form :o(> >

According to "The Mathematical Experience", by Davis and Hersh, page 334,

> > Russell and Whitehead proved that 1+1=2, using only logic.

>

>1+1=2 by definition. They proved 2+2=4 though

>(technically (1+1)+(1+1)=((1+1)+1)+1)

"Russell and Whitehead ... after 362 pages, the arithmetic proposition

1+1=2 is established." and they show part of that page which says "From

this proposition it will follow, when arithmetical addition has been

defined, that 1+1=2."

+--------------------------------------------------------+

| Jud McCranie |

| |

| ... algorithms are concepts that have existence apart |

| from any programming language. The word "algorithm" |

| denotes an abstract method of computing some output |

| from some input ... -- Donald Knuth, CACM, 1966 |

+--------------------------------------------------------+