Author Topic: Math and Platonism (excerpt from TISv.I p. 9-13)  (Read 270 times)

Xepera maSet

Math and Platonism (excerpt from TISv.I p. 9-13)
« on: November 22, 2017, 07:25:08 pm »
One  of  the  fundamental  questions  of  the  philosophy  of  math  is  “does  math  itself  exist  as  a  part  of  the  universe,  wherein it  is  merely  discovered  by  humans,  or  is  math  invented  by  humans?”   The  area  of  Mathematical  Realism  claims  that  yes, mathematics  exists  independent  of  the  human  mind.    Clearly,  a  necessary  first  step  would  be  to  determine  whether  the  natural numbers  exist  (proving  that  the  primes  exist  would  suffice,  but  it  turns  out  it  is  a  little  easier  to  simply  construct  the  naturals together  with  zero).    The  naturals,  together  with  zero,  are  the  building  blocks  upon  with  all  other  numbers  are  defined,  and  while mathematics  certainly  can  be  done  without  the  use  of  numbers,  arguably  the  construction  of  mathematical  identities  without numbers  as  first principles  is  not  possible.

 To  this  end,  think  of  a  number,  say,  for  example,  3.    Does  3  exist?    Certainly  one  could  have  a  collection  of  3  objects, or, to put it in a Platonic  sense, one  could have  a  collection of  objects  having the  Form  of  3.   It is  under  this  abstraction that we claim  the  natural  numbers  exist  at  all,  and  –  at  a  basic  level,  one  could  say  that  it  would  suffice  to  describe  the  forms  of  only  the prime  numbers.    The  number  8,  for  instance,  is  constructed  from  forms  of  2  (2*2*2  =  8.)  and  arguably  3  (since  2^3  =  8.).    The fundamental  theorem  of  arithmetic  tells  us  that  any  natural  number  greater  than  one  can  be  written  as  a  finite  product  of  primes.

One of  the  most useful areas  of  mathematics  is  set theory  (no relation to  Set the  Egyptian Neter or  Form of isolate consciousness).  A  set is  merely  a  collection of  objects called  elements, with no  structure  or  order.  Sets  have  a cardinality, the  number  of  elements  they  contain, and we  can talk about  subsets, that is, a  set whose  elements  can all be  found in some  other  set (the naturals  are  a  subset of  the integers, for instance).  We  can use  set  theory  to construct the  natural  numbers.     

A critical note:  a  set itself  can be an element of  another  set.  In this  case, the  set is  counted as  a  single  element, regardless  of  its  cardinality.

The  most basic,  and arguably  trivial, set is  called the empty set: { }.   It is  usually  denoted as  Ø.    The  first axiom of  set theory  is  that  the empty set  exists.  It contains  no elements, and  since  every  set  has a  cardinality  by  its  definition,  we  call this cardinality  zero. Next, consider  the  set containing the empty set: {  Ø }.   This  set has  a  cardinality  that is  not the  same  as  the  cardinality of  the empty set  ( it has  an element ), so we  call its  cardinality  1.  This  construction  of  one will be  the block  on which all the other  naturals  are  constructed. Then, we  take  the  following  set: { Ø, {  Ø} }, and we  call  its  cardinality  2  (it contains  two elements  –  the  empty set, and the  set containing  the empty  set). In other  words, we  take  the  previous  set, and  then add  that set itself   as  an element  to an otherwise  identical set.   If  we call the  cardinality  of  the  original  set  c, then  our  new  set  necessarily  has  a  cardinality  of  c +1.    Next, we  would construct the  set { Ø, {  Ø}, { Ø,  { Ø} }  }, and call this  cardinality  3  (its  elements  are  the  empty  set, the set containing the empty  set, and the  set  from the  previous step).   By  continuing  this  way, we  can construct  the  natural numbers  using increasingly  nested sets  of  sets  of  empty  sets. As far  as  constructing the  natural numbers  goes, this  is  just  about the  most  abstract way  to do  it, and to  return to  apple-counting, in a  deep sense, we  count apples  by  comparing the  cardinality  of  a  set  of  apples  to  the  cardinal numbers  associated with these nested empty sets.   These  cardinal numbers  we  call the  naturals.

Does this  answer  the  question  “does  3 exist?”    It certainly  answers  the question  “can  3 be  constructed?”   However, pi is  a  transcendental number,  meaning that it  cannot  be  constructed from first principles  (in this  case, constructed from straightedge  and compass, a  la  Euclid), and yet if  numbers  exist,  pi indisputably  does.   

Think, then, about the  Form  of  each natural  number.  We  can think of  these  cardinal  numbers  as  having  a  Platonic form (for  example, the  form  of  3), then it is  not  hard  to  see that if  an abstract  structure  with the  form of  a  natural  number can be constructed,  then  numbers  must therefore exist  –  in  that  a form  which  does not  exist cannot  be constructed.    In this  way  we  can surmise  from the  Theory  of Forms  that  constructibility  implies existence  (the converse, however, is  not  true  –  existence  does  not  imply  constructibility, as  is  the case  of  pi and  the other  transcendentals).

Mathematical Platonism  refers  to a metaphysical interpretation of  mathematics  concerning the  existence  of  mathematical ontology.  It is  based,  primarily, on three  theses:

 1) Existence, that is,  some mathematical ontology  does  exist.

2) Abstractness,  that is,  mathematical ontology  is  abstract.

3) Independence, that is,  mathematical ontology  is  independent of  the  activities  of  rational  beings.

Items  which are  fundamental to mathematical ontology  are  called objects.   In  the sense  of  Platonism,  these objects  must then exist,  be  abstract, and be  independent of  rational activity.  Do the natural numbers  (together with  zero)  satisfy  these conditions?

Since  the natural numbers  are  constructible, they  exist, as  previously  discussed, which  satisfies  the  first condition.   Are they  abstract?   This  is  a  complex term, and there  are  a  number of  conditions  by  which abstractness  is  determined:

1) Non-spatio-temporaility.    Another  way  to state  this  is  that an abstract object does  not exist in the  Objective Universe  (OU). Numbers do  not  exist in the  OU. You cannot  pick up  “3”  and hold it.   However,  relations  between abstract objects  may  hold in the  OU,  such as in the  case  of  binary  operations, even when they  are  not  directly demonstrable.

2) Acausality.  An abstract object does not exert a causal influence  over  any  other  object, nor  does  any  other  object have  a  causal influence  over  it.  One can change  the wavelength of  light by  which the  precise  color  “royal blue”  is  defined, or  the  number  of carbon atom widths  that determine  the length of  a  meter, or  the oscillations  of  a  cesium atom  defining one second  –  none of  these things change  the meaning of  the  numbers  themselves.   “3”  always  means  the  same  thing, even if  you pick up a  fourth  apple.

3) Eternality.   This  may  either  mean that the  abstract object exists  for  all of  time, or  the  object exists  outside  temporal relations.  This is  an interesting  question: is  the dimension of  time  necessary  for  mathematics  to exist?  It is  certainly necessary  for  the practice  of  mathematics  -  as  any  student of  math can attest to, there  never  seems  to be  enough of  it.   

Since  mathematics  has  been  used since  the dawn of  human history, and even outside  the context of  human history  –  for  example,  when  the solar  system was  formed  from  a  collapsing  solar  nebula, a  central protostar  and a number  of  planetessimals  formed from the  collapsing gas  and dust.  Even though this  lies  well outside  the realm of  human  history, and we  may  never  know  the precise  number  of  planetessimals  initially  formed in the  solar nebula, reason  tells  us  that  this  must  have  been a natural  number.  Planetessimals  are  discrete objects, and the cardinality  of  a  set of  them  can be compared to cardinal  numbers  of  sets.

For  this  reason and  many  others, it is  clear  to  me  that the  natural numbers  either exist for  all time, or  that  they exist  outside  time.  Personally, I  lean toward the  former  as  being the  more  accurate  statement, but an argument can be made  for  the latter.

4) Changelessness.  That is, the  intrinsic  properties  of  the  item will not change. As far  as  we  can tell, going back to  Babylonian mathematics, and even farther  back to  earliest systems  of counting, while  the  names  for  the  numbers  may  have  changed, the  number  that we  understand as  “3”  has  always been of  the  same  form.   Moreover, a  statement  such as  “3  is  prime”  has always  been true, and shall always  be true.

5) Necessary existence.   The  abstract object could not have  failed to exist. So long as  the OU can be  reduced to any  sort  of  discrete  components  (clusters, galaxies, stars, planets,  molecules, atoms, particles),  the natural numbers  cannot  fail to exist.   Note  that  this  is  not  contingent upon atomism  –  the components  need not be  irreducible  to be  countable.   For  example, let  S be  a  set of  infinitely  many  elements.   A set containing  S, and  only S,  has  finitely  many  elements;  namely, one.

6) Independent existence. The  most common way  to explain independence  is  to say  that  “X is  independent of  Y.”   That is, X  would continue  to exist  even if  Y  did not.   In the  case  of  the  natural numbers, an example  would  be  that the  natural  numbers  would continue  to exist  even without  humans  to do arithmetic  with  them.   Even deeper, the  independence  thesis  states  that  the properties  of  these  mathematical objects  would continue  to  exist independent of  rational  activity.  The  statement  that  “3  is  prime”  would be  true  without anyone to attempt to  find some other factor  for  it  using Euler’s  formula  or  some other  method.  8,  similarly, would always  contain the properties  of  being equivalent to 2 * 2  *  2,  or  4+4,  or  any  number  of  other  methods  of  arriving  at the  number, even  if  no rational mind existed  to perform the calculation.  That is  to  say, these  relations  are  not  reliant upon  mathematical activities  to  reveal them.  To  use  another astronomical example, the  Chandrasekhar  limit is  the upper  limit of  the  mass  a  white  dwarf  can have  and not result in either  a nova, or  a  supernova  explosion.  If  a  white dwarf  is  gravitationally  gaining mass  from  a  nearby  red dwarf, for  instance, one does  not need to compute this  limit  in order  for  the  inevitable  result  to occur  once  it is  reached.

By  these  features, we  can  say  that the natural numbers,  together with zero, are  an example  of  a  mathematical ontology.   From these, we  can summarily  construct the  integers, and  from  the integers,  the rationals, and from these, we  can construct the ordered  field we  call the  real numbers, and from  these, the  complex.  Kronecker  once  famously  stated  that  “God created the natural numbers, all else  is  the work of  man.”   I  cannot concur  with his  attribution, but that  the natural numbers  exist outside  of  the influence  of  reason seems  clear.   

THEREFORE

P1. The existence  of  the natural numbers  is  a  consequence  of  the set-theoretic  axiom  that the  empty  set exists.   

P2. Therefore, since  the  natural  numbers  can be constructed  from  the empty set alone,  the  natural numbers  exist.

P3. If  the  natural  numbers  exist,  they  are  abstract objects, independent of  rational activities.

P4. This is, by  definition, arithmetic-object Platonism, and we  may  conclude  that mathematical ontologies  up to  and including  the complex numbers, along with all  related binary  operations, do exist.   

CLEARING MY HEAD, WILL LIKELY RETURN

pi_rameses

Re: Math and Platonism (excerpt from TISv.I p. 9-13)
« Reply #1 on: April 16, 2018, 01:53:15 pm »
Reminds me of something that I saw. Why is time even considered a dimension as it isn't really Euclidean but more hyperbolic? From Zeno's time up until now, it's elusive to some extent. It is a bit confusing because whenever someone mentions higher dimensions, it cannot be assumed that we are starting at 4D. Perhaps in their mind, it's 5D to start.

Anyway, here's the old video that I saw that brought this response seemingly or of nowhere:

Pro omnis dominos viae sinistra, sic itur ad astra
Nylfmedli14