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Many logical concepts which we take for granted, might not be meaningful in the realm of infinite. For example, something being equal to itself, some statement
by zkmon 5y ago
Many logical concepts which we take for granted, might not be meaningful in the realm of infinite. For example, something being equal to itself, some statement being true or false (no third state), things having unique identity or location, the concept or possibility of multiple things (and distinguishability), being able to compare one thing with another etc. All these make sense only in finite contexts but not in infinite or infinitesimal contexts.
If we accept infinity as a participant in the mathematical logic, the infinitesimal also deserves a role as a counter-party.
- thaumasiotes 5y ago> If we accept infinity as a participant in the mathematical logic, the infinitesimal also deserves a role as a counter-party. Infinitesimals have a much bigger role in widely-known mathematics than infinities do; they are the basis for calculus.
- qweqwweqwe-90i 5y agoSeries play a big role in all of math...
- jyounker 5y agoInfinitesimals are just a different aspect of mathematical infinity. This is the magic that makes calculus work (from a formal standpoint).
- im3w1l 5y agoWhile you can build a calculus using infinitesimals, the usual construction does not use them. https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis
- anikan_vader 5y agoThe original construction did though, if you look at the writings of Newton/Leibniz. But their work was totally hand-wavey. Cauchy discovered limits in the 1800s, and it wasn’t until the 20th century that the original idea of infinitesimals was formalized. In some sense we have come full circle. I do think that infinitesimals are a much more intuitive way to understand calculus. It’s a shame it took so long to find positive numbers less than any real.