On Apr 23, 12:27 pm, Aatu Koskensilta<aatu.koskensi...@xortec.fi
wrote:
On 2008-04-21, in sci.logic, LauLuna wrote:

"Mathematics is what mathematicians do".
I'd prefer something a bit less sociological. What about "mathematics
is any theoretical activity which can rely on theorem proving" ?
As a definition of mathematics "Mathematics is what mathematicians do"
is of course rather poor. I offered it merely as a reminder that if we
propose a definition or a characterisation of mathematics there is
surely something wrong with it if it bears no apparent relation to
what it is mathematicians actually go about doing.

Your proposal, that "mathematics is any theoretical activity which can
rely on theorem proving" conveys something important about
mathematics, but is itself in need of further elucidation of just what
counts as "theorem proving". Not just any logically correct
argumentation is mathematics, even in theoretical contexts.

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Of course, simply carrying out formally correct reasoning is not
mathematics but it is neither theorem proving.

I'd say one needs to start from necessarily true premises or axioms to
get a theorem.

This seems to come nearer to what mathematicians actually do, at least
if we grant that what mathematicians do need not always be especially
interesting.

Regards

Aatu Koskensilta wrote:
On 2008-04-24, in sci.logic, Wolf Kirchmeir wrote:
1) Your proof fails for n=1. You need to add "and n=>2" to the premises.

Quite so.

2) We need rules of inference. One way to define a "rule of inference"
is "any rule that specifies the rewriting and/or combination of premises
in such a way that the result is true IFF the premises are true." The
result is termed a theorem. We define "premise" as "an axiom, or any
theorem, or any conjunction of axioms and theorems proven true."

In order to characterise or define mathematics it must be explained
what counts as an axiom and why. In this context we may take it as
unproblematic what is logically correct reasoning.

3) You don't need the necessary truth of the premises. Inferred truth is
sufficient. (I don't think "necessary truth" is a useful term, except as
a synonym of "inferred truth.")

But what sort of premises are acceptable?

Any that are "axioms, theorems, or conjunctions of axioms and theorems
proven true."

In order to make some use of
the definition of mathematics as "any theoretical activity which can
rely on theorem proving" we must explain just what "theorem proving"
amounts to. Any logically correct argumentation from random premises
certainly does not count as "theorem proving".


Theorem proving amounts to writing a series of statements, each of which
is produced by inference rules applied to one or more of the premises.
NB that by this definition, any such statement in turn becomes a premise.

NB also that by the given definition, in mathematics "true" is a
property of statements. Any theorem is true by virtue of proof. Axioms
are true by definition.

Whether some such statements can be interpreted to apply to or be
meaningful in some universe of discourse outside mathematics is a
different issue. "Truth" outside of mathematics is also a different
issue. But "truth" is always a property of representations, whether
those representations are statements or something else. More precisely,
we say a representation is true if it resembles the thing represented.

"Resemble" varies with the universe of discourse. That's one of the
facts of human existence that makes life interesting.

Anyhow, the above makes sense to me. YMMV.

LauLuna wrote:
On Apr 23, 12:27 pm, Aatu Koskensilta<aatu.koskensi...@xortec.fi
wrote:
On 2008-04-21, in sci.logic, LauLuna wrote:

"Mathematics is what mathematicians do".
I'd prefer something a bit less sociological. What about "mathematics
is any theoretical activity which can rely on theorem proving" ?
As a definition of mathematics "Mathematics is what mathematicians do"
is of course rather poor. I offered it merely as a reminder that if we
propose a definition or a characterisation of mathematics there is
surely something wrong with it if it bears no apparent relation to
what it is mathematicians actually go about doing.

Your proposal, that "mathematics is any theoretical activity which can
rely on theorem proving" conveys something important about
mathematics, but is itself in need of further elucidation of just what
counts as "theorem proving". Not just any logically correct
argumentation is mathematics, even in theoretical contexts.

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Of course, simply carrying out formally correct reasoning is not
mathematics but it is neither theorem proving.

I'd say one needs to start from necessarily true premises or axioms to
get a theorem.

In the case of the Axiom of Choice, is it true, or simply "assumed to
be true"?

I'm not too clear on questions like that.

David Bernier


This seems to come nearer to what mathematicians actually do, at least
if we grant that what mathematicians do need not always be especially
interesting.

Regards

On 2008-04-24, in sci.logic, Lester Zick wrote:
On Thu, 24 Apr 2008 07:14:17 GMT, Aatu Koskensilta
aatu.koskensilta@xortec.fi> wrote:

But do you count just any demonstration of truth as mathematics?

Of course. No other way to settle issues.

So if someone demonstrates he was, in fact, at home yesterday that's
mathematics?

On 2008-04-24, in sci.logic, Wolf Kirchmeir wrote:
Theorem proving amounts to writing a series of statements, each of which
is produced by inference rules applied to one or more of the premises.
NB that by this definition, any such statement in turn becomes a premise.

