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Study Guide

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It's easy to get into this club. All you need is

- an astronomically high IQ;
- to have produced one of the all-time most important mathematical results;
- to believe that mathematical entities really,
*really*exist, just like tables and chairs (except abstract).

It turns out they had to turn away lots of applications for the club. Here are the lucky few who were admitted.

*President*

You remember Kurt, right? World-renowned 20th-century mathematician and logician, he's best known for the Gödel Incompleteness Theorem. We're sure that most Shmoopers can run through this result in their sleep, so we won't bore you with the details right here.

But was he a Platonist? It's enough to quote this little passage from one of his more philosophical works: "…The assumption of [mathematical] objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence" (source). Get the picture?

*Vice-President*

This guy was a little too smart for his own good. Not only did he co-found the most influential school of philosophy in the 20th century—analytic philosophy—he also co-founded mathematical logic. The *Principia Mathematica* is his masterpiece in the latter area. You should definitely give it a read some day, as long as you have 10 years or so of spare time on your hands.

Bertie didn't just believe that logical and mathematical objects existed. He also thought that the task of philosophy was to bring you to have the same sort of direct acquaintance with such objects as one might have with…the taste of a pineapple.

*Co-vice-president* (since he and Russell share everything else)

Gotty was the *other *guy who co-founded both analytic philosophy and mathematical logic. His *Basic Laws of Arithmetic* is just as unreadable as Russell's *Principia*. And he's as committed to the reality of logical and mathematical objects as his buddy. (Don't get the wrong idea about Frege and Russell, though—they hardly knew each other.)

*Treasurer*

This guy, the creator of set theory, probably makes it on most Top Ten lists of All Time Great Mathematicians. And of course, the greater the mathematician, the more extravagant his claims (feel free to express that as a function, Shmoopers). Cantor believed that his theory of transfinite numbers was communicated to him by God. In fact, he thought his notion of the absolute infinite *was* God.

So it's not enough for Georg just to say that the familiar numbers are out there and exist. No, a whole new realm of numbers—the transfinite numbers that he discovered— are also said to exist. There is, in fact, an entire previously unknown *universe *of very real mathematical objects that Cantor has unveiled. It's Platonism on steroids.

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