Re: Galileo's Paradox

Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:
Tony Orlow wrote:

What,in mathematics, has a solution which is neither a real measure, or
the measure of truth of a statement, 0, 1, or somewhere in between?

"Find all pairs of distinct naturals (x,y) such that x^y = y^x". The
solution to which is the set {(2,4), (4,2)}, which doesn't appear to be
a "real measure", nor a "measure of the truth of a statement" (as far
as I can understand your meaning of the terms).

Those are specifications for two points in a 2D array of naturals, the
values within each pair denoting the distance of each point in each of
the two directions, from the origin.

That's one way of imagining it. Another interpretation would be that
each pair (x,y) is a specification for the point on the real line whose
distance from the origin is denoted by 2^x/3^y (this is a dense set).

Yet another interpretation is that the pair (x,y) is a specification of
the set of all x by y chessboard positions containing 4 non-attacking
rooks.

Are you saying that the "solution" of {(2,4), (4,2)} in these
interpretations would be somehow different from the solution in your
interpretation? Why is an interpretation required at all?

Are you saying x and y and 2 and 4
are not taken to be quantities? That's a rather strange position to take.


Well, your phrase "real measure" is a little vague. I really don't
understand what you mean by it.

Sure, I understand that 2 and 4 are quantities in the usual sense of
the word: "2 apples", "4 apples". And I understand that when I add 1/2
cup of apple juice to 1/2 cup of apple juice, I get 1 cup of apple
juice; "1/2" is a "real measure".

But the "solution" to the problem given is not 2 or 4 or 1/2; it is a
set of ordered pairs of naturals.

Why is the set {(2,4), (4,2)} a "quantity" or a "real measure"? Can I
have "{(2,4), (4,2)} apples", or "{(2,4), (4,2)} cups of apple juice"?


I assume that you offer more than the trivial observation that all
mathematical statements P, including the statement "2^4=4^2", are
examples of the (boolean) truth valued statement "it can be proved that
P". If so, I would claim that the mathematical question is "Find a
proof of P, or proof of not P". And the solution is not "0" or "1"; the
solution is either a proof of P, or a proof of not P.

First of all, that trivial observation is plenty. All logic is subsumed
under math as a calculation of truth values between 0 and 1.

I think you are confusing boolean algebra with mathematical proof.

Further, a
proof is precisely this calculation of truth value for a given
statement. To say "find a proof" is to say "define a sequence of
operations on the given statements to produce another such that its
value is 1".

A proof is a series of calculations such as: from the strings "(p->q)"
and "p" use the rule modus ponens to produce the string "q". It ends
with a step which produces the string corresponding to the assertion of
the theorem.

These are not calculations producing the value 1, except in the trivial
sense that the statement "It is the case that George W Bush is the
current president of the US" can also be called "a calculation
producing the value 1".

In which case, the "solution" to just about any question you like is "a
measure of truth of a statement, 0, 1, or somewhere in between"; so I
don't see how it's a particular property of /mathematical/ questions as
opposed to other types of questions.

Truth is a form of quantity.

In different contexts there are different axioms and different rules of
inference; so "truth" is certainly not a "quantity" that is fixed for
some statement such as "there exists x such that 2*x = 3".



"Find all finite groups G having a maximal subgroup S, and having a
subgroup T which is isomorphic to S but not maximal". This question was
asked in sci.math a few days ago. The groups in question are not
"numbers" at all; and a set of them possesses no particularly natural
total orderings. They can be partially ordered by "size" (number of
elements); but there are, in general, multiple distinct groups on a set
of any given size.

Assign each element a bit, and every group corresponds to a binary
string, which corresponds to a value.

#1: There is 1 group of order 1.
#2: There is 1 group of order 2.
#3: There is 1 group of order 3.

There are two groups of order 4, both are commutative. I will
arbitrarily number them as

#4 : 0 + x = x; x + x = 0; 1 + 2 = 3; 1+ 3 = 2; 2 + 3 = 1
#5 : 0 + x = x; 1 + 1 = 2; 1 + 2 = 3; 1+ 3 = 0.

But how does this numbering help me figure out what the groups of order
5 are?

I'm not arguing that it is impossible to count these groups (i.e.,
assign a unique natural to each one). I'm arguing that /before/ you can
count them, you must produce them. /Producing/ them is then the
"solution" to the problem, not counting them /after/ they have been
produced.

I think you haven't been exposed to much mathematics where this is the
case; and I'm guessing that that's why you seem to assume that
mathematics is about "measure".

For example, "find all squares of naturals", is a case where we /can/
produce them by /counting/: "the first is 1*1 = 1, the second is 2*2 =
4, ..., the nth one is n*n = n^2, ..."

A more complicated example: we can produce the Fibonacci numbers (1, 1,
2, 3, 5, 8, ...) by counting them via the function F(n) = (phi^n - (1 -
phi^n))/sqrt(5).

But there is no similar "closed form function" that maps naturals onto
groups in this way. To find the groups with 5 elements up to
isomorphism, one can examine one by one each element of the set of 5^25
(approximately 10^17) binary functions on 5 elements, which can
certainly be enumerated.

But it's far faster (and of more mathematical interest) to use a proof
to show that there is exactly 1 such group (up to isomorphism), because
5 is a prime number.

These letters on your screen are
numbers.


And so every statement communicated through written language is a
number, and therefore everything is mathematics? This again becomes
true of just about any statement. And yet, I feel that there is a
difference between the branches of human endeavor "history" and
"mathematics".

<snip>


That's fine. I am still of the opinion that math boils down to measure,
the language of measure, and the operations allowed on that language.


Your statements would be easier to understand if you explained what you
mean by "measure", "the language of measure", and the operations you
allow on that language.

Cheers - Chas

.

Relevant Pages

  • Re: Galileos Paradox
    ... Those are specifications for two points in a 2D array of naturals, ... I'm just saying it boils down to quantity (including truth values), language, and operators. ... Remember, I included the language of expression, and the operators allowed in the language, in mathematics. ...
    (sci.math)
  • Re: An uncountable countable set
    ... required to list the naturals in binary defies classification in this ... you will have at most n-1 unused strings. ... Truth is the value put on a statement, ... with reality" that apply to mathematics, ...
    (sci.math)
  • Re: An uncountable countable set
    ... so there must be a cardinality for the set of bit positions in the naturals. ... number of strings, so if you have gotten to the least bit string necessary for n naturals, you will have at most n-1 unused strings. ... Truth is the value put on a statement, ... with reality" that apply to mathematics, truth in mathematics appears to be entirely independent of reality. ...
    (sci.math)
  • Re: Galileos Paradox
    ... Those are specifications for two points in a 2D array of naturals, ... "measure of the truth of a statement", but that does not mean that I ... But one "embed" logic within mathematics without having logic a priori ... the language of measure, and the operations allowed on that language. ...
    (sci.math)
  • Re: Semantics of First-Order Languages
    ... It just means there are open problems in mathematics. ... false on an interpretation. ... assigning a truth value where we can not? ...
    (sci.logic)
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