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I recently stumbled upon the Wikipedia entry on finitism (there is even ultrafinitism). However, the article on ultrafinitism mentions that no satisfactory development in this field exists at present. I'm wondering in which way the limitation to finite mathematical objects (say a set of natural numbers with a certain largest number n) would limit 'everyday' mathematics. What kind of mathematics would we still be able do (cryptography, analysis, linear algebra …)?
Bishop built real analysis constructively right? Jayne's probability theory is from finite sets as well.