Re: Galileo's Paradox

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


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. 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". Truth is a form of quantity.


"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. These letters on your screen are numbers.


You might look at the following threads currently active in sci.math
within the last 24 hours (believe it or not, not everyone posts to
argue about Cantor/infinity):

"some advance algebra q's. please help"
"Group on arbitrary ordinal"
"Another two universal algebra questions"
"? how subspaces are like"
"large ordinals, help!"
"Continuous injection from a subset of R^n to R"
"Smith normal form in a binary field (F2), symbolic"
"Infinite width Moebius strip"
"notation of fields and rings"
"(Universal Algebra) The function determined by a term..."
"Intermediate Fields in Galois Theory"
"Mapping of integer functions into reals"
"Sets of quaternions with no repeats in infinite set of products:"
"please help me explain: Counter-Example for Riesz representation
Theorem"
"how to deduce the algebra structure"
"Infinite simple groups and their proper subgroups"
"Right cosets intersection"
"Valuations in field extensions"
"Groups and commutators"
"Compactness"

... and so on.

There are many mathematical questions of the forms: "does there exist X
such that P(X)?", "characterize all X such that P(X)", and "find an X
such that P(X)" that do not use a "real measure" to describe X, and are
not about "measuring" things with "numbers" in the sense I'm guessing
you mean. In particular on sci.math, there's topology, number theory,
abstract algebra, linear algebra, and graph theory.

These areas exist because of a combination of two factors: they are
sometimes useful, and sometimes interesting in their own right to its
practitioners.

As far as /I'm/ concerened, these are the only actually /interesting/
parts of mathematics. Face it - Calculus is boring! I don't like adding
up columns of numbers, so I have a calculator. Nor do I like plodding
through many certainly quite well-known transformations; so I look it
up in a table if I need it.

On the other hand, given my interests, I don't need it that often; so I
also don't personally find it very "useful" :).

Cheers - Chas


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.

Tony
.

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