The Sum is Greater Than It’s Parts
In a fit of true geekiness, I was thinking about the saying “the sum is greater than it’s parts“ while taking a shower earlier. When I got back to my desk, I quickly scribbled the inequality using summations and some topos theory. I’m not too sure about my presentation but maybe I’ll come back to this and clear it up with a better presentation if it comes to me. Here’s what it looks like:
$$\{\phantom{\rule{thinmathspace}{0ex}}\mathrm{\Omega}\phantom{\rule{thinmathspace}{0ex}}|\phantom{\rule{thinmathspace}{0ex}}{\omega}_{1},{\omega}_{2},...,{\omega}_{n}\phantom{\rule{thinmathspace}{0ex}}\};\text{}\coprod \mathrm{\Omega}\sum _{1}^{n}\omega $$The inequality, in plain english, says that the sum of a series of numbers is greater than the sum of all those numbers. How this could possibly come about in mathematics, I have the faintest idea. I can’t think of any possible example that could prove this particular inequality. So I guess it’s one of those sayings that has no particular application to mathematics.
That’s it, end of procrastination. Still have a long day of reading ahead!
Update
On second thought, maybe I shouldn’t use the disjoint union $\coprod \mathrm{\Omega}$ but rather $\coprod {\omega}_{i}$ instead. Maybe this is cleaner and simpler:
$$\mu ([0,1])>\sum _{x\in [0,1]}\mu (\{x\})$$Thoughts?
- Tags: codomain, coproduct, mathematics, omega, sigma, summation, topos theory
Comments (4)
techim
I don't think I understood any of that.
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Saint Elsewhere
Why use Topos theory? Can't you come up with the summation without it?
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Raymond
Nevermind lol. This is less about writing something properly and more of showing off the new MathML functionality on the site ;)
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Raymond
I'm not really sure why I put that in there in the first place. It feels right and wrong at the same time.
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