Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: LOGIC & MATHEMATICS
Replies: 96   Last Post: Jun 6, 2013 5:19 AM

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: LOGIC & MATHEMATICS
Posted: May 28, 2013 8:56 PM

On 5/28/2013 8:17 AM, Julio Di Egidio wrote:
> "Zuhair" <zaljohar@gmail.com> wrote in message

>> On May 27, 6:51 pm, Charlie-Boo <shymath...@gmail.com> wrote:
>>> On May 26, 2:52 am, Zuhair <zaljo...@gmail.com> wrote:
>>>

>>> > Frege wanted to reduce mathematics to Logic
>>>
>>> What does it mean to "reduce mathematics to Logic"?

>>
>> It means that any mathematical discourse can be interpreted within a
>> logical discourse.
>>
>> Logic is responsible for laying down a set of rules that results in
>> generation of non contradictory statements in the most general manner.

>
> Logic is not just mathematical logic: case in point, you can make of
> identity an axiom, but the notion of identity itself presupposes an
> existential stance, so it is not a purely logical notion (in the sense
> you have just stated). My take is that mathematics uses logic (as the
> language of mathematics is as well logical) and logic uses mathematics
> (as the logical calculus is mathematical), but neither can be reduced to
> the other. Mathematics is the study and applications of "numbers", which
> is not a purely logical endeavour, while logic is the study and
> applications of "rational discourse", which is not an essentially
> mathematical endeavour.

Frege concluded that his attempt had been an error. In
"Numbers and Arithmetic" he retracted his logicism with
the statement,

"The more I have thought the matter
over, the more convinced I have become
that arithmetic and geometry have
developed on the same basis -- a
geometrical one in fact -- so that
mathematics in its entirety is
really geometry"

Frege

Although generally ignored, Kant invoked a strict demarcation
between logic and mathematics. Frege's logicism is essentially
the idea that one can have "logical objects" and this had been
one of the things that Kant had rejected. There is a nice
discussion of Kant and Frege in sections 4 and 5 of John
MacFarlane's dissertaion,

http://johnmacfarlane.net/dissertation.pdf

Of course, there have been significant efforts at
recovering Frege's program. I can give you links
if you are interested, but MacFarlane's paper gives
an analysis similar to your own statements. So,
I thought you might find it interesting.

>
> I'd think the only way to operate the "unification" you have in mind is
> by reducing *everything* to just its operational side, the calculus:
> then there is not even any difference left between logic and
> mathematics, indeed you'd have destroyed the very nature of both.
>

This is exactly how I feel when I run into views
that reject model theory. I am at a loss of how
a "syntax only" view of mathematics is coherent.
But, that is one of the developments in the history
of mathematics as different "number systems"
evolved.

Among classical writers, De Morgan seems to have
seen this as an outcome of adopting these "new"
arithmetical systems as part of mathematics:

"As soon as the idea of acquiring
symbols and laws of combination,
without given meaning, has become
familiar, the student has the notion
of what I will call a symbolic
calculus; which, with certain symbols
and certain laws of combination, is
symbolic algebra: an art, not a
science; and an apparently useless
art, except as it may afterwards
furnish the grammar of a science.
The proficient in a symbolic calculus
would naturally demand a supply
of meaning. Suppose him left without
the power of obtaining it from
without: his teacher is dead, and he
must invent meanings for himself.
His problem is: Given symbols and
laws of combination, required meanings
for the symbols of which the right
to make those combinations shall be
a logical consequence. He tries,
and succeeds; he invents a set of
meanings which satisfy the conditions.
Has he then supplied what his teacher
would have given, if he had lived?
In one particular, certainly: he has
turned his symbolic calculus into a
significant one. But it does not
follow that he has done it in a way
which his teacher would have taught
if he had lived. It is possible
that many different sets of meanings
may, when attached to the symbols,
make the rules necessary consequences."

Augustus De Morgan

In other words, each of us interprets
the language of mathematics individually
according to our experiences with it.

Date Subject Author
5/26/13 Zaljohar@gmail.com
5/26/13 namducnguyen
5/26/13 Zaljohar@gmail.com
5/26/13 namducnguyen
5/26/13 Peter Percival
5/26/13 namducnguyen
5/26/13 Peter Percival
5/26/13 namducnguyen
5/26/13 Zaljohar@gmail.com
5/28/13 Charlie-Boo
5/28/13 Charlie-Boo
5/26/13 Zaljohar@gmail.com
5/27/13 zuhair
5/27/13 fom
5/27/13 Zaljohar@gmail.com
5/27/13 fom
5/28/13 namducnguyen
5/28/13 Zaljohar@gmail.com
5/28/13 namducnguyen
5/29/13 Peter Percival
5/30/13 namducnguyen
5/30/13 Peter Percival
5/30/13 Peter Percival
5/30/13 namducnguyen
5/31/13 Peter Percival
5/30/13 Bill Taylor
5/30/13 Peter Percival
5/30/13 Zaljohar@gmail.com
5/30/13 Zaljohar@gmail.com
5/30/13 namducnguyen
5/31/13 Peter Percival
5/31/13 Zaljohar@gmail.com
5/31/13 LudovicoVan
5/31/13 fom
5/28/13 Peter Percival
5/28/13 namducnguyen
5/27/13 Charlie-Boo
5/27/13 fom
5/28/13 Charlie-Boo
5/28/13 fom
6/4/13 Charlie-Boo
6/4/13 fom
6/5/13 Zaljohar@gmail.com
5/28/13 Zaljohar@gmail.com
5/28/13 LudovicoVan
5/28/13 ross.finlayson@gmail.com
5/28/13 LudovicoVan
5/28/13 LudovicoVan
5/28/13 fom
5/29/13 LudovicoVan
5/29/13 fom
5/30/13 LudovicoVan
5/29/13 fom
5/30/13 LudovicoVan
5/30/13 fom
5/31/13 LudovicoVan
5/31/13 Zaljohar@gmail.com
5/31/13 LudovicoVan
5/31/13 ross.finlayson@gmail.com
6/1/13 LudovicoVan
6/1/13 namducnguyen
6/1/13 ross.finlayson@gmail.com
6/2/13 LudovicoVan
6/2/13 ross.finlayson@gmail.com
6/3/13 Shmuel (Seymour J.) Metz
6/3/13 ross.finlayson@gmail.com
6/4/13 LudovicoVan
6/4/13 namducnguyen
6/4/13 Peter Percival
6/5/13 Shmuel (Seymour J.) Metz
6/5/13 fom
6/6/13 Peter Percival
5/31/13 fom
6/1/13 LudovicoVan
6/1/13 fom
6/2/13 ross.finlayson@gmail.com
6/2/13 fom
6/2/13 Herman Rubin
6/2/13 fom
6/2/13 LudovicoVan
6/3/13 Herman Rubin
6/3/13 Peter Percival
6/4/13 Herman Rubin
6/4/13 Peter Percival
6/4/13 Peter Percival
6/1/13 fom
6/1/13 LudovicoVan
6/1/13 namducnguyen
6/5/13 Peter Percival
6/1/13 fom
6/2/13 LudovicoVan
6/2/13 fom
5/28/13 Zaljohar@gmail.com
5/28/13 Charlie-Boo
5/27/13 Zaljohar@gmail.com
5/28/13 Charlie-Boo
5/30/13 Zaljohar@gmail.com