On Platonism.

[Antti Karttunen]

Henry Gould wrote:

> Dear Antti,
>
> I am so pleased to learn that you are a Platonist, as many 
> mathematicians are.
> The six Platonic solids are beautiful. Indeed, there are six not five!
> They come in three pairs: Hexahedron (cube) and Octahedron are dual - 
> the one has as man faces as the other has edges, both having the same 
> number of vertices. The Icosahedron and Dodecahedron are likewise dual 
> the same way. Finally the Tetrahedron is dual to itself, to its 
> Doppelgänger you see. Thus there are six regular solids, but five 
> distinct ones.

In a way I see this. When I was younger, I was obsessed with the Platonic
solids, and also knew about this "embedding of duals inside each other".
I also took photocopies of many pages of Kepler's Harmonices Mundi 
(Latin original),
and my intention was to learn Latin to be able to read it. Like many of
my projects, it remains to be completed (even started), but fortunately,
when I heard that there was an English translation, I ordered it 
immediately:
http://www.amazon.com/Harmony-Memoirs-American-Philosophical-Society/dp/0871692090/sr=1-2/qid=1166749821/ref=sr_1_2/002-0183964-2915202?ie=UTF8&s=books

> The ideas (forms) are eternal and exist whether we mortal, material 
> Sequence Searchers are here or not.


I think there are at least two good points in Platonic attitude towards 
mathematics:

First, it is psychologically much more rewarding to grasp for something 
real from "out there",
than just to engage in some empty formalistic games played on the paper.

Secondly, it can be psychologically alleviating, when we realize the 
things we have found,
but for which somebody else has collected the fame, exist already, have 
always existed,
thus we just (re)discover, do not invent anything, the names do not matter,
somebody has already discovered them in some another civilization,
either in this solar system, or somewhere else.

So, should we have Fibonacci numbers or Pingala numbers? Backus-Naur or
Panini-Backus form? Does it matter?
(Read http://en.wikipedia.org/wiki/Indian_mathematics , with normal amount
of skepticism reserved for Wikipedia...
E.g. First voice: "Back in c. 300 BC, Pingala was playing around with 
some numbers,
       and guess what he came up with? Sequence A000045!! He started
       with 0 and next to it he wrote 1."
       Second voice: "Gee, what was he thinking of?"....

But..., I'm not a platonist for merely utilitarian reasons, my conviction
is deeper than that. How hard it is to mentally grasp these objects from the
Platonic realm, depends of course on the person. For me, being a programmer
who can read just enough mathematical notation to be able to convert
inductive definitions to recursive functions, it is easy to get assured
about the existence of enumerable sets. Essentially anything that
can be programmed, and converted to OEIS-sequence certainly exists
(in Platonic plane) for me.

But what about structures like factor group R*/Q*
where R* and Q* are multiplicative groups defined on sets
R\{0} and Q\{0} (Non-zero reals and rationals with the usual
multiplication as their operation). If I remember correctly, this
involves a thing called "Hamel base" (I call it "Hameln's base",
after the musician, just for fun), which seems much harder
to grasp on, essentially because it involves the Axiom of Choice.
If N -> N functions are the concrete objects to which we bounce against
in the Platonic realm, then the latter are something akin to dreaming there.
Thing like this would exists, provided that this kind of axiom would hold...
(But so far I'm not assured. Maybe I should have more imagination,
Unfortunately, I'm just a dumb constructivist coder.)


> Ah Glaucon, of these things we may speak again someday.
>
> If we demote enough "planets" we could return to Kepler's 
> Cosmographicum model of the region we inhabit.

Maybe we could even (at some later date, when the technology is 
sufficiently advanced) tweak
the orbits of the classical planets, to get the real world match more 
exactly with the model?

>
> Platonically,
>
> Henry Gould
>

Plutonically,

Antti

> Antti Karttunen wrote:
>
>> Jonathan Post wrote:
>>
>>> "Astrological object" an interesting notion, which occurred to me 
>>> also.  Given the the Greeks consider the dodecahedron as iconic of 
>>> the Cosmos, and the icosahedron is its Platonic solid dual, I think 
>>> they were making a categorical statement of morphisms between 
>>> cosmologies. The Greeks saw the icosahedron as iconic of the element 
>>> "air." Coincidently, "Where does it all end: Search for the shape of 
>>> space" is the cover story in the 9-15 December New Scientist.
>>
>>
>>
>> And it is claimed that the truncated icosahedron is responsible for 
>> the "Self-organized breakup of Gondwana":
>> http://www.mantleplumes.org/EarthTess2.html
>> (nice application, at least...)
>>
>> I will quote from John Baez, from 
>> http://math.ucr.edu/home/baez/platonic.html
>>
>> "Everything sufficiently beautiful is connected to all other 
>> beautiful things!
>> Follow the beauty and you will learn all the coolest stuff.
>> The Platonic solids are a nice place to start."
>>
>>
>>
>> Platonically,
>>
>> Antti
>>
>>