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        riddles >> general problem-solving / chatting / whatever >> Mathematics: Concrete or Discrete? 
        
(Message started by: KeyBlader01 on Oct 9^th, 2008, 10:18am)

Title: Mathematics: Concrete or Discrete?
Post by KeyBlader01 on Oct 9^th, 2008, 10:18am

Everyone got so excited (haha) from my last post, I thought I'd put this question under a new topic since everyone's forgotten about it including me until I was overlooking my questions booklet XD

I was talking to my classmate on how mathematics everything is rational, fact and possesses concrete ideas. He goes and dissents that mathematics is discrete not concrete. Then I abruptly end the conversion by saying, wait what's discrete again and he says look it up. Now, I've looked it up and I'm still not sure. I think it's both because according to the definition of concrete, it says it's an actual thing or existent. Mathematics is existent in our lives such as counting numbers. Also it can be discrete (besides that it says in mathematics under the definition =P) such as an ideal gas equation which is not present in reality. I don't think I really understanding what discrete particularly in mathematics means. Defined only for an isolated set of points... like the figures for sin or cosine? Let me know if I'm completely missing the point or definition of concrete and discrete esp. in mathematics.

From dictionary.com :
Concrete(n.):
1.constituting an actual thing or instance; real. (Or the way I put it)
2. a concrete idea or term; a word or notion having an actual or existent thing or instance as its referent.

Discrete(n.):
1.apart or detached from others; separate; distinct:
Mathematics:
a.(of a topology or topological space) having the property that every subset is an open set.
b.defined only for an isolated set of points: a discrete variable.
c.using only arithmetic and algebra; not involving calculus: discrete methods.

Title: Re: Mathematics: Concrete or Discrete?
Post by towr on Oct 9^th, 2008, 10:49am

I would contrast discrete with continuous.
The integers are discrete, real numbers are continuous.

Concrete, as far as I know, is not a mathematical term.
I suppose you can contrast concrete with abstract. And maths is pretty much the science of abstraction.

In any case, I wouldn't mix and match just because both words end in crete.

Title: Re: Mathematics: Concrete or Discrete?
Post by ThudanBlunder on Oct 9^th, 2008, 11:10am

on 10/09/08 at 10:49:35, towr wrote:

Concrete, as far as I know, is not a mathematical term.

I agree, although one of Donald Knuth's text books is called Concrete Mathematics, with 'Concrete' being a conflation of 'continuous' and 'discrete'.

Title: Re: Mathematics: Concrete or Discrete?
Post by KeyBlader01 on Oct 9^th, 2008, 11:22am

Oh so mathematics is never concrete but why?

Also, how is mathematics discrete? Give me some examples so I understand it better besides integers.

Thanks a lot guys! Honestly, thanks!

Title: Re: Mathematics: Concrete or Discrete?
Post by towr on Oct 9^th, 2008, 11:49am

on 10/09/08 at 11:22:21, KeyBlader01 wrote:

Oh so mathematics is never concrete but why?

I wouldn't say it's never concrete. I mean, money is pretty concrete; you give some amount of money, get some amount of change back. That's all arithmetic going on in the register. When you handle actual objects, it's fairly concrete.

But the real art behind it is abstract. Rather than adding with coins or other token, we use number. 1+1=2, rather than "one coin and another coin makes two coins".

Quote:

Also, how is mathematics discrete? Give me some examples so I understand it better besides integers.

Throw a pair of dice, what number of eyes can you get? Can you get any real number between 2 and 12? No; you can only get 2,3,4,5,6,7,8,9,10,11 and 12; but not http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif, or e or http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(101). You can only get a select number of distinct values; hence it's a discrete problem. (Under definition b)

If on the other hand you take a measurement of someone's length; you can, in principle, get any value between the maximum and minimum length for a human. You might get 180+pi centimeters, or http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(27313) or anything. Length is a continuous variable. People don't come in discrete lengths, where they're, say, a whole number (or for something completely different, a fibonacci number) of centimeters or millimeters long; any value is possible, in principle.

Title: Re: Mathematics: Concrete or Discrete?
Post by JohanC on Oct 9^th, 2008, 12:46pm

You might want to check out Doron Zeilberger (http://scienceworld.wolfram.com/biography/Zeilberger.html)'s paper showing that "Real" Analysis is a Degenerate Case of Discrete Analysis (http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf).

Title: Re: Mathematics: Concrete or Discrete?
Post by towr on Oct 9^th, 2008, 1:37pm

Seems rather arbitrary to me to call one a degenerate case of the other or vice versa. And reality has little to do with it. Reality might be quantized, but the position of each particle is a continuous (and complex!) function.