FOM: Meaning vs Significance Nailed Down, Work for Logic and Rev.Math.
Robert S Tragesser
RTragesser at compuserve.com
Wed Dec 10 02:51:21 EST 1997
MEANING AND SIGNIFICABCE SORT OF NAILED DOWN
Contents: Making exact the distinction between meaning and significance by
following out the distinction between elementary and nonelementary proofs
of theorems, and the bearing of this on CH?.
An open characterizational problem for logic (and a problem set for
Reverse Mathematics?).
How significance can force a change of meaning.
Since Neil Tennant has found useful the distinction between the
meaning and significance of a theorem/problem, and since 'significance' is
as wildly wobbly a signifier as 'meaning', it does seem worthwhile to
point to some examples from a category of mathematical activity which,
borne in mind, would serve to (ostensively) fix (by example) the sense of
both 'meaning' and 'significance', and allow us to articulate the way in
which the question of the significance of CH is anomalous.
Proofs of difficult theorems are often first attained by
non-elementary means, and then the task is to discover an elementary
proof. "Elementary" does not of course mean simple. It means roughly:
proved by means native to the subject with which the statement of the
theorem has greatest affinity, e.g., an elementary proof that there are
infinitely many prime should involve only concepts whose "elements" are
native to the concept "prime natural number".
Example 1. Euclid's proof that there are infinitely many prime numbers is
elementary while Euler's proof (a seed proof for analytic number theory) is
not.
Example 2. (nonelementary) Proof using Pythagorean geometric-figurate
arrays that the sum of the first n positive integers is n(n+1)/2, whereas
an elementary proof would be by mathematical induction [someone asked:
unless one believes that the natural numbers just are fuigural point
arrays?].
Example 3. The Prime Number Theorem (pi(x) squiggle x/logx). The first
proofs were nonelementary (Hadamard/Poussin) because employing the theory
of functions of a complex variable. Erdos/Selberg gave an elementary
proof ("uses only elementary estimates of the relative magnitudes of
primes, relying only on cocepts intrinsic to the conception of prime
number").
Example 4. The Fundamental Theorem of Algebra. It has not (as far as I
know) been given a purely algebraic proof. Here there is some ambiguity
about "elementary proof". Would only a purely algebraic proof be
elementary? Some say not because analytical ideas are inherent to the
sense of the theorem -- "because the field R, and consequently the
extension field C, is a construct belonging to analysis".
Example 5. (Fundamental Theorem continued) Hopf's Theorem &
Gelfand-Mazur theorem.
One way to explore the possibility of an elementary proof might be
to look for more abstract theorems which have the Fundamental Theorem of
Algebra as a consequence. E.g., Hopf's theorem (Every finite dimensional
real commutative division algebra A = (V,.) is at most 2-dimensional).
No elementary proof seems to be known.--a proof involving topological
mappings of projective spaces P(n) into sphere S(n); a proof from
algebraic geometry (I don't know where).
Gelfand-Mazur Theorem: Every commutative banach division algebra is
isomorphic to field R or C. Apparently GM is a consequence of Hopf; no
elementary proof of GM is apparently known (one apparently popular proof
involves power series whose terms are also power series).
Both Hopf and G-M entail the Fundamental Theorem of Algebra. Are
they somehow part of the deep meaning of the FTA?
SIGNIFICANCE & MEANING: I want to say (by way of anchoring those two
terms): a (proposed) theorem has SIGNIFICANCE beyond MEANING insofar as a
nonelemetary proof (or disproof) of it is given. A (proposed) theorem is
weak in MEANING insofar as an elementary proof (or disproof) of it cannot
be given.
It might be, then, that CH is weak in meaning, where elementary
proof (or disproof) means proof (or disproof) in pure, transfinite set
theory. There are two scenarios WHICH SUGGEST THAT MEANING AND
SIGNIFICANCE CAN VARY (AS IT WERE) INDEPENDENTLY OF ONE ANOTHER:
(1) CH can only be given a nonelemetary proof or disproof (as for
example by appeal to some sort of geometric insight -- perhaps this is what
Goedel had in mind when he spoke of a spatial aspects of sets).
(2) CH has essentially (say) geometric significance (having to do
with intuitive continuity and measure), but that its meaning (in pure
transfinite set theory) is out of wack because pure transfinite set theory
as currently conceived is not right for capturing/representing continuity.
Clearly, Brouwer thought that (2) is the case [AND NOT BECAUSE HE
WAS AN ANTI-REALIST -- AS I ARGUED ABOVE, IN THE END, B'S SUPPOSED
ANTI-REALISM IS OF NO IMPORTANCE WHATSOEVER TO THE EMERGENT REPRESENTATION
OF FLUIDIC CONTINUA!].
Can a similar situation ever occur for a proposed theorem of the
sorts that occur in the five examples? That the SIGNIFICANCE OF A THEOREM
CAN CAUSE US TO REVISE THE TERMS OF ITS STATEMENT IN SUCH A FUNDAMENTAL
WAY, isn't that interesting about set theory???
Could other mathematical theories behave in this way?
Well, isn't this what happened to the Eulerian infinitesimal
calculus -- the significance of the theorems forced a fundamental, radical
change in the terms in which they were cast (fundamentally altering their
meaning, but clarifying their significance)??????????
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