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All of math is connected

Posted: Thu Mar 22, 2018 7:18 pm
by Philosophy Explorer
There are no separate subject areas or branches in math.
That is my belief based on the Langlands program.

The implications are wide ranging. If there is a single foundation for math, then this belief would make it a foundation for all of math. Here is a Wikipedia article with more on this topic:

https://en.m.wikipedia.org/wiki/Robert_Langlands

Note: a closely related question is "What is math?"

PhilX πŸ‡ΊπŸ‡Έ

Re: All of math is connected

Posted: Thu Mar 22, 2018 9:54 pm
by Science Fan
After you figure out what math is then you'll have to figure out what applied math is as well. I don't know if definitions for subjects like math, applied math, physics, electrical engineering, make a whole lot of sense. The only real way to know what these subjects are about is to learn them. I don't think any definition can capture any discipline completely and informatively.

Re: All of math is connected

Posted: Fri Mar 23, 2018 12:15 am
by vegetariantaxidermy
Thilly thubject.

Re: All of math is connected

Posted: Fri Mar 23, 2018 12:59 am
by Philosophy Explorer
The entire category would be silly to those who don't know advanced math.

PhilX πŸ‡ΊπŸ‡Έ

Re: All of math is connected

Posted: Fri Mar 23, 2018 4:28 pm
by Science Fan
Phil: Veggie probably doesn't realize that the Langland's project is the equivalent of the mathematicians' search for a unified theory of mathematics, like physicists look for a unified theory in physics.

Re: All of math is connected

Posted: Fri Mar 23, 2018 7:02 pm
by Philosophy Explorer
Definitions is a tricky business because you never know when it needs to be adjusted for new concepts. The best you can hope for is to anticipate a trend so that the definition can cover any forthcoming concepts.

PhilX πŸ‡ΊπŸ‡Έ

Re: All of math is connected

Posted: Fri Mar 23, 2018 8:13 pm
by wtf
What about disconnected sets? Are they connected?

Re: All of math is connected

Posted: Fri Mar 23, 2018 8:41 pm
by Philosophy Explorer
wtf wrote: ↑
Fri Mar 23, 2018 8:13 pm
What about disconnected sets? Are they connected?
Depends on what you mean by disconnected. They could be connected. For example the sets of even and odd numbers are connected through the set of natural numbers.

PhilX πŸ‡ΊπŸ‡Έ

Re: All of math is connected

Posted: Fri Mar 23, 2018 8:48 pm
by wtf
Philosophy Explorer wrote: ↑
Fri Mar 23, 2018 8:41 pm
Depends on what you mean by disconnected.
A set is disconnected if it can be partitioned into two disjoint open sets. For example the union of the intervals [0,1] and [2,3] is a disconnected set.

Is it connected? No, it's disconnected. Connected and disconnected are technical terms in general topology.

https://en.wikipedia.org/wiki/Connected_space

Perhaps a better word for you to use would be "related," since that's not a technical term in common use. I had a math professor who was a bit of a wildman type who said he'd like to walk into class and start lecturing on whatever happened to be on the blackboard from the previous class, on the theory that everything's related. Or maybe he said connected.

Just funnin' ya as usual.

Re: All of math is connected

Posted: Fri Mar 23, 2018 11:14 pm
by Philosophy Explorer
My main interest is recreational math. Many regard it as trivial or unimportant even though it can entertain or amuse.

I think many would be surprised that there's a narrow line between recreational and practical and applied math. An outstanding example is the Seven Bridges of Konisberg puzzle which Leonhard Euler solved in the 18th century
(leading to the creation of a whole new branch of math,
topology which Poincare expanded on in the 19th century).

Another example is magic squares which my book shows has practical uses. It takes imagination and vision
to expand the boundaries of math.

This article from Wikipedia gives more details:

https://en.m.wikipedia.org/wiki/Topology

PhilX πŸ‡ΊπŸ‡Έ