FOM: NATURE of mathematics

[Julio Gonzalez Cabillon]

Dear List,

Professor Silver was very kind in addressing my questions a few
days ago, and he privately encouraged me to direct them to the
whole list again. I will add further remarks and queries.

Although these issues, most probably, exceed the scope of FOM, I
do think the exchange of these ideas might be of interest to the
forum. At any rate, I gladly accept Steve's authority to censor
this post if he finds it off-topic.

What I would like to ask you about here and now, and I would be most
grateful for your answers/thoughts/misgivings/whatever, is not what
you are *against* in someone else's philosophy, but what you are *for*
in your own conception of the NATURE of mathematics. 

As F. Xavier Noria nicely put it yesterday "Internet is really great",
and I'd add that it could well be even much better. At least, I hope so.


For instance:

1) Should mathematics on other planets (of other galaxies) be entirely
   different from ours -- provided its existence? Would you expect it
   to be similar? ... What do you think? ... No kidding, please.


2) Do you definitely feel comfortable with the idea conveyed in the
following well-known remark?

     "mathematical reality lies outside us, that our function
     is to discover or observe it, and that the theorems which
     we prove, and which we describe grandiloquently as our
     'creations', are simply our notes of our observations."

If so, I would appreciate to know why? ... Should it be any locus of
mathematical reality? ...


3) In the following incomplete sentences,

      i)  "Conway .......... what Knuth christened _surreal numbers_",

      ii)  "Nobody knows who .......... _number pi_",

   assume we ought to fill in the blanks with one word.
         
      [discovered / invented / constructed / found / conceived / ...]

   What would be your choice in each case? Why? ...


4) In what *sense* can we say that mathematics is (timelessly) true? ...
   Is there any absolute concept of truth in Mathematics? Is the notion
   of proof weaker than the notion of truth?...
   
   What is "truth", and what is "true"?


5) In the year 2098 (say, in case our fragile planet still be there),
   will mathematicians believe that there *really* are infinitely many
   natural numbers?... OK, without doing futurology, do you believe
   here and now that there *really* are infinitely many natural numbers?
   If so, where are they located?...

   Do the lack of a definite location for the natural numbers force them
   to *exist* somewhere within the Cosmos? ... What *kind* of existence
   is that? ... Faith? ...

   Do you really believe that it is conceivable that these infinitely
   many natural numbers did exist on a Jurassic park despite the fact
   that there were no mathematicians around to imagine them?

   Do you really believe that it is conceivable that these infinitely
   many natural numbers *really* existed already on a Jurassic park
   (just) because they were "sleeping truths"?


6) Do you think that the probability that there's another Mersenne prime
   is greater than that that there's a mistake in the proof that the one
   recently discovered is prime?
 

7) In the words of a contemporary famous mathematician: "Computers
   are important, but not to mathematics". Assume that we read this
   apothegm as "mathematics must be regarded just as a human activity",
   how computers fit in this scheme when they handle, for instance,
   primality tests? Any reactions? ...


8) "The simple fact that mathematical activity is social is obvious".
   [For further context cf. FOM list, Martin Davis, 12 Mar 1998 12:35:50]
   
   Assume that on March 18, 2098, World War IV occurs on our planet, and
   obliterates all human life; but one brand modern thought-engine escapes
   destruction. This silicon-based brain, working alone, finds a pattern
   of the "structure" of prime numbers (but there is nobody left to
   "celebrate" this remarkable achievement). Will mathematical activity
   still be __social__ on March 19, 2098? ...

Again, many thanks for your comments (privately or on-list).

Greetings from cloudy Montevideo,
                                  Julio Gonzalez Cabillon