On the definition of Mathematics and its completeness

[Zuhair]

In post titled "mathematics is" I had presented a definition of
mathematics, to re-iterate:

Mathematics is the investigation of behaviors that are formalizable in
consistent formal systems extending logic.

Although the concept is correct yet it is immense. There are
infinitely many consistent formal systems that can extend logic, and
some of them can be effectively generated in a very complex manner,
some actually may not be attainable to human capability. Also a lot of
behaviors that we think they are not formalizable in a manner
extending logic like human behavior for example might be in reality
formalizable but the formal systems in which they are formalized are
utterly complex to the degree of being non describable using our
limited human resources. So we need to ease a little bit on this
definition.

My general feeling is that mathematics is ought to be the next tier
after logic in simplicity. So mathematics much be formalized in
systems that can be effectively generated in somewhat simple manner.
What decides the degree of simplicity necessary for mathematics to be
defined after and how can we determine it and measure it is a
legitimate question of course, however I wont discuss these details
and I shall concentrate my attention on discussing the possibility of
COMPLETELY generating mathematics, which is possible as I would show
here.

So to redefine mathematics:

Mathematics is the investigation of behaviors that are formalizable in
consistent formal systems extending logic that are effectively
generated in a simple manner.


Now according to this definition the possibility of defining an
effectively generated formal system that can capture ALL mathematical
statements is not refutable. Something that goes against the general
impression Godel's incompleteness theorems imparted.

We can define an effectively generated consistent formal system that
can capture all mathematical statements yet that system itself is NOT
effectively generated in a simple manner! Truly this formal system
would be "incomplete" in the strict logical sense of not deciding on
all sentences in its language, yet all those sentences can be non
mathematical (not formalizable by some consistent formal system
effectively generated in a simple manner). Of course neither this
formal system nor its proof of consistency would be parts of
mathematics since they are too complex, they are parts of some ultra-
complex formalizable behavior, I like to call that part: ULTRA-
FORMATICS.

So the "whole" of mathematics in the sense defined above can be
interpreted in an effectively generated consistent ultra-formatical
system.

The real challenge here is of setting the needed simplicity
criterion.

Zuhair

[Ross A. Finlayson]

Hello,

It's a worthy exercise to work toward expressive definitions.

Mathematics is the integers.

Then, what don't we get from that?

Now the 19th century proclamation "God made the integers, the rest is
the work of man" may be read instead "given the integers mathematics
follows".

Then, the integers are a(n) "effectively generated consistent ultra-
formatical system".

I think the word is good, to use format-ical or forma-t-ical, I think
you mean "ultra-formal", or even as it were "ultra-formalatical", even
formulatical, sui pro forma. Then though "Formatics" is interesting,
here I think you could develop that.

Well I'm not really caught up on the conversation in the other thread
there, mathematics is. Still though I think this is a reasonable
point. This most complex structure is quite simple.

While that may be so, I don't want to distract you from your course so
please continue.

Regards,

Ross Finlayson