Then we are back to treating just any logically correct argumentation
as mathematics.

NB also that by the given definition, in mathematics "true" is a
property of statements. Any theorem is true by virtue of proof. Axioms
are true by definition.

We can find a logically correct derivation of "0=1" from any number of
premises. Surely that doesn't mean that 0=1 is true.

On 2008-04-24, in sci.logic, mitch.nicolas.raem...@gmail.com wrote:

The complex plane is based on one negative root quantity called i.
These simply do not exist as qauntities.

The complex plane is not based on any quantities. It's just a
mathematical structure. What do you have in mind with "believing in
the complex plane"?

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

mitch.nicolas.raem...@gmail.com wrote:
On Apr 23, 11:20 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi
wrote:
On 2008-04-24, in sci.logic, mitch.nicolas.raem...@gmail.com wrote:

The complex plane is based on one negative root quantity called i.
These simply do not exist as qauntities.
The complex plane is not based on any quantities. It's just a
mathematical structure. What do you have in mind with "believing in
the complex plane"?

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

If a mathematics doesn't apply in real physical quantities it is just
imagination.

Stephen Hawkings imaginary time is just that. It requires an Imaginary
clock.

"As far as the laws of mathematics refer to reality, they are not certain,
as far as they are certain, they do not refer to reality."

Albert Einstein- Hide quoted text -

- Show quoted text -

mitch.nicolas.raem...@gmail.com wrote:

[...]

Mitch Raemsch Twice Nobel Laureate 2008

Kindly refer me to the Nobel Laureate list on which your name appears.

--
wolf k.

On Apr 23, 11:20 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi
wrote:
On 2008-04-24, in sci.logic, mitch.nicolas.raem...@gmail.com wrote:

The complex plane is based on one negative root quantity called i.
These simply do not exist as qauntities.
The complex plane is not based on any quantities. It's just a
mathematical structure. What do you have in mind with "believing in
the complex plane"?

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

If a mathematics doesn't apply in real physical quantities it is just
imagination.

Stephen Hawkings imaginary time is just that. It requires an Imaginary
clock.

On 2008-04-24, in sci.logic, Wolf Kirchmeir wrote:
Theorem proving amounts to writing a series of statements, each of which
is produced by inference rules applied to one or more of the premises.
NB that by this definition, any such statement in turn becomes a premise.

Then we are back to treating just any logically correct argumentation
as mathematics.

NB also that by the given definition, in mathematics "true" is a
property of statements. Any theorem is true by virtue of proof. Axioms
are true by definition.

We can find a logically correct derivation of "0=1" from any number of
premises. Surely that doesn't mean that 0=1 is true.

mitch.nicolas.raemsch@gmail.com wrote:
On Apr 23, 11:20 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi
wrote:
On 2008-04-24, in sci.logic, mitch.nicolas.raem...@gmail.com wrote:

The complex plane is based on one negative root quantity called i.
These simply do not exist as qauntities.
The complex plane is not based on any quantities. It's just a
mathematical structure. What do you have in mind with "believing in
the complex plane"?

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

If a mathematics doesn't apply in real physical quantities it is just
imagination.

Stephen Hawkings imaginary time is just that. It requires an Imaginary
clock.

"As far as the laws of mathematics refer to reality, they are not certain,
as far as they are certain, they do not refer to reality."

Albert Einstein

Mitch Raemsch Twice Nobel Laureate 2008

On 2008-04-28, in sci.logic, Occidental wrote:

The question is - what is the demarcation criterion between those
things about which man nicht sprechen kann and everything else? I'm
quite prepared to be totally schweigen, but I need a good reason.

A list of subjects on which we must remain forever silent can be found
in Wittgenstein's famous mauve and black books, dating from his
post-Norwegian period.

--
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

The question is - what is the demarcation criterion between those
things about which man nicht sprechen kann and everything else? I'm
quite prepared to be totally schweigen, but I need a good reason.

On 2008-04-29, in sci.logic, Occidental wrote:
But a list is not a criterion; is the criterion itself one of the
things about which we must remain silent?

Alas Wittgenstein was never very explicit on this issue in his
writing. The closest he comes is on p. 342 of the Hello-Kitty book --
so named because his notes were at the time circulated in the form of
indecipherable scribbling in a Hello-Kitty notebook -- where he
says, in Heikki Nyman's translation:

127. It might happen that someone asks us a question about our own
past work. Suppose, for example, that someone asks you "Why did
you choose gardening?". Just how are we to answer? We are
naturally drawn to certain ways of answering, but thinking of
that gives me a strange tingly feeling in my toe. Here we must
ask: why the toe? Why should the question give rise to such a
strange occurrence specifically in the toe? (NB. Consider the
claim "The dog is nowhere to be seen but my toe feels quite
tingly indeed". Why would anyone ever say such a thing?)