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Graduate Pizza Seminar

Analysis without the Analysis

Brian Pinsky - Rutgers University

Location:  GSL
Date & time: Friday, 09 November 2018 at 1:40PM - 2:40PM

Abstract:  I'd like to share a definitely true story. Once upon a time, there lived a logician named Abraham Robinson. Overall, he had a happy life, but there was one problem: the analysts of his department were constantly being friendly and telling him about their research. This led to far too many conversations involving the letter epsilon for Robinson to live a happy and successful life. Then, one day, Robinson had an idea. "What if I reengineered all of real analysis so that it required a whole course in model theory to understand even the most basic definitions? Surely," he thought, "that would make the analysts stop trying to talk to me".

And with that, he set to work. To succeed with his plan, he knew he would first have to learn how to do analysis. For months, he read paper after paper, and filled entire chalkboards, only to find himself lost in a sea of epsilons and sign errors. As he chased references backwards, he started reading older and older works, until eventually, he found himself all the way back at the beginning, in Leibniz's first formulations of calculus. Back then, before the idea of epsilons, people had been making do with just the idea of "infinitesimal numbers". Most sensible mathematicians of Robinson's day would criticize this as informal and lacking in rigor; and with good reason. Nevertheless, this proved to be the one paper Robinson could read, and so here he began his work.

Some years later, Robinson had perfected an analyst-proof formulation of calculus. Unfortunately for him, it turned out better than standard calculus in many ways, and some of the open minded analysts wanted to actually learn this horrible system he had derived, and they made him spend the rest of his career explaining it to them.

These days, the framework he built is called "non-standard analysis". In my talk, I plan to introduce you to this wonderful world where definitions make intuitive sense without 5 layers of quantifiers, and all the proofs feel like you're definitely cheating. My agenda will be to define the hyperreals properly, then spend the remaining 5 minutes spewing out definitions, and perhaps actually proving a theorem.