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Experimental Mathematics Seminar

Rethinking the Foundations of Mathematics and its Practical Consequences

Jonathan Lenchner, IBM T.J. Watson Research Center, Yorktown Heights, NY

Location:  via Zoom: link [password: 6564120420 ]
Date & time: Thursday, 12 November 2020 at 5:00PM - 6:00PM

Abstract: In this talk I shall argue that we should recast the foundations of mathematics using the foundational notion of a bit, rather than the much more ambiguous notion of a set, and the axiomatic assumption that infinite sets exist, whatever that might mean. I shall argue that mathematics and physics are inextricably connected and that treating mathematics axiomatically, and believing that mathematical objects like points, lines or even simple infinite decimal expansions like 0.1111 exist in some Platonic idealist sense, is just coercing mathematics into some meaningless ideal form. I will try to convince you that the deepest questions are not about some large cardinals that are inexpressible in the language of set theory (so-called inaccessible cardinals), but about extremely large natural numbers that we are unable to express in any way given a finite universe and so finite rewritable memories and finite computational power. Forget about infinite cardinal numbers, do these finite numbers exist or not? What precisely do we mean by expressibility, specifically expressibility in N bits? What are the limits of our expresibility? What do these questions say about the limitations of what is provable about arithmetic, or about finite structures like graphs? 

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