1. HISTORICAL NOTES
History [of foundations of math]. Babylonian and Egyptian mathematics emphasized
arithmetic and the idea of explicit calculation. But Greek mathematics tended to
focus on geometry, and increasingly relied on getting results by formal deduction.
For being unable to draw geometrical figures with infinite accuracy this seemed the
only way to establish anything with certainty. And when Euclid around 330 BC did
his work on geometry he started from 10 axioms (5 "common notions" and 5
"postulates") and derived 465 theorems. Euclid’s work was widely studied for more
than two millennia and viewed as a quintessential example of deductive thinking. But
in arithmetic and algebra - which in effect dealt mostly with discrete entities - a largely
calculational approach was still used. In the 1600s and 1700s, however, the
development of calculus and notions of continuous functions made use of more
deductive methods. Often the basic concepts were somewhat vague, and by the mid-
1800s, as mathematics became more elaborate and abstract, it became clear that to
get systematically correct results a more rigid formal structure would be needed.
The introduction of non-Euclidean geometry in the 1820s, followed by various
forms of abstract algebra in the mid-1800s, and transfinite numbers in the 1880s,
indicated that mathematics could be done with abstract structures that had no
obvious connection to everyday intuition. Set theory and predicate logic were
proposed as ultimate foundations for all of mathematics (see note below). But at the
very end of the 1800s paradoxes were discovered in these approaches. And there
followed an increasing effort - notably by David Hilbert - to show that everything in
mathematics could consistently be derived just by starting from axioms and then
using formal processes of proof.
Gödel’s Theorem showed in 1931 that at some level this approach was flawed. But
by the 1930s pure mathematics had already firmly defined itself to be based on the
notion of doing proofs - and indeed for the most part continues to do so even today
(see page 859). In recent years, however, the increasing use of explicit computation
has made proof less important, at least in most applications of mathematics.

2. Buhay ni sec.jesse manalastas robredo
Si Jesse Manalastas Robredo ay isang politikong Pilipino na
naglingkod bilang Kalihim ng Kagawaran ng Interyor at Pamahalaang
Lokal (DILG) sa administrasyon ni Pangulong Benigno Aquino
III mula 2010 hanggang sa kanyang kamatayan ngayong 2012. Dati
ring kasapi si Robredo ng Partido Liberal.
Simula noong 1988, naglingkod nang anim na taning si Robredo bilang
alkalde ng Lungsod ng Naga sa Camarines Sur, mula 1988 hanggang
1998, at muli mula 2001 hanggang 2010, noong siya'y ipinili ni
Pangulong Aquino bilang Kalihim ng Interyor at Pamahalaang Lokal.
Bilang pagkilala sa kaniyang tungkulin bilang alkalde ng Lungsod ng
Naga, ginawaran si Robredo ng Gawad Ramon Magsaysay para sa
Panunungkulan sa Pamahalaan noong 2000, ang unang alkalde mula sa
Pilipinas na ginawaran ng ganitong parangal.
Noong 18 Agosto 2012, ay nag Crash ang eroplanong sinasakyan ni
Kalihim Robredo sa may pampang ng Lungsod ng
Masbate.[2]Nakatakda siyang umuwi sa Naga upang dalawin ang
kaniyang anak na lalahok sa isang paligsahan sa paglalangoy. Ayon sa
DILG, sumaklolo ang piloto ng eroplano sa Paliparan ng
Masbate upang humiling ng pahintulot na magsagawa ng
pangkagipitang paglapag ng eroplano. Gayunpaman, hindi naabutan ng
eroplano ang paliparan at lumapag ito sa dagat.[3][ 4] Nakunan naman
ang bangkay ni Robredo noong 21 Agosto, tatlong araw makatapos
ang aksidente, sa lalim na 54 metro (180 talampakan) sa ilalim ng antas
ng dagat.

3. You would not believe your eyesIf here)Why do I tire of counting
ten million firefliesLit up the world sheep(Please take me away from
as I fell asleep here)When I'm far too tired to fall
'Cause they'd fill the open airAnd asleep
leave teardrops everywhereYou'd To ten million firefliesI'm weird
think me rudeBut I would just stand 'cause I hate goodbyesI got misty
and stare eyes as they said farewell
I'd like to make myself believeThat But I'll know where several areIf my
planet Earth turns slowlyIt's hard to dreams get real bizarre'Cause I saved
say that I'd rather stayAwake when a few and I keep them in a jar
I'm asleep'Cause everything is never I'd like to make myself believeThat
as it seems planet Earth turns slowlyIt's hard to
'Cause I'd get a thousand hugsFrom say that I'd rather stayAwake when
ten thousand lightning bugsAs they I'm asleep'Cause everything is never
tried to teach me how to dance as it seemsWhen I fall asleep
A foxtrot above my headA sock hop I'd like to make myself believeThat
beneath my bedA disco ball is just planet Earth turns slowlyIt's hard to
hanging by a thread say that I'd rather stayAwake when
I'd like to make myself believeThat I'm asleep'Cause everything is never
planet Earth turns slowlyIt's hard to as it seemsWhen I fall asleep
say that I'd rather stayAwake when I'd like to make myself believeThat
I'm asleep'Cause everything is never planet earth turns slowlyIt's hard to
as it seemsWhen I fall asleep say that I'd rather stayAwake when
Leave my door open just a I'm asleepBecause my dreams are
crack(Please take me away from bursting at the seams
here)'Cause I feel like such an
insomniac(Please take me away from