Re: Revised Tautology FAQ - Thread-2

On Aug 18, 2:46 am, j...@xxxxxxxxxxxxx (John S. Wilkins) wrote:
Steven L. <sdlit...@xxxxxxxxxxxxx> wrote:
Friar Broccoli wrote:
On Aug 17, 5:14 am, nos...@xxxxxxxxxxxxxxxx (J. J. Lodder) wrote:
David Hare-Scott <sec...@xxxxxxxxxx> wrote:
On the whole it's good.  If you are aiming at an explanation for the
non-technical but educated layman (tha's me) the tone and level seems
right. One problem that I noticed.  I have snipped the rest to make it
clearer.
This bit:
 Tautologies are True by definition and
may be characterised as follows:
.....
- not dependent on empirical testing (observations from the real
 world)
......
contradicts this bit
Thus, being a tautology, does not immunize a
statement from being falsified by observation.
I think I know what you are driving at but as it stands it's just a
contradiction.
The friar is not clear in his mind on the disticttion
between mathematical equations and physical ones.

F = ma as an axiom in a mathematical formulation of Newton's theory
and F = ma as an economical description of thing observble
are entirely different things conceptually,
even though they look just the same.

(I'll try to come back to this point)

Yes, I would VERY much like more input on this point/issue.
The distinction between physical reality and the formulations
that people use to describe them is central to many of the points
I am trying to make.

An algebraic equation, devoid of interpretation (semantics), cannot be
falsified.  Indeed, it cannot even be meaningfully employed in mathematics.

Devoid of interpretation, F = m a is no more falsifiable than X = y z,
or r = s t, or any one of same structure without a supplied semantics..
In fact, without an interpretation, even the equals sign "=" is
meaningless.  (Would intelligent aliens from another planet, though
scientifically far more advanced than humans, know what it means without
being told?)

In mathematical logic, we learn how to provide an interpretation
(semantics) for each term  and operator in a mathematical statement,
that maps each term to its allowed set, and each operator to the axioms
that define it.  To begin with, F, m, and a are considered by physicists
to be real numbers, and "=" to mean equality.  The axioms of equality
are taken to be the Peano postulates for the equality relation.

Science provides interpretations that go further and ostensibly match up
with quantities observable in the real world.  So they interpret F to
mean physical force, m to mean mass, etc.  Then they assert that F = m a
accurately depicts the behavior of phenomena in the real world.  But
that assertion can be falsified, and in fact the Theory of Relativity
falsified it for very high speeds and masses.

Electrical engineers freely work with complex numbers to describe
real-world phenomena, showing that it's highly important to get the
semantics well defined.

The clue as to why NS is not a tautology when used as an explanation is
exactly that point - the theorem[s] need to be interpreted. Used another
way: f=ms is a tautology (it is true by definition). When used to
explain why a given body is accelerating, it is not.

We need to distinguish several things that get called tautologies:

1. Definitional tautologies; bachelors are unmarried males.

2. Logical tautologies: the RHS is formally implied by the LHS of the
equation

3. Mathematical tautologies: terms either side of an equals sign are
coterminous

As a result of comments by Burkhard I stumbled across a
circular reasoning "tautology":

"Women write the best novels because men do not write novels as well."

would this fit into the class of Logical tautologies?





Definitional tautologies explain nothing but what is meant by the
defined term. Logical tautologies are implications of theorems or
axioms. They explain nothing but unravel what may be otherwise
unrealised in a set of premises. Mathematical tautologies are basically
your garden variety equations, and when interpreted, they are
explanations of one side's term by the other side's terms. For instance,
the ideal gas law PV=nRT explains why temperature and pressure covary,
although this is a simple tautology. When you interpret P as pressure,
measured in a physical manner, V as volume, measured in a physical
manner, and T as temperature, measured in a physical manner, n as the
number of gas molecules, and R as the gas constant for each gas, you
have an explanation.

The Price equation of natural selection is

wDELTAz = cov(wi, zi)+E(wi,DELTAzi)

The Wiki article <http://en.wikipedia.org/wiki/Price_equation> states,
correctly:

"Price's equation is, importantly, a tautology. It is a statement of
mathematical fact between certain variables, and its value lies in the
insight gained by assigning certain values encountered in evolutionary
genetics to the variables. For example, the statement "if every pair of
birds has two offspring, then among ten pairs of birds there will be
twenty offspring" is a tautology. It doesn't really impart any new
information about birds so much as it organizes our concepts about birds
and their offspring. The Price equation is much more sophisticated than
the above statement, but at its core, it too is a mathematically
provable tautology."

It is a mathematical tautology, which we use to compute what will happen
under physically realisable circumstances.
--
John S. Wilkins, Philosophy, University of Sydneyhttp://evolvingthoughts.net
But al be that he was a philosophre,
Yet hadde he but litel gold in cofre- Hide quoted text -

- Show quoted text -

.

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