Theos-Talk Archives (January 1997 Message tl00154)

theos-l@vnet.net wrote:
> 
> Date: Wed Jan  8 10:02:30 1997
> From: John Straughn <JTarn@envirolink.org>
> To: theos-l@vnet.net
> Subject: Re: The Limits of Free Will
> Message-ID: <199701081502.KAA24250@envirolink.org>
> 
> Bart Lidofsky writes:
> >Tom Robertson wrote:
> >> I would be curious to know the name of the individual who arbitrarily
> >> decided that 2+2=4.  But I'm flattered that you told my idea to several of
> >> your friends.  I'd like to see a "formal proof" that 2+2 might not be 4,
> >> also.  When might 2+2 become 5?
> >
> >       Without the use of mathematics (i.e. without circular reasoning),
> >please explain the meaning of 2 + 2 = 4. Mathematics is meaningless
> >without mathematics; it is a self-contained, self-referential system.
> >
> >       Bart Lidofsky
> There is a phsics equation in existence, I have seen it and tested it, which
> allows 1 + 1 to equal 1.  I'm going to talk to a physics instructor in the
> area soon and see if I can get it for you.  As far as explaining 2 + 2 = 4
> without using mathematics ...well I'll take a crack at it.  First of all, 2
> and 4 are merely symbols used to define a certain number.  And "number" is a
> symbol used to define and to help the psyche better understand quantity.  2
> represents a certain quantity, however, that specific quantity is not always
> equal, whether it is represnted by the two or not.  Two plus two equals four
> means absolutely nothing by itself.  All numbers are not nouns, even though
> they may be thought of in that way mathematically.  They are in actuality
> adjectives which qualify, perhaps a better word would even be quantify, a
> noun.
> 
> For instance, mathematically, 2 + 2 = 4 seems logical.  However, that is only
> illusionary logic because when I say two plus two equals four, I could be
> talking about an entirely different quantity than when you say it.  I can tell
> you right now that I figured out the radius of the cosmos and can prove it
> mathematically.  For info, it's 2.  2 what?  I'm sure you can figure out how i
> did it.  Anyhow back to the point.
> 
> Like I said, numbers are qualifiers of nouns, not nouns.  In my hand I have an
> apple.  In my other hand I just happen to have another apple.  These two
> apples look nothing alike.  One is twice as big as the other and one has green
> skin and one has red.  Nevertheless, I choose to call them apples.  Notice I
> said two just now.  I could have said three if I had wanted to, it really
> doesn't matter.  But in order for my to call it three I have to change my
> whole concept of three.  Three would no longer be able to represent what I
> have let it represent most of the years of my life.  So, to avoid confusion
> and rediculous nonsense, I chose to represent the apples as a quantity by a
> symbol called two.
> 
> Oh my GOSH!  You'll never believe this, but each apple just self-replicated
> right before my eyes!  Please excuse me for a second while I pick the clones
> up off of the floor...
> 
> Ok.  Because my apples just cloned themselves, I have realized that I now have
> increased the quantity of apples by exactly the amount of apples I had before.
>  Now, logically, I can name the newly replicated apples' quantity with the
> same symbol that I named their parents.  I shall represent the new quantity
> with the number 2.
> 
> OKAY!  Here goes.  I just lined up the two quantities of apples on the table
> in front of me and I have decided to take these two quantities and make them
> one quantity.  To do this, I need to name the new quantity.  Once again, to
> avoid confusion, I will name the new quantity with a different symbol than the
> one I used to represent the smaller quantities.  I'll call it four.  Four
> sounds good.  Now that I have defined my quantities, I can come to a reasonble
> conclusion that two apples plus two apples equals four apples.  However, like
> all things, this is only a relative deduction, for you can symbolize your
> quantities in any way you want to.
> 
> In answer to Tom's question:
> by using your free will to decide that the symbol four should be replaced by
> the symbol five, 2 + 2 can equal 5.
> 
> Oh yeah, and if you can convince the masses to do the same, a different method
> of learning may evolve.  Who knows?
> ---
> The Triaist
> 
> ------------------------------


Bertrand Russell defined Pure Mathematics as the class of all statements 
of the form "If p then q".  

The question of when is 2+2=5 is not so much a propos as the question of 
what is the formal structure which characterizes a mathematical system. 
 From the point of view of the pure mathematician, mathematics is a game 
played out with symbols, derived from arbitrary postulates, which 
characterize the nature of the system.

For example, in the class of all integers, there exists an operation 
(addition) which is commutative and associative, with an identity 
element (zero), and in which every element (integer) possesses an 
additive inverse.  Whitehead and Russell attempted to show that the 
mathematics (arithmetic and algebra) of the integers could be derived 
from pure logic alone, and they wrote a monumental 3 volume treatise 
attempting to do so.  Only well into volume II did they finally prove 
that 1 + 1 = 2.  Now if one can derive cardinality from logic, then 
ordinary arithmetic has a well defined meaning derivable from logic 
(pure reason).  Then the statement 2 + 2 = 5 is false (within such a 
structure).

As another example, sets (classes of objects) satisfy a formal structure 
known as Boolean algebra.  We know that the set of all subsets of a 
finite set has a cardinality which is a power of 2.  But how can we 
prove this statement from the postulates of Boolean algebra only?  The 
most straightforward way in my opinion is to derive operations from 
(union) and (intersection) which satisfy the group theoretic postulates 
(with a little cleverness, this is not hard to do).  This group is a 
"binary group" in the sense that each element is its own additive 
inverse.  This is quite a different result than obtains for the 
integers.  Then using some well known group theoretic results, one can 
show that the cardinality of the structure must be a power of 2.  This 
result can thus be obtained from the postulates of Boolean algebra 
alone, and without reference to their realization in set theory.  Any 
finite Boolean algebra has a cardinality which is a power of two.

Applied mathematics is a discipline which takes the structures/patterns 
of pure mathematics and finds real world realizations.  (In practice, 
the process of discovery/invention in mathematics is quite the converse; 
one solves a real-world problem, and then abstracts the solution to a 
form of pure mathematics).  Mathematical physics is a branch of applied 
mathematics in which mathematics is applied to the discipline of 
physics.  Thus finite groups, space groups, tensor calculus, sherical 
harmonics, differential equations, topological fiber bundles, and so on, 
are are employed in mathematical physics.

Truth in the discipline of mathematical physics is very different from 
truth in pure mathematics.  The idea of truth in pure mathematics is 
something akin to the idea of logical consistency.  But truth in 
mathematical physics entails questions of the correctness of 
representing the physical world, as well as consistency.  However in 
mathematical physics, a wrong theory can be useful if it illuminates the 
process of developing real world mathematical models.  But the bottom 
line in physics is that reference is made to experience and experiment 
to determine questions of truth of theoretical hypotheses.

The relevance of pure mathematics to theosophy goes back to Plato.  
Plato assumed that the structures of pure mathematics actually exist in 
an ideal realm, and that this ideal realm is the true world of 
existence, of which our world of experience is only a shadaow.  Thus, 
mathematics as a discipline leads to awareness in the pure realms of 
existence, the realm of the Soul.  Hence, above the entrance to the 
academies of philosophy was written not only "Man, Know Thyself", but 
also "Let no man ignorant of mathematics enter herein